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Particle transport in microfluidic environments : particle adsorption at the polydimethylsiloxane-water… Mustin, Benjamin 2015

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  Particle Transport in Microfluidic Environments: Particle Adsorption at the Polydimethylsiloxane-Water Interface and the Effects of Flow Field and Image Processing on the Measurement Depth in Micro Particle Image Velocimetry by Benjamin Mustin Dipl.-Ingenieur (TH) Technische Universität Dresden, 2010  A THESIS SUBMITTED IN PARTIAL FULLFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  Doctor of Philosophy in The Faculty of Graduate and Postdoctoral Studies  (Mechanical Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) September 2015 © Benjamin MustinAbstract This thesis investigates several research questions associated with the transport of microscopic particles in microfluidic environments. This includes an investigation of the motion and deposition of micron sized polystyrene particles at the planar polydimethylsiloxane–water interface. Particle tracking shows that particles near the substrate can be immobilized to different degrees. Careful analysis of the more weakly immobilized particles reveals that there is a buildup of a particle accumulation layer near the substrate in which particle motion parallel to the substrate is hindered by non-hydrodynamic effects. The presence of lateral surface interaction forces resulting from charge heterogeneity of the PDMS substrate is found to be the most plausible explanation for the hindered particle transport across the substrate. The two other problems are concerned with micro particle image velocimetry (µPIV), which is a particle transport-based optical method for the characterization of flow fields in microfluidic devices. One limitation µPIV is the finite measurement depth associated with the optical setup, which can lead to bias errors in the measured flow-velocity. Analytical and numerical models are developed that describe the effect of common image pre-processing filters on the measurement depth in µPIV. Further, previous models are revisited that describe the effect of flow velocity gradients on the measurement depth in µPIV.    ii Preface This PhD thesis is original work by the author, Benjamin Mustin. The research presented in this thesis was planned and conducted by myself.  I acknowledge general guidance and excellent feedback on my manuscripts by my supervisor Dr. Boris Stoeber. Financial support for this work was provided through research grants from the Natural Science and Engineering Research Council (NSERC) of Canada. A version of chapter 2 is currently under review with a peer reviewed journal.  A version of chapter 3 has been published as B. Mustin and B. Stoeber, "Effect of linear image processing on the depth of correlation in micro PIV," Experiments in fluids, vol. 55:1817, 2014. A version of chapter 4 has been submitted to a peer reviewed journal.    iii Table of Contents Abstract ................................................................................................................................... ii Preface .................................................................................................................................... iii Table of Contents ................................................................................................................... iv List of Tables ......................................................................................................................... vii List of Figures ...................................................................................................................... viii List of Supplementary Materials ......................................................................................... xiv 1 Introduction ...................................................................................................................... 1 1.1 Motivation .................................................................................................................... 1 1.2 Research Objectives ................................................................................................... 5 2 Single layer deposition of polystyrene particles onto planar Polydimethylsiloxane substrates ......................................................................................................................... 7 2.1 Introduction ................................................................................................................. 7 2.2 Materials and methods ................................................................................................ 9 2.2.1 Impinging jet cell .................................................................................................. 9 2.2.2 Fabrication of flow cell and substrate preparation .............................................. 13 2.2.3 Preparation of fluids ........................................................................................... 14 2.2.4 Particle characterization ..................................................................................... 15 2.2.5 Experimental setup and procedure .................................................................... 16 2.2.5.1 Assembling, filling and calibration of the impinging jet cell .......................... 16 2.2.5.2 Particle deposition experiment .................................................................... 16 2.2.5.3 Removal of particles from substrate ............................................................ 17 2.2.5.4 Image acquisition and particle tracking ....................................................... 18 2.3 Results and discussion .............................................................................................. 20 2.3.1 Characterization of particle tracking ................................................................... 20 2.3.2 Surface interaction potentials and predicted concentration profiles .................... 23 2.3.2.1 DLVO interaction potentials ........................................................................ 24 iv 2.3.2.2 Extended DLVO interaction potentials ........................................................ 27 2.3.2.3 Concentration profiles and particle flux ....................................................... 31 2.3.3 Trajectory types and deposition criteria .............................................................. 32 2.3.4 Effect of ionic strength on the initial deposition rate............................................ 38 2.3.5 Displacement distributions ................................................................................. 43 2.4 Conclusions ............................................................................................................... 49 3 Effect of linear image processing on the depth of correlation in micro PIV .............. 52 3.1 Introduction ............................................................................................................... 52 3.2 Theoretical framework ............................................................................................... 54 3.2.1 Image plane intensity distribution of a single micro particle ................................ 54 3.2.2 Micro PIV displacement correlation peak ........................................................... 56 3.2.3 Weighting functions and depth of correlation ..................................................... 59 3.2.4 Influence of linear image processing on the DoC ............................................... 61 3.2.5 Measurement of the weighting function .............................................................. 62 3.2.6 Numerical computation of the weighting function and depth of correlation ......... 62 3.3 Materials and methods .............................................................................................. 64 3.3.1 Materials ............................................................................................................ 64 3.3.2 Test setup .......................................................................................................... 65 3.3.3 Imaging and image averaging ............................................................................ 65 3.3.4 Image processing .............................................................................................. 66 3.3.5 Determination of the weighting functions ............................................................ 66 3.4 Results and discussion .............................................................................................. 67 3.4.1 Assessment of the model for micro PIV images ................................................. 67 3.4.2 Influence of linear image processing on the DoC ............................................... 69 3.4.2.1 Band-pass filtering ...................................................................................... 69 3.4.2.2 High pass filtering ....................................................................................... 77 3.4.2.3 Low-pass filtering ........................................................................................ 81 3.5 Conclusions ............................................................................................................... 84 4 On the effect of velocity gradients on the depth of correlation in micro PIV ............. 85 v 4.1 Introduction ............................................................................................................... 85 4.2 Theoretical framework ............................................................................................... 86 4.2.1 Image plane intensity distribution of a single micro particle ................................ 86 4.2.2 Displacement correlation peak ........................................................................... 86 4.2.3 Weighting functions............................................................................................ 89 4.3 Weighting function and DoC for constant velocity gradients ...................................... 93 4.3.1 Weighting functions............................................................................................ 93 4.3.2 Depth of correlation.......................................................................................... 100 4.4 Weighting function, DoC and velocity bias for common flow fields in micro channels ....   ............................................................................................................................... 101 4.4.1 Bias error ......................................................................................................... 102 4.4.2 Weighting functions and Depth of correlation ................................................... 106 4.5 Extension to different particle image models ........................................................... 108 4.6 Conclusions ............................................................................................................. 113 5 Conclusions ................................................................................................................. 115 5.1 Limitations ............................................................................................................... 115 5.2 Recommendations for future work ........................................................................... 117 Bibliography ........................................................................................................................ 119 Appendix ............................................................................................................................. 141 A1 Simulation method and parameters: ........................................................................ 141 A2 Implications of other choices for the Hamaker constant A132 .................................... 143 A3 Displacement distributions with standard deviation .................................................. 145 A4 Relationship between curvature of the local correlation function and the weighting function ................................................................................................................... 145    vi List of Tables Table 1 Surface energy data. All surface energies are in mJ/m2, the unretarded Hamaker constants Aii are displayed in 10-20J. ..........................................    30 Table 2 Standard deviation of substrate zeta potentials based on the DLVO interaction potential eq (14) and based on the extended DLVO interaction potential eq (15). ......................................................................    42 Table 3 Comparison of the measured and expected numbers of particles near the substrate. ............................................................................................    49 Table 4 Summary of objective lens parameters ......................................................   69 Table 5 Simulation parameters ..............................................................................   142 Table 6 Minimum (dimensionless) separation distances H0 and expected particle counts Npe within the field of view obtained by employing the DLVO theory and the extended DLVO theory in combination with different Hamaker constants and for different salt concentrations CNaCl .....      144   vii List of Figures   Figure 1 a) negative of the photomask used to fabricate mold masters for the flow cells; b) cross section view of the deposition chamber (cut plane is indicated in a)); c) flow cell on substrate without clamping; d) schematic of the clamped flow cell. .................................................   10   Figure 2 Schematic of the experimental setup a) for the calibration step and b) for the particle deposition experiment. The thin lines represent Tygon tubing and the black dots to steel needle tips that serve as tubing-tubing interface as well as flow-cell-tubing interface; c) schematic of particle removal from the substrate by passing a moving contact line over the substrate; d) schematic of particle removal from the substrate by using a sonicator. ................................   18   Figure 3 Excerpts of a particle image a) obtained from optical setup and b) corresponding computer generated image. Both images show the same field of view and consist of the same number of pixels and are based on the same particle concentration. Furthermore, both images are displayed by using the same color-map. ...........................   22   Figure 4 a) Tracking efficiencies βm(ze) and βd and fraction of invalid trajectories βb; b) precision for deposited particles; c) precision valid trajectories of mobile particles within z < 3 µm; d) Precision for all valid trajectories ........................................................................   23   Figure 5 Zeta Potentials over salt concentration: Polystyrene particles (circles) and interpolant (solid line). The error bars (where available) represent the standard deviation of the particle zeta potential distribution across the particle population; PDMS zeta potential (dashed line) according to literature [121]. The pH of the all particle suspensions used in the current work was between 6.3 and 6.6. ...............................................................................................   26   Figure 6 DLVO interaction potential for different concentrations of sodium chloride: a) energy barriers for CNaCl < 500 mM, b) secondary minima for CNaCl = 1 mM, 10 mM and 100 mM. H = h/a represents the ratio of the particle interface gap h and the particle radius a = 500 nm. ...............................................................................................   27   viii   Figure 7 Extended DLVO interaction potential for different concentrations of sodium chloride: a) energy barriers for CNaCl < 100 mM, b) secondary minima for CNaCl = 1 mM and 10 mM; The curves corresponding to 100 mM and 500 mM are indistinguishable. H = h/a represents the ratio of the particle interface gap h and the particle radius a = 500 nm. ..................................................................   30   Figure 8 a) Steady state concentration profiles based on the extended DLVO interaction potential eq (15). b) corresponding steady state normal component of the particle flux. H-H1 = (h-h1)/a represents the ratio of the gap width (h-h1) between particle and primary minimum and the particle radius a = 500 nm. The gap width for particles in the primary minimum is h1 = 0.158 nm...............................   32   Figure 9 Typical trajectories observed during a laboratory experiment at CNaCl = 0.01 M: a) a particle undergoes significant displacement (not shown in histogram) before it comes to rest. b) and c) particle undergoes significant Brownian motion but with almost zero mean displacement. All displacements in a)-c) were calculated from eq (24) with p = 1. ....................................................................................                                           37   Figure 10 Mean square displacement (MSD) of a particle as a function of the time interval ∆t. The error bars represent the expected standard deviation of the MSD based on the given sample size and ∆t [72]. a) and b) show MSD-plots for the particles in Figure 9b and c, respectively. ........................................................................................   37   Figure 11 a) Transition to steady state concentration profiles for different salt concentrations with Λ as defined in eq (11). The extended DLVO interaction potential was used for the computation: b) Control experiments at CNaCl = 0.1 mM demonstrating that the particle removal procedure described in section 2.2.5.3 has a negligible effect on the deposition rates. .............................................................   42   Figure 12 a) Number of deposited particles over time for one substrate; b) steady-state deposition rates for three substrates and for different salt concentrations: the error bars represent the standard deviation of the deposition rates for three different substrates. The dashed line and solid line represent numerical predictions based on the DLVO interaction potential eq (14) and the extended DLVO interaction potential eq (15), respectively. ...........................................   42   ix   Figure 13 Distributions of particle displacements: a) measured overall distribution with a 1 pixel bin size; b) corresponding close up view of the region of small displacements with a 0.1 pixel bin size; c) predicted distributions corresponding to the concentration profiles predicted by the convective diffusion theory; d)  predicted distributions accounting for anomalously accumulated particles with hydrodynamic interactions. .................................................................   48   Figure 14 a) Schematic of the model-optical system (single thin lens) with the distance s0 between lens and object plane, image distance si, position of the object plane zf, position of the lens zl, coordinates x,y,z in the physical domain and coordinates X,Y in the image plane. b) Schematic of the integrand in eq. (47); grey regions correspond to support of Ĵo�X�⃗ -Mx�⃗ ; z� and of Ĵo�X�⃗ -Mx�⃗ + s⃗-M∆x�⃗ ; z +∆z� ......................................................................................................   54   Figure 15 Weighting functions obtained by using the lenses a 20× NA 0.5, b 40× NA 0.8, c 40× NA 1.3 with 1 µm particles (squares) and 3.1 µm  particles (circles). The solid lines represent the model eq. (54) after introducing an effective NA = CdNAl for each lens, with Cd listed in table 1. ................................................................................................   68   Figure 16 Particle intensity profiles for the 1 µm particles imaged through different lenses and different focus positions: a) 20× Plan Fluor the symbols (circles squares triangles) correspond to z = [0.4 3 6.4] µm; b) 40× Fluor; z = [0 3 6] µm. c) 40× Plan Fluor; z = [0 1.8 3.6] µm. The solid lines represent the model eqs (37)-(40) with eq. (65). ...   68   Figure 17 Comparison of model eq. (68) (solid lines) to measured weighting function (symbols) for DoG-filtered images for different values of KDoG  =  σG1de (z=zf) and kDoG = σG2σG1 (in legend KDoG; kDoG): a) M = 20× – dp  = 1 µm; b) M  = 40× -  dp  = 1 µm ; c) M = 20× –  dp  = 3.1 µm; d) M = 40× –  dp  = 3.1 µm. .................................................................   71   Figure 18 Contour of zDoCzDoC0 for particle images subject to the difference of Gaussians filter eq. (66) with KDoG  =  σG1de (z=zf) and kDoG = σG2σG1 for ɛ = 0.01 and Δz = 0 valid for zDoC ≪ s0: a for a wider range of kDoG and KDoG; b close-up view of the parameter range leading to a reduction of the DoC.  The difference between neighbouring contour levels is 1 in a and 0.1 in b and the highest contour levels are equal to the highest values shown in the color maps. ...................   72   Figure 19 Comparison of model eq (72) (solid lines) to measured weighting function (symbols) for LoG-filtered images for different values of  KLoG =  σde (z=zf) (in legend): a) M = 20× – dp  = 1 µm; b) M = 40× -  dp  = 1 µm ; c) M = 20× –  dp  = 3.1 µm; d) M = 40× –  dp  = 3.1 µm. ..   74 x   Figure 20 Effect of the Laplacian of a Gaussian filter on the depth of correlation eq. (74) with filter parameter KLoG = σde(z=zf) for Δz = 0 and ɛ = 0.01, valid for zDoC ≪ s0 ..........................................................   75   Figure 21 a) and b): Contour of zDoCzDoC0 for particle images subject to the difference of moving averages filter eq. (75) with KDoMA =  L1de (z=zf) and kDoMA = L2L1  for ɛ = 0.01 and Δz = 0 valid for zDoC ≪ s0. The difference between neighbouring contour levels is 0.5 in a and 0.1 in b) and the highest contour levels are equal to the highest values shown in the color maps; c) zDoCzDoC0 as a function of (de (z = zf)) when following the recommendations by [177]. ............................................   77   Figure 22 Comparison of model eq. (78) (solid lines) to measured weighting function (symbols) for different values of KGHP = σG3de(z=zf) (in legend): a) M = 20× – dp  = 1 µm; b) M = 40× -  dp  = 1 µm; c) M = 20× –  dp  = 3.1 µm; d) M = 40× –  dp  = 3.1 µm. .................................................   79   Figure 23 Relative decrease of the depth of correlation after high-pass filtering with filter parameters KGHP = σG3de(z=zf) (for Gaussian based filter eq. (77)), KMAHP = L3de(z=zf) (for Moving-average based filter eq. (80)) and KMAHP' = KMAHP3.5  for ɛ = 0.01 and valid for zDoC ≪ s0 : ‘Exact solution’ zDoCzDoC0(KGHP)  for Gaussian based filter (solid line) and corresponding approximate expression eq. (79) (dashed line) as well as numerical solution (open circles). Numerical solution zDoCzDoC0(KMAHP) for moving average based filter eq. (80) (crosses). Scaled numerical solution zDoCzDoC0(KMAHP' ) for moving average based filter (dash-dotted line). .......................................................................   81   xi   Figure 24 Effect of low pass filtering with a Gaussian filter with standard deviation σG = KGLPde (z = zf) on the weighting function WLP. The values for KGLP are shown in legend. Solid lines are equation (82) and symbols are experimentally determined weighting functions for a) M = 20× – dp  = 1 µm; b) M = 40× -  dp  = 1 µm ; c) M = 20× –  dp  = 3.1 µm; d) M = 40× –  dp  = 3.1 µm. ...........................................   82   Figure 25 Effect of low-pass filtering on the depth of correlation for ɛ = 0.01 and for zDoC ≪ s0 : zDoCzDoC0 (KGLP) for Gaussian low-pass filter (solid line) and corresponding numerical solution (open circles). Numerical solution zDoCzDoC0(KMALP) for moving average low-pass filter eq. (85) (crosses). Scaled numerical solution zDoCzDoC0(KMALP' ) for moving average low-pass filter (dash-dotted line). Smoothing parameters are KGLP = σGde(z=zf) (for Gaussian based filter eq. (81)), KMALP = L3de(z=zf) (for Moving-average based filter eq. (85)) and KMALP' = KMALP3.5  for scaled Moving-average based filter. .......................   84   Figure 26 a) Effect of constant out-of plane shear B on the weighting functions W�x and W�y for zero in-plane shear A = 0 and b) visual interpretation. c) Effect of in plane shear A on the weighting functions W�x and W�y and d visual interpretation for non-zero out-of-plane shear. The circles in b) and d) represent example particle patterns at t = t1 (lines) and at t = t2 (filled circles without lines). Solid lines represent in-focus particles and dashed lines out-of-focus particles. Out-of-focus particles are shown larger as they appear larger in an image. Also shown, are schematics of the contributions of in-focus (solid lines) and out-of-focus particles (dashed lines) to the overall displacement correlation peak 〈RD〉 . ......   99   Figure 27 Effect of dimensionless shear rates A = asLyM∆tde (z=zf) and B = bzDoC0M∆tde (z=zf)  on the relative depth of correlation: a) zDoCx(A,B)zDoC0, b) zDoCy(A,B)zDoC0. ..............   101   Figure 28 Relative measured channel center velocity for large channel aspect ratios wdhd> 10 as a function of a 1Hd= zDoC0hd  (legend shows values of Xmax = umaxM∆tde (z=zf)) and b as a function of the ratio Xmax  between particle displacement to particle image size (legend shows values of Hd). .................................................................................................   105   xii   Figure 29 Relative measured channel center velocity as a function for different aspect ratios: a) wdhd= 1, b) wdhd= 0.2. Legends show values of Xmax. ................................................................................................   106   Figure 30 Weighting functions for large aspect ratios WdHd> 10 for a) Hd = 10, b) Hd = 1, c) Hd = 0.01. d) Relative depth of correlation as a function of Xmax for different Hd. The legends in a), b) and c) show values of Xmax. The legend in d shows values of Hd. ...........................   108   Figure 31 a) Peak intensity and b) particle image diameter as a function of the separation z-zf between particle and object plane with zf = 0. The symbols represent experimental data for 1.28 µm diameter particles seen through an upright combi microscope at M = 25× by [199]. The lines represent different empirical models (see text for explanation). c) Predicted bias error (lines) for two different particle image models and with Xmax = 0.53 and hd = 74 µm. Data-points show data by [199]. .............................................................................   113   Figure 32 Figure 4 of [98] reproduced. jn is the dimensionless normal component of the particle flux as defined in [98]. ................................   141   Figure 33 Average displacement distributions from Figure 13a) with associated standard deviation shown as error bars for different salt concentrations CNaCl: a) 0.1 mM, b) 1mM, c) 10 mM, d) 100 mM. ........   145    xiii List of Supplementary Materials Trajectory-Fig9b.avi Video file corresponding to the particle trajectory in Figure 9b of the thesis ..................................................   http://hdl.handle.net/2429/54802 Trajectory-Fig9c.avi Video file corresponding to the particle trajectory in Figure 9c of the thesis ..................................................  http://hdl.handle.net/2429/54802   xiv 1 Introduction 1.1 Motivation Microfluidic devices are small fluidic systems that allow precise manipulation of fluids in channels with dimension in the order of tens of micrometers [5]. The small characteristic channel dimensions of microfluidic devices give rise to interesting physical phenomena [6] that have been exploited by an increasing number of applications in many fields of biology [7], [8], [9], [10], [11], [12], chemistry [13], medicine [14], [15], [16] and optics [17].  One particularly interesting feature of the small characteristic channel dimensions of micro-fluidic devices is that microparticles such as biological cells [18], [19] or functionalized latex particles [20], [21] can be transported through a tailored environment that is of similar size as the particles themselves. These environments can further be enriched by electrical and optical components as well as by integrated flow control elements that allow for well-defined flow conditions. Thus, microfluidic devices are an ideal environment for the fabrication, manipulation, and investigation, of microscopic particles and not surprisingly, the transport of microparticles is an important factor in many microfluidic devices. The ability to precisely control adjacent immiscible fluid streams at the micron scale has been utilized for fabricating monodisperse polymer microparticles in a variety of shapes [22], [23] in microfluidic reactors. These microfluidic devices mainly follow the same principle: A fluid stream of monomer is brought into contact with other streams of fluid that are immiscible with the monomer. Surface tension then leads to breakup of the monomer stream, and a monomer emulsion is formed. Subsequently the monomer droplets are crosslinked in the microfluidic reactor, e.g. through photopolymerization before the resulting microparticles are transported out of the device. Variations of this method have been used to fabricate polymer micro-capsules [24] as well as microparticles for drug delivery [25], [26]. Particles fabricated in this manner may then serve as moving parts in passive micro-flow control elements such as fluidic check valves for micro pumps [27], [28], [29]. In these flow control elements, fluid flow first drags the particles towards an aperture or an array of channel constrictions where the particles subsequently block the flow passage until the direction of flow is reversed. Other microfluidic devices aim at separating different particle species. These devices have been extensively reviewed in [30], [31], [32], [33]. Interesting example applications of particle separation in microfluidic devices can be found here at the University of British Columbia. I.e., 1 cancer cells are separated from leukocytes by exploiting differences in size and deformability of these two cells [34] and magnetic micro particles [35] and aerosol particles [36] are separated by size. Furthermore, microfluidic devices with integrated electrodes or optical elements are often used for continuous counting of dispersed particles [37], [38]. The two functionalities, particle separation and particle counting, can also be combined in one microfluidic device [39], [40], [41], which may open the door to hand-held particle dispersion characterization devices.  Much effort has been devoted to the development of methods for controlled and reversible trapping of particles in specific regions of a microfluidic device [42]. Example applications for particle trapping are sample pre-concentration [43], [44] and controlled pairing and fusion of cells [45]. Controlled retention of cells is promising for basic research that investigates the response of cell cultures to the flow of different analytes [46], [47]. In general, the means to separate, count and trap individual microscopic particles make microfluidic devices an ideal environment for the manipulation and analysis of individual cells or populations of cells [48], [49], [50]. Particle trapping is also commonly utilized for particle-based immunoassays. Here, functionalized polymer microparticles or magnetic microparticles are first trapped in a certain region of the immuno-assay device, before the analyte is immobilized on the surface of the microparticles [21], [38], [51], [52], [53], [54]. All of the above examples of particles in microfluidic devices exploit the small characteristic dimensions of microfluidic devices. The small dimensions of microfluidic devices also imply that microparticles contained in the device are typically relatively close to a device wall. Thus, particle-wall interactions can have a decisive impact on the transport of particles in microfluidic environments. For example, when a microparticle is transported through a straight micro-channel by the flow field, hydrodynamic particle-wall interactions can lead to significant particle migration in the direction perpendicular to the channel axis [55], [56], [57] and this effect has been harnessed for particle separation in microfluidic devices [58]. Specific particle-wall interactions are frequently employed for the immobilization of specific cells and proteins to surfaces for subsequent targeted analysis or separation of different species [59], [60]. On the other hand, non-specific particle-wall interactions such as Lifshitz-van der Waals, electrostatic double layer and polar acid-base interactions can lead to non-specific adsorption at the solid liquid-interface, impeding targeted analysis of biological mechanisms in microfluidic devices [61]. Non-specific surface interactions can also lead to successive adsorption of micron-sized cells or polymer microspheres in narrow regions of a microchannel [62], [63], [64] until the flow passage is clogged. Over the past 10 years, microfluidic devices made from Polydimethyl-siloxane (PDMS) have been frequently employed for the systematic investigation of successive 2 particle deposition at microchannel walls from flowing aqueous suspensions of polymer microparticles. The focus of these investigations was either on the time period from initialization of a (constant-) pressure driven flow until blockage of a flow passage due to successive particle deposition, or on the morphology of the deposits. More specifically, the time to channel blockage has been characterized with respect to the particle concentration, the (constant) driving pressure, the ratio of particle size to channel cross sectional dimension [62], [63] and the particle size distribution [63]. The morphology of the particle deposits have been shown to be sensitive to the ionic strength of the liquid phase, pre-conditioning of the PDMS walls [65], [66] and to the flow conditions [67], [66]. In addition to these laboratory experiments, recent numerical simulations of channel clogging based on the force coupling method have shed important insight into the complex mechanisms leading to channel blockage [68], [69]. However, the current understanding of the complex interplay between particle transport, particle-wall interactions and particle-particle interactions leading to channel clogging is incomplete and clogging can at best be described qualitatively by currently available theoretical models [70], [68], [69]. Although successive deposition of polystyrene particles leading to channel blockage in microfluidic devices made from PDMS has been reported frequently, no one has systematically investigated the initial stages of clogging in which the deposition of particles is mainly governed by particle transport and interactions between dispersed particles and the bare PDMS walls. Thus, the present work aims at characterizing the motion and deposition of polystyrene particles in the vicinity of planar PDMS substrates by employing particle tracking (see research objective 1 in the following sub-section).  The motion of particles in microfluidic devices is most frequently characterized by optical methods such as micro particle tracking velocimetry (µPTV) [71] and micro particle image velocimetry (µPIV) [72]. Both methods determine particle velocities from digital particle image pairs or sequences. In contrast to particle tracking, where trajectories of individual particles are sought, µPIV employs cross correlation to quantify the velocity of small ensembles of particles in so called interrogation windows. Although µPTV and µPIV inherently determine the velocities of particles, as the name of these methods suggest, they are most commonly applied to the measurement of flow fields in microfluidic devices. Application of these particle-based velocimetry methods to the measurement of flow fields in microfluidic devices requires the fluid in the device to be seeded with carefully selected particles so that these particles can follow the flow field in good approximation. Thus, µPTV and µPIV not only harness particle transport to infer the flow field in microfluidic devices, but they are also important tools for the design of particle-transport based microfluidic devices discussed before. As particle transport based velocimetry methods, µPTV is the method of choice when the particle image density is 3 relatively low and when flow velocity gradients are large. On the other hand µPIV typically outperforms µPTV when flow velocity gradients are small and the particle image density is high [73]. Furthermore, µPIV is less sensitive to image noise than µPTV and can provide more accurate velocity measurements than µPTV when the signal to noise ratio of the particle images is low. For flow scenarios where the particle image density is high, flow velocity gradients are large and the signal to noise ratio in the images in not too small, hybrid methods have been developed that aim at combining the advantages of µPTV and µPIV [74], [75]. The present work utilizes well-established particle tracking methods in order to characterize particle motion and deposition at the water-PDMS interface (see research objective 1). Furthermore, this work aims at contributing to the current understanding of the theory of cross correlation for classical µPIV. In the classical µPIV method, particle images are acquired by a single camera connected to a epifluorescence microscope that is also equipped with an appropriate illumination source that illuminates the entire measurement volume [72], [76]. Since particle motion in the direction of the optical axis of the microscope cannot be detected directly by a single camera, only the projection of the flow velocity onto the object plane can be measured directly. However, for steady flows, the velocity component perpendicular to the object plane can be inferred from the principle of mass conservation [77]. Over the years, several modifications of the classical µPIV setup have been proposed and demonstrated [78]. Nevertheless, the classical µPIV setup is still in wide-spread use due to its excellent spatial resolution in combination with the simplicity of the setup as compared to other methods.  One limitation of the classical µPIV setup is associated with the finite depth of the measurement volume in the direction of the optical axis of the microscope [79]. In other words, particle images acquired with the classical µPIV setup typically contain (partially) out-of-focus particles that are some distance away from the object plane and move with a different velocity as compared to (in-focus) particles in the object plane. It has been shown, that the velocity estimate obtained from cross-correlation of (µPIV) particle image pairs can be described as a weighted average of the true fluid velocity over a measurement volume of finite depth termed the depth of correlation [79]. The corresponding weighting functions describe the relative influence of particles at a certain depth in the fluid on the measured velocity. Since it was recognized that µPIV-velocity estimates can be biased as a result of the finite depth of correlation, a variety of image pre-processing methods have been applied to address the problem [80], [81]. However, the effect of commonly used image filter operations such as low-pass, high-pass and band-pass filters on the depth of correlation has never been investigated 4 systematically, which motivates research objective 2 in the following sub-section of this dissertation. The depth of correlation is not only affected by the optics of the imaging system, particle size and image pre-processing, but it ultimately depends on the flow field to be measured. The effect of flow velocity gradients on the depth of correlation has been described by previous work [82], [83] based on the assumption, that the corresponding weighting functions are directly related to the curvature of the local correlation function [84]. The local correlation function is defined as the contribution of particles at a certain distance from the object plane to the overall-correlation function. Research objective 3 is motivated by the fact, that the relationship between the weighting function and curvature of the local correlation function has never been confirmed rigorously for the case where the flow field to be measured is non-uniform. 1.2 Research Objectives Important note on research objectives: Although all of the following research objectives involve the transport of microparticles in microfluidic environments, the first research objective is not closely related to research objectives 2 and 3. Objective 1: The first research objective is to investigate particle motion and deposition of polystyrene particles near the planar PDMS-water interface for different ionic strengths of the liquid phase by using a suitable experimental setup. It will be tested if the measured initial deposition rates are captured by a commonly employed mass-transport model for the initial stages of particle adsorption at the solid liquid interface. This mass transport model can account for surface interaction forces. The surface interactions between polystyrene particles in aqueous suspension and PDMS will be probed by considering different models for non-specific surface interaction forces and by comparing the corresponding predicted deposition rates to the measured deposition rates. The overall idea is identify chemical conditions (i.e. ionic strength) for which the initial adsorption of particles on PDMS is predictable. Such conditions should then be employed for future research on the clogging of microchannels due to particle deposition.  Objective 2: The second research objective is to investigate the effect of commonly employed pre-processing methods of the particle images for µPIV. The first sub-goal is to identify means for incorporating the effect of linear image processing into the existing theoretical framework for the depth of correlation in µPIV. Next, theoretical models that describe the effect of low-pass, 5 high-pass and band-pass filtering on the depth of correlation will be developed and verified through appropriate experiments.   Objective 3: The third objective is to revisit the relationship between the weighting functions and the local correlation functions that has previously been used to derive models that describe the influence of flow-velocity gradients on the depth of correlation in µPIV. This relationship has previously only been confirmed for the case when the flow field is uniform and it is not obvious to us why this relationship should hold for the case of non-uniform flow. Depending on the outcome of the theoretical analysis of the relationship between the weighting functions and the local correlation functions the goal is to either confirm previous models for the influence of gradients on the depth of correlation or to correct them. Another goal of this work is to study the depth of correlation for the commonly encountered flow field in a straight microchannel with rectangular cross section.    6 2 Single layer deposition of polystyrene particles onto planar Polydimethylsiloxane substrates 2.1 Introduction The emerging area of microfabricated lab-on-a-chip systems has resulted in an increasing number of microfluidic devices, many of which transport microscopic particles through their flow channels [85], [32], [86]. However, previous research [87], [88], [62], [63] has demonstrated that the presence of particles in the fluid can compromise the reliable operation of microfluidic devices; particles have been observed to deposit successively onto channel walls and onto previously deposited particles until they block a flow passage, which then leads to device failure. Polydimethylsiloxane (PDMS) is one of the most commonly used materials for fabricating microfluidic devices [89], [90] and the majority of previous research on clog formation in microfluidic devices was performed using devices made from PDMS. Kusaka et al. [67] investigated the deposition of polystyrene particles from unstable suspensions onto a PDMS cylinder for different particle sizes and flow conditions. After exposing the PDMS cylinder to the flow of a particle suspension, particles started to deposit on the bare PDMS cylinder. As the experiment proceeded, particles deposited onto previously deposited particles to form a particle aggregate on the cylinder. The size and shape of these particle aggregates was characterized by means of the Péclet number which describes the relative influence of convective and diffusive particle transport toward the cylindrical collector. The authors concluded that a full description of the resulting particle aggregates on the cylinder has to account for shear-stress induced aggregate-breaking mechanisms. Other research investigated the deposition of polystyrene particles from stable suspensions near the entrance of a narrow PDMS microchannel. It was shown, that particles from stable suspensions can deposit on previously deposited particles to form particle aggregates that can eventually cover the whole cross section of the channel [65], [66]. The time until a channel is blocked was found to be proportional to the particle concentration and inversely proportional to the constant driving pressure [62], [63]. A small fraction of particles, particle-aggregates or contaminants, that are much larger than the nominal mean particle size, can have a predominant effect on the evolution of a channel blockage [63], [91]. Bacchin et al. [65], [66] investigated the influence of the ionic strength of the liquid phase on the particle aggregate growth rate and aggregate morphology. The higher the salt concentration, the less significant were repulsive forces between particles and between particles and wall and the faster was the 7 observed aggregate growth rate. Interestingly, the morphology of the resulting aggregates was strongly influenced by the salt concentration. Furthermore, it was suggested that channels can only clog if the particle flux density through the microchannel entrance surpasses a critical particle flux density [66]. Although the clogging of PDMS microchannels due to successive deposition of polystyrene particles has been frequently reported, no one has systematically investigated the initial stages of clogging in which the deposition of particles is mainly governed by interactions between dispersed particles and the bare PDMS walls.  Extensive reviews of the theory of single layer particle adsorption as well as corresponding experimental methods and results can be found in [92], [93], [94]. In general, the kinetics of particle adsorption is well understood and can be quantitatively predicted as long as short-ranged interaction forces between particle and substrate are purely attractive across the entire substrate. However, currently available theories fail to predict initial particle adsorption kinetics when particles experience repulsive electrostatic double layer interaction with the substrate [92]. More specifically, significantly more particles are observed to adsorb on the substrate than is expected by theory and the observed deposition rates are less sensitive to particle size than is predicted by theory based on the convective diffusion equation. The discrepancy between predicted and observed deposition rates has been attributed to several factors. It has been shown that the presence of surface roughness on the substrate and particles tends to weaken the repulsive interactions between substrate and particles. Hence, deposition of particles is facilitated on rough surfaces as compared to smooth surfaces [95] [96] [97]. Another potential source of the discrepancy between theoretical and observed deposition kinetics under repulsive interaction conditions has been attributed to charge heterogeneity on the substrate. If the surface charge is heterogeneously distributed across the substrate, the magnitude of the repulsive interactions between particles and substrates will also vary across the substrate. It has been shown that experimentally observed particle adsorption rates can be explained based on the assumption that the repulsive interactions between particle and substrate vanish over a relatively small fraction of the entire substrate as a result of surface charge heterogeneity [98]. A third mechanism that potentially contributes to the discrepancy between predicted and observed particle deposition rates has been termed secondary minimum deposition and is based on the fact that there is a region at intermediate distances from the substrates where (especially larger) particles can experience a significant force in the direction normal to the substrate that tends to keep particles in that region. The magnitude of this normal force at intermediate distances from the substrate is expected to decrease with increasing ionic 8 strength of the liquid phase. This is in line with break-through experiments that were performed with packed bed columns where it has been observed that a significant fraction of the retained colloids is released upon flushing the column with a fluid with low electrolyte concentration [99], [100]. However, particles at intermediate distances from the substrate should remain mobile in the direction parallel to the substrate if the adhesive forces only act in the direction normal to the substrate as it is typically assumed. Indeed, experimental evidence has been presented that indicates that particles associated with secondary minimum deposition can roll along the surface of a grain of a packed bed column under repulsive interaction conditions until they come to rest in regions of near zero flow velocity [101]. However, it has been argued [102] that the typically observed amounts of particle retention and the particle release rates upon flushing with a low-ionic strength fluid under repulsive interaction conditions suggest that particles associated with secondary minimum may also be immobilized in any direction parallel to the interface. To the best knowledge of the authors of the present work, the question whether a particle associated with secondary minimum deposition can be immobilized with respect to its motion in the direction parallel to the substrate is currently still subject of debate.  The present work utilizes the impinging jet flow cell [103] to investigate the initial stages of deposition of polystyrene particles onto planar PDMS substrates for different concentrations of sodium chloride. Furthermore, the use of well-established particle tracking methods enables us to provide evidence that supports the view that particles near the substrate may be partially immobilized by the presence of surface interaction forces that act in the direction parallel to the substrate. This indicates that particles associated with the secondary minimum of the corresponding surface interaction potential may not always be freely mobile in the direction parallel to the substrate. The results of this work are important for future research on the clogging of microfluidic devices. 2.2 Materials and methods 2.2.1 Impinging jet cell Dabros et al. were the first to utilize the impinging jet cell for particle deposition experiments [103]. In the impinging jet cell (illustrated in Figure 1), a confined jet of a particle suspension is directed towards a substrate so that a stagnation point flow field develops. Direct optical access through the transparent substrate allows straight forward enumeration of deposited particles near the stagnation point. This section will give a brief review of convective diffusion equation 9 that is commonly used to describe particle deposition in the impinging jet cell. The impinging jet cell has been described more in detail elsewhere [103], [104], [92], [93].   a) b) c)  d)  Figure 1: a) negative of the photomask used to fabricate mold masters for the flow cells; b) cross section view of the deposition chamber (cut plane is indicated in a)); c) flow cell on substrate without clamping; d) schematic of the clamped flow cell.  At low and moderate Reynold’s number  𝑅𝑅𝑅𝑅 =  𝑉𝑉∞𝑅𝑅i𝜈𝜈, (1) with the kinematic viscosity of the suspension 𝜈𝜈 and with the mean fluid velocity 𝑉𝑉∞,  in the chamber inlet with radius 𝑅𝑅i  and a chamber height hc, the flow field near the stagnation point (i.e. for z/Ri < 0.2 and r/Ri < 0.5 if Re < 16) is well approximated by [103], [104]   𝑉𝑉r = 𝛼𝛼r(𝑅𝑅𝑅𝑅,ℎc𝑅𝑅i)𝑉𝑉∞𝑅𝑅i2 𝑧𝑧𝑧𝑧 (2)  𝑉𝑉h = −𝛼𝛼r(𝑅𝑅𝑅𝑅,ℎc𝑅𝑅i)𝑉𝑉∞𝑅𝑅i2 𝑧𝑧2 (3) where αr is the flow parameter depending on the cell geometry and Reynolds number. For the experimental conditions used in this work (Re = 0.75 and hc/Ri = 1.91), the flow parameter αr = 0.9975 was obtained by fitting equations (2) and (3) to a numerical solution of the flow field in the impinging jet cell near the stagnation point. The numerical solution was obtained by solving 10 the Navier Stokes equation with appropriate boundary conditions for our impinging jet cell geometry by using the FEM software package Comsol 4.3b. The obtained value for the flow parameter is very close to the documented value αr = 1 for similar conditions hc/Ri = 2 and  Re = 1 [93].  For the low Reynolds number used in this work, theory predicts a homogeneous particle adsorption flux in the region r/Ri < 0.5 around the stagnation point, given that the substrate can be considered homogeneous [103], [104], [92], [93]. In the current work, initial deposition rates are estimated based on the convective diffusion equation with perfect sink boundary condition. However, this equation is typically very stiff in a numerical sense when short ranged colloidal forces and hydrodynamic particle-wall interactions are taken into account, and solving it requires a special coordinate transformation. The current work employs the coordinate transformation proposed by [4]. In the notation of Weronski et al. the governing convective diffusion equation reads  exp(2𝑥𝑥1)𝐹𝐹1(𝑥𝑥1) 𝜕𝜕𝑦𝑦1𝜕𝜕𝜕𝜕 = 𝜕𝜕2𝑦𝑦1𝜕𝜕𝑥𝑥12 + 𝜕𝜕𝑦𝑦1𝜕𝜕𝑥𝑥1 �𝜕𝜕𝑦𝑦1𝜕𝜕𝑥𝑥1 + 𝑏𝑏1(𝑥𝑥1)�+ 𝑐𝑐1(𝑥𝑥1), (4) where 𝑦𝑦1 = ln (𝑐𝑐) is the logarithm (naturalis) of the dimensionless particle concentration 𝑐𝑐 = 𝑛𝑛/𝑛𝑛b  with the bulk particle concentration 𝑛𝑛b, 𝜕𝜕 = 𝐷𝐷0𝑎𝑎2 𝑡𝑡 is the dimensionless time with the particle bulk diffusion coefficient 𝐷𝐷0 and the particle radius a, and  𝑥𝑥1 = ln (𝐻𝐻 − 𝐻𝐻1), (5) where 𝐻𝐻 = ℎ/𝑎𝑎 is the non-dimensional particle-interface gap width and H1 = h1/a is the non-dimensional gap width corresponding to the primary energy minimum. Expressions for the coefficients  F1(x1), b1(x1) and c1(x1)  in eq (4) can be found in [4]. Note that although Weronski et al. [4] only derived expressions for the steady state version of eq (4), the coefficient associated with the time derivative in eq (4) directly follows from the application of the coordinate transformation proposed by these authors. Eq (4) was solved by using the boundary conditions [4]  lim𝑥𝑥1→−∞ 𝜕𝜕2𝑦𝑦1𝜕𝜕𝑥𝑥12 = 0 (perfect sink) (6)  lim𝑥𝑥1→∞ 𝑦𝑦1 = 0, (7) and the initial condition 𝑦𝑦1 = 0 at 𝜕𝜕 = 0.  11 The method used for solving this boundary-initial value problem as well as all relevant simulation parameters are summarized in the Appendix A1. In general, equation (4) with conditions eq (6) and (7) predicts transient concentration profiles until a steady state concentration profile is reached after a certain time-period has passed [92], [93]. With known steady state solutions for 𝑐𝑐 and 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕, the normal component of the particle flux was determined from [4]  𝑗𝑗n(𝐻𝐻) = − 𝐷𝐷0𝑎𝑎𝑛𝑛b 𝐹𝐹1(𝐻𝐻) �𝑑𝑑𝜕𝜕𝑑𝑑𝜕𝜕 + [𝑃𝑃𝑅𝑅𝐹𝐹2(𝐻𝐻)(𝐻𝐻 + 1)2 − (𝐹𝐹𝜕𝜕(𝐻𝐻) + 𝐹𝐹𝑒𝑒)𝑐𝑐(𝐻𝐻)]�, (8) where   𝑃𝑃𝑅𝑅 = 𝛼𝛼r(𝑅𝑅𝑅𝑅, ℎ𝑅𝑅i) 𝑉𝑉∞𝑎𝑎3𝐷𝐷0𝑅𝑅𝑖𝑖2  (9) is the Péclet number, 𝐹𝐹𝜕𝜕 is the normal component of the (dimensionless) short ranged colloidal interaction force, 𝐹𝐹2 is a dimensionless hydrodynamic correction function and 𝐹𝐹𝑒𝑒 is the (dimensionless) external force due to gravity. All forces are made dimensionless by the factor a/kT. The colloidal force 𝐹𝐹𝜕𝜕 will be discussed in more detail in section 2.3.2 and expressions for 𝐹𝐹2 and 𝐹𝐹𝑒𝑒 can be found in [4]. Note however, that the expression for 𝐹𝐹𝑒𝑒  given in [4] assumes gravity to be directed away from the substrate while in the setup employed by the current work, gravity is directed towards the substrate. The deposition rate was determined as  𝑗𝑗dep =  −𝑗𝑗n(𝐻𝐻𝑚𝑚𝑖𝑖𝑛𝑛), (10) where 𝐻𝐻min is the lower integration limit for H. For the case where the colloidal force 𝐹𝐹c is purely attractive, one obtains the mass transfer limited deposition rate, which is denoted by 𝑗𝑗dep0. Chemical conditions for which the colloidal force 𝐹𝐹c is purely attractive are called favourable conditions for deposition and chemical conditions for which the colloidal force 𝐹𝐹c also has a repulsive component are called unfavourable chemical conditions. Finally, we define the time period 𝑡𝑡steady from initialization of the flow until establishment of steady state concentration profiles as  12  Λ�𝑡𝑡steady � ≡ ∫ �𝑐𝑐�𝑥𝑥1, 𝑡𝑡steady � − 𝑐𝑐steady(𝑥𝑥1)�𝑑𝑑𝑥𝑥1∞−∞∫ 𝑐𝑐steady(𝑥𝑥1)𝑑𝑑𝑥𝑥1∞−∞ ≤ 0.1%, (11) where 𝑐𝑐(𝑥𝑥1, 𝑡𝑡) is the transient solution and  𝑐𝑐steady is the corresponding steady state solution of equations (4),(6), (7), respectively.  2.2.2 Fabrication of flow cell and substrate preparation  The flow cell was fabricated by using standard PDMS soft lithography. Micromold masters for the flow cell were produced in a cleanroom using standard SU-8 (MicroChem, Newton, MA, US) photolithography with typical feature heights of 100 µm.  A negative of the mask that was used for the process is shown in Figure 1a. In order to achieve the desired flow chamber dimensions (see Figure 1b) and inlet hole, small disk shaped parts with a pin in the center were machined from aluminum and bonded on the micromold. In the next fabrication step, Sylgard 184 PDMS base and hardener were combined at a base : hardener ratio 10:1 and subsequently mixed in two steps: First manual mixing was performed for 3 minutes by using a clean metal spatula. Subsequently, the pre-mixed PDMS was put in a AR250 (Thinky) vortex mixer (3 minutes mixing and 1:30 minutes degassing). The mixture was poured over the mold master and then put in a desiccator for further degassing. Subsequently, the PDMS covered mold was put in an oven to cure at 65 °C for 5 hrs. Finally, the cured PDMS was peeled from the mold, cut into shape and the outlet hole was punched. Substrates were prepared by pouring Sylgard 184 in 10:1 (base : hardener, same mixing procedure) over a microscope slide and subsequent degassing. The work was performed under a laminar flow hood to minimize dust deposition on the substrate. The substrates were then cured at 65 °C for 5 hrs. The thickness of the PDMS layer over the glass slide varied between 0.5 – 1 mm. At such film thickness, the cross-linking of PDMS is unaffected by the presence of the microscope slide [105]. Subsequently, the substrates were put in sealed petri dishes and stored in a dark cabinet until they were used for a deposition experiment. None of the substrates used for the experiments in this work was older than 3 days. We did not observe any correlation between substrate age and deposition rates. Although the surface roughness of the substrates used in this work was not characterized experimentally, cross-linked PDMS films are known to be very smooth with a typical surface roughness of Ra < 2.5 nm [106], [107], 13 [108]. It is worth mentioning that cross-linked PDMS is nanoporous and water as well as small molecules can be absorbed by PDMS [109], [110], [106]. The impinging jet cell was completed by placing the flow cell on the substrate and subsequent mechanical clamping of these two parts as shown in Figure 1d. By assuming uniform compression of the PDMS device and by treating the clamp-plates as perfectly rigid, the typical resulting compression of the deposition chamber height h due to clamping was estimated to be close to 1%.  2.2.3 Preparation of fluids An aqueous suspension of monodisperse, fluorescent polystyrene particles with a nominal mean diameter of 1 µm were purchased (Thermo Scientific, Waltham, MA, US). This suspension was dispersed in deionized water at 1% solid content and, according to the manufacturer, contained a small concentration in the order of 0.01 wt% of a surfactant consisting of a 12-carbon chain with a sulfate group at one end. Small amounts of the suspension provided by the manufacturer were first diluted with distilled water and then washed before they were further diluted by water and saline solutions to obtain the desired chemical conditions for the experiment. Before washing, a small amount of the suspension provided by the manufacturer was diluted with Milli-Q ultrapure water to obtain a solid concentration of 0.01 wt%.  Subsequently, the diluted suspension was washed by using an accuspin 1R centrifuge (Fisher Scientific). Several washing steps were performed and after each step, the electrophoretic mobility of the suspension was measured by using a Zetasizer nano ZS 90. After three washing steps, the mobility decreased from -6.3 µm∙cm/(V∙s) to -3.5 µm∙cm/(V∙s)  and subsequent washing steps did not result in a further change of the mobility. Thus it was concluded that all surfactant up to an insignificant amount had been removed by the three step washing procedure. Some of the particles were accidentally removed from the suspension during the washing process. Since initial deposition rates are directly proportional to the particle concentration, the particle concentration had to be determined before the washed particle suspension could be used to prepare the suspensions used for the experiments. Therefore small amounts of Tween 20 surfactant was added to samples of the washed suspension, which were then injected into a straight microchannel with known volume. The amount of surfactant added was kept small enough so that it had a negligible influence on the particle concentration to be measured. However, the surfactant did prevent particles from depositing in the concentration 14 measurement channel. The microchannel used for concentration measurement was 400 µm wide and 11 µm deep.  For each sample, 60 particle images were acquired as the suspension was flowing through the channel by using the Eclipse Ti microscope (Nikon) and Imager ProX2M camera (LaVision). The time between two image captures was large enough for all particles in the previous image to exit the field of view before the next image was taken. Subsequently, particles in each image were counted and the particle concentration was calculated using a custom Matlab script based on digital image processing methods that will be described in section 2.2.5.4 of this thesis. In order to minimize Poisson noise, the particle counts determined from all 60 images per sample were averaged. Furthermore, it was ensured that the depth of field exceeded the channel height so that all particles in the channel appeared approximately in focus on the image.  With increasing particle concentration and channel height, the probability for the view on a particle to be blocked by another particle increases. However, it was ensured through numerical experiments that the concentration of particles and the channel height were low enough so that this effect could be neglected.  By using the procedure described above, 2 liters of washed particle suspension with a solid concentration of 0.0065 wt% was prepared. The washed particle suspension was stored in a Nalgene bottle in a refrigerator at 5°C. It was then used to prepare all of the suspensions used in this work. All experiments were conducted during the course of two months.  During this time, the stability of the washed suspension was monitored by performing visual inspections with our microscope setup (described further below). Particle agglomeration was not observed during the visual inspections However, visual inspection of the Nalgene container walls (using a microscope) showed that particles deposited to the walls of the Nalgene container and, as a result, the concentration of dispersed particles in the container slightly decreased with increasing storage time. Hence, the concentration of the washed particle suspension was measured each time before it was used to prepare the suspensions for the deposition experiments. Before each particle deposition experiment, the suspensions used for the experiment were prepared by diluting the washed suspension further with biological certified sodium chloride (Fisher Scientific) saline solutions. The final solid concentration used for all experiments in this work was 0.001 wt% (𝑛𝑛b = 1.91 ∙ 107 1cm3).   2.2.4 Particle characterization The electrophoretic mobility of the particle suspensions used in this work was measured by using a Zetasizer Nano ZS-90 (Malvern) and DTS-1061 cuvettes. The Zetasizer measures the electrophoretic mobility of particles based on the electrophoretic light scattering technique. For 15 high conductivity samples, the Zetasizer measures the mean electrophoretic mobility of the ensemble of particles in a given test volume. For low conductivity samples (here for 𝐶𝐶𝑁𝑁𝑎𝑎𝑁𝑁𝑁𝑁 ≤10−2𝑀𝑀), the Zetasizer provides the distribution of electrophoretic mobilities in the test volume, which is characterized by the corresponding standard deviation.  The mobility was measured for different salt concentrations. For each salt concentration, three suspensions were prepared and subsequently one sample of each suspension was characterized three times. The pH of all samples was measured before starting the characterization by using an In-lab-flex micro pH electrode (Mettler Toledo). The pH of all samples varied between 6.3 and 6.6. The repeatability of the mean mobility measurement was excellent (<5% deviation from mean mobility across all measurements) for all salt concentrations except for CNaCl = 0.5 M. At this salt concentration, the mobility was very low and it was difficult to measure low mobilities accurately.  2.2.5 Experimental setup and procedure The experimental procedure used in this work can be subdivided in 3 steps 2.2.5.1 Assembling, filling and calibration of the impinging jet cell The flow cell was formed by clamping the PDMS device to the substrate as shown in Figure 1d. The impinging jet cell was filled with ultra-pure water; then the cell was connected to Tygon tubing that has also been filled with the same ultra-pure water.  AFD steel needle tips served as interface between the impinging jet cell and tubing as indicated by the filled circles in Figure 2a. The needle tips were also used to connect the syringe pump to the setup. Before each experiment, the flow resistance of the impinging jet cell was determined by using the setup shown in Figure 2a. A syringe-pump (KD Scientific) drove fluid at a flow rate of 0.15 ml/min through the device while the pressure drop was measured by a ASDXRRX015PDAA5 differential pressure sensor (Honeywell). A typical pressure drop across the device at this flow rate was 360mbar +/- 20%. 2.2.5.2 Particle deposition experiment Once the desired pressure drop over the device was determined, the flow was stopped and the syringe pump was exchanged for a pressure source and a reservoir containing the particle suspension as shown in Figure 2b. Instead of driving the flow with a syringe pump which allows specifying flow rate in a direct manner, the flow was driven by a pressure source in order to 16 avoid unwanted pressure fluctuations caused by the stepper motor of the syringe pump [111]. The flow cell was placed on the stage of an inverted epi-fluorescence microscope equipped with CCD camera. The flow-rate was set to Q = 0.15 ml/min (corresponds to Re = 0.75 and Pe = 1.3∙10-4) by adjusting the applied pressure until the desired pressure drop over the device was achieved. The CCD camera was then set to display images of the particles as they were transported towards the substrate by the flow and the field of view (0.658 mm × 0.877 mm) was adjusted until it contained the stagnation point, which is easily identified from the displayed image sequence. The first particle that deposited on the substrate was used as guide for focusing on the substrate, which was otherwise difficult to identify in the images. After initialization of the flow introducing the particle suspension into the flow cell, the particle concentration profile and the associated deposition rate are expected to change over a certain time period until a steady state is reached and the concentration profile as well as the deposition rate become constant. The current work aims at comparing measured deposition rates to predicted deposition rates under steady state conditions. According to eq (11), steady state conditions were expected 15 minutes after initialization of the flow. Thus, image acquisition was initialized 15 minutes after initialization of the flow and a series of images was recorded at a constant frame rate of 0.1 fps while particles deposited on the substrate. After the experiment, these images were evaluated by a custom Matlab script that determines the initial particle deposition rate from the image sequence. The image processing procedure used in this work is described in section 2.2.5.4. After 15-30 minutes, the flow and the acquisition was stopped, then the whole system was flushed with particle free, ultra-pure water. 2.2.5.3 Removal of particles from substrate In the current work, several deposition experiments were performed on the same substrate. Before a substrate could be re-used, deposited particles had to be removed from the substrate. In order to do so, the impinging jet cell (filled with particle-free fluid) was disconnected from the tubing and carefully removed from the clamp without displacing the PDMS flow cell relative to the substrate.  Particles were removed from the substrate by passing a liquid-air interface over the substrate [112], [113], [114]. This was achieved by pressing one end of the flow cell to the substrate and lifting the other end by using tweezers as shown in Figure 2c. Subsequently, the substrate was briefly (< 10 seconds) dipped into a beaker that was immersed in a sonicator bath (Figure 2d). The beaker contained the same ultra-pure water that was used to prepare all other fluids in this work and the water was replaced after each use. The particle removal procedure described here removed the majority of deposited particles from the substrate without altering the position 17 of the flow cell relative to the substrate. The small number of particles (<100) that remained on the substrate after the cleaning procedure were expected to have negligible influence on the next deposition experiment as these remaining particles covered less than 0.02% of the substrate area in the field of view. After each particle removal procedure, the water in the beaker was replaced by fresh, ultra-pure water. a)  b)  d)  Figure 2: Schematic of the experimental setup a) for the calibration step and b) for the particle deposition experiment. The thin lines represent Tygon tubing and the black dots to steel needle tips that serve as tubing-tubing interface as well as flow-cell-tubing interface; c) schematic of particle removal from the substrate by passing a moving contact line over the substrate; d) schematic of particle removal from the substrate by using a sonicator. 2.2.5.4 Image acquisition and particle tracking All images were acquired under identical conditions. The optical system used in this work consisted of an Eclipse Ti inverted epi-fluorescence microscope (Nikon) equipped with a 20x Plan Fluor NA = 0.5 objective lens (Nikon), an imager ProX2M (LaVision) CCD camera and a 41002-TRITC filter cube (Chroma Technology Corp). The excitation light was provided by Intensilight (Nikon) light source that was set to the lowest intensity possible (ND = 1) in order to minimize photobleaching during the course of an experiment. The framerate was 0.1 frames per second (fps) and the exposure time was 600 ms for all experiments. In the obtained 14 bit gray-scale images with a resolution of 1600 pixel  × 1200 pixel, particles appear as bright and the background as dark. The pixel size was 0.548 µm/pixel and the stagnation point was in the center of the images so that the area covered by the field of view was within r/Ri < 0.5 where the approximations equations (2)-(3) are valid. The intensity distribution of individual particles    c)  18 obtained from the optical system used in this work is well approximated by a Gaussian (see section 3.4.1). The image processing and particle tracking procedure used in this work follows classical methodology [115]. All images were smoothed by using a 3 x 3 pixel Gaussian blur filter (standard deviation 0.5 pixel) and all intensity values below 550 (au) were set to zero by using Matlab (Matworks). This particular intensity threshold value was chosen to remove the dark current and particles that are strongly out of focus from the images. Next, the positions of particles corresponding to local intensity maxima were detected by using the Matlab function imregionalmax. Subsequently, subpixel accuracy was achieved by finding the brightness weighted centroid for each particle as recommended in [115]. It will be demonstrated in section 2.3.1 that this method allows determining particle positions with an accuracy of +/- 0.1 pixel.Note, that subpixel accuracy can also be achieved by using other methods. For example, particle images can be interpolated by using different interpolants such as Gaussians or parabolic functions [116]. Once a specific interpolant has been determined, a subpixel estimate of the particle positions can be derived from the interpolant.  The current work utilizes the particle tracking method by [117]. In brief, a proximity matrix containing Gaussian weighted distances for all possible links between located particles of an image pair is constructed. The singular value decomposition (SVD) of the proximity matrix then contains the information of a set of i particle links with length δi, which effectively represents a minimum length mapping [117]. The only free parameter in this tracking algorithm is the standard deviation (= 24 pixel) used for creating the proximity matrix. In order to maximize the success rate for tracking a deposited (immobile) particle, a simple proximity search is applied before the SVD-based algorithm is used: For each particle at image position ?⃗?𝑋𝑖𝑖 in image one, we assign the closest neighbor in image two in a 3 x 3 pixel neighborhood around ?⃗?𝑋𝑖𝑖 as a match. As it will be shown further below, this pre-tracking step does not introduce a significant number of bad matches at the low image densities of moving particles given in this work. Each particle tracking step is finalized by simple outlier detection where all displacements larger 90 pixel are discarded as invalid matches. 19 2.3 Results and discussion 2.3.1 Characterization of particle tracking The performance of our image processing and tracking procedure was tested by using computer generated particle image pairs that closely resemble a pair of sequential experimental images.  As shown in section 3.4.1, images of individual particles at any distance from the object plane are quantitatively captured by a Gaussian image intensity distribution. An image of many particles is then obtained based on the simplifying assumption that images of individual particles at different locations in the physical domain are additive. Furthermore, for each pixel with intensity I a random intensity value was first sampled from a normal distribution with standard deviation 𝜎𝜎𝑁𝑁(𝐼𝐼) and subsequently added to the intensity value of that pixel to represent camera noise. The noise characteristic 𝜎𝜎𝑁𝑁(𝐼𝐼av) of the camera had been determined experimentally from uniform grey images with average intensity 𝐼𝐼av. For a given set of particle coordinates in the physical domain, the model outlined here produced synthetic particle images that match the corresponding experimental particle images well in terms of particle image size, particle image intensity, depth of view, pixel resolution and noise. A qualitative comparison of a particle image obtained from our optical setup and a corresponding synthetic particle image is shown in Figure 3. Sets of particle coordinates in a virtual physical domain above the substrate were produced by a simple dynamic simulation. First, a realistic number 𝑛𝑛b𝑉𝑉𝑉𝑉 of mobile particles was distributed inside the virtual physical domain with Volume 𝑉𝑉𝑉𝑉 starting 1 µm above the substrate. In addition to the mobile particles, a certain number of virtual deposited particles are distributed across the substrate. All particle positions were generated using the Matlab function rand to achieve the desired concentration of randomly distributed particles. Next, the positions of the mobile particles were advanced over a total time period of Δt = 10 seconds (with ∆𝑡𝑡step = 0.1 𝑠𝑠 time steps) by using a simple explicit Euler forward time stepping scheme. At each time step, each coordinate of the position of a particle was advanced by a portion resulting from convection based on equations (2)-(3) and a portion due to Brownian motion that was sampled from a normal distribution with standard deviation �2𝐷𝐷0∆𝑡𝑡step. Particles penetrating the virtual substrate were forced back to the substrate after each time step. Initial and final positions of the particles were stored in an array of reference trajectories. A synthetic particle image pair was then generated from the initial and final coordinates of all particles in the virtual domain. 20 Subsequently, the tracking algorithm described in the previous section was applied to find particle trajectories in the synthetic image pairs. Particle tracking is most challenging at high particle image densities. Hence, the performance of the algorithm was tested for the worst case scenario where 2000 deposited particles have already deposited on the substrate. The number of deposited particles did not exceed 2000 in any of the laboratory experiments performed in this work. The tracking efficiency for mobile particles 𝛽𝛽𝑚𝑚(𝑧𝑧e) is defined as the number of successful matches between tracking results and reference trajectories of mobile particles in z < ze divided by the number of reference trajectories of mobile particles in z < ze. A match between an identified trajectory and a reference trajectory is considered successful if each coordinate of the tracking result falls within a 3x3 pixel window around the corresponding coordinate of the reference trajectory. The tracking efficiency for deposited particles 𝛽𝛽𝑑𝑑 is defined as the number of successful matches between tracking results and reference trajectories of deposited particles divided by the number of deposited particles. The ratio of the number of invalid trajectories and the total number trajectories of mobile particles is represented by 𝛽𝛽𝑏𝑏.  Evaluation of 1000 image pairs gave a fraction of invalid matches of only 𝛽𝛽𝑏𝑏 = 0.51%. The tracking efficiency for deposited particles is 𝛽𝛽𝑑𝑑 = 98.1%. It may seem surprising that 𝛽𝛽𝑑𝑑 <100%, considering the simple pre-tracking procedure described in section 2.2.5.4. However, in our computer generated particle images, particles were allowed to overlap and a small fraction of the deposited particles is expected to be too close to a neighboring particle in order to resolve the individual particle positions. Figure 4a shows the tracking efficiency for mobile particles 𝛽𝛽𝑚𝑚(𝑧𝑧e). For ze< 10 µm, we find 𝛽𝛽𝑚𝑚 > 90%. The decreasing trend of 𝛽𝛽𝑚𝑚 in 2 µm < ze< 10 µm is expected, as particles appear larger and less bright in an image the further away they are from the substrate. In other words, tracking of partially out-of-focus particles is more challenging than tracking of in-focus particles. The sudden change of the slope in Figure 4a around ze = 10 µm indicates that the relevant depth of field (after image processing) is close to 10 µm. Since the intensity distributions of individual particles in the synthetic images as a function of the particle distance from the object plane closely matches the corresponding particle image distribution of the experimental images, the depth of field in the experiments should also be close to 10 µm.  The precision of the detected particle location is evaluated by taking the difference between the reference particle position and the found particle positions for all successfully matched trajectory pairs (tracking results and reference trajectory). The resulting differences in the 21 horizontal components and in the vertical components of the particle image positions are all combined into one histogram for all trajectory pairs. For deposited particles (Figure 4b and for mobile particles that are within z < 3 µm (Figure 4c), the precision is about +/- 0.1 pixel (+/-54.8 nm) as expected [115]. For all valid trajectories of mobile particles, an overall precision of about +/-0.25 Pixel (+/-137  nm)  is achieved (Figure 4d).   a) b)  Figure 3: Excerpts of a particle image a) obtained from optical setup and b) corresponding computer generated image. Both images show the same field of view and consist of the same number of pixels and are based on the same particle concentration. Furthermore, both images are displayed by using the same color-map.   22 a) b)      c)  d)  Figure 4: a) Tracking efficiencies 𝛽𝛽𝑚𝑚(𝑧𝑧e) and 𝛽𝛽𝑑𝑑  and fraction of invalid trajectories 𝛽𝛽𝑏𝑏; b) precision for deposited particles; c) precision valid trajectories of mobile particles within z < 3 µm; d) Precision for all valid trajectories 2.3.2 Surface interaction potentials and predicted concentration profiles The deposition of particles is strongly influenced by short range colloidal forces ?⃗?𝐹c between substrate and particles. These forces (here made dimensionless by the factor a/kT) are considered conservative, i.e.   ?⃗?𝐹c = − 𝑎𝑎𝑘𝑘𝑘𝑘 ∇�⃗ 𝐸𝐸tot, (12) where 𝐸𝐸tot represents the total surface interaction energy (or potential). In commonly used models, only colloidal forces   𝐹𝐹c = − 𝑎𝑎𝑘𝑘𝑘𝑘 𝜕𝜕𝐸𝐸tot𝜕𝜕ℎ  (13) acting in the direction normal to the substrate are considered. The colloidal forces are most often modeled within the framework of the DLVO theory [118], which assumes that the total surface interaction energy  23  𝐸𝐸totDLVO = 𝐸𝐸EDL + 𝐸𝐸LW (14) is comprised of the sum of the contributions of the electrical double layer interaction energy 𝐸𝐸EDL and the Lifshitz-van der Waals interaction energy 𝐸𝐸LW. The electrostatic double layer interaction energy describes the force between two charged interfaces in an aqueous electrolyte solution that results from the interaction of charges (i.e. ions) that accumulate near the charged interfaces. The Lifshitz-van der Waals interaction energy captures interactions between the fluctuating component of electrical dipoles of interacting molecules. More recently it has been suggested [119] that the extended DLVO (xDLVO) theory according to [120] may provide a more complete description of the short ranged colloidal forces. In the xDLVO theory, the total interaction energy   𝐸𝐸totxDLVO = 𝐸𝐸EDL + 𝐸𝐸LW + 𝐸𝐸AB (15) contains one additional contribution, the acid base interaction energy 𝐸𝐸AB, resulting from differences between the polar properties of the constituents. Depending on the materials, the acid-base interaction force can be repulsive or attractive, in which case it is often referred to as hydration interaction and hydrophobic interaction, respectively. The extended DLVO theory has been applied in numerous studies to the adhesion and deposition of bacteria [121], [122], proteins [108] and model colloids [119] on hydrophilic and hydrophobic substrates.  2.3.2.1 DLVO interaction potentials  The current work utilizes the expression for the electrical double layer interaction potential based on the linear superposition approximation [123], [93], [124], [4]   𝐸𝐸EDL =  4𝜋𝜋𝜀𝜀0𝜀𝜀𝑎𝑎 �𝑘𝑘𝑘𝑘𝑅𝑅c�2𝑌𝑌p𝑌𝑌i exp(−𝜅𝜅𝑎𝑎𝐻𝐻), (16) where   𝑌𝑌p = 8tanh (𝜉𝜉p𝑒𝑒c4𝑘𝑘𝑘𝑘 )1+�1−2𝜅𝜅𝜅𝜅+1(𝜅𝜅𝜅𝜅+1)2𝑡𝑡𝑎𝑎𝑛𝑛ℎ2(𝜉𝜉p𝑒𝑒c4𝑘𝑘𝑘𝑘 ) , (17) and 24  𝑌𝑌𝑖𝑖 = 4 tanh (𝜉𝜉i𝑅𝑅c4𝑘𝑘𝑘𝑘) (18) are the effective (constant) surface potentials of the particle and PDMS substrate, respectively, as given by [125] with the particle zeta potential 𝜉𝜉p and the PDMS zeta potential 𝜉𝜉i, and  𝜅𝜅 = �2𝑅𝑅c2𝑁𝑁A𝐶𝐶NaCl𝜀𝜀0𝜀𝜀𝑘𝑘𝑘𝑘 (19) is the inverse Debye length with the Avogadro constant 𝑁𝑁A and the electrolyte concentration 𝐶𝐶NaCl. As before, H = h/a in eq (16) represents the ratio of the particle interface gap h and the particle radius a. Eq (16) is technically only valid for 𝜅𝜅ℎ > 1 and it tends to overpredict the electrical double layer interaction force for 𝜅𝜅ℎ < 1 as compared to exact solutions of the nonlinear Poisson Boltzmann equation with constant potential boundary condition, especially when the surface potential of substrate and particle differ [126]. However, in the region 𝜅𝜅ℎ < 1, the interaction potential assumes very large absolute values and thus, a rather large uncertainty of 𝐸𝐸EDL (resulting from inadequacy of the model) can be tolerated in this region [126].  The particle zeta potentials were derived from the measured electrophoretic mobilities (described in section 2.2.4 by solving Ohshima’s expression for the electrophoretic mobility [127] which provides a good approximation for the range of zeta potentials and double layer thicknesses met in the current work. All samples showed a negative electrophoretic mobility, which indicates the presence of negative surface charge on the particle surfaces. According to the manufacturer, the surface charge most likely originates from sulfate groups on the surface of the particles. The particle zeta potentials are shown as solid circles in Figure 5, where the error bars represent the standard deviation of the zeta potential across one sample. The zeta potential of the PDMS substrates is estimated by using the empirical correlation equation proposed by [1] (shown as dashed line in Figure 5). The correlation equation was derived from experiments on Sylgard 184 PDMS (Dow Corning 10/1 (base/hardener)) that were performed at 6.5 < pH < 7. The negative zeta potential of PDMS indicates the presence of negative surface charges on PDMS surfaces. The origin of the surface charge on hydrophobic, chemically inert, PDMS is still under debate. It seems commonly accepted that the surface of PDMS is mainly populated by the dimethyl groups −Si(CH3)2 − O −of PDMS [128], [129] (for more information on the chemical composition of cross-linked PDMS see [130]). These surface 25 groups render PDMS surfaces hydrophobic and chemically inert and should not give rise to a negative surface charge when exposed to water. It is currently believed that the negative surface charge at the PDMS-water interface originates from hydroxyl ion adsorption at the PDMS-water interface, which is facilitated by the structure of water near the PDMS-water interface [131].   Figure 5: Zeta Potentials as a function of salt concentration: Polystyrene particles (circles) and interpolant (solid line). The error bars (where available) represent the standard deviation of the particle zeta potential distribution across the particle population; PDMS zeta potential (dashed line) according to literature [1]. The pH of the all particle suspensions used in the current work was between 6.3 and 6.6. The present work employs Gregory’s expression for the retarded van der Waals interaction potential [132]   𝐸𝐸LW = −𝐴𝐴1326𝜕𝜕 � 𝜆𝜆r𝜆𝜆r+14𝜕𝜕�, (20) where 𝐴𝐴132 is the unretarded Hamaker constant for a particle (1) interacting with the PDMS substrate (2) through water (3) and 𝜆𝜆r = 𝜆𝜆/𝑎𝑎 is the characteristic wavelength (𝜆𝜆  = 100 nm) of the dispersion interaction scaled with the particle radius. Expression (20) is a good approximation for separation distances H < 0.2 [132]. The Hamaker constant 𝐴𝐴132 can be derived from the Hamaker constants 𝐴𝐴𝑖𝑖𝑖𝑖 ,i = 1,2,3 of material i interacting with the same material through vacuum by employing the commonly used approximation [92], [93]  𝐴𝐴132 = ��𝐴𝐴11 − �𝐴𝐴22���𝐴𝐴22 − �𝐴𝐴33�, (21) despite the fact that it may not be accurate, especially when two materials are interacting through highly polar medium such as water [133], [134], [93]. Some (by far not all) values for the Hamaker constants 𝐴𝐴𝑖𝑖𝑖𝑖 that have been reported or used in literature are compiled in Table 1. Depending on the choice of the values for 𝐴𝐴𝑖𝑖𝑖𝑖, Hamaker constants between 𝐴𝐴132 = −1.42 ∙10−21𝐽𝐽 and 𝐴𝐴132 = 1.91 ∙ 10−21 J can be obtained, where the limits correspond to either weak 26 repulsive and weak attractive van der Waals interaction, respectively. A significantly larger value of the Hamaker constant (𝐴𝐴132 = 5.2 ∙ 10−20𝐽𝐽) for the polystyrene-water-(Sylgard 184) PDMS system has been determined from direct force measurements between a polystyrene microsphere and a PDMS substrate in aqueous saline solution by using an atomic force microscope [135]. We conclude that severe discrepancies exist between different reported values for the Hamaker constant 𝐴𝐴132 for the polystyrene-water-PDMS system. Until more accurate data is available we simply choose some of the more commonly used values for the Hamaker constants, i.e. 𝐴𝐴11 = 7.9 ∙ 10−20𝐽𝐽 (for polystyrene) [136], [108], 𝐴𝐴22 = 4.4 ∙ 10−20𝐽𝐽 (for PDMS) [137], [138], [139], and 𝐴𝐴33 = 3.7 ∙ 10−20𝐽𝐽 (for water) [140], [136], [138], [141], [142]. Inserting these values in eq (21) yields 𝐴𝐴132 = 1.54 ∙ 10−21𝐽𝐽, which indicates rather weak attractive van der Waals forces. This particular choice for the Hamaker constant will facilitate the discussion of our experimental results in the remainder of this work. Admittedly, the chosen value may seem arbitrary considering the large range of reported values. Thus, we discuss implications of other choices for 𝐴𝐴132 on the conclusions that will be derived from the remainder of this chapter in Appendix A2. Figure 6 shows the total surface interaction potential 𝐸𝐸totDLVO based on the DLVO theory for different concentrations of sodium chloride (at pH 6.3-6.6). Figure 6a predicts substantial energy barriers >500 kT are predicted for all salt concentrations under consideration and Figure 6b indicates the presence of shallow secondary minima for 10 mM and 100 mM sodium chloride. a) b)  Figure 6: DLVO interaction potential for different concentrations of sodium chloride: a) energy barriers for CNaCl < 500 mM, b) secondary minima for CNaCl = 1 mM, 10 mM and 100 mM. H = h/a represents the ratio of the particle interface gap h and the particle radius a = 500 nm. 2.3.2.2 Extended DLVO interaction potentials Where the DLVO theory discussed in the previous subsection only considers Lifshitz van der vaals interactions resulting from the interaction of apolar molecules, and  27 Van Oss [120] derived expressions for the free energy of interaction per unit area resulting from Lewis acid-base interactions ∆𝐸𝐸𝑑𝑑AB between two infinite plates (1,2) immersed in medium (3). For two infinite plates at (minimum) separation 𝑑𝑑0 = 0.158 𝑛𝑛𝑛𝑛 , the interaction energies  ∆𝐸𝐸𝑑𝑑0AB = 2��𝛾𝛾1+𝛾𝛾3− + �𝛾𝛾2+𝛾𝛾3− + �𝛾𝛾1−𝛾𝛾3+ + �𝛾𝛾2−𝛾𝛾3+ − �𝛾𝛾1+𝛾𝛾2− − �𝛾𝛾1−𝛾𝛾2+ − 2�𝛾𝛾3+𝛾𝛾3−�, (22) the (non-additive) electron acceptor (acid) 𝛾𝛾𝑖𝑖+  and electron-donor (base) 𝛾𝛾𝑖𝑖− components of the Lewis acid-base component of the surface tension.  Application of the model eq (22) requires knowledge of the surface tension parameters, 𝛾𝛾𝑖𝑖+ and 𝛾𝛾𝑖𝑖−, which can be obtained from contact angle measurement by using three probe liquids with known surface tension parameters by employing the extended  Young’s equation (see i.e. [120]). The acid base component of the interaction potential   𝐸𝐸AB = 2𝜋𝜋𝑎𝑎𝜆𝜆0𝐸𝐸𝑑𝑑0ABexp �𝑑𝑑0 − 𝑎𝑎𝐻𝐻𝜆𝜆0 � (23) with the characteristic decay length 𝜆𝜆0 was derived from the corresponding expression for plate-plate interaction potential [120] by using the Deryaguin approximation [143]. The Deryaguin approximation is valid for 𝑎𝑎/𝜆𝜆0 > 5 and H << 1.  Assigning an appropriate value for the decay length 𝜆𝜆0 ideally requires the direct measurement of the interaction forces for the given material combination by using a surface force apparatus [136] or atomic force microscope [144]. According to van Oss [120], appropriate values are in the range 0.6 𝑛𝑛𝑛𝑛 <𝜆𝜆0 < 1 𝑛𝑛𝑛𝑛 and Israelachvili et al. suggests 0.3 𝑛𝑛𝑛𝑛 < 𝜆𝜆0 < 2 𝑛𝑛𝑛𝑛  [145] while specifically reporting 𝜆𝜆0 = 1.6 𝑛𝑛𝑛𝑛 for the interaction of two (un-crosslinked) PDMS monolayers. Note, that much larger decay lengths can be observed for the interaction of hydrophobic substrates when the fluids have not been degassed [145]. Newby et al. [108] used 𝜆𝜆0 = 0.6 𝑛𝑛𝑛𝑛 to model the AB-interaction between bovine serum albumin and crosslinked PDMS. In the current work, we arbitrarily set 𝜆𝜆0 = 1 𝑛𝑛𝑛𝑛. Measured values for the surface tension parameters 𝛾𝛾𝑖𝑖+ and 𝛾𝛾𝑖𝑖− were taken from literature and some values found in literature are summarized in Table 1. For all possible combinations of parameters sets in Table 1, represent the interaction energy eq (22) at contact ranges from to −7.5 ∙ 104 kT to −5 ∙ 104 kT depending on the combination of parameter sets (the parameter set of [108] was chosen arbitrarily.) Thus, for the polystyrene-water-PDMS system, eqs (22) and (23) represent an attractive interaction (sometimes termed hydrophobic interaction).  28 In order to gain a qualitative understanding of the origin of the attractive force between two hydrophobic interfaces in aqueous media, consider a hydrophobic interface in an aqueous medium. Far away from the interface, the water will have an unperturbed structure resulting from hydrogen bonds between the highly polar water molecules. Near the hydrophobic interface this natural structure of water is perturbed and some of the hydrogen bonds between water molecules are lost. This perturbation of the water structure requires work to be performed when the hydrophobic interface is placed in the aqueous medium and thus it increases the free energy of the system. When another hydrophobic interface is introduced into the same aqueous medium, there is a tendency for these interfaces to aggregate (i.e. attract each other) in order to reduce the hydrophobic interface area that is exposed to water as this will lead to less water being perturbed and thus a smaller free energy of the system. Figure 7 shows the resulting extended DLVO interaction potentials for different concentrations of sodium chloride. The additional hydrophobic interaction mainly decreases the height of the energy barrier for all salt concentrations. Significant energy barriers > 100 kT are still predicted for salt concentrations CNaCl < 10 mM, no energy barrier is predicted for CNaCl = 100 mM. Very shallow secondary minima can be observed in Figure 7b for CNaCl = 1 mM and 10 mM.   29 Table 1: Surface energy data. All surface energies are in mJ/m2, the unretarded Hamaker constants Aii are displayed in 10−20𝐽𝐽. Material Aii 𝛾𝛾𝑖𝑖+ 𝛾𝛾𝑖𝑖− Reference PDMS  4.31 0 3.05 [108] PDMS  3.46 0.64 0.32 [146] PDMS  4.4 - - [137] [138] [139]  PDMS  4.5 - - [141] [147] PS 7.89 0.08 0.15 [108]  PS 8.34 0.46 2.22 [148]  PS 7.98 1.3 3.1 [149] PS 7.07 0.57 5.27 [119] PS 6.6-7.9   [136] water 4.10 25.5 25.5 [120] [146] [108] [148] [149] [119]  water 3.7 - - [140] [136] [138] [141] [142]  a)  b)  Figure 7: Extended DLVO interaction potential for different concentrations of sodium chloride: a) energy barriers for CNaCl < 100 mM, b) secondary minima for CNaCl = 1 mM and 10 mM; The curves corresponding to 100 mM and 500 mM are indistinguishable. H = h/a represents the ratio of the particle interface gap h and the particle radius a = 500 nm.  30 2.3.2.3 Concentration profiles and particle flux The current section briefly discusses some aspects of the expected concentration profiles and the particle flux that are obtained from numerical solutions of eq (4) based on the extended DLVO interaction potential eq (15). Figure 8a and b show expected steady state particle concentration profiles and the normal component of the particle flux in the vicinity of the PDMS substrate for different salt concentrations, respectively. When conditions are unfavourable for deposition (CNaCl < 0.1 M) theory predicts a sudden decrease of the particle concentration and the normal component of the particle flux down to zero at separation distances that are associated with the energy barrier in the corresponding interaction potentials in Figure 7. In other words, under unfavourable conditions there is a region close to the substrate that is inaccessible to the particles. For each salt concentration associated with unfavourable conditions (CNaCl < 0.1 M) a minimum separation distance 𝐻𝐻0 is implicitly defined as 𝑐𝑐(𝐻𝐻0) =10−10. Furthermore, the concentration profiles for CNaCl < 0.1 M show a characteristic concentration peak (2.5 < c < 1) in front of the energy barrier extending up to about ℎ ≈ 15 𝜇𝜇𝑛𝑛. This peak results from the balance between the normal particle flux due to gravity and convection which transports particles towards the energy barrier and the radial particle flux due to convection, which washes the repelled particles away from the stagnation point1 [150]. Also note, that an additional small concentration peak at gap distances associated with the secondary minimum of the interaction potential is predicted for CNaCl = 0.01 M as it can be seen in the inset of Figure 8a.  The concentration profiles and particle fluxes corresponding to favourable conditions CNaCl > 0.1 M were indistinguishable and thus, only the case CNaCl = 0.1 M is shown in Figure 8. When conditions are favourable for depositions, there is no energy barrier or secondary minimum and particles can readily pass through to the primary minimum (H0 = H1).  1 Note, that the model outlined in section 2.2.1 accounts for a lateral particle flux even though it is not explicitly mentioned. This becomes clear when following a detailed derivation of the governing equations as i.e. presented in [90] (pp 517-526) or in [100], [101], [89], [90]. In short, the governing equations are simplified forms of a particle transport equation which satisfies the conservation of particle mass. With a non-uniform normal component of the particle flux as shown in Figure 8b, mass conservation can only be satisfied by a non-zero lateral component of the particle flux.  31                                                 a)  b)  Figure 8: a) Steady state concentration profiles based on the extended DLVO interaction potential  eq (15). b) corresponding steady state normal component of the particle flux. H-H1 = (h-h1)/a represents the ratio of the gap width (h-h1) between particle and primary minimum and the particle radius a = 500 nm. The gap width for particles in the primary minimum is h1 = 0.158 nm.  2.3.3 Trajectory types and deposition criteria Let us now examine some interesting types of trajectories, denoted by  ?⃗?𝑋𝑖𝑖�𝑡𝑡𝑗𝑗� = �𝑋𝑋𝑖𝑖�𝑡𝑡𝑗𝑗� 𝑌𝑌𝑖𝑖�𝑡𝑡𝑗𝑗��𝑘𝑘 , observed during a laboratory experiment at CNaCl = 0.01 M. It is interesting to examine the displacement of particle i   ∆?⃗?𝑋𝑖𝑖�𝑡𝑡𝑗𝑗� = ?⃗?𝑋𝑖𝑖�𝑡𝑡𝑗𝑗� − ?⃗?𝑋𝑖𝑖�𝑡𝑡𝑗𝑗+𝑝𝑝� (24) between two frames acquired at time tj and tj+p, respectively. Note, that this corresponds to the displacement of particle I after the time period ∆𝑡𝑡 = 𝑝𝑝 ∙ 10 𝑠𝑠. Figure 9a shows a typical trajectory where the particle initially undergoes large displacements before it comes to rest. The displacement histograms (for p = 1) in Figure 9a show, that once the particle comes to rest, its position fluctuates within the precision limit of the particle tracking method (i.e. +/-0.1 Pixel). In other words, the particle shown in Figure 9a is deposited on the substrate. 32 Another interesting type of trajectory that can be frequently observed at low and intermediate salt concentrations (𝐶𝐶NaCl ≤ 10−2𝑀𝑀) is shown in Figure 9b and c (the video sequences corresponding to these trajectories can be found in the supplementary material).  A particle initially undergoing large displacements enters a random walk with a mean displacement close to zero. When observing the particle images by eye, a particle following a trajectory of this type can easily be misidentified as a deposited particle as it typically moves less than the size of a particle image over the course of 20 minutes. In order to find out if these trajectories are to be expected as a result of the low Péclet number conditions chosen for this work, the mean particle drift velocity   𝑉𝑉�p𝑖𝑖 = 1∆𝑡𝑡 �∑ ∆𝑋𝑋�⃗ 𝑖𝑖(𝑡𝑡𝑗𝑗)𝑁𝑁𝑁𝑁𝑗𝑗=1 �, (25) where p = 1 in eq (24) and N = N1 = 100 is the number of available samples, is compared to theory. For the argument to be made here, it is sufficient to assume that the direction of the mean drift particle velocity (25) is identical to the direction of the lateral component of the fluid velocity eq (2) 𝑉𝑉r𝑖𝑖(𝑧𝑧 = ?̅?𝑧𝑖𝑖, 𝑧𝑧 = 𝑎𝑎), where ?̅?𝑧𝑖𝑖 is the average lateral position of the particle i (while it is undergoing random walk). For now, i = 1,2 refers to the particle in Figure 9b and c, respectively. For the particle in Figure 9b,  𝑉𝑉�p1𝑉𝑉r1=0.029, where 𝑉𝑉r1 =1.5 pixel/∆𝑡𝑡. For the particle in Figure 9c,  𝑉𝑉�p2𝑉𝑉r2=0.013, where 𝑉𝑉r2 =1.4 pixel/∆𝑡𝑡. It can be concluded that these particles move, on average, significantly slower than the fluid velocity at a particle radius distance away from the substrate. The observed low mean particle drift velocity cannot be explained solely based on hydrodynamic particle-wall interactions (within the framework of continuum theory) as such low particle drift velocities would only be expected for particles that are much closer to the wall than the size of a water molecule [151]. On the other hand, if these particles were repelled by the energy barrier shown in Figure 6 or Figure 7 (CNaCl = 0.01 M), it would be expected that 𝑉𝑉�p𝑖𝑖𝑉𝑉r𝑖𝑖≈ 0.6 [151]. It can be concluded that these particles experience some degree of hindrance in their lateral motion that cannot be explained solely based on hydrodynamics.  The same conclusion can be drawn from an analysis of the effective lateral diffusion coefficient   𝐷𝐷II𝑖𝑖 = ∆𝑋𝑋MSD𝑖𝑖24∆𝑡𝑡  (26) of the particles, which is estimated from the mean square displacement (MSD) 33  ∆𝑋𝑋MSD𝑖𝑖2 = 1𝑁𝑁��∆?⃗?𝑋𝑖𝑖(𝑡𝑡𝑗𝑗)�2𝑁𝑁𝑗𝑗=1 (27) of particles [115]. Of course, the particles in Figure 9b and c might not be undergoing free diffusion during the whole observation time, so that application of eq (27) on the entire trajectory data set would not be strictly meaningful. However, for the sake of the argument to be made here, eq (27) is applied to the entire trajectory data set and and eq (26) yields an effective diffusion coefficient. Also note, that the relatively small drift velocities of the particles has been neglected in the determination of the diffusion coefficient from eq (26) [152]. For p = 1 in eq (24) so that ∆𝑡𝑡 = 10 𝑠𝑠 (and N = N1 ≈ 100); this simplifying assumption is valid as diffusion dominates over drift on such a small timescale. The resulting effective lateral diffusion coefficients are 𝐷𝐷II1𝐷𝐷0= 0.0042 and 𝐷𝐷II2𝐷𝐷0= 0.0026  for the particle in Figure 9b and c, respectively, where 𝐷𝐷0 = 0.485 𝜇𝜇𝑚𝑚2𝑠𝑠  is the diffusion coefficient of the particle in the bulk (as calculated from the Stokes Einstein equation). These values can be compared to well established predictions [93]  𝐷𝐷II𝐷𝐷0= 1𝐾𝐾H2, (28) where 𝐾𝐾H2 is a non-dimensional friction coefficient that accounts for hydrodynamic particle-wall interactions [153], [154]. However, similar to the particle drift velocity discussed above, such low values 𝐷𝐷II𝐷𝐷0 would only be expected for particles that are much closer to the wall than the size of a water molecule [154]. Thus it can be concluded that the low observed effective lateral diffusion coefficients cannot be explained solely based on hydrodynamic particle-wall interactions. Note, that the Brownian motion of a particle could potentially also be hindered by the presence of the electrical double layer around the particle [155]. However, for an isolated particle with a double layer thickness 𝜅𝜅𝑎𝑎 > 10 (as is the case in this work), 𝐷𝐷II𝐷𝐷0 is only reduced by less than 2% due to the presence of the double layer [155]. Thus it can be concluded that the observation made up to here indicates that the particles in Figure 9b and c are experiencing some degree of additional hindrance in their lateral motion.   Further experiments were carried out on a pre-cleaned microscope slide as substrate at CNaCl = 0 and 0.015 M and otherwise identical conditions to investigate if similar anomalous motion of particles can be observed on other substrate materials. Subsequent evaluation of the particle trajectories either showed particle trajectories of type Figure 9a or particles that were 34 clearly moving across the substrate driven by the flow. However, similar anomalous diffusion of particles near a substrates has already been observed for a 5 µm silica particle in the vicinity of a glass substrate in absence of fluid flow [156] although the seemingly random displacements of the silica particle were much smaller (i.e. in the order of 60 nm at CNaCl = 10 mM) as compared to the displacements observed in the current work. This suggests that the same anomalous diffusion mechanism may be active for the polystyrene-water-glass system but the resolution of the optical system used in the current work was insufficient to resolve the anomalous motion of the particles.  At this point, the underlying hindrance mechanism of the lateral motion of the particles can only be hypothesized. One potential explanation is that the particles in Figure 9b and c deposit and detach rapidly (on a timescale smaller than the 10 second inter-frame time) in and from the primary energy minimum. Numerous experimental studies on the particle adsorption at interfaces suggest that particles that deposit at the primary minimum do not exhibit any lateral mobility [93] as a result of unknown lateral interaction forces that keep them in place. However, deposition at the primary minimum at constant chemical conditions is currently believed to be irreversible [157] so that it is unlikely that the particles in Figure 9b and c switch back and forth between deposited state and free state.  Specific interactions between the particle surface and collector can potentially lead to tethering of the particle [158], [159]. Although there is evidence for a hairy layer consisting of polymer chains extruding from the surface of polystyrene particles [160], it is unclear if the hairy layer can bind to the PDMS surface. However, the length of the extruding polymer chains (depending on the pH and ionic strength of the liquid phase) is typically significantly shorter than the random walk displacements of the particles observed in Figure 9b and c, which indicates that particle tethering is not the mechanism leading to hindrance of lateral particle motion in the present work.  Another potential explanation for the observed hindered lateral particle motion are lateral surface interaction forces acting in a plane parallel to the substrate [156], [161]. Such lateral forces are expected to arise from surface charge heterogeneity and surface roughness of the collector [162], [163], [157], [161]. Indirect evidence for the surface charge of PDMS being inhomogeneously distributed has already been reported in literature [164]. Furthermore, it has been shown that lateral colloidal forces arising from charge heterogeneity of the collector can become significant at separation distances corresponding to the secondary minimum [162]. Surface roughness and charge heterogeneity can potentially lead to a landscape of DLVO interaction energy wells in the x-y plane (parallel to the substrate), effectively hindering particle 35 motion across the substrate [157], [156], [161]. This hypothesis suggests to examine the MSD as a function of ∆𝑡𝑡. Thus we set p = 1,2,3 in eq (24) so that ∆𝑡𝑡 = 𝑝𝑝 ∙ 10 𝑠𝑠 and 𝑁𝑁 = 𝑁𝑁1 − 𝑝𝑝 + 1 in (27). Figure 10a and b show the resulting plots of MSD over ∆𝑡𝑡 for the particles in Figure 9b and c, respectively. In these plots, the error bars represent the expected standard deviation for the MSD based on the given number of samples in a trajectory [72]. A linear increase of the MSD with ∆𝑡𝑡 indicates free diffusion of the particles [165]. If the particles are trapped, i.e. in a potential well, the MSD is expected to reach a limiting value for large ∆𝑡𝑡 [165]. Although the trends of the MSD over ∆𝑡𝑡 would support the hypothesis of confined lateral particle motion, the high statistical uncertainty for the MSD at large ∆𝑡𝑡 due to the relatively low number of samples for each trajectory does not allow making this conclusion [72]. To conclude this section up to here, some particles near the substrate showed anomalous lateral mobility. It was hypothesized that the low mobility of particles may be associated with lateral surface interaction forces arising from charge heterogeneity on the PDMS substrate which is in line with previous interpretations of the phenomenon [156], [161]. Particle deposition rates onto substrates exhibiting charge heterogeneity show a much more gradual increase of the deposition rate with respect to the electrolyte concentration than predicted by the convective diffusion theory for homogeneous substrates [98]. Thus, before the anomalously low particle mobility will be further characterized as a function of the electrolyte concentration in a following section of this chapter, measured deposition rates will be compared to predictions based on the convective diffusion equation outlined in section 2.2.1. However, the observations described in this section raise the question when to consider a particle as deposited and the chosen criteria for deposition is expected to greatly affect the determined deposition rates. The model described in section 2.2.1 only considers irreversible deposition in the primary energy minimum and it is currently believed that particles associated with the primary minimum will be completely and irreversibly immobilized by strong lateral specific interaction forces that keep them in place [93], [155]. Thus we consider a particle as deposited at time t = t0 when for all t> t0 (until the end of the experiment) the absolute particle position changes by less than a small critical distance �∆?⃗?𝑋�c compared to its position at time t0. Critical distances in the range 0.25 pixel < �∆?⃗?𝑋�c < 1 pixel were considered. The value  �∆?⃗?𝑋�c= 0.5 pixel was chosen, as this gave the best agreement between the resulting largest deposition rates with respect to the salt concentration (at CNaCl = 0.1 M) and the predicted mass transfer limited rate. As an example, for a given experiment at CNaCl = 0.1 M where the largest deposition rates were found, jdep/ jdep0= 84% for �∆?⃗?𝑋�c = 0.25 pixel, jdep/ jdep0= 103% for �∆?⃗?𝑋�c = 0.5 pixel and jdep/ jdep0= 110% for �∆?⃗?𝑋�c = 1. It is worth noting, 36 that according to the chosen deposition criteria with �∆?⃗?𝑋�c= 0.5 pixel, the particles in Figure 9b and c are not being considered as deposited. a)  b)  c)  Figure 9: Typical trajectories observed during a laboratory experiment at CNaCl = 0.01 M: a) a particle undergoes significant displacement (not shown in histogram) before it comes to rest. b) and c)particle undergoes significant Brownian motion but with almost zero mean displacement. All displacements in a)-c) were calculated from eq (24) with p = 1. a) b)  Figure 10: Mean square displacement (MSD) of a particle as a function of the time interval ∆𝑡𝑡. The error bars represent the expected standard deviation of the MSD based on the given sample size and ∆𝑡𝑡 [72]. a) and b) show MSD-plots for the particles in Figure 9b and c, respectively. 37 2.3.4 Effect of ionic strength on the initial deposition rate Sets of experiments were conducted in order to investigate the effect of salt concentration in the range 0.1 mM <  CNaCl < 500 mM on the initial deposition rates. For three substrates, a series of experiments at different salt concentrations was performed, starting at the lowest salt concentration 0.1 mM and subsequently increasing the salt concentration up to 0.1 M. After each experiment at constant salt concentration, particles were removed from the substrate as described in section 2.2.5.3 before the substrate was re-used for the next experiment at a different salt concentration.  This procedure was repeated for all salt concentrations up to CNaCl = 0.1M. The experiments at CNaCl = 0.5M were conducted on three fresh substrates. Figure 11a visualizes the expected transition to steady state concentration profiles after initialization of the flow at t = 0 through eq (11). These curves were computed by assuming the extended DLVO interaction profile eq (15). For experiments performed under unfavourable and favourable conditions, steady state conditions can be expected 15 minutes and 5 minutes after initialization of the flow, respectively. The same computation was performed by employing the DLVO interaction potential eq (14) which predicts unfavourable conditions for salt concentrations CNaCl < 0.3 M. The DLVO theory predicts a significantly slower transition to steady state for CNaCl = 0.1 M as compared to the extended DLVO theory. However, for all interaction potentials and salt concentrations considered in this work, a 15 minutes wait time from the time of initialization of the flow until initialization of image acquisition was found to be sufficient to achieve steady state conditions. In order to ensure that the particle removal procedure does not alter (or contaminate) the substrate to a degree that the deposition rates are affected, a control experiment was performed by using the same substrate and salt concentration (CNaCl = 0.1 mM) for three consecutive experiments. Between each experiment, particles were removed from the substrate following the procedure described in section 2.2.5.3. The resulting deposition rates are shown in Figure 11b, where the expected mass transfer limiting rate jdep0 was determined by solving the convective diffusion equation outlined in section 2.2.1. Figure 11b suggests that the particle removal procedure does not have a significant effect on the initial deposition rates. Figure 12a shows a typical result of a deposition experiment for one substrate. The number of deposited particles increases linearly in time, as expected for steady state conditions and for low surface coverages investigated in the current work. Figure 12b shows the (constant) deposition rates for different concentrations of sodium chloride derived from data as in Figure 12a for 3 substrates. The deposition rates increase gradually with increasing salt concentration 38 until the theoretically predicted mass transfer limit rate is observed at CNaCl = 0.1M. The measured deposition rate at CNaCl = 0.5M was slightly lower than the predicted mass transfer rate, which may seem unexpected. However, this observation may be related to the lack of stability of the particle suspension at such large salt concentration, which was supported by the fact that some small particle aggregates were found to deposit at such large salt concentration.  The measured deposition rates are in strong contrast to predictions made by the model outlined in section 2.2.4 as shown in Figure 12b by the dashed line for the DLVO interaction potential eq (14) and by the solid line for the extended DLVO interaction potential eq (15). Where theory predicts a sudden increase of the deposition rate at a critical salt concentration, the measured deposition rates increase much more gradually with the salt concentration. The same discrepancy between predicted and measured deposition rates under chemically unfavourable conditions has been observed for other substrate and particle materials [166], [167], [168], [92]. Evidence has been presented that the more gradual increase of the deposition rates with the salt concentration can be attributed to surface charge heterogeneity [98] and secondary minimum deposition [169] [170] or a combination of both effects [171]. However, as a result of the relatively low Hamaker constant, all interaction potentials discussed in section 2.3.2.2 show very shallow secondary minima (<0.5 kT) for CNaCl < 0.1 M and it is questionable whether such shallow secondary minima can contribute to the relatively large measured deposition rates at these salt concentrations [170]. Let us for now assume that every particle arriving at gap widths associated with the secondary minimum is held there. Considering that the particle flux into the regions associated with the secondary minimum is about two orders of magnitude smaller than the mass transfer limited deposition rate into the primary minimum (see Figure 8b) and noting that the measured deposition rate at CNaCl = 10 mM was about 30% relative to the mass transfer limited deposition rate, it can be concluded that the effect of the secondary minimum on the observed deposition rates is rather negligible. Consequently, the authors of the present work believe, that the observed rate of deposition events as defined in the previous section is mainly governed by heterogeneity of the substrate when conditions are unfavourable for deposition. Next, the models by Song et al. [98] were used in order to quantify the degree of surface charge heterogeneity for our substrates and for the given chemical conditions.  The first model considers a continuous distribution of local substrate zeta potentials 𝜉𝜉?̅?𝑖. As no information on the distribution of the local zeta potentials is available, it is generally assumed to be distributed according to the normal distribution 39  𝑝𝑝𝑖𝑖�𝜉𝜉?̅?𝑖, 𝜉𝜉𝑖𝑖� = 1𝜎𝜎𝑖𝑖√2𝜋𝜋 exp �− �𝜉𝜉�𝑖𝑖−𝜉𝜉𝑖𝑖�22𝜎𝜎𝑖𝑖2 �, (29) where 𝜉𝜉?̅?𝑖, 𝜉𝜉𝑖𝑖 and 𝜎𝜎𝑖𝑖 are the local substrate zeta potential, mean zeta potential (as determined from electrokinetic measurements) and standard deviation of the substrate zeta potential, respectively. Furthermore, the electrokinetic measurements described in section 2.2.4 suggest that the particle zeta potentials are normally distributed across the particle population, i.e.  𝑝𝑝𝑝𝑝�𝜉𝜉?̅?𝑝, 𝜉𝜉𝑝𝑝� = 1𝜎𝜎𝑝𝑝√2𝜋𝜋 exp �− �𝜉𝜉�𝑝𝑝−𝜉𝜉𝑝𝑝�22𝜎𝜎𝑝𝑝2 �, (30) where 𝜉𝜉?̅?𝑝, 𝜉𝜉𝑝𝑝 and 𝜎𝜎𝑝𝑝 are the particle zeta potential, mean particle zeta potential and standard deviation of the particle zeta potential, respectively. Values for 𝜎𝜎𝑝𝑝 were taken from the electrokinetic measurements described in section 2.2.4 if available and for all other salt concentrations it was assumed to be 10% of the mean zeta potential 𝜉𝜉𝑝𝑝 [92]. The local deposition rate for a given local surface zeta potential 𝜉𝜉?̅?𝑖 was calculated from [92]  𝑗𝑗loc�𝜉𝜉?̅?𝑖, 𝜉𝜉𝑝𝑝� = � 𝑝𝑝𝑝𝑝�𝜉𝜉?̅?𝑝, 𝜉𝜉𝑝𝑝�∞−∞𝑗𝑗dep�𝜉𝜉?̅?𝑖, 𝜉𝜉?̅?𝑝�𝑑𝑑𝜉𝜉?̅?𝑝, (31) where 𝑗𝑗dep was determined from the numerical solution of the steady state version of the convective diffusion equation described in section 2.2.1. The expected effective deposition rate   𝑗𝑗eff�𝜉𝜉𝑖𝑖, 𝜉𝜉𝑝𝑝� = ∫ 𝑝𝑝𝑖𝑖�𝜉𝜉?̅?𝑖 , 𝜉𝜉𝑖𝑖�𝑗𝑗loc(𝜉𝜉?̅?𝑖, 𝜉𝜉𝑝𝑝)𝑑𝑑𝜉𝜉?̅?𝑖∞−∞ , (32) then accounts for both, substrate charge heterogeneity and the distribution of particle zeta potential. The standard deviation 𝜎𝜎𝑖𝑖 of the surface zeta potential was determined by matching eq (32) to the measured deposition rate for each salt concentration under investigation.  Another model for describing surface heterogeneity proposed by Song et al. [98] assumes that the substrate consists of two types of surface patches:  patches at which deposition occurs at the mass transfer limited rate (favourable patch) and patches where no deposition occurs (unfavourable patch). The fraction of patches 𝜆𝜆i = 𝑗𝑗dep𝑗𝑗dep0 onto which particles deposit at the mass transfer limited rate is then simply the ratio of the observed deposition rate and the theoretically predicted mass transfer limited rate.  40 Table 2 compiles the resulting standard deviations of the substrate zeta potential and patch fractions 𝜆𝜆i for different salt concentrations and for the two interaction potentials considered in this work. Here it can be seen, that for both interaction potentials under consideration rather large and nearly identical standard deviations are required to predict the measured deposition rates for CNaCl < 0.1M. However, at CNaCl = 0.1M where the DLVO theory predicts no deposition, the observed deposition rate cannot be explained based on the distribution of surface potentials eq (29). If the patch model is used, a patch fraction of 98.9% has to be assumed in order to explain the observed deposition rates based on the DLVO interaction potential. In contrast, the extended DLVO theory captures the observed deposition at the mass transfer limited rate at CNaCl = 0.1M as the repulsive EDL interaction is completely screened by the Acid-Base interaction at CNaCl = 0.1M (see interaction potentials in section 2.3.2.2). Thus, it can be concluded that the extended DLVO theory in combination with the assumption of surface charge heterogeneity based on the patch model yields a better description of the observed deposition rates as compared to the DLVO theory in combination with surface charge heterogeneity. However, it is important to note, that the models by Song et al. are rather simplified as the spatial distribution of heterogeneity in not taken into account. This is so, as typically no information on the spatial distribution of surface charge is available. In the model, each patch is considered as homogeneous collector for itself so that concentration fields above each patch is assumed to be governed by the equations described in section 2.2.1. In a more realistic scenario, the overall concentration field above the heterogeneous collector and the deposition rate on individual favourable patches is expected to depend strongly on radial particle transport processes. Indeed, it has been demonstrated, that the actual deposition rate on a favourable patch surrounded by unfavourable patches can be much larger as compared to the deposition rate on a homogeneous substrate under favourable conditions, which essentially renders the assumption  𝜆𝜆i = 𝑗𝑗dep𝑗𝑗dep0 as inaccurate [172]. Nevertheless, the discussion provided in this section suggests that the PDMS substrates used in this work exhibit some degree of charge heterogeneity.   41 a) b)  Figure 11: a) Transition to steady state concentration profiles for different salt concentrations with Λ as defined in eq (11). The extended DLVO interaction potential was used for the computation: b) Control experiments at CNaCl = 0.1 mM demonstrating that the particle removal procedure described in section 2.2.5.3 has a negligible effect on the deposition rates. a) b)  Figure 12: a) Number of deposited particles over time for one substrate; b) steady-state deposition rates for three substrates and for different salt concentrations: the error bars represent the standard deviation of the deposition rates for three different substrates. The dashed line and solid line represent numerical predictions based on the DLVO interaction potential eq (14) and the extended DLVO interaction potential eq (15), respectively. Table 2: Standard deviation of substrate zeta potentials based on the DLVO interaction potential eq (14) and based on the extended DLVO interaction potential eq (15).  CNaCl (mM) 𝜉𝜉p(mV) 𝜉𝜉i (mV) 𝜎𝜎𝑖𝑖 (mV) (DLVO) 𝜎𝜎𝑖𝑖 (mV) (xDLVO) 𝜆𝜆i (%) 0.1 -68.62 -112.25 42.86 42.85 0.452 1 -86.6 -82.5 57.44 57.43 7.65 10 -85.1 -52.75 98.3 98.29 29.74 100 -37.1 -23 Not applicable Not determined 98.9 500 -9.2 -2.2 Not determined Not determined Not determined 42 2.3.5 Displacement distributions  The observations made in section 2.3.3 suggest that particles may exhibit anomalous low lateral mobility in the vicinity of the substrate. This hypothesis is tested further and in a more systematic fashion in the current section. The tool chosen for this visualization is the distribution of particle displacements that were derived from the particle trajectories. In the first step, all trajectories were trimmed to the same length and trajectories starting in the first image were discarded as these might correspond to particles that deposited before the experiment started. Next, absolute inter-frame displacements of the particle �∆?⃗?𝑋𝑖𝑖�𝑡𝑡𝑗𝑗�� (see eq (24)) were determined for all remaining trajectories. Subsequently, the displacement information of all trajectories under consideration was combined into histograms 𝑓𝑓𝑝𝑝��∆?⃗?𝑋�, 𝑡𝑡𝑗𝑗� by using a bin width of 0.1 pixel. The resulting histograms were rather noisy. However, since the experiments were performed under steady state conditions, where the normal component of the particle flux is constant in time (at any distance from the substrate), it is meaningful to average the histograms over the entire duration of the experiment. The resulting average histogram 𝑓𝑓p̅��∆?⃗?𝑋�� then reflects the actual displacement histogram half way through the experiment. As described in the previous section, three experiments were conducted for each salt concentration (each time on a different substrate) by keeping all other parameters constant. All histograms available for a given salt concentration were averaged. However, at the highest salt concentration (0.5 M) two of the three data sets were significantly shorter compared to the other data sets so that these were excluded from the analysis 2. Note, that the histograms are expected to contain velocity information of particles that are up to 10 µm away from the substrate, as it was deduced from the characterization of the particle-tracking procedure described in section 2.3.1. Figure 13 shows the resulting averaged particle displacement histograms for each salt concentration under consideration. In Appendix A3, each of these histograms is shown with the corresponding standard deviations. The standard deviations can be rather large, which is most 2 It was difficult to run experiments for longer times at the highest salt concentration (0.5 M) as the particle suspensions were very unstable at this salt concentration. Thus, particle aggregates ended up partially clogging the outlet channels of the impinging jet cell, resulting in a shift of the stagnation point position for longer experimentation times.  43                                                 likely due to the fact that the stagnation point was not always perfectly in the center of the image and the position of the object plane may also vary to some degree from experiment to experiment 3. In Figure 13a, displacement distributions corresponding to unfavourable and favourable conditions for deposition are represented by thin lines and thick lines, respectively. In the region �∆?⃗?𝑋� > 5 pixel, the histograms look qualitatively similar in shape, although the histograms for CNaCl = 1 mM and 0.1 mM indicate more particles within 5 pixel < �∆?⃗?𝑋� < 30 pixel compared to the other histograms. The histograms show a significant increase of the particle count for �∆?⃗?𝑋� → 0 , which correspond to slow, potentially deposited particles very close to the interface. A close up view of the peak-region 0.25 pixel < �∆?⃗?𝑋� < 5 pixel is shown in Figure 13b.  Displacements �∆?⃗?𝑋�  < 0.25 pixel were discarded from the histograms in order to focus on those particles that are moving to a degree that can be resolved by our particle tracking algorithm. The histograms in Figure 13b corresponding to CNaCl ≥ 0.1 M look very similar. This is in line with the indistinguishable concentration profiles that are predicted by theory at these salt concentrations (shown in Figure 8a). All histograms, except for the one corresponding to the lowest salt concentration CNaCl = 0.1 mM, show significantly increasing count of particles with decreasing associated (small) displacement. The onset of the increase is at significantly smaller displacements for the histograms corresponding to favourable conditions as compared to the histograms corresponding to unfavourable conditions. Furthermore, in the histograms corresponding to unfavourable conditions for deposition (0.1 mM < CNaCl < 0.1 M), an increasing number of slow particles with �∆?⃗?𝑋� < 1.5 pixel is found for increasing salt concentration. However further discussion will be required in order to attribute the increasing number of slow particles for increasing salt concentration seen in the histograms to anomalous low particle mobility. More specifically, it has to be shown that the observed trends of the number of slow particles (�∆?⃗?𝑋� < 5 pixel) cannot be attributed solely to hydrodynamic particle-wall interactions. A good starting point for this discussion are the expected displacement distributions for the case when particles do not exhibit anomalously low mobility. A simple numerical model was developed in order to estimate the expected displacement distributions. In short, a large 3 More specifically, the true position of the stagnation point is estimated to be somewhere in a circular region centered on the center of the image, where the diameter of this region is about 160 pixels 44                                                 number of virtual particles are distributed in a virtual region above the substrate. We then assign an expected displacement after Δt = 10 s to each particle only based on its initial position in the flow field and with respect to the substrate while completely neglecting any dynamical aspects occurring within the Δt = 10 seconds. Finally, the expected displacements are combined into histograms for qualitative comparison with the histograms obtained from the laboratory experiments. In order to distribute virtual particles in the model domain x- and y- coordinates  (in the plane parallel to the substrate) were generated by using the Matlab function rand to achieve a random distribution at the desired concentration. The expected non-uniform concentration profiles in the direction perpendicular to the substrate can be taken into account by sampling the z-positions from a probability density function that is directly proportional to the concentration profiles. This is achieved in approximate manner by dividing the region z < 10 µm of interest above the substrates into 1000 small sub-slices with thickness Δzs. The planes delimiting the i’th slice are denoted by 𝑧𝑧b𝑖𝑖 = 𝑎𝑎 + 𝐻𝐻0𝑎𝑎 + (𝑖𝑖 − 1)∆𝑧𝑧𝑠𝑠 and  𝑧𝑧e𝑖𝑖 = 𝑎𝑎 + 𝐻𝐻0𝑎𝑎 + 𝑖𝑖∆𝑧𝑧𝑠𝑠 and the center of the slice is at 𝑧𝑧c𝑖𝑖 = 𝑎𝑎 + 𝐻𝐻0𝑎𝑎 + �𝑖𝑖 − 12�∆𝑧𝑧𝑠𝑠, where 𝐻𝐻0 is the minimum separation distance as defined in section 2.3.2.3 (and listed in Table 3). In each slice, a large number 𝑁𝑁s𝑖𝑖 = 2000𝑐𝑐(𝑧𝑧c𝑖𝑖) of particle positions are generated to achieve the corresponding distributions.  As a result, the concentration of virtual particles was close to the predicted particle concentration everywhere in the model domain. Let the positions of the virtual particles be denoted as ?⃗?𝑋v𝑖𝑖 = [𝑥𝑥v𝑖𝑖 𝑦𝑦v𝑖𝑖] and 𝑧𝑧v𝑖𝑖. Next, the radial flow velocity eq (2) is evaluated at each virtual particle position and transformed into Cartesian coordinates 𝑣𝑣vx𝑖𝑖,𝑣𝑣vy𝑖𝑖. For each particle, the expected displacement after Δt =10 seconds is defined as  �∆?⃗?𝑋v�𝑖𝑖 = �(𝐾𝐾H1(ℎv𝑖𝑖)𝑣𝑣vx𝑖𝑖Δ𝑡𝑡 + ∆𝑥𝑥D𝑖𝑖)2 + �𝐾𝐾H1(ℎv𝑖𝑖)𝑣𝑣vy𝑖𝑖Δ𝑡𝑡 + ∆𝑦𝑦D𝑖𝑖�2 (33) where 𝐾𝐾H(ℎv𝑖𝑖) is a dimensionless hydrodynamic correction factor accounting for hydrodynamic particle-wall interactions, ℎv𝑖𝑖 = 𝑧𝑧v𝑖𝑖 − 𝑎𝑎 is the separation distance of the particle from the wall and ∆𝑥𝑥D𝑖𝑖 and ∆𝑦𝑦D𝑖𝑖 are random displacements due to Brownian motion. The random displacements were sampled from normal distributions with zero mean and a standard deviation of    𝜎𝜎D𝑖𝑖 = � 2𝐷𝐷0∆𝑡𝑡𝐾𝐾H2(ℎv𝑖𝑖), (34) where 𝐾𝐾H2(ℎv𝑖𝑖) is a hydrodynamic correction factor. Values for the correction factors 𝐾𝐾H1 and 𝐾𝐾H2 were interpolated from numerical data provided by [151] and [153], respectively. 45 Figure 13c shows the predicted displacement distributions 𝑓𝑓pv��∆?⃗?𝑋v�� for the different salt concentrations under consideration. These predicted distributions all decrease monotonically for small displacements �∆?⃗?𝑋� <5 pixel thus showing a fundamentally different trend as compared to the measured distributions in this region. The predicted histograms corresponding to unfavourable conditions all look very similar which is a consequence of nearly identical predicted concentration profiles at these salt concentrations. This is in contrast to the measured histograms for unfavourable conditions, where the histogram for CNaCl = 10 mM deviates significantly from the other histograms corresponding to unfavourable conditions. Furthermore, the predicted histogram for favourable conditions differ significantly relative to the histograms for unfavourable conditions as do the concentration profiles in Figure 8a.  This trend is reflected nicely in the measured histograms. Let us first interpret the observed increase of the measured histograms in the region �∆?⃗?𝑋� < 5 pixel. If particles are held on the substrate by additional (lateral) forces that are not considered by the model, one would expect to find more particles near the substrate than is predicted by theory. Thus, we compare the predicted number   𝑁𝑁pe = 𝑛𝑛b𝐴𝐴� 𝑐𝑐(𝑧𝑧)𝑑𝑑𝑧𝑧10 𝜇𝜇𝑚𝑚0 (35) of mobile particles (in the field of view A) within the region in which particles are expected to contribute to the measured histograms (i.e. z < 10 µm) to the total number 𝑁𝑁pexp of mobile particles (i.e. �∆?⃗?𝑋� > 0.25 pixel) in the experimentally determined histograms. As described in section 2.3.1, the relevant depth of field is expected to be close to 10 µm. From the corresponding values in Table 3 it can be seen, that anomalous particle accumulation described by the factor  �𝑁𝑁pexp − 𝑁𝑁pe�/𝑁𝑁pe increases from about 0.5 at the lowest salt concentration up to 1.3 at the highest salt concentration corresponding to unfavourable conditions (CNaCl = 10 mM). Upon increasing the salt concentration further, anomalous particle accumulation becomes less significant when conditions are favourable. In order to test the hypothesis of anomalously low particle mobility, we add an additional ∑ 𝑁𝑁s𝑖𝑖𝑖𝑖 �𝑁𝑁pexp − 𝑁𝑁pe�/𝑁𝑁pe virtual particles to the numerical model, where each particle is assumed to be placed as close to the energy barrier as possible (ℎv𝑖𝑖 = 𝐻𝐻0𝑎𝑎) so that they exhibit the lowest possible mobility that can be explained based on hydrodynamic interactions. The resulting histograms in Figure 13d clearly show a decreasing trend for �∆?⃗?𝑋� < 2.5 pixel for all salt concentrations under investigation. Thus it can finally be concluded that at least a fraction of these anomalously accumulated particles must exhibit anomalously low mobility that cannot be explained by hydrodynamic particle-wall interactions. Figure 13b then suggests that particles are bound 46 “tighter” to the substrate at larger salt concentrations thus indicating that the influence of the unknown lateral forces responsible for the reduced mobility seems to increase with increasing salt concentration. This trend is in line with the expected trend of lateral surface interaction forces resulting from charge heterogeneity of the substrate within the framework of the DLVO theory [162]. Next, we look at the fraction  Υ =  𝑁𝑁pexp5 − 𝑁𝑁pe5𝑁𝑁pexp − 𝑁𝑁pe (36) of the number 𝑁𝑁pexp5 − 𝑁𝑁pe5 of anomalously accumulated particles in the region 0.25 pixel < �∆?⃗?𝑋� < 5 pixel and the total number of accumulated particles 𝑁𝑁pexp − 𝑁𝑁pe of the measured histograms. The expected number 𝑁𝑁pe5 of particles in the region 0.25 pixel < �∆?⃗?𝑋� < 5 pixel was simply determined by multiplying 𝑁𝑁pe by the ratio of the particle counts within �∆?⃗?𝑋v� < 5 pixel and the total particle counts of the corresponding predicted histograms in Figure 13c. For all salt concentrations under investigation, the fraction Υ is smaller than one which suggests that some of the particles initially ‘captured’ by lateral forces may subsequently escape into regions further away from the substrate where they do not exhibit anomalous mobility anymore but might later return closer to the substrate where they are captured again. This would correspond to an adsorption – desorption mechanism that increases the effective concentration of particles near the substrate. This was supported by direct observation of video sequences which clearly showed that, in between two frames, particles can effectively move in the opposite direction of the flow due to Brownian motion (in both directions, parallel and normal to the surface) as a consequence of the low Péclet number and gravity number conditions in this work. As it can be seen from the values of Υ listed in Table 3, significantly more of the accumulated particles are associated with anomalously slow mobility at intermediate salt concentrations (CNaCl = 1 mM – 10 mM) as compared to the lowest salt concentration (0.1 mM). This observation may be related to the (extremely shallow) secondary minima that can be seen in the interaction potentials corresponding to intermediate salt concentrations in Figure 7. Thus, particles are experiencing (weak) normal forces that make it less likely for a particle to escape into regions further away from the substrate at intermediate salt concentrations as compared to the lowest salt concentration.  Where the observations made up to here are all very much in line with the idea that particles are experiencing lateral surface interaction forces that increase in magnitude with increasing 47 salt concentration, the histograms corresponding to favourable conditions for deposition and the corresponding fraction Υ of anomalously slow particles are somewhat puzzling. More specifically, in view of the mechanism discussed here, it is surprising that only a relatively small fraction Υ = 53% − 56% of the anomalously accumulated particles exhibit anomalously slow mobility as this suggests that a large fraction of the particles initially experiencing hindrance due to lateral forces close to the interface are subsequently released into regions further away from the interface where they do not experience these forces anymore. This is in contrast with the strong normal forces that are expected to keep the particles close to the primary minimum when conditions are favourable for deposition thus making particle escape into regions further away from the substrate much less likely as compared for the case CNaCl = 10 mM where Υ = 78% . Another potential problem with the qualitative analysis performed in this section is that none of the predicted concentration fields account for charge heterogeneity and lateral surface interaction forces and at present it is unclear how these factors are affecting the concentration fields quantitatively.  a)  b)  c)  d)  Figure 13: Distributions of particle displacements: a) measured overall distribution with a 1 pixel bin size; b) corresponding close up view of the region of small displacements with a 0.1 pixel bin size; c) predicted distributions corresponding to the concentration profiles predicted by the convective diffusion theory; d)  predicted distributions accounting for anomalously accumulated particles with hydrodynamic interactions. 48 Table 3: Comparison of the measured and expected numbers of particles near the substrate. 𝐶𝐶NaCl (mM)  𝐻𝐻0  𝑁𝑁pe   𝑁𝑁pexp �𝑁𝑁pexp − 𝑁𝑁pe�/𝑁𝑁pe Υ 0.1 0.3271 110 164 0.49 13% 1 0.1027 112 220 0.96 77% 10 0.0299 113 263 1.33 78% 100 0.0003 58 81 0.40 56% 500 0.0003 58 90 0.55 53% 2.4 Conclusions The present work studied initial deposition rates of 1 µm polystyrene particles onto PDMS substrates at relatively low Peclet number conditions for different concentrations of sodium chloride in solution by using an impinging jet flow cell.  For the first time, particle tracking has been applied to the investigation of particle adsorption at the solid-liquid interface and particle tracking revealed an interesting phenomenon: Some particles near the substrate were found to exhibit anomalously low mobility, which makes the definition of a deposition event somewhat difficult. Nevertheless, an attempt was made to identify those particles that are associated with the primary minimum based on the assumption that they should practically move less than half a pixel until the end of the experiment. These particles were considered as strongly immobilized particles. Subsequently, the resulting  rates at which the strongly immobilized particles deposit were compared to predictions based on the convective diffusion theory for particle adsorption while considering two different models (DLVO and extended DLVO) for the surface interaction potentials. Measured deposition rates match the corresponding predictions based on the extended DLVO theory well when conditions are favorable for deposition. On the other hand, the DLVO theory predicts that no deposition occurs at salt concentrations where the mass transfer limited deposition rate was observed in the corresponding experiments. In general it was found that the convective diffusion theory for particle adsorption on homogeneous substrates fails to predict initial deposition rates of polystyrene particles onto PDMS substrates when conditions are unfavourable for deposition. More specifically, theory predicts vanishing deposition rates when conditions are unfavorable for deposition but significant deposition rates were observed in the corresponding experiments. Where previous 49 work reported the same discrepancy between predicted and measured deposition rates for other substrate and particle materials, the present work is the first report to confirm this for the polystyrene-water-PDMS system. The discrepancy between measured and predicted deposition rates were subsequently interpreted in terms of secondary minimum deposition and surface charge heterogeneity of the collector, the latter being the more plausible explanation for the presence of strongly immobilized particles when conditions are unfavorable for deposition.  Next, the motion of the more weakly immobilized particles near the substrate was characterized for different salt concentrations in terms of displacement distributions. Some of the observed trends in the displacement distributions may be explained by the presence of lateral surface interaction forces of electrostatic and dispersive nature that hinder particle motion parallel to the substrate. However, the experiments performed under favourable conditions for deposition did not quite fit into this picture. The extremely short ranged lateral surface interaction forces are difficult to quantify at present as a tool for their direct measurement has yet to be developed. Furthermore, to the best of our knowledge, substrate heterogeneity can currently not be characterized to a degree that would allow predicting the resulting lateral forces in a quantitative fashion [173]. Considering the lack of direct experimental evidence for lateral surface interaction forces, it is not surprising that the views on whether a particle can be immobilized at the secondary minimum have been divided [102], [174]. Most real-world substrates exhibit some degree of heterogeneity and when a particle interacts with a heterogeneous substrate, lateral interaction forces are ultimately expected [161] , [162], [175]. We believe, that secondary minimum deposition is ultimately linked to the surface charge heterogeneity and surface roughness of the substrate and is not expected to occur on a homogeneous, perfectly smooth planar collector subject to flow as a result of the lack of lateral forces that keep particles in place. A recent experimental investigation of the Brownian motion of a single silica particle in the vicinity of a glass substrate in absence of fluid flow suggests that this particle can experience lateral surface interaction forces [156]. The present work supports this view and clearly shows that the hindrance of the lateral particle motion does not result from hydrodynamic interactions between particle and substrate. Furthermore, the current work visualizes implications of the hindrance of lateral particle motion for the particle concentration profiles near the substrate within the framework of particle adsorption at the solid liquid interface, namely, the presence of a particle accumulation layer near the substrate. Particles in the accumulation layer can neither be considered as adsorbed in the classical sense nor be considered as “freely-mobile”. The presence of the observed accumulation layer is not predicted within the framework of the convective diffusion theory when the perfect sink boundary condition is used. In view of the results presented here, a more adequate boundary 50 condition may be based on the non-penetration boundary condition in combination with an appropriate immobilization model [155], [93]. In view of the question whether particles can be immobilized at separation distances corresponding to the secondary minimum it can be concluded that particle immobilization at the secondary minimum is indeed possible. However, the mechanism leading to the immobilization of particles is expected to compete with viscous drag forces that tend to drag particles across the substrate and thus, the phenomenon is expected to become less significant at larger Péclet numbers. This work was originally motivated by numerous studies on the onset of multi-layer deposition leading to channel blockage in microfluidic devices made from PDMS. As such, this work gives some insight into mechanisms in place that affect the deposition of a first particle layer onto a PDMS surface. However, care needs to be taken when applying these results to microfluidic devices made from PDMS as the method used for sealing such devices often involves exposing the PDMS to oxygen plasma which drastically changes the temporary surface characteristics of the material [176]. Furthermore, the PDMS is often cast from mold masters that partially consist of photoresist and a recent study showed that commonly used photoresists tend to contaminate the PDMS [177].   51 3 Effect of linear image processing on the depth of correlation in micro PIV 3.1 Introduction Micro particle image velocimetry (µPIV) was developed in the late 1990s [178] and has become a well-established tool for measuring flow fields in microfluidic environments. The standard μPIV setup consists of an inverted epifluorescence microscope that is connected to a digitial camera and a double pulsed Nd:YAG laser as light source. To perform a μPIV measurement, a microfluidic chip is mounted onto the stage of the inverted microscope. The fluid in the microfluidic chip is seeded with fluorescent particles and the fast imaging digital camera captures a series of image pairs of the seed particle as they flow through the microfluidic chip. Conventional PIV computer algorithms based on 2D-cross correlation are then used to calculate the flow field from the particle images. As a single camera can only register particle motion within the image plane directly, the original µPIV setup was intended to measure only two components of a flow field in a 2D plane (2D-2C) – not for the entire flow field.  In contrast to macroscale PIV where the particles are illuminated by a thin laser sheet, in µPIV the whole depth of the microchannel is illuminated [76]. Consequently, not only particles in the object plane contribute to the cross correlation analysis but also particles that are some distance away from the object plane. The depth of the volume in which particles contribute to the cross correlation analysis is called depth of correlation (DoC). A finite DoC can lead to a bias error in the measured velocity when the flow has a non-zero velocity gradient along the optical axis z. One way to reduce the depth of correlation is to decrease the tracer particle size and to increase the numerical aperture (NA) of the objective lens [79]. However the particle size cannot be chosen arbitrarily small as with decreasing particle size the signal to noise ratio of the particle images decreases and the Brownian motion of the particles becomes more significant. Alternatively, the DoC can be reduced via image processing. Various attempts have been made to remove out-of-focus particles from the particle images or to emphasize in focus particles prior to the cross correlation analysis. All these methods make use of the fact that the appearance of a particle in an image depends on its distance from the object plane. Briefly, the particle image diameter increases and the particle image intensity decreases with the distance of the particle from the object plane.  52 An effective method to modify the DoC is the power filter method [80] which emphasizes bright particles in an image and has been shown to reduce the DoC by a factor of two if the brightness value of each pixel of an image is squared prior to cross correlation. The image overlapping method is another strategy to reduce the DoC. Here, several image pairs are combined to one image pair by choosing for each pixel only the brightest pixel value of all image pairs [81], [179]. Nuygen et al. [81]  claim that the image overlapping method reduces the DoC as dark particles that are further away from the object plane will be replaced by brighter particles that are closer to the object plane.  Another approach to reduce the DoC through image processing is to use thresholding methods.  The simplest and most commonly applied method is the so called “base clipping method”, where intensity values below a certain threshold are set to zero [179], [180], [181], [182], [183]. Alternatively a threshold can be applied to the normalized cross-correlation function of the particle image with an image of an in-focus particle. The latter method is called particle mask correlation method [184]. While this method was originally intended to identify particles in an image [185] it has also been applied to remove out-of-focus particles from particle images [184]. Finally, linear low-pass, high-pass and band-pass filters are commonly applied to µPIV images in order to emphasize the displacement peak over noise contributions to the correlation function and to increase the number of valid vectors. Low pass filters are applied to reduce the contribution of single pixel thermal noise of the imaging chip to the correlation function [2], [186]. High pass filtering of µPIV images [2], [187], [188], [189] aims at reducing low frequency noise originating from the larger out-of-focus particles [2], [81] and non-uniform background illumination (i.e. resulting from unwanted light reflection). High-pass filtering is also implemented in commercially available PIV software (DaVis 8.1.5 LaVision) as optional image pre-processing step. Band-pass filtering of µPIV images [2], [190], [187], [182], [81], [191], [192] combine the effects of low-pass filters and high-pass filters and can suppress both, single pixel noise and lower frequency noise [2]. Although it has been suggested that high-pass and band-pass filtering can reduce the influence of the out-of-focus particles on the cross correlation [81], [2], [193], the effect of the filter operations on the depth of correlation has not been systematically investigated yet. The first part of the current work revisits the theory of depth of correlation before it is extended to account for linear image processing. The cross-correlation will be evaluated in a slightly different way than previously in order to recover the in-plane-loss of correlation. A scheme that can be used to compute the depth of correlation numerically is discussed before analytical 53 models and numerical solutions for the effect of linear image processing on the depth of correlation are developed and verified by experiment.  3.2 Theoretical framework 3.2.1 Image plane intensity distribution of a single micro particle The theory of micro PIV cross correlation that will be summarized in the following section requires a description of how a single micro-particle appears in an image. As it is typically assumed that the image of many particles is a superposition of the images of individual particles, it is sufficient to model the image of a single particle.  The current work employs the commonly used model for the image intensity function (per unit illumination) for a particle that is imaged through a single thin lens as proposed by Adrian and Olsen [79], [194]. The simple model is illustrated in Figure 14a. Here, we consider the image distance 𝑠𝑠i as fixed. Focusing (i.e. changing the position 𝑧𝑧𝑓𝑓 of the object plane) is achieved by translating both, lens and image plane along the optical axis z (i.e. by turning the focus knob of the microscope). a)  b)  Figure 14: a) Schematic of the model-optical system (single thin lens) with the distance 𝑠𝑠0 between lens and object plane, image distance 𝑠𝑠𝑖𝑖, position of the object plane 𝑧𝑧𝑓𝑓, position of the lens 𝑧𝑧l, coordinates x,y,z in the physical domain and coordinates X,Y in the image plane. b) Schematic of the integrand in eq. (47); grey regions correspond to support of 𝐽𝐽o�?⃗?𝑋 − 𝑀𝑀?⃗?𝑥; 𝑧𝑧� and of 𝐽𝐽o�?⃗?𝑋 − 𝑀𝑀?⃗?𝑥 + 𝑠𝑠 − 𝑀𝑀∆?⃗?𝑥; 𝑧𝑧 + ∆𝑧𝑧� For this case, the intensity distribution of a particle can be approximated by the Gaussian  𝐽𝐽0𝐺𝐺 = 𝐴𝐴c(𝑠𝑠0 + 𝑧𝑧 − 𝑧𝑧𝑓𝑓)2𝑑𝑑e2(𝑧𝑧 − 𝑧𝑧𝑓𝑓) exp�− 4𝛽𝛽2�?⃗?𝑋�2𝑑𝑑e2(𝑧𝑧 − 𝑧𝑧𝑓𝑓)� (37) 54 where ?⃗?𝑋 and z are Cartesian coordinates as depicted in Figure 14a [194]. Here, de is the particle image diameter, 𝛽𝛽2 = 3.67 [195], s0 is the distance from lens to object plane, and 𝑧𝑧 − 𝑧𝑧𝑓𝑓 is the distance of the particle from the object plane.  The constant Ac is given by  𝐴𝐴c = 𝐽𝐽p𝐷𝐷a2𝛽𝛽24𝜋𝜋 , (38) where 𝐷𝐷a is the aperture of the lens and 𝐽𝐽p is the flux of light emitted by the particle per unit of illuminating intensity. For most micro PIV setups, the distance 𝑠𝑠0 ≫ 𝑧𝑧 − 𝑧𝑧𝑓𝑓 from lens to object plane is much larger than the distance of the particle from the object plane. Thus, eq (37) will be simplified by noting that (𝑠𝑠0 + 𝑧𝑧−𝑧𝑧𝑓𝑓)2 ≈ 𝑠𝑠02. The particle image diameter   𝑑𝑑e = 𝑀𝑀�𝑑𝑑p2 + 1.49𝜆𝜆2 � 𝑛𝑛02𝑁𝑁𝐴𝐴2 − 1� + 4(𝑧𝑧 − 𝑧𝑧𝑓𝑓)2 � 𝑛𝑛02𝑁𝑁𝐴𝐴2 − 1�−1�1/2 (39) corresponds to the width of the intensity distribution where the intensity is 0.0255 times the maximum intensity and depends on the distance 𝑧𝑧 − 𝑧𝑧𝑓𝑓 between particle and the object plane, magnification M, the physical particle diameter 𝑑𝑑p, the wavelength 𝜆𝜆 of the light emitted by the particle, the numerical aperture NA of the lens and on the refractive index 𝑛𝑛0 of the lens immersion medium . Note, that in the limit of 𝑑𝑑p → 0 𝜇𝜇𝑛𝑛 and 𝑧𝑧 → 𝑧𝑧𝑓𝑓, the particle image diameter eq (39) reduces to the diameter of the point response function of a diffraction limited lens measured at the first dark ring of the Airy disk intensity distribution [79], [194]. If the refractive index 𝑛𝑛0 of the lens immersion medium differs from the refractive index 𝑛𝑛l of the liquid phase surrounding the particle then refraction needs to be taken into account. In this case, s0 depends on the position 𝑧𝑧l of the lens and a scaling factor   𝑞𝑞 = 𝛥𝛥𝑧𝑧l𝛥𝛥𝑧𝑧𝑓𝑓 (40) relating the distance 𝛥𝛥𝑧𝑧l scanned by the objective along the optical axis and the corresponding distance 𝛥𝛥𝑧𝑧𝑓𝑓 actually scanned by the focused spot within the sample space has to be determined experimentally [196], [197], [194].  We employed the method described in [194] to determine the q-value for each of the objectives used in this work (see table 1). Note, that the coverslip was assumed to be infinitely thin. 55 However, we expect that a coverslip with finite thickness would only introduce a constant offset between 𝛥𝛥𝑧𝑧l and 𝛥𝛥𝑧𝑧𝑓𝑓 (as long as the object plane is not within the coverslip). 3.2.2 Micro PIV displacement correlation peak The derivation for an expression of the depth of correlation is based on the spatial cross correlation function [79], [198], [83], [73]  𝑅𝑅(𝑠𝑠) = ∬𝐼𝐼1�?⃗?𝑋�𝐼𝐼2�?⃗?𝑋 + 𝑠𝑠�𝑑𝑑𝑋𝑋𝑑𝑑𝑌𝑌, (41) of two PIV images 𝐼𝐼1 and 𝐼𝐼2, with the Cartesian image coordinates ?⃗?𝑋 = (𝑋𝑋 𝑌𝑌)𝑘𝑘 and the coordinates in the correlation plane 𝑠𝑠 = (𝑠𝑠𝑥𝑥 𝑠𝑠𝑦𝑦)𝑘𝑘. The bounds for all integrals in this work are -∞ and ∞ unless specified otherwise. It is typically assumed that images of individual particles are additive so that a PIV image can be described as   𝐼𝐼1�?⃗?𝑋� = 𝑊𝑊I1�?⃗?𝑋 − ?⃗?𝑋1�∭𝐼𝐼o1�𝑥𝑥�⃗�𝐽𝐽o�?⃗?𝑋 −𝑀𝑀?⃗?𝑥; 𝑧𝑧�∑ 𝛿𝛿�𝑥𝑥�⃗ − 𝑥𝑥�⃗𝑖𝑖�𝑖𝑖 𝑑𝑑𝑥𝑥𝑑𝑑𝑦𝑦𝑑𝑑𝑧𝑧, (42) where 𝑊𝑊I1�?⃗?𝑋 − ?⃗?𝑋1� is a weighting function defining the interrogation window, 𝐼𝐼o1 defines the illumination pulse, 𝐽𝐽o is the particle image intensity function per unit illumination intensity, 𝛿𝛿�𝑥𝑥�⃗ − 𝑥𝑥�⃗𝑖𝑖� is the Dirac delta function, 𝑥𝑥�⃗𝑖𝑖 are the positions of the particles in the measurement domain at the time the image was taken and ?⃗?𝑥 = (𝑥𝑥 𝑦𝑦)𝑘𝑘 and 𝑥𝑥�⃗ = (𝑥𝑥 𝑦𝑦 𝑧𝑧 )𝑘𝑘 are position vectors in the physical domain with Cartesian coordinates x,y,z. This work considers the simple 𝑊𝑊I1�?⃗?𝑋 − ?⃗?𝑋1� = 1 for all ?⃗?𝑋 ∈ Ω𝐼𝐼1, and 𝑊𝑊I1 = 0 for all other ?⃗?𝑋, where Ω𝐼𝐼1 is a rectangular region in the image plane centered around ?⃗?𝑋1. It is important to note that the continuous cross correlation eq. (41) of particle images of type eq. (42) is only a good approximation of the discrete cross correlation typically employed for digital PIV systems when min (𝑑𝑑e ) 𝑑𝑑px� > 2, with the pixel pitch 𝑑𝑑𝑝𝑝𝑥𝑥 [199], [73]. In other words, a particle image has to span over at least 9 pixels, assuming a pixel fill factor equal to one. Any cross correlation function   𝑅𝑅(𝑠𝑠) = 𝑅𝑅𝑁𝑁(𝑠𝑠) + 𝑅𝑅𝐹𝐹(𝑠𝑠) + 𝑅𝑅𝐷𝐷(𝑠𝑠) (43) obtained by cross correlating two particle images consists of a contribution 𝑅𝑅𝑁𝑁(𝑠𝑠) representing the correlation of the mean background intensity over the interrogation windows 𝑊𝑊I1 and 𝑊𝑊I2, a 56 contribution 𝑅𝑅𝐹𝐹(𝑠𝑠) representing the correlation of the fluctuating intensities in the first window with the mean intensity in the second window (and vice versa) and a contribution 𝑅𝑅𝐷𝐷(𝑠𝑠) representing the cross correlation of the fluctuating intensities in both interrogation windows [73]. 𝑅𝑅𝑁𝑁(𝑠𝑠) and 𝑅𝑅𝐹𝐹(𝑠𝑠) can be eliminated by subtracting the mean image intensities from the image data [73]. 𝑅𝑅𝐷𝐷(𝑠𝑠) contains the desired displacement information as well as random peaks originating from single pixel thermal noise and random peaks originating from the random positions of the particles. Let us assume we have NP correlation functions 𝑅𝑅𝑖𝑖(𝑠𝑠) = 𝑅𝑅𝐷𝐷,𝑖𝑖(𝑠𝑠) of independent image pairs that were recorded under identical conditions. We can then determine the ensemble averaged correlation function 〈RD� (𝑠𝑠)〉 = 1𝑁𝑁P ∑ 𝑅𝑅𝐷𝐷,𝑖𝑖(𝑠𝑠)𝑁𝑁P𝑖𝑖=1 , where 〈 〉�  is the ‘real’ ensemble average operator. As the number of averaged correlation functions NP increases, the signal to noise ratio of 〈RD� (𝑠𝑠)〉 increases [200]. Given that for 𝑁𝑁P  → ∞, we have one 𝑅𝑅𝐷𝐷,𝑖𝑖(𝑠𝑠) for each possible tracer particle configuration, then 〈RD� (𝑠𝑠)〉 → 〈RD(𝑠𝑠)〉, where the latter has infinite signal to noise ratio. 〈RD(𝑠𝑠)〉 is called the displacement correlation peak [73]. Following the evaluation of the correlation function eq. (41)  with micro PIV images of the form eq. (42) as presented by [79] (i.e. eq 29), [198], [73], the micro PIV displacement correlation peak reads  〈𝑅𝑅𝐷𝐷(𝑠𝑠 − 𝑀𝑀∆?⃗?𝑥)〉 = � 〈𝑅𝑅𝐷𝐷(𝑠𝑠 − 𝑀𝑀∆?⃗?𝑥), 𝑧𝑧〉 𝑑𝑑𝑧𝑧 (44) with the local correlation function  〈𝑅𝑅𝐷𝐷(𝑠𝑠 −𝑀𝑀∆?⃗?𝑥), 𝑧𝑧〉 = ∬ �∬ 𝑓𝑓�𝑥𝑥�⃗,∆𝑥𝑥�⃗�𝑊𝑊I1�?⃗?𝑋 − ?⃗?𝑋1�𝑊𝑊I2�?⃗?𝑋 + 𝑠𝑠 − ?⃗?𝑋2�𝐽𝐽o�?⃗?𝑋 −𝑀𝑀?⃗?𝑥; 𝑧𝑧�𝐽𝐽o�?⃗?𝑋 −𝑀𝑀?⃗?𝑥 + 𝑠𝑠 − 𝑀𝑀∆?⃗?𝑥; 𝑧𝑧 + ∆𝑧𝑧�𝑑𝑑𝑥𝑥𝑑𝑑𝑦𝑦 � 𝑑𝑑𝑋𝑋𝑑𝑑𝑌𝑌, (45) representing the contribution of particles at a certain depth z in the fluid to the overall correlation function and with  𝑓𝑓�𝑥𝑥�⃗,∆𝑥𝑥�⃗� =  𝐶𝐶�𝑥𝑥�⃗, 𝑡𝑡1�𝐼𝐼o1�𝑥𝑥�⃗�𝐼𝐼o2�𝑥𝑥�⃗ + ∆𝑥𝑥�⃗�. (46) In eqs (44) and (45), ∆?⃗?𝑥 = (∆𝑥𝑥 ∆𝑦𝑦)𝑘𝑘 and ∆𝑥𝑥�⃗ = (∆𝑥𝑥 ∆𝑦𝑦 ∆𝑧𝑧 )𝑘𝑘 are vectors containing the displacements ∆𝑥𝑥(𝑧𝑧, 𝑡𝑡), ∆𝑦𝑦(𝑧𝑧, 𝑡𝑡) and ∆𝑧𝑧(𝑧𝑧, 𝑡𝑡) of the ideal seed particles in the flow field  𝑢𝑢��⃗ (𝑧𝑧, 𝑡𝑡) =∆𝑥𝑥�⃗ ∆𝑡𝑡⁄  (that is considered to be steady during the ensemble averaging process). The current 57 work considers the special case of zero out-of-plane velocity, i.e. ∆𝑧𝑧(𝑧𝑧, 𝑡𝑡) = 0 (see [198] for a discussion of this topic). It has to be emphasized that, although the displacements ∆?⃗?𝑥(𝑧𝑧 =𝑐𝑐𝑐𝑐𝑛𝑛𝑠𝑠𝑡𝑡) at a certain depth in the fluid may vary in an interrogation window, the variation need to be vanishingly small in order for eq. (45) to be meaningful [201], [73], [82], [83].  The current work choses a slightly different route to simplify eq. (45), as compared to the original work [79], [198]. Given that the concentration and illumination are uniform so that 𝑓𝑓�𝑥𝑥�⃗,∆𝑥𝑥�⃗� →  𝑓𝑓 = 𝑐𝑐𝑐𝑐𝑛𝑛𝑠𝑠𝑡𝑡, the inner integral in equation (45) becomes   𝑓𝑓�𝐽𝐽o�?⃗?𝑋 − 𝑀𝑀?⃗?𝑥; 𝑧𝑧�𝐽𝐽o�?⃗?𝑋 − 𝑀𝑀?⃗?𝑥 + 𝑠𝑠 − 𝑀𝑀∆?⃗?𝑥; 𝑧𝑧 + ∆𝑧𝑧�𝑑𝑑𝑥𝑥𝑑𝑑𝑦𝑦 (47) if ?⃗?𝑋  ∈ Ω𝐼𝐼1 and if ?⃗?𝑋 + 𝑠𝑠 ∈ Ω𝐼𝐼2, or it becomes 0 for all other ?⃗?𝑋 and 𝑠𝑠. Obviously, we are only interested in the non-zero solution, which puts bounds on ?⃗?𝑋 and 𝑠𝑠. For a given z, a physically meaningful 𝐽𝐽o is well approximated by a function with limited support. The integrand of eq (47) represents two similar functions of ?⃗?𝑥 with similar (approximately limited) support centered around ?⃗?𝑋 and shifted by 𝑠𝑠 −𝑀𝑀∆?⃗?𝑥, as illustrated in Figure 14b.  Given that the particle intensity distribution 𝐽𝐽o is well approximated by a function with limited support ?⃗?𝑋 in eq (47) can be omitted as the integration over x and y are performed from minus infinity to infinity and since ?⃗?𝑋 and 𝑠𝑠 stay finite (as mentioned above). Although 𝐽𝐽o may not be well approximated by a function with limited support as  �𝑧𝑧 − 𝑧𝑧𝑓𝑓� → ∞, with physically realistic particle intensity distributions 𝐽𝐽o, particles only contribute to the local correlation function (45) if their distance from the object plane is finite. Hence, we conclude that for any physically meaningful particle image intensity distribution 𝐽𝐽o, the solution of the inner integral eq. (47) will (in good approximation) be independent of ?⃗?𝑋 so that eq. (45) can be written as  〈𝑅𝑅𝐷𝐷(𝑠𝑠 −𝑀𝑀∆?⃗?𝑥), 𝑧𝑧〉 = 𝑓𝑓𝐹𝐹𝐼𝐼(𝑠𝑠)∬ 𝐽𝐽o(−𝑀𝑀?⃗?𝑥; 𝑧𝑧)𝐽𝐽o(−𝑀𝑀?⃗?𝑥 + 𝑠𝑠 −𝑀𝑀∆?⃗?𝑥; 𝑧𝑧 + ∆𝑧𝑧)𝑑𝑑𝑥𝑥𝑑𝑑𝑦𝑦 , (48) with the in-plane loss of correlation [73]  𝐹𝐹𝐼𝐼(𝑠𝑠) = � 𝑊𝑊I1�?⃗?𝑋 − ?⃗?𝑋1�𝑊𝑊I2�?⃗?𝑋 + 𝑠𝑠 − ?⃗?𝑋2�𝑑𝑑𝑋𝑋𝑑𝑑𝑌𝑌. (49) Finally, evaluation of eq. (48) for the case of the Gaussian particle intensity distributions eqs (37)-(40) gives the local correlation function  58  〈𝑅𝑅𝐷𝐷(𝑠𝑠 −𝑀𝑀∆?⃗?𝑥), 𝑧𝑧〉 = 𝐽𝐽p2𝐷𝐷a4𝛽𝛽2?̂?𝑓128𝜋𝜋𝑀𝑀2 𝐹𝐹𝐼𝐼(𝑠𝑠)(𝑠𝑠0 +𝑧𝑧−𝑧𝑧𝑓𝑓)4𝑑𝑑e2 exp �− 2𝛽𝛽2|𝑠𝑠 −𝑀𝑀∆𝑥𝑥|2𝑑𝑑e2 �. (50) In contrast to the corresponding result by [198], eq. (50) accounts for the skewing of the correlation function by the in-plane loss of correlation 𝐹𝐹𝐼𝐼(𝑠𝑠) [199], [73].  3.2.3 Weighting functions and depth of correlation For the remainder of the current work, skewing of the correlation function due to in-plane correlation loss is neglected, i.e. 𝐹𝐹𝐼𝐼(𝑠𝑠) = 1. In practice the bias error of the measured velocity introduced by skewing can be eliminated by shifting the second interrogation [202] window and/or by using interrogation windows of appropriate different sizes [199], [73]. Since 𝐹𝐹𝐼𝐼(𝑠𝑠) is known the correlation function can be divided by 𝐹𝐹𝐼𝐼(𝑠𝑠) prior to peak detection in order to eliminate skewing completely [73].  As originally suggested by Olsen, the measured velocities   𝑢𝑢0𝑥𝑥(𝑡𝑡) = ∫𝑊𝑊�𝑥𝑥�𝑢𝑢��⃗ ,𝑢𝑢�⃗ 0,𝑧𝑧�𝑢𝑢𝑥𝑥(𝑧𝑧,𝑡𝑡)𝑑𝑑𝑧𝑧∫𝑊𝑊�𝑥𝑥�𝑢𝑢��⃗ ,𝑢𝑢�⃗ 0,𝑧𝑧�𝑑𝑑𝑧𝑧  ,  𝑢𝑢0𝑦𝑦(𝑡𝑡) = ∫𝑊𝑊�𝑦𝑦�𝑢𝑢��⃗ ,𝑢𝑢�⃗ 0,𝑧𝑧�𝑢𝑢𝑦𝑦(𝑧𝑧,𝑡𝑡)𝑑𝑑𝑧𝑧∫𝑊𝑊�𝑦𝑦�𝑢𝑢��⃗ ,𝑢𝑢�⃗ 0,𝑧𝑧�𝑑𝑑𝑧𝑧  (51) obtained from micro PIV cross correlation can be interpreted as the weighted average of the actual velocities 𝑢𝑢𝑥𝑥 and 𝑢𝑢𝑦𝑦 over the depth of the fluid (z). In eq. (51), 𝑊𝑊�𝑥𝑥 and 𝑊𝑊�𝑦𝑦 are weighting functions describing the relative influence of all particles in the interrogation window at a certain depth z in the fluid to the measured velocities 𝑢𝑢0𝑥𝑥 and 𝑢𝑢0𝑦𝑦. In addition to vanishing in-plane flow velocity gradients, the current work considers negligible out-of-plane flow velocity gradients. For this case, the weighting functions  𝑊𝑊�𝑥𝑥�𝑢𝑢��⃗ ,𝑢𝑢�⃗ 0, 𝑧𝑧� = 𝑀𝑀∆𝑡𝑡 𝜕𝜕2𝜕𝜕𝑠𝑠𝑥𝑥2 〈𝑅𝑅𝐷𝐷(𝑠𝑠 − 𝑀𝑀∆?⃗?𝑥), 𝑧𝑧〉�𝑠𝑠=𝑀𝑀∆𝑥𝑥, 𝑊𝑊�𝑦𝑦�𝑢𝑢��⃗ ,𝑢𝑢�⃗ 0, 𝑧𝑧� = 𝑀𝑀∆𝑡𝑡 𝜕𝜕2𝜕𝜕𝑠𝑠𝑦𝑦2 〈𝑅𝑅𝐷𝐷(𝑠𝑠 − 𝑀𝑀∆?⃗?𝑥), 𝑧𝑧〉�𝑠𝑠=𝑀𝑀∆𝑥𝑥,  (52) are directly related to the curvature of the local correlation function 〈𝑅𝑅𝐷𝐷(𝑠𝑠 − 𝑀𝑀∆?⃗?𝑥), 𝑧𝑧〉 evaluated at the (local) signal peak (up to a constant factor) [84], the latter one being a measurable quantity. As physically meaningful weighting functions always approach 0 for 𝑧𝑧 − 𝑧𝑧𝑓𝑓 → + −⁄ ∞, they can be used to define a depth of the measurement volume [79]; the depth of correlation 𝑧𝑧DoC. 𝑧𝑧DoC is 59 defined as the larger one of the width of the weighting functions where they fall below a certain threshold. In order to determine the DoC from the weighting functions one first has to solve for all the real roots 𝑧𝑧corrx,i of  𝑊𝑊�𝑥𝑥�𝑧𝑧 = 𝑧𝑧corrx,i� − 𝜀𝜀𝑊𝑊�𝑥𝑥�𝑧𝑧 =  𝑧𝑧𝑓𝑓� = 0, (53) where 0.01 is the typical value for ɛ ( [79] Olsen and Adrian 2000). Then determine 𝑧𝑧DoCx =max�𝑧𝑧corrx,i� − 𝑛𝑛𝑖𝑖𝑛𝑛�𝑧𝑧corrx,i�. The same procedure needs to be repeated for 𝑊𝑊�𝑦𝑦 to determine 𝑧𝑧DoCy. The depth of correlation follows from 𝑧𝑧DoC = max (𝑧𝑧DoCx, 𝑧𝑧DoCy).  Evaluation of eq. (52) with the local correlation function (50) and with ∆?⃗?𝑥 = 𝑐𝑐𝑐𝑐𝑛𝑛𝑠𝑠𝑡𝑡 (uniform flow) yields the well-known result   𝑊𝑊�𝑥𝑥 = 𝑊𝑊�𝑦𝑦 = 𝑊𝑊(𝑧𝑧) = 1(𝑠𝑠0 + 𝑧𝑧 − 𝑧𝑧𝑓𝑓)4𝑑𝑑e4 (54) for the weighting function that is valid for vanishing (in-plane and out-of-plane) shear [79], [198]. For most micro PIV setups, particles only contribute significantly to the correlation function if 𝑧𝑧 − 𝑧𝑧𝑓𝑓 ≪ 𝑠𝑠0  so that (𝑠𝑠0 + 𝑧𝑧 − 𝑧𝑧𝑓𝑓)4  ≈ 𝑠𝑠04 is a good approximation [79], [198]. Solving eq. (53) with the weighting function eq.(54) yields two real roots and the corresponding expression for the depth of correlation [79], [198]  𝑧𝑧DoC0 = �1 − √𝜀𝜀√𝜀𝜀�𝑛𝑛02𝑁𝑁𝐴𝐴2− 1��𝑑𝑑p2 + 1.49𝜆𝜆2 � 𝑛𝑛02𝑁𝑁𝐴𝐴2 − 1�� (55) is valid if, in addition to the assumptions made for obtaining the weighting function eq. (54), 𝑧𝑧DoC0 ≪ 𝑠𝑠0 holds. Note, for the simple case of a uniform flow field with zero out-of-plane component (Δz = 0) considered here, the weighting function eq. (54) and depth of correlation eq. (55) only depend on the optics of the imaging system, particle size [79], [84], [203], [194]. In general, these quantities are also affected by image processing [80], velocity gradients [82], [83], and Brownian motion [204]. Finally, it is worth mentioning that eq. (54) (with (𝑠𝑠0 + 𝑧𝑧 −𝑧𝑧𝑓𝑓)4 ≈ 𝑠𝑠04) becomes independent of the optical parameters when expressed in terms of 𝑧𝑧 = 𝑧𝑧−𝑧𝑧𝑓𝑓𝑧𝑧DoC0. 60 3.2.4 Influence of linear image processing on the DoC The current work considers µPIV images that are subject to a linear shift invariant image processing operation  𝐼𝐼F�?⃗?𝑋� = �𝐼𝐼(𝜕𝜕)𝐺𝐺(?⃗?𝑋 − 𝜕𝜕)𝑑𝑑𝜕𝜕𝑥𝑥𝑑𝑑𝜕𝜕𝑦𝑦 (56) with the filter 𝐺𝐺 and the Cartesian vector 𝜕𝜕 = (𝜕𝜕𝑥𝑥 𝜕𝜕𝑦𝑦)𝑘𝑘. More specifically, a filtered PIV image reads  𝐼𝐼F�?⃗?𝑋� = 𝑊𝑊I�?⃗?𝑋 − ?⃗?𝑋1��𝐼𝐼o1 (?⃗?𝑥)𝐽𝐽oF�?⃗?𝑋 − 𝑀𝑀?⃗?𝑥; 𝑧𝑧��𝛿𝛿�?⃗?𝑥 − ?⃗?𝑥𝑗𝑗�𝑗𝑗𝑑𝑑𝑥𝑥𝑑𝑑𝑦𝑦𝑑𝑑𝑧𝑧 (57) with the filtered particle image intensity function per unit illumination density  𝐽𝐽oF�?⃗?𝑋 − 𝑀𝑀(𝑥𝑥,𝑦𝑦); 𝑧𝑧� = �𝐽𝐽o(𝜕𝜕; 𝑧𝑧)𝐺𝐺(?⃗?𝑋 − 𝑀𝑀?⃗?𝑥 − 𝜕𝜕)𝑑𝑑𝜕𝜕𝑥𝑥𝑑𝑑𝜕𝜕𝑦𝑦. (58) Applying a linear filter operation to the model for a PIV image eq. (42) does not change its form so that the equivalent of the local correlation function eq. (48) for filtered images   〈𝑅𝑅𝐷𝐷(𝑠𝑠 −𝑀𝑀∆?⃗?𝑥), 𝑧𝑧〉 = 𝑓𝑓𝐹𝐹𝐼𝐼(𝑠𝑠)∬ 𝐽𝐽oF(−𝑀𝑀?⃗?𝑥; 𝑧𝑧)𝐽𝐽oF(−𝑀𝑀?⃗?𝑥 + 𝑠𝑠 − 𝑀𝑀∆?⃗?𝑥; 𝑧𝑧 + ∆𝑧𝑧)𝑑𝑑𝑥𝑥𝑑𝑑𝑦𝑦 , (59) is obtained by replacing 𝐽𝐽o with 𝐽𝐽oF (eq. (58)) and the development of analytical expressions for the weighting function of a filtered image can follow the same path as for unfiltered images.  As for the continuous cross correlation eq. (41), the continuous convolution eq. (56) is only a good approximation of the discrete convolution typically employed in praxis when 𝑑𝑑G 𝑑𝑑𝑝𝑝𝑥𝑥� > 2, with the pixel pitch 𝑑𝑑𝑝𝑝𝑥𝑥 and the extent 𝑑𝑑G of the kernel of G [199], [73]. This means that not only the particle images have to cover at least 9 pixel but also the filter kernels have to span at least 9 pixels (i.e. 3 × 3 pixel box filter). The present work assumes that particle image plane intensity distributions of unfiltered images are Gaussian (i.e. 𝐽𝐽o = 𝐽𝐽0𝐺𝐺). Linear image processing is applied to micro PIV images in order to increase the signal to noise ratio of the correlation function R eq. (43). Depending on the filter parameters, the filter operations discussed in the following can have a great effect on the relative height of the displacement peak 〈RD〉 and the random correlation peaks RD − 〈RD〉 of the displacement correlation function RD (the latter one also including the peaks resulting from single pixel thermal noise).  61 The current work focusses on 〈RD〉 only, which by definition, has an infinite signal to noise ratio. It is not subject of the current work to find filter parameters for which the corresponding filter operation has an enhancing effect on the correlation function 𝑅𝑅 (enhancing means that the relative height of displacement peak compared to other contributions increases due to filter operation). Instead, we will focus on how the filter operations affect the depth of correlation and refer to previous work that investigated the effects of these filter operations on the quality of 𝑅𝑅.  In praxis, one typically measures an approximation 〈RD〉�  ≈ 〈RD〉, with finite signal to noise ratio, as only a finite number of correlation functions RD can be averaged. In that case, the filter parameters have an effect on the number 𝑁𝑁P of correlation functions to be averaged in order to maintain a certain signal to noise ratio for 〈RD〉�   and we will comment on this in the corresponding sections of the current work. 3.2.5 Measurement of the weighting function  The commonly used approach to measure the weighting function W is based on the observation of Bourdon et al. [84] that the weighting function is related to the curvature of the local correlation function evaluated at its peak. The local correlation function 〈𝑅𝑅𝐷𝐷(𝑠𝑠 − 𝑀𝑀∆?⃗?𝑥), 𝑧𝑧〉 is a measurable quantity so that equation (52) can be used to verify models for the weighting function 𝑊𝑊 for negligible out-of-plane velocity gradients experimentally [84], [203], [84], [194]. However, the weighting function cannot be determined directly from µPIV images as the separations z-zf of the particles from the object plane are unknown in µPIV images. A setup that allows controlling the separation of the particles from the object plane is required. This is typically achieved by fixing particles onto a microscope slide and subsequent imaging of these particles at different (controlled) focus positions [203], [84], [194]. The local correlation function can then be determined from the cross correlation of particle images at a certain distance z from the object plane with subsequent ensemble averaging. The details of this procedure are described in the materials and methods section of this chapter. 3.2.6 Numerical computation of the weighting function and depth of correlation Equations (52) not only give access to measuring weighting functions but also allow computing their values numerically. The current work focuses on weighting functions for negligible flow gradients. For the sake of brevity we simply focus on the computationally most efficient method for calculating weighting functions that is applicable only to separable particle image plane 62 intensity distributions 𝐽𝐽o(𝑋𝑋,𝑌𝑌; 𝑧𝑧) = 𝐽𝐽ox(𝑋𝑋; 𝑧𝑧)𝐽𝐽oy(𝑌𝑌; 𝑧𝑧) fulfilling the symmetry 𝐽𝐽o(𝑋𝑋,𝑌𝑌; 𝑧𝑧) =𝐽𝐽o(𝑌𝑌,𝑋𝑋; 𝑧𝑧) and for filters of the type 𝐺𝐺(𝑋𝑋,𝑌𝑌) = 𝐺𝐺(𝑌𝑌,𝑋𝑋) = 𝐺𝐺1(𝑋𝑋,𝑌𝑌) − 𝐺𝐺2(𝑋𝑋,𝑌𝑌) , where 𝐺𝐺1(𝑋𝑋,𝑌𝑌) =𝐺𝐺x1(𝑋𝑋)𝐺𝐺y1(𝑌𝑌) and 𝐺𝐺2(𝑋𝑋,𝑌𝑌) = 𝐺𝐺x2(𝑋𝑋)𝐺𝐺y2(𝑌𝑌) are separable filters. Under these conditions    𝐽𝐽oF(𝑋𝑋,𝑌𝑌; 𝑧𝑧) = 𝐽𝐽oF(𝑌𝑌,𝑋𝑋; 𝑧𝑧) = 𝐽𝐽oF1(𝑋𝑋,𝑌𝑌, 𝑧𝑧) − 𝐽𝐽oF2(𝑋𝑋,𝑌𝑌, 𝑧𝑧) (60) as a result of the distributivity of convolution eq. (58), where   𝐽𝐽oF1(𝑋𝑋,𝑌𝑌, 𝑧𝑧) = 𝐽𝐽oF1(𝑌𝑌,𝑋𝑋, 𝑧𝑧) = 𝐽𝐽oF1x(𝑋𝑋, 𝑧𝑧)𝐽𝐽oF1y(𝑌𝑌, 𝑧𝑧) (61) with   𝐽𝐽oF1x(𝑋𝑋, 𝑧𝑧) = ∫ 𝐽𝐽ox(𝜕𝜕𝑥𝑥; 𝑧𝑧)𝐺𝐺𝑥𝑥1(𝑋𝑋 − 𝜕𝜕𝑥𝑥)𝑑𝑑𝜕𝜕𝑥𝑥;   𝐽𝐽oF1y(𝑌𝑌, 𝑧𝑧) = ∫ 𝐽𝐽oy�𝜕𝜕𝑦𝑦; 𝑧𝑧�𝐺𝐺𝑦𝑦1(𝑌𝑌 − 𝜕𝜕𝑦𝑦)𝑑𝑑𝜕𝜕𝑦𝑦,  (62) and with analogous relations for 𝐽𝐽oF2(𝑋𝑋,𝑌𝑌, 𝑧𝑧). Since flow gradients are neglected weighting functions can be derived from eq. (52) and since the filtered particle images eq. (60) considered in this work are symmetric with respect to Y = X, both weighting function in eq. (52) will be identical. Evaluation of eq. (52) with the local correlation function (59) (and with  𝐹𝐹𝐼𝐼(𝑠𝑠) = 1) gives  𝑊𝑊�𝑥𝑥(𝑧𝑧) = 𝑊𝑊(𝑧𝑧) = ∬𝐽𝐽oF(𝑝𝑝; 𝑧𝑧) 𝜕𝜕2𝐽𝐽oF(?⃗?𝑝;𝑧𝑧+∆𝑧𝑧)𝜕𝜕𝑝𝑝𝑥𝑥2 𝑑𝑑𝑝𝑝𝑥𝑥𝑑𝑑𝑝𝑝𝑦𝑦. (63) Taking into account (60)-(62), the double integral in equation (63) can be reduced to the sum of products of single integrals  𝑊𝑊(𝑧𝑧) = �𝐽𝐽oF1x2(𝑝𝑝𝑥𝑥; 𝑧𝑧)𝑑𝑑𝑝𝑝𝑥𝑥 � 𝐽𝐽oF1x(𝑝𝑝𝑥𝑥; 𝑧𝑧)𝜕𝜕2𝐽𝐽oF1x(𝑝𝑝𝑥𝑥; 𝑧𝑧)𝜕𝜕𝑝𝑝𝑥𝑥2 𝑑𝑑𝑝𝑝𝑥𝑥+ �𝐽𝐽oF2x2(𝑝𝑝𝑥𝑥; 𝑧𝑧)𝑑𝑑𝑝𝑝𝑥𝑥 � 𝐽𝐽oF2x(𝑝𝑝𝑥𝑥; 𝑧𝑧) 𝜕𝜕2𝐽𝐽oF2x(𝑝𝑝𝑥𝑥; 𝑧𝑧)𝜕𝜕𝑝𝑝𝑥𝑥2 𝑑𝑑𝑝𝑝𝑥𝑥− �𝐽𝐽oF1x(𝑝𝑝𝑥𝑥; 𝑧𝑧)𝐽𝐽oF2x(𝑝𝑝𝑥𝑥; 𝑧𝑧)𝑑𝑑𝑝𝑝𝑥𝑥 � 𝐽𝐽oF2x(𝑝𝑝𝑥𝑥; 𝑧𝑧)𝜕𝜕2𝐽𝐽oF1x(𝑝𝑝𝑥𝑥; 𝑧𝑧)𝜕𝜕𝑝𝑝𝑥𝑥2 𝑑𝑑𝑝𝑝𝑥𝑥− �𝐽𝐽oF1x(𝑝𝑝𝑥𝑥; 𝑧𝑧)𝐽𝐽oF2x(𝑝𝑝𝑥𝑥; 𝑧𝑧)𝑑𝑑𝑝𝑝𝑥𝑥 � 𝐽𝐽oF1x(𝑝𝑝𝑥𝑥; 𝑧𝑧)𝜕𝜕2𝐽𝐽oF2x(𝑝𝑝𝑥𝑥; 𝑧𝑧)𝜕𝜕𝑝𝑝𝑥𝑥2 𝑑𝑑𝑝𝑝𝑥𝑥 . (64) For the filter operations and particle intensity distribution that will be considered in this work, analytical expressions for the eight integrands 𝐼𝐼(N) (N = 1,…,8) in eq.(64) are easily obtained. Alternatively, the integrands can be determined numerically by evaluating the discrete part of 63 the convolutions eq. (62) and by approximating the derivatives in (64) with finite differences if the corresponding analytical expressions are unavailable. Below is an outline of the simple numerical procedure used to evaluate eq. (64) to obtain a set of values Wj = W�zj� of the weighting function at specified positions 𝑧𝑧𝑗𝑗. I. With analytical expressions for the integrands 𝐼𝐼(𝑁𝑁) (N = 1,…,8) available, compute 𝐼𝐼𝑖𝑖,𝑗𝑗(𝑁𝑁) = 𝐼𝐼(𝑁𝑁)�𝑝𝑝𝑥𝑥,𝑖𝑖; 𝑧𝑧𝑗𝑗�, where 𝑝𝑝𝑥𝑥,𝑖𝑖; 𝑧𝑧𝑗𝑗 are on a uniform grid (i = 1,…,Nx, j = 1,…,Nz).  II. For every 𝑧𝑧𝑗𝑗 compute the corresponding value of the weighting function by approximating the integrals as sum of the values of the discrete integrands (constant grid-spacing factor can be omitted).  III. Increase the width of the computational domain max(𝑝𝑝𝑥𝑥,𝑖𝑖) – min�𝑝𝑝𝑥𝑥,𝑖𝑖� as well as Nx and repeat step 1-3 until the computed values for the weighting function do not change significantly upon further increase of Nx or max(𝑝𝑝𝑥𝑥,𝑖𝑖) – min�𝑝𝑝𝑥𝑥,𝑖𝑖� In the current work we consider the special case of Gaussian particle image plane intensity distributions 𝐽𝐽o(𝑋𝑋,𝑌𝑌; 𝑧𝑧) = 𝐽𝐽oG(𝑋𝑋,𝑌𝑌; 𝑧𝑧)  eq. (37). The weighting functions for zero-out-of plane shear considered in the remainder of this work are all symmetric with respect to 𝑧𝑧 = 𝑧𝑧𝑓𝑓 and are monotonically decreasing with �𝑧𝑧 − 𝑧𝑧𝑓𝑓�. Hence eq. (53) has two solutions 𝑧𝑧corrx,i which satisfy 𝑧𝑧corrx,1 − 𝑧𝑧𝑓𝑓 = 𝑧𝑧𝑓𝑓 − 𝑧𝑧corrx,2 so that the depth of correlation is 𝑧𝑧DoC = 𝑧𝑧corrx,2 − 𝑧𝑧corrx,1. In order to determine 𝑧𝑧corrx,1 defined by eq. (53), we interpolate 𝑊𝑊𝑗𝑗 = 𝑊𝑊�𝑧𝑧𝑗𝑗� linearly.  The numerical procedure described above will be validated in sections 3.4.2.2 and 3.4.2.3.  3.3  Materials and methods 3.3.1 Materials Aqueous suspensions of Fluoro Max fluorescent polystyrene microspheres in two different sizes (1 µm and 3.1 µm) were purchased from Thermo Fisher. The particles with a nominal mean diameter of 1 µm were loaded with a red fluorescent dye (ex/em) = (542 nm / 612 nm) red and the 3.1 µm particles were loaded with a green dye (ex/em) = (468 nm / 508nm), where ex/em represent peak excitation and emission wavelengths respectively. The particle suspensions were diluted with distilled water to a solid concentration of 0.1% and then washed 3 times via centrifugation. The washing steps removed surfactant from the suspensions which facilitated subsequent deposition of particles onto microscope cover slips.  64 3.3.2 Test setup The particles were deposited onto coverslips (Fisherbrand coverglasses, 170 µm thickness) by sedimentation. A small glass beaker was filled with washed and sonicated suspension and the coverslip was placed at the bottom of the beaker. Over time, particles deposited onto the coverslip driven by sedimentation. In the next step, the coverslip was placed in a beaker with distilled water to remove all particles that were not attached to the coverslip.  Particle images were acquired with a Flow Master MicroPIV system (LaVision) which consists of an imager sCMOS camera, a double pulsed ND-YAG laser (50 mJ/pulse at 100%), a Nikon TE-2000 epifluorescence microscope and a data acquisition computer. The microscope was equipped with three objective lenses made by Nikon; A 20× Plan Fluor NA = 0.5 air immersion lens, a 40× Fluor NA = 0.8 water immersion lens and a 40× Plan Fluor NA = 1.3 oil immersion lens. Camera and laser were synchronized by a personal timing unit (LaVision) and the camera was connected to the computer via a Framegrabber (LaVision). The microscope was equipped with optical filters that only allow the light emitted by the fluorescent particles to reach the camera (excitation light is rejected). The reader is referred to [205] for a more detailed description of our microPIV system.  The setup described above was used to acquire images of the particles fixed on the coverslip at different focus positions zf. The coverslips covered with particles were placed on the microscope stage and the particles were immersed in water by depositing a droplet of water on the coverslip. Turning the focus knob of the microscope translates the objective turret along the optical axis z of the microscope and a scale on the focus knob indicates the z-position of the objective turret in 1 µm steps. The scale was used to adjust the z-position of the lens with a resolution of 0.5 µm when imaging the 1 µm particles and with a resolution of 1 µm when imaging the 3.1 µm particles. In both cases the uncertainty for the z-postion of the objective lens is estimated to be +/- 0.5 µm. 3.3.3 Imaging and image averaging  For the 1 µm particles, 20 image pairs were recorded in “double frame/single exposure” mode [205] at each z-position by using the ND-YAG laser as light source. The laser power was set to 30% in order to keep photo bleaching of the particles to a minimum. For the 3.1 µm particles, a sequence of 40 images was recorded at each z-position by using a mercury arc lamp as light source. In each case, the area of interest was set to regions that show more than 15 particles and no particle agglomerates.  65 In order to reduce the contribution of image noise to the correlation functions, image averaging was performed with DaVis 8.1.5. For each z-position, Frames 1 and 2 of the set of 20 image pairs of the 1 µm particles were averaged to obtain 20 single frames. In the next step two sets of 10 subsequent single frames were averaged so that two particle images remain for each z-position. Similarly, two sets of 20 subsequent single frames of the 3.1 µm particles were averaged to obtain two particle images for each z-position. 100 dark frames were recorded while a cap blocked all light from the camera. Subsequently, the average of the 100 dark frames was subtracted from the averaged particle images. The resulting images are referred to as raw images in the remainder of this work.  3.3.4 Image processing  In the next step, linear image processing procedures were performed on the images by using the conv2 function of Matlab R2011a (Mathworks). The Gaussian filter kernels were generated manually and the kernel size was chosen to be an uneven number close to 10× the standard deviation of the correpsonding Gaussian in order to minimize truncation errors. 3.3.5 Determination of the weighting functions  In order to calculate the weighting functions, one interrogation window was centered on each of the particles. The interrogation window size was chosen large enough to fit the image of a highly defocussed particle and small enough so that they only contain a single particle. Furthermore, it was ensured that no boundary artifacts resulting from filtering were contained in the interrogation windows. The interrogation window sizes used for the 1 µm particles were 64 × 64 pixels and 128 × 128 pixels for the 20× NA = 0.5 lens and for the 40× NA = 0.8 lens respectively. The interrogation window sizes used for the 3.1 µm particles were 128 × 128 pixels and 183 × 183 pixels for the 20× NA = 0.5 lens and for the 40× NA = 0.8 lens respectively. At each z-position, the cross correlation of an interrogation region in the first image with the same interrogation region in the second image was performed by using the xcorr2 function of Matlab. After calculating the correlation function for each correlation window, the correlation functions of all interrogation windows of each image pair were averaged. A Gaussian was fitted to the averaged correlation functions. We used the lsqcurvefit routine provided by Matlab to solve the nonlinear least square problem associated with data-fitting. The lsqcurvefit routine uses the so-called ‘trust-region-reflective method’ to solve the nonlinear-least square problem and more detailed information can be found in the documentation for Matlab. The curvature at the peak was determined from the coefficients returned by this routine. For 66 each of the determined weighting functions, the quality of the fit was confirmed manually at three different z-positions.  3.4 Results and discussion 3.4.1 Assessment of the model for micro PIV images The performance of models for the weighting function W based on the framework presented in the previous sections of this chapter ultimately depends on the capability of the model for micro PIV images eqs (37)-(40) to describe real images [79], [84]. Although synthetic images based on equations (37)-(40) can agree very well with experimental images [84], [203], this is not always the case [3] and especially when objective lenses with high NA are used, the synthetic images can deviate significantly from experimental images [194]. It has been suggested that the reason for the discrepancy is the single thin lens approximation which fails to describe the optics of a complex microscope [3], [194] quantitatively.   As the current work focuses on the depth of correlation, it is important that the synthetic images capture the relative influence of particles at a certain distance from the object plane on the correlation function. We follow [194] and introduce an effective numerical aperture   NA = CdNAl, (65) where NAl is the nominal numerical aperture of the lens and Cd is a constant that is determined from experiment. The constant Cd is chosen so that the weighting functions determined from experimental images of the 1 µm particles match the model for the weighting function eq. (54) derived by Olsen [79] (see table 1 for Cd values) well. In Figure 15a and b it can be seen that the model for the weighting function with corrected NA agrees well with the measured weighting functions for both particle sizes under consideration.   Figure 16 compares measured images of the 1 µm particles for different levels of defocusing with the corrected model eqs. (37)-(40) with eq. (65). The agreement between the measured images and the model images is excellent for the 20× NA 0.5 and 40× NA 0.8 lenses (Figure 16a and b). Agreement between the model for particle images and real images was only tested for the 1 µm particles. However, the good agreement of the weighting functions for the 3.1 µm  particles in Figure 15a and b suggests that model images and measured images match equally well for the 3.1 µm particles for the 20× NA 0.5 and 40× NA 0.8 lenses. This assumption is reasonable, as the weighting functions are directly proportional to the curvature of the 67 autocorrelation of the particle images (for a given particle distance z from the object plane) evaluated at the peak of the autocorrelation function. Although the weighting function in Figure 15c for the 40× NA 1.3 lens matches the model with corrected NA, there is a significant discrepancy between the model for the particle images and the measured particle image as it can be seen in Figure 16c. Hence, it is expected that neither the corrected model for the weighting function, nor the corrected model for the particle images will match with experimental results. Furthermore, the models for the weighting function of pre-processed µPIV images developed in the remainder of the work are not expected to agree with measured weighting functions for this lens. The remainder of the work thus focuses on the first two lenses in Table 1.  a)  b)   c) Figure 15: Weighting functions obtained by using the lenses a 20× NA 0.5, b 40× NA 0.8, c 40× NA 1.3 with 1 µm particles (squares) and 3.1 µm  particles (circles). The solid lines represent the model eq. (54) after introducing an effective NA = CdNAl for each lens, with Cd listed in table 1.  a)  b)  c) Figure 16: Particle intensity profiles for the 1 µm particles imaged through different lenses and different focus positions: a) 20× Plan Fluor the symbols (circles squares triangles) correspond to z = [0.4 3 6.4] µm; b) 40× Fluor; z = [0 3 6] µm. c) 40× Plan Fluor; z = [0 1.8 3.6] µm. The solid lines represent the model eqs (37)-(40) with eq. (65).    68 Table 4: Summary of objective lens parameters Description Immersion medium NAl Cd q 𝑧𝑧DoC0 (dp = 1 µm) 𝑧𝑧DoC0 (dp = 3.1 µm) 20× Plan Fluor Air 0.5 0.65 0.74 19 µm 32 µm 40× Fluor Water 0.8 0.7 1 11.5 µm 22 µm 40× Plan Fluor Oil 1.3 0.8 1.23 4.4 µm 11 µm 3.4.2 Influence of linear image processing on the DoC 3.4.2.1 Band-pass filtering  Band-pass filtering is often performed to micro PIV images in order to attenuate single pixel noise and signals from large particle agglomerates and large out-of-focus particles in particle images [2], [187], [182], [81], [191], [192].  When the band-pass parameters are chosen properly, the displacement correlation peak of the correlation function can be emphasized by band-pass filtering [2] and the number of valid vectors can be increased [190]. Some groups prefer to use the difference of two moving average filters as band-pass filters [2], [187], [182]. Other groups prefer to use the Difference of Gaussians (DoG) filter and the Laplacian of a Gaussian filter [190]. Other (nonlinear) filters such as the difference of two medians filter [81], [191] have also been used for this purpose. Difference of Gaussians Let us consider the Difference of Gaussians  𝐺𝐺�?⃗?𝑋� = 12𝜋𝜋𝜎𝜎𝐺𝐺12 exp�− ?⃗?𝑋22𝜎𝜎𝐺𝐺12 � − 12𝜋𝜋𝜎𝜎𝐺𝐺22 exp �− ?⃗?𝑋22𝜎𝜎𝐺𝐺22 � (66) band-pass filter with the standard deviations 𝜎𝜎𝐺𝐺1 < 𝜎𝜎𝐺𝐺2. Evaluation of eqs (44), (59) and (58) with eq (66) and with 𝐽𝐽o = 𝐽𝐽0𝐺𝐺 gives the displacement correlation peak  69  〈𝑅𝑅𝐷𝐷(|𝑠𝑠 − 𝑀𝑀∆?⃗?𝑥|2)〉= 𝐽𝐽p2𝐷𝐷a4𝛽𝛽2𝑓𝑓4𝜋𝜋 256 𝑀𝑀2� 1(𝑠𝑠0 + 𝑧𝑧 − 𝑧𝑧𝑓𝑓)4⎣⎢⎢⎡�𝑑𝑑e28𝛽𝛽2 + 𝜎𝜎𝐺𝐺12 �−1 exp�− |𝑠𝑠 −𝑀𝑀∆?⃗?𝑥|2𝑑𝑑e22𝛽𝛽2 + 4𝜎𝜎𝐺𝐺12 �− 4� 𝑑𝑑e24𝛽𝛽2 + 𝜎𝜎𝐺𝐺12 + 𝜎𝜎𝐺𝐺22 �−1 exp�− |𝑠𝑠 − 𝑀𝑀∆?⃗?𝑥|2𝑑𝑑e22𝛽𝛽2 + 2𝜎𝜎𝐺𝐺12 + 2𝜎𝜎𝐺𝐺22 �+ � 𝑑𝑑e28𝛽𝛽2 + 𝜎𝜎𝐺𝐺22 �−1 exp�− |𝑠𝑠 −𝑀𝑀∆?⃗?𝑥|2𝑑𝑑e22𝛽𝛽2 + 4𝜎𝜎𝐺𝐺22 �⎦⎥⎥⎤𝑑𝑑𝑧𝑧. (67) With the correlation function eq. (67), the weighting functions   𝑊𝑊BP�𝑥𝑥�⃗� = 1(𝑠𝑠0 + 𝑧𝑧−𝑧𝑧𝑓𝑓)4 �� 𝑑𝑑e24𝛽𝛽2 + 2𝜎𝜎𝐺𝐺12 �−2 − 2� 𝑑𝑑e24𝛽𝛽2 + 𝜎𝜎𝐺𝐺12 + 𝜎𝜎𝐺𝐺22 �−2+ � 𝑑𝑑e24𝛽𝛽2 + 2𝜎𝜎𝐺𝐺22 �−2� (68) obtained from eq. (52) are identical. In the limit 𝜎𝜎𝐺𝐺1 → 0 and 𝜎𝜎𝐺𝐺2 → ∞, the filter eq. (66) subtracts a constant from the original image. Subtracting a constant from the particle images has no effect on the part of the correlation containing the particle displacement information and consequently eq. (68) reduces to the weighting function eq. (54) previously derived by [198] for raw images. For 𝜎𝜎𝐺𝐺1 =  𝜎𝜎𝐺𝐺2 eq. (66) has a zero band-gap and both the displacement correlation peak and weighting function vanish for this case. Figure 17 compares measured weighting functions to the model eq. (68) for band-pass filtered micro PIV images for two objective lenses and two particle sizes. Depending on the choice of the dimensionless band-pass filter parameters 𝐾𝐾DoG  =  𝜎𝜎𝐺𝐺1𝑑𝑑e �𝑧𝑧=𝑧𝑧𝑓𝑓� and 𝑘𝑘DoG = 𝜎𝜎𝐺𝐺2𝜎𝜎𝐺𝐺1  , the weighting functions can become narrower or wider through band-pass filtering. We calculated the RMS error for all data sets in Figure 17. The model eq. (68) and the measured weighting functions agree well. The largest RMS errors are found for the data sets represented by square symbols. 70 More specifically, the largest RMS errors are 0.038, 0.038, 0.036 and 0.045 in Figure 17 a,b,c and d, respectively.  a)  b)   c)  d) Figure 17: Comparison of model eq. (68) (solid lines) to measured weighting function (symbols) for DoG-filtered images for different values of 𝐾𝐾DoG  =  𝜎𝜎𝐺𝐺1𝑑𝑑e �𝑧𝑧=𝑧𝑧𝑓𝑓� and 𝑘𝑘DoG = 𝜎𝜎𝐺𝐺2𝜎𝜎𝐺𝐺1 (in legend 𝐾𝐾DoG; 𝑘𝑘DoG): a) M = 20× – 𝑑𝑑p  = 1 µm; b) M  = 40× -  𝑑𝑑p  = 1 µm ; c) M = 20× –  𝑑𝑑p  = 3.1 µm; d) M = 40× –  𝑑𝑑p  = 3.1 µm. In order to determine an analytical expression for the depth of correlation, we approximate (𝑠𝑠0 + 𝑧𝑧−𝑧𝑧𝑓𝑓)4  ≈ 𝑠𝑠04 in eq. (68) and then solve eq. (53) with eq. (68). However, this requires us to solve for the roots of a 12th order polynomial and despite some effort, we were only able to find extremely cumbersome expressions for 𝑧𝑧DoC (even for Δz = 0) with the help of the symbolic math toolbox of Matlab (Mathworks). However, with (𝑠𝑠0 + 𝑧𝑧 − 𝑧𝑧𝑓𝑓)4  ≈ 𝑠𝑠04, the weighting function eq. (68) becomes independent of the optical parameters (up to an arbitrary constant) when expressed in terms of 𝑧𝑧̅ = 𝑧𝑧−𝑧𝑧𝑓𝑓𝑧𝑧DoC0, 𝐾𝐾DoG  and 𝑘𝑘DoG and thus, the ‘relative’ depth of correlation 𝑧𝑧DoC𝑧𝑧DoC0= 𝑓𝑓(𝐾𝐾DoG,𝑘𝑘DoG, 𝜀𝜀) will also be independent of optical parameters. A map of 𝑧𝑧DoC𝑧𝑧DoC0 =𝑓𝑓(𝐾𝐾DoG,𝑘𝑘DoG, 𝜀𝜀 = 0.01)  is shown in Figure 18. It is important to note that as a consequence of approximating (𝑠𝑠0 + 𝑧𝑧−𝑧𝑧𝑓𝑓)4  ≈ 𝑠𝑠04, the map in Figure 18 is only valid if the depth of correlation 𝑧𝑧DoC ≪ 𝑠𝑠0  is much smaller than the working distance of the lens.  71 In Figure 18 it can be seen that for any fixed  𝑘𝑘DoG > 1, the DoC increases with increasing 𝐾𝐾DoG so that for sufficiently large 𝐾𝐾DoG, 𝑧𝑧DoC ≪ 𝑠𝑠0will not hold anymore and the data in Figure 18 becomes invalid. As discussed earlier, with increasing 𝑘𝑘DoG, the DoC will approach the DoC of low-pass filtered images. For a given 𝐾𝐾DoG > 0, the minimum depth of correlation is always achieved in the limit of zero band gap, i.e. for 𝑘𝑘DoG → 1. There is a theoretical minimum for the depth of correlation 𝑧𝑧DoC𝑧𝑧DoC0→ 0.49 as 𝐾𝐾DoG → 0 and 𝑘𝑘DoG → 1. However, this limit can never be achieved in praxis as the band-pass filter eq. (66) has zero band gap and the filtered particle images eq. (57) correlation function eq. (43) and displacement correlation peak eq. (44) are equal to zero. Furthermore, when approaching the limit 𝐾𝐾DoG → 0 and 𝑘𝑘DoG → 1, the influence of single pixel noise on the correlation function increases and the contributions from the (larger) particles are increasingly attenuated. Hence, in order to maintain a certain signal to noise ratio for the correlation function 〈RD〉�   a larger number of images is required as 𝐾𝐾DoG → 0 and 𝑘𝑘DoG → 1. Tian et al. [190] investigated the effect of the difference of Gaussians filter on the ‘quality’ of the correlation function eq. (43) and recommends 𝐾𝐾DoG = 0.81 and 𝑘𝑘DoG = 1.6 for an enhancing effect of the filter on the correlation function. According to Figure 18, this recommendation increases the DoC by a factor of 2.75.  a)  b) Figure 18: Contour of 𝑧𝑧DoC𝑧𝑧DoC0 for particle images subject to the difference of Gaussians filter eq. (66) with 𝐾𝐾DoG  =  𝜎𝜎𝐺𝐺1𝑑𝑑e �𝑧𝑧=𝑧𝑧𝑓𝑓� and 𝑘𝑘DoG = 𝜎𝜎𝐺𝐺2𝜎𝜎𝐺𝐺1 for ɛ = 0.01 and Δz = 0 valid for 𝑧𝑧DoC ≪ 𝑠𝑠0: a for a wider range of 𝑘𝑘DoG and 𝐾𝐾DoG; b close-up view of the parameter range leading to a reduction of the DoC.  The difference between neighbouring contour levels is 1 in a and 0.1 in b and the highest contour levels are equal to the highest values shown in the color maps. 72 Laplacian of Gaussian As shown by Marr [206], in the limit of 𝜎𝜎𝐺𝐺1 →  𝜎𝜎𝐺𝐺2 ≡ 𝜎𝜎 the Difference of Gaussians filter eq. (66) behaves like the Laplacian of a Gaussian (LoG)  𝐺𝐺�?⃗?𝑋� = 1𝜋𝜋𝜎𝜎4�1 − ?⃗?𝑋2𝜎𝜎2� exp �− ?⃗?𝑋22𝜎𝜎2 � (69) (up to a constant factor 2𝜎𝜎). Tian et al. [190] suggest to pre-filter the images with the LoG in order to obtain reliable and accurate measurements in low SNR conditions, where cross-correlation of raw-images fails to produce a detectable displacement peak. Solving the integrals in eq. (59) with the filter eq. (69) is rather difficult. However, a closer look at eq. (52) with (59) reveals that   𝑊𝑊�𝑥𝑥 = 𝑊𝑊�𝑦𝑦 = 𝑊𝑊𝐿𝐿𝐿𝐿𝐺𝐺(𝑧𝑧) = ∬ 𝐽𝐽oF(−𝑀𝑀?⃗?𝑥; 𝑧𝑧) 𝜕𝜕2𝐽𝐽oF(−𝑀𝑀𝑥𝑥+𝑠𝑠−𝑀𝑀∆𝑥𝑥;𝑧𝑧+∆𝑧𝑧)𝜕𝜕𝑠𝑠𝑥𝑥2 �𝑠𝑠=𝑀𝑀∆𝑥𝑥 𝑑𝑑𝑥𝑥𝑑𝑑𝑦𝑦. (70) For Δz = 0 this yields  𝑊𝑊𝐿𝐿𝐿𝐿𝐺𝐺(𝑧𝑧) = 𝐽𝐽p2𝐷𝐷a4𝑀𝑀∆𝑡𝑡256𝜋𝜋2(𝑠𝑠0 + 𝑧𝑧)4� exp �−𝑀𝑀2(𝑥𝑥2 + 𝑦𝑦2)𝜎𝜎𝐿𝐿 ��𝑀𝑀2(𝑥𝑥2 + 𝑦𝑦2)2𝜎𝜎𝐿𝐿6 − 1𝜎𝜎𝐿𝐿4�� 1𝜎𝜎𝐿𝐿6−𝑀𝑀2𝑥𝑥24𝜎𝜎𝐿𝐿8 + 𝑀𝑀4𝑥𝑥44𝜎𝜎𝐿𝐿10 + 𝑀𝑀4𝑥𝑥2𝑦𝑦24𝜎𝜎𝐿𝐿10 − 𝑀𝑀4𝑦𝑦24𝜎𝜎𝐿𝐿8 �𝑑𝑑𝑥𝑥𝑑𝑑𝑦𝑦 (71) where 𝜎𝜎𝐿𝐿2 = 𝑑𝑑e28𝛽𝛽2 + 𝜎𝜎2 . The solutions for the different integrals in eq. (71) were taken from [207] and the final solution is  𝑊𝑊𝐿𝐿𝐿𝐿𝐺𝐺�𝑥𝑥�⃗� = 1(𝑠𝑠0 +𝑧𝑧−𝑧𝑧𝑓𝑓)4 𝜎𝜎𝐿𝐿−8 = 1(𝑠𝑠0 +𝑧𝑧−𝑧𝑧𝑓𝑓)4 � 𝑑𝑑e28𝛽𝛽2 + 𝜎𝜎2 �−4. (72) Figure 19 compares measured weighting functions to the model eq. (72) for micro PIV images subject to the Laplacian of a Gaussian filter eq. (69) for two objective lenses and two particle sizes. Depending on the choice of the dimensionless band-pass filter parameters 𝐾𝐾𝐿𝐿𝐿𝐿𝐺𝐺  = 𝜎𝜎𝑑𝑑𝑒𝑒 �𝑧𝑧−𝑧𝑧𝑓𝑓�, the weighting function can become wider or narrower. The agreement between the model and the measured weighting functions is good. The largest RMS errors are 0.047, 0.038 and 0.034 for the square symbols in Figure 19a,b and c, respectively. In Figure 19d, the RMS error of 0.054 is the largest for the dataset represented by triangles. 73 a)  b) c)  d) Figure 19: Comparison of model eq (72) (solid lines) to measured weighting function (symbols) for LoG-filtered images for different values of  𝐾𝐾𝐿𝐿𝐿𝐿𝐺𝐺 =  𝜎𝜎𝑑𝑑e �𝑧𝑧=𝑧𝑧𝑓𝑓� (in legend): a) M = 20× – 𝑑𝑑p  = 1 µm; b) M = 40× -  𝑑𝑑p  = 1 µm ; c) M = 20× –  𝑑𝑑p  = 3.1 µm; d) M = 40× –  𝑑𝑑p  = 3.1 µm. After approximating (𝑠𝑠0 + 𝑧𝑧−𝑧𝑧𝑓𝑓)4  ≈ 𝑠𝑠04, the analytical expression for the depth of correlation  𝑧𝑧DoC = �1 − √𝜀𝜀4√𝜀𝜀4 �𝑛𝑛02𝑁𝑁𝐴𝐴2− 1��𝑑𝑑p2 + 1.49𝜆𝜆2 � 𝑛𝑛02𝑁𝑁𝐴𝐴2 − 1� + 8𝜎𝜎2𝛽𝛽2𝑀𝑀2 � (73) is determined by solving eq. (53) with weighting function eq. (71). To find a dimensionless representation for eq. (73), we divide it by the depth of correlation 𝑧𝑧DoC0 (eq. (55)) of unprocessed images and express the result as a function of the dimensionless smoothing parameter KLoG = 𝜎𝜎𝑑𝑑𝑒𝑒(𝑧𝑧=𝑧𝑧𝑓𝑓) to obtain  𝑧𝑧DoC𝑧𝑧DoC0= � √𝜀𝜀41+ √𝜀𝜀4 (1 + 8𝛽𝛽2𝐾𝐾LoG2). (74) Eqs (73) and (74) are valid for 𝑧𝑧DoC ≪ 𝑠𝑠0 . Figure 20 shows the non-dimensional depth of correlation as a function of the dimensionless LoG-filter parameter KLoG. It is identical to Figure 18 for 𝑘𝑘DoG  =  1. However, for 𝑘𝑘DoG = 1, the difference of Gaussians filter eq. (66) has a zero 74 response (no signal passes through) while the LoG filter has a non-zero response. Consistent with the observation in Figure 18, the minimum  𝑧𝑧DoC𝑧𝑧DoC0= 0.49 is achieved for KLoG  →  0. As before, one cannot simply choose KLoG without considering other effects of the LoG filter, i.e. as KLoG  →  0, the filter becomes increasingly sensitive to single pixel noise that is typically present in micro PIV images. Hence, in order to maintain a certain signal to noise ratio for the ensemble averaged correlation function 〈RD〉�   more correlation functions need to be averaged as KLoG  →  0.  Figure 20: Effect of the Laplacian of a Gaussian filter on the depth of correlation eq. (74) with filter parameter KLoG = 𝜎𝜎𝑑𝑑𝑒𝑒(𝑧𝑧=𝑧𝑧𝑓𝑓) for Δz = 0 and ɛ = 0.01, valid for 𝑧𝑧DoC ≪ 𝑠𝑠0 Difference of moving averages Let us consider the difference of moving averages band-pass filter  𝐺𝐺�?⃗?𝑋� = 1𝐿𝐿12 𝑧𝑧𝑅𝑅𝑐𝑐𝑡𝑡 �𝑋𝑋𝐿𝐿1� 𝑧𝑧𝑅𝑅𝑐𝑐𝑡𝑡(𝑌𝑌𝐿𝐿1) − 1𝐿𝐿22 𝑧𝑧𝑅𝑅𝑐𝑐𝑡𝑡 �𝑋𝑋𝐿𝐿2� 𝑧𝑧𝑅𝑅𝑐𝑐𝑡𝑡( 𝑌𝑌𝐿𝐿2) (75) Where 𝐿𝐿1 and 𝐿𝐿2  are the widths of the square support of the first moving average filter and the second moving average filter in eq. (75) respectively and   𝑧𝑧𝑅𝑅𝑐𝑐𝑡𝑡(𝑋𝑋) = �1 for |𝑋𝑋| < 120 elsewhere  (76) is the rectangular function.  Despite some effort, we were unable to find analytical solutions for the local correlation function and for the weighting function. Instead we compute the weighting functions and the depth of correlation for particle images subject to the filter eq. (75) numerically by following the strategy 75 described in section 3.2.6 (the numerical procedure will be validated in following subsections) and by approximating (𝑠𝑠0 + 𝑧𝑧 − 𝑧𝑧𝑓𝑓)2 ≈ 𝑠𝑠02 in eq. (37). It was verified by extensive testing that the ratio 𝑧𝑧DoC𝑧𝑧DoC0 becomes independent of the physical parameters when expressed as a function of 𝐾𝐾DoMA = 𝐿𝐿1𝑑𝑑𝑒𝑒(𝑧𝑧=𝑧𝑧𝑓𝑓) and 𝑘𝑘DoMA = 𝐿𝐿1𝐿𝐿2 and a map of 𝑧𝑧DoC𝑧𝑧DoC0= 𝑓𝑓(𝐾𝐾DoMA,𝑘𝑘DoMA) valid for 𝑧𝑧DoC ≪ 𝑠𝑠0 is shown in Figure 21a-b.  In general, 𝑧𝑧DoC𝑧𝑧DoC0(𝐾𝐾DoMA,𝑘𝑘DoMA) follows the same trend as 𝑧𝑧DoC𝑧𝑧DoC0 (𝐾𝐾DoG,𝑘𝑘DoG) for particle images subject to the Difference of Gaussians filter eq. (66). As for the other band-pass filters, there is a theoretical minimum for the relative change of the depth of correlation 𝑧𝑧DoC𝑧𝑧DoC0→ 0.49 as 𝐾𝐾DoMA →  0 and 𝑘𝑘DoMA →  1. According to [2], the difference of moving averages band-pass filter has an ‘enhancing effect’ on the correlation function and the displacement correlation peak if the width of the smaller averaging window 𝐿𝐿1𝑑𝑑px= 2𝑧𝑧1 + 1 =  3 and if the width of the larger averaging window 𝐿𝐿2𝑑𝑑px =2𝑧𝑧2 + 1 >  2 𝑑𝑑e �𝑧𝑧=𝑧𝑧𝑓𝑓�𝑑𝑑px + 1.  When following these guidelines by [2], band-pass filtering can increase or decreases the DoC as compared to the unfiltered case. As shown in Figure 21c, the relative change of the DoC depends on the particle image size and the DoC is decreased due to filtering when particles appear larger in an image. In [2]  𝑑𝑑e �𝑧𝑧 = 𝑧𝑧𝑓𝑓� ≈ 5 𝑑𝑑px so that according to Figure 21 with 𝐾𝐾DoMA = 0.6 and 𝑘𝑘DoMA = 3.67 the depth of correlation was slightly increased by a factor of 1.1 through band-pass filtering.    76 a) b) c)  Figure 21: a) and b): Contour of 𝑧𝑧DoC𝑧𝑧DoC0 for particle images subject to the difference of moving averages filter eq. (75) with 𝐾𝐾DoMA =  𝐿𝐿1𝑑𝑑e �𝑧𝑧=𝑧𝑧𝑓𝑓� and 𝑘𝑘DoMA = 𝐿𝐿2𝐿𝐿1  for ɛ = 0.01 and Δz = 0 valid for 𝑧𝑧DoC ≪ 𝑠𝑠0. The difference between neighbouring contour levels is 0.5 in a and 0.1 in b) and the highest contour levels are equal to the highest values shown in the color maps; c) 𝑧𝑧DoC𝑧𝑧DoC0 as a function of (𝑑𝑑e �𝑧𝑧 = 𝑧𝑧𝑓𝑓�) when following the recommendations by [2]. 3.4.2.2 High pass filtering High pass filtering of PIV images aims at removing a non-uniform image background intensity (e.g. resulting from unwanted reflections in the experimental setup), as well as at reducing the influence of out-of-focus particles [73], [2], [188], [208], [193], [189], [209], [210].  High pass filtering is achieved by subtracting a low-pass filtered version of the image (often called background) from the original image. Again, the moving average filter is often used as low-pass filter [2], [208], [188], [210]. High-pass filtering via subtraction of a sliding average from the particle images is even offered in the standard image pre-processing menu of the commercial PIV software DaVis. However, other (non-linear) filters such as a median filter [188] have also been used for this purpose.   77 Gaussian-based high pass filter  The Gaussian-based high pass filter   𝐺𝐺HP�?⃗?𝑋� =  𝛿𝛿�?⃗?𝑋� − 12𝜋𝜋𝜎𝜎𝐺𝐺32 exp�− ?⃗?𝑋22𝜎𝜎𝐺𝐺32 � (77) is a special case of the Difference of Gaussians filter eq. (66) for 𝜎𝜎𝐺𝐺1 → 0 and 𝜎𝜎𝐺𝐺2 ≡  𝜎𝜎𝐺𝐺3. The corresponding weighting function   𝑊𝑊HP�𝑥𝑥�⃗� = 1(𝑠𝑠0 + 𝑧𝑧 − 𝑧𝑧f)4 �� 𝑑𝑑e24𝛽𝛽2�−2 − 2� 𝑑𝑑e24𝛽𝛽2 + 𝜎𝜎𝐺𝐺32 �−2 + � 𝑑𝑑e24𝛽𝛽2 + 2𝜎𝜎𝐺𝐺32 �−2� (78) follows from eq. (68) for 𝜎𝜎𝐺𝐺1 → 0 and 𝜎𝜎𝐺𝐺2 ≡  𝜎𝜎𝐺𝐺3.  Figure 22 compares measured weighting functions to the model eq. (78) for high-pass filtered micro PIV images for two objective lenses and two particle sizes. For small values of the dimensionless high-pass filter parameter 𝐾𝐾GHP  =  𝜎𝜎𝐺𝐺3𝑑𝑑e �𝑧𝑧=𝑧𝑧𝑓𝑓� < 1, the weighting functions becomes slightly narrower as a result of high-pass filtering. The agreement between model eq. (78) and measured weighting functions is acceptable. The largest RMS errors are 0.040, 0.051 and 0.049 for the square symbols in Figure 22a,c and d, respectively. In Figure 22b, the RMS error of 0.038 is the largest for the dataset represented by circles.   78 a)  b)        c)       d) Figure 22: Comparison of model eq. (78) (solid lines) to measured weighting function (symbols) for different values of 𝐾𝐾GHP = 𝜎𝜎𝐺𝐺3𝑑𝑑𝑒𝑒(𝑧𝑧=𝑧𝑧𝑓𝑓) (in legend): a) M = 20× – 𝑑𝑑p  = 1 µm; b) M = 40× -  𝑑𝑑p  = 1 µm; c) M = 20× –  𝑑𝑑p  = 3.1 µm; d) M = 40× –  𝑑𝑑p  = 3.1 µm. After approximating (𝑠𝑠0 + 𝑧𝑧−𝑧𝑧𝑓𝑓)4  ≈ 𝑠𝑠04 an analytical expression for the depth of correlation of high-pass filtered micro PIV is determined by solving the equivalent of eq. (53) for the weighting function eq. (78) by using the symbolic math toolbox of Matlab. As before, the resulting expression is rather cumbersome and will not be presented here. Again, the weighting function eq. (78) and the ratio 𝑧𝑧DoC𝑧𝑧DoC0 become independent of the physical parameters when expressed as a function of 𝐾𝐾GHP = 𝜎𝜎𝐺𝐺3𝑑𝑑𝑒𝑒(𝑧𝑧 = 𝑧𝑧𝑓𝑓). For ɛ = 0.01, the approximate expression   𝑧𝑧DoC𝑧𝑧DoC0= 1.008 − 11.947 + 10.86𝐾𝐾GHP2  (79) can be used for 0 < 𝐾𝐾GHP < 5 and 𝑧𝑧DoC𝑧𝑧DoC0≈ 1 for 𝐾𝐾GHP >5 as long as 𝑧𝑧DoC ≪ 𝑠𝑠0 . The ‘exact’ solution (valid for 𝑧𝑧DoC ≪ 𝑠𝑠0 ) (solid line) as well as the expression eq. (79) (dashed line) for the depth of correlation are shown in Figure 23. As the latter approximates the exact solution very well, the two lines appear almost as one. For larger values of 𝐾𝐾GHP  >  1, the 79 depth of correlation is unaffected by high-pass filtering. For 0 <  𝐾𝐾GHP  <  1, the DoC decreases with decreasing 𝐾𝐾GHP until the theoretical minimum 𝑧𝑧DoC𝑧𝑧DoC0→ 0.49 at 𝐾𝐾GHP  → 0 is reached. However, as 𝐾𝐾GHP  → 0, the high-pass filter will increasingly attenuate the displacement correlation peak relative to peaks resulting from single pixel random noise so that more correlation functions need to be averaged in order to maintain a certain signal to noise ratio of 〈RD〉� . Moving-average-based high pass filter The moving-average based high-pass filter   𝐺𝐺�?⃗?𝑋� = 𝛿𝛿�?⃗?𝑋� − 1𝐿𝐿32 𝑧𝑧𝑅𝑅𝑐𝑐𝑡𝑡 �𝑋𝑋𝐿𝐿3� 𝑧𝑧𝑅𝑅𝑐𝑐𝑡𝑡( 𝑌𝑌𝐿𝐿3) (80) is a special case of the Difference of Moving-average band-pass filter eq. (75) for 𝐿𝐿1 → 0 and 𝐿𝐿2 ≡  𝐿𝐿3. As before, weighting functions and depth of correlation for particle images subject to the filter eq. (80) are calculated numerically by following the strategy described in section 3.2.6 and by approximating (𝑠𝑠0 + 𝑧𝑧 − 𝑧𝑧𝑓𝑓)2 ≈ 𝑠𝑠02 in eq. (37). It was verified by extensive testing that the ratio 𝑧𝑧DoC𝑧𝑧DoC0 becomes independent of the physical parameters when expressed as a function of 𝐾𝐾MAHP = 𝐿𝐿3𝑑𝑑𝑒𝑒(𝑧𝑧=𝑧𝑧𝑓𝑓) and the results for 𝑧𝑧DoC𝑧𝑧DoC0 = 𝑓𝑓(𝐾𝐾MAHP) (valid for 𝑧𝑧DoC ≪ 𝑠𝑠0) are shown in Figure 23.  In general, 𝑧𝑧DoC𝑧𝑧DoC0(𝐾𝐾MAHP) follows the same trend as 𝑧𝑧DoC𝑧𝑧DoC0 (𝐾𝐾GHP) for particle images subject to the Gaussians based high-pass filter eq. (77) so that the discussion does not need to be repeated here.  It is interesting to note, that after scaling 𝐾𝐾MAHP′ = 𝐾𝐾MAHP3.5 , expression (79) for the Gaussian-based high-pass filter is a good approximation for  𝑧𝑧DoC𝑧𝑧DoC0(𝐾𝐾MAHP′ ) (dash dotted line in Figure 23) for 𝐾𝐾MAHP′ < 0.6. Note, that our numerical procedure was validated against known analytical solutions for the depth of correlation for particle images subject to the Gaussian high-pass filter. In figure 10, the numerical solution (open circles) match the corresponding analytical expression eq. (79) (solid line) well.   80  Figure 23: Relative decrease of the depth of correlation after high-pass filtering with filter parameters 𝐾𝐾GHP = 𝜎𝜎𝐺𝐺3𝑑𝑑𝑒𝑒(𝑧𝑧=𝑧𝑧𝑓𝑓) (for Gaussian based filter eq. (77)), 𝐾𝐾MAHP = 𝐿𝐿3𝑑𝑑𝑒𝑒(𝑧𝑧=𝑧𝑧𝑓𝑓) (for Moving-average based filter eq. (80)) and 𝐾𝐾MAHP′ = 𝐾𝐾MAHP3.5  for ɛ = 0.01 and valid for 𝑧𝑧DoC ≪ 𝑠𝑠0 : ‘Exact solution’ 𝑧𝑧DoC𝑧𝑧DoC0 (𝐾𝐾GHP)  for Gaussian based filter (solid line) and corresponding approximate expression eq. (79) (dashed line) as well as numerical solution (open circles). Numerical solution 𝑧𝑧DoC𝑧𝑧DoC0(𝐾𝐾MAHP) for moving average based filter eq. (80) (crosses). Scaled numerical solution 𝑧𝑧DoC𝑧𝑧DoC0(𝐾𝐾MAHP′ ) for moving average based filter (dash-dotted line). 3.4.2.3 Low-pass filtering Low-pass filtering (smoothing) of PIV images aims at reducing the contribution of single pixel noise to the correlation function [2], [186].  In previous work a moving average filter was used for smoothing [2], [3]. Gaussian low-pass filter The Gaussian low-pass filter   𝐺𝐺𝐿𝐿𝐿𝐿�?⃗?𝑋� =  12𝜋𝜋𝜎𝜎𝐺𝐺2 exp�− ?⃗?𝑋22𝜎𝜎𝐺𝐺2� (81) follows as special case of eq. (66) for of  𝜎𝜎𝐺𝐺2 → ∞ and 𝜎𝜎𝐺𝐺1 ≡ 𝜎𝜎𝐺𝐺 . The corresponding weighting function   𝑊𝑊LP�𝑥𝑥�⃗� = 1(𝑠𝑠0 + 𝑧𝑧 − 𝑧𝑧𝑓𝑓)4 �� 𝑑𝑑e24𝛽𝛽2 + 2𝜎𝜎𝐺𝐺2�−2� (82) is derived from eq. (68) in the same limit. Figure 24 compares measured weighting functions and model eq. (82) for low-pass filtered micro PIV images. Here it can be seen that the width of the weighting function increases with 81 increasing low-pass filter parameter 𝐾𝐾GLP  =  𝜎𝜎𝐺𝐺𝑑𝑑e �𝑧𝑧=𝑧𝑧𝑓𝑓�. The agreement between model eq. (82) and measured weighting functions is excellent. The largest RMS errors are 0.038 for the triangles in Figure 24a, 0.043 for the square symbols in Figure 24b, 0.021 for the circles in Figure 24c and 0.026 for the circles in Figure 24d. a)  b)  c)  d)  Figure 24: Effect of low pass filtering with a Gaussian filter with standard deviation 𝜎𝜎𝐺𝐺 = 𝐾𝐾GLP𝑑𝑑e (𝑧𝑧 = 𝑧𝑧𝑓𝑓) on the weighting function 𝑊𝑊LP. The values for 𝐾𝐾GLP are shown in legend. Solid lines are equation (82) and symbols are experimentally determined weighting functions for a) M = 20× – 𝑑𝑑p  = 1 µm; b) M = 40× -  𝑑𝑑p  = 1 µm ; c) M = 20× –  𝑑𝑑p  = 3.1 µm; d) M = 40× –  𝑑𝑑p  = 3.1 µm. Solving equation (53) for the depth of correlation with the weighting function eq. (82), ∆𝑧𝑧 =0 and (𝑠𝑠0 + 𝑧𝑧−𝑧𝑧𝑓𝑓)4  ≈ 𝑠𝑠04 gives  𝑧𝑧DoC𝑧𝑧DoC0= �1 + 8𝛽𝛽2𝐾𝐾𝐺𝐺𝐿𝐿𝐿𝐿2 . (83) For 𝐾𝐾GLP = 0 this reduces to the well-known result eq. (55) of [79]. The evaluation of eq. (83) for the relative increase of the DoC with the dimensionless smoothing parameter 𝐾𝐾GLP is shown in Figure 25. For small values of 𝐾𝐾GLP, the DoC is almost unaffected by smoothing. I.e., smoothing will increase the DoC by less than 10% as long as 82  𝜎𝜎𝐺𝐺 < 0.18𝑑𝑑𝑒𝑒(𝑧𝑧 = 𝑧𝑧𝑓𝑓), (84) while an accurate estimate of 𝑑𝑑𝑒𝑒(𝑧𝑧 = 𝑧𝑧𝑓𝑓) can easily be obtained (by eye) from any microPIV image. As a consequence of approximating (𝑠𝑠0 + 𝑧𝑧 − 𝑧𝑧𝑓𝑓)4  ≈ 𝑠𝑠04, eq. (83) is only valid for 𝑧𝑧DoC ≪ 𝑠𝑠0 . Moving-average low-pass filter The moving-average based low-pass filter   𝐺𝐺�?⃗?𝑋� = 1𝐿𝐿2𝑧𝑧𝑅𝑅𝑐𝑐𝑡𝑡 �𝑋𝑋𝐿𝐿� 𝑧𝑧𝑅𝑅𝑐𝑐𝑡𝑡(𝑌𝑌𝐿𝐿) (85) is a special case of the Difference of Moving-average band-pass filter eq. (75) for 𝐿𝐿2 → ∞ and 𝐿𝐿1 ≡  𝐿𝐿. Weighting functions and depth of correlation for particle images subject to the filter eq. (85) are calculated numerically by following the strategy described in section 3.2.6 and by approximating (𝑠𝑠0 + 𝑧𝑧 − 𝑧𝑧𝑓𝑓)2 ≈ 𝑠𝑠02 in eq. (37). It was verified by extensive testing that the ratio 𝑧𝑧DoC𝑧𝑧DoC0 becomes independent of the physical parameters when expressed as a function of 𝐾𝐾MALP = 𝐿𝐿𝑑𝑑𝑒𝑒(𝑧𝑧=𝑧𝑧𝑓𝑓) and the results for 𝑧𝑧DoC𝑧𝑧DoC0 = 𝑓𝑓(𝐾𝐾MALP) (valid for 𝑧𝑧DoC ≪ 𝑠𝑠0) are shown in Figure 25 (crosses).  After introducing the scaling 𝐾𝐾MALP′ = 𝐾𝐾MALP3.5 , expression (83) for the Gaussian low-pass filter is a good approximation for  𝑧𝑧DoC𝑧𝑧DoC0(𝐾𝐾MALP′ ) (dash dotted line in Figure 25) for Moving-average low-pass filter for 𝐾𝐾MALP′ < 0.6. Wereley et al. [2] recommend to use a rectangular moving average filter with  𝐿𝐿1𝑑𝑑px= 2𝑧𝑧1 + 1 = 3 to reduce the influence of single pixel random noise on the correlation function. In Figure 25 it can be seen that, when following this recommendation by choosing 𝐾𝐾MALP = 3 𝑑𝑑px𝑑𝑑𝑒𝑒(𝑧𝑧=𝑧𝑧𝑓𝑓), the increase of the depth of correlation due to low-pas filtering decreases with increasing particle (image) size. Again, the numerical procedure was validated against known analytical solutions for the depth of correlation for particle images subject to the Gaussian low-pass filter. In figure 12, the 83 numerical solution (open circles) match the corresponding analytical expression eq. (83) (solid line) well.   Figure 25 Effect of low-pass filtering on the depth of correlation for ɛ = 0.01 and for 𝑧𝑧DoC ≪ 𝑠𝑠0 : 𝑧𝑧DoC𝑧𝑧DoC0(𝐾𝐾GLP) for Gaussian low-pass filter (solid line) and corresponding numerical solution (open circles). Numerical solution 𝑧𝑧DoC𝑧𝑧DoC0(𝐾𝐾MALP) for moving average low-pass filter eq. (85) (crosses). Scaled numerical solution 𝑧𝑧DoC𝑧𝑧DoC0(𝐾𝐾MALP′ ) for moving average low-pass filter (dash-dotted line). Smoothing parameters are 𝐾𝐾GLP = 𝜎𝜎𝐺𝐺𝑑𝑑𝑒𝑒(𝑧𝑧=𝑧𝑧𝑓𝑓) (for Gaussian based filter eq. (81)), 𝐾𝐾MALP = 𝐿𝐿3𝑑𝑑𝑒𝑒(𝑧𝑧=𝑧𝑧𝑓𝑓) (for Moving-average based filter eq. (85)) and 𝐾𝐾MALP′ = 𝐾𝐾MALP3.5  for scaled Moving-average based filter. 3.5 Conclusions  This work investigated how the depth of correlation is affected by image processing methods that are commonly applied to micro PIV images in order to improve the quality of the correlation functions. It was shown that smoothing tends to increase the DoC while high-pass filtering tends to decrease the DoC. Consequently, Band-pass filtering of micro PIV images can either decrease or increase the DoC, depending on the choice of filter parameters. The general results for different filter methods also include special cases previously discussed in literature. It was shown that all band-pass filters considered by this work can only decrease the depth of correlation up to a theoretical minimum of approximately 𝑧𝑧DoC𝑧𝑧DoC0= 0.5. However, in this limit, the filters have zero band-gap so that this limit can never be reached in praxis. Finally, this work presented the evaluation of the integrals describing micro PIV cross correlation in a slightly different way as compared to previous work. As a result, we were able to recover the in-plane loss of correlation known from the theory of classical PIV.    84 4 On the effect of velocity gradients on the depth of correlation in micro PIV 4.1 Introduction As described in section 3.1, µPIV is a well-established tool for measuring flow fields in microfluidic environments. One limitation of micro PIV is the finite measurement depth (Depth of Correlation) that is associated with the micro PIV method. A finite Depth of Correlation can lead to a bias error in the measured velocity when the fluid velocity varies in the direction of the optical axis of the microscope.  In the landmark paper by M.G. Olsen [79], an analytical expression for the DoC for the case when flow velocity gradients can be neglected over the extent of the measurement domain has been developed. The model derived by this work has gained widespread acceptance. Later, Olsen proposed extensions to this model that account for the effects of Brownian motion [204], out-of-plane velocity [198] and power filtering [80] on the DoC.  In closely related work, [84] made a connection between the expression for the DoC for uniform flow derived by Olsen [79] and the curvature at the peak of the local correlation function. The curvature of the local correlation function is a measurable quantity and has been utilized [84], [203], [194] to verify the model for the depth of correlation for approximately uniform flow [79]. The concept of curvature of the local correlation function has also been utilized to derive expressions for the depth of correlation that account for the effect of velocity gradients in the interrogation domain [82], [83]. However, it has never been verified if the concept of curvature of the local correlation function is valid to determine the DoC for the case when flow velocity gradients are present. The current work re-visits the connection between correlation function curvature and DoC and clearly identifies scenarios under which these quantities are related. Furthermore, the current work re-evaluates the previous definition of the DoC for the case when in-plane velocity gradients are present. A slightly different definition of the DoC is proposed that can take in-plane velocity gradients into account. The effect of constant in- and out-of plane flow gradients on the DoC is re-evaluated and corrections to previous results are proposed. This work derives dimensionless parameters that allow defining more general criteria for which the effect of flow gradients on the DoC can be neglected. We then move on to more complicated flow scenarios where a finite DoC introduces a bias error to the measured velocity and we show that additional dimensionless parameters are required to describe the problem.  85 4.2 Theoretical framework 4.2.1 Image plane intensity distribution of a single micro particle The theory of µPIV cross correlation that will be summarized in the following section requires a model for the particle images. The commonly used model is summarized in section 3.2.1.  The validity of the results presented in the current chapter ultimately depends on the capability of the model for µPIV images eqs (37)-(40) to describe real images [79], [84]. More specifically, the results presented in the following two sections of this work require that the particle images are well described by eq. (37) with a particle image diameter of the form  𝑑𝑑e = �𝐶𝐶1 + 𝐶𝐶2(𝑧𝑧 − 𝑧𝑧𝑓𝑓)2�1/2, (86) Where C1 and C2 are constants that depend on the parameters describing the optical system. Although synthetic images based on equations (37)-(40) can agree very well with experimental images [84], [203], this is not always the case [3], [194]. It has been suggested that the reason for the discrepancy is the single thin lens approximation which fails to describe the optics of a complex microscope [3], [194] quantitatively.   However, in many cases, agreement between the model for particle images and experimental images can be established by using eq. (86) instead of eq. (39) with 𝐶𝐶1 and 𝐶𝐶2 determined from experiments [194], [3]. Previous work indicates that eq. (86) is appropriate for most setups based on inverted microscopes [84], [203], [194], [3] but becomes invalid when other, specialized microscopes are used [3]. 4.2.2 Displacement correlation peak In this section an expression for the micro-PIV displacement correlation peak is derived from the underlying framework. A detailed description of the framework can be found in [211], [199], [82], [201], [83], [73].  It is commonly assumed that a µPIV image pair can be described as  𝐼𝐼1�?⃗?𝑋� = 𝑊𝑊I1�?⃗?𝑋 − ?⃗?𝑋1�∭ 𝐼𝐼o1�𝑥𝑥�⃗�𝐽𝐽o�?⃗?𝑋 − 𝑀𝑀?⃗?𝑥; 𝑧𝑧�∑ 𝛿𝛿�𝑥𝑥�⃗ − 𝑥𝑥�⃗𝑗𝑗(𝑡𝑡)�𝑗𝑗 𝑑𝑑𝑥𝑥𝑑𝑑𝑦𝑦𝑑𝑑𝑧𝑧, 𝐼𝐼2�?⃗?𝑋� = 𝑊𝑊I2�?⃗?𝑋 − ?⃗?𝑋2�∭ 𝐼𝐼o2�𝑥𝑥�⃗′�𝐽𝐽o�?⃗?𝑋 − 𝑀𝑀?⃗?𝑥′; 𝑧𝑧′�∑ 𝛿𝛿�𝑥𝑥�⃗′ − 𝑥𝑥�⃗𝑗𝑗(𝑡𝑡′)�𝑗𝑗 𝑑𝑑𝑥𝑥′𝑑𝑑𝑦𝑦′𝑑𝑑𝑧𝑧′, (87) 86 where 𝑊𝑊I1and 𝑊𝑊I2 are weighting functions defining the interrogation windows, 𝐼𝐼o1 and  𝐼𝐼o2 define the illumination pulses, 𝐽𝐽o is the particle image intensity function per unit illumination density, 𝛿𝛿�𝑥𝑥�⃗ − 𝑥𝑥�⃗𝑖𝑖� is the Dirac delta function, 𝑥𝑥�⃗𝑗𝑗(𝑡𝑡) and 𝑥𝑥�⃗𝑗𝑗(𝑡𝑡′) are the positions of the particles in the physical domain at the time the images were taken and ?⃗?𝑥 = (𝑥𝑥 𝑦𝑦)𝑘𝑘, 𝑥𝑥�⃗ = (𝑥𝑥 𝑦𝑦 𝑧𝑧 )𝑘𝑘 , ?⃗?𝑥′ =(𝑥𝑥′ 𝑦𝑦′)𝑘𝑘 and 𝑥𝑥�⃗′ = (𝑥𝑥′ 𝑦𝑦′ 𝑧𝑧′ )𝑘𝑘 are Cartesian position vectors in the physical domain (𝑧𝑧 and 𝑧𝑧’ point in the direction of the optical axis of the microscope). This work considers the simple windowing function 𝑊𝑊I1�?⃗?𝑋 − ?⃗?𝑋1� = 1 for all ?⃗?𝑋 ∈ Ω𝐼𝐼1, and 𝑊𝑊I1 = 0 for all other ?⃗?𝑋, where Ω𝐼𝐼1 is a rectangular region in the image plane centered around ?⃗?𝑋1 (𝑊𝑊I2 analogous). All integrals in this work are to be evaluated from –infinity to infinify if not stated otherwise. Furthermore, the solutions to the integrals in this work were taken from [207], if not stated otherwise. As shown in previous work, the corresponding PIV displacement correlation peak follows from [211], [82], [83]  〈𝑅𝑅𝐷𝐷(𝑠𝑠)〉𝑢𝑢�⃗ = ∬∭∭𝑊𝑊I1�?⃗?𝑋 − ?⃗?𝑋1�𝑊𝑊I2�?⃗?𝑋 + 𝑠𝑠 − ?⃗?𝑋2�𝐼𝐼o1�𝑥𝑥�⃗�𝐼𝐼o2�𝑥𝑥�⃗′�𝐽𝐽o�?⃗?𝑋 − 𝑀𝑀?⃗?𝑥; 𝑧𝑧�𝐽𝐽o�?⃗?𝑋 +𝑠𝑠 −𝑀𝑀?⃗?𝑥′; 𝑧𝑧′�𝐶𝐶�𝑥𝑥�⃗, 𝑡𝑡�𝑓𝑓�𝑥𝑥�⃗, 𝑡𝑡, 𝑥𝑥�⃗′, 𝑡𝑡′�𝑢𝑢�⃗𝑑𝑑𝑥𝑥′𝑑𝑑𝑦𝑦′𝑑𝑑𝑧𝑧′𝑑𝑑𝑥𝑥 𝑑𝑑𝑦𝑦 𝑑𝑑𝑧𝑧 𝑑𝑑𝑋𝑋 𝑑𝑑𝑌𝑌, (88) with the Cartesian coordinates 𝑠𝑠 = (𝑠𝑠𝑥𝑥 𝑠𝑠𝑦𝑦)𝑘𝑘 and particle concentration 𝐶𝐶�𝑥𝑥�⃗, 𝑡𝑡�. In eq. (88), the conditional transition probability 𝑓𝑓�𝑥𝑥�⃗, 𝑡𝑡, 𝑥𝑥�⃗′, 𝑡𝑡′�𝑢𝑢�⃗𝑑𝑑𝑥𝑥′𝑑𝑑𝑦𝑦′𝑑𝑑𝑧𝑧′ represents the probability that a particle at (𝑥𝑥�⃗, 𝑡𝑡) moves into the differential volume 𝑑𝑑𝑥𝑥′𝑑𝑑𝑦𝑦′𝑑𝑑𝑧𝑧′ at (𝑥𝑥�⃗′, 𝑡𝑡′), given they are randomly distributed and follow the incompressible flow field 𝑢𝑢�⃗  ideally [211], [82], [73].  The current work considers the steady flow field   𝑢𝑢�⃗ = �∆𝑥𝑥∆𝑡𝑡∆𝑦𝑦∆𝑡𝑡�𝑘𝑘 = (𝑢𝑢𝑥𝑥 𝑢𝑢𝑦𝑦)𝑘𝑘 = (𝑢𝑢 + 𝑎𝑎s(𝑦𝑦 − 𝑦𝑦0) + 𝑔𝑔(𝑧𝑧 − 𝑧𝑧0) 𝑣𝑣)𝑘𝑘, (89) and ∆𝑧𝑧∆𝑡𝑡= 0 in a rectangular interrogation region centered around 𝑥𝑥0′ ,𝑦𝑦0′  with dimensions 𝐿𝐿𝑥𝑥,𝐿𝐿𝑦𝑦. In eq. (89), 𝑢𝑢,𝑣𝑣 and 𝑎𝑎s are constants, g is a function of 𝑧𝑧′  and ∆𝑡𝑡 = 𝑡𝑡′ − 𝑡𝑡. Following the methodology in [82], [83], the corresponding conditional probability density   𝑓𝑓�𝑥𝑥�⃗, 𝑡𝑡, 𝑥𝑥�⃗′, 𝑡𝑡′�𝑢𝑢�⃗= �𝛿𝛿(𝑧𝑧′ − 𝑧𝑧)𝛿𝛿(𝑦𝑦′ − 𝑦𝑦 − ∆𝑦𝑦) 1𝑎𝑎s𝐿𝐿𝑦𝑦∆𝑡𝑡 , for 𝑥𝑥 + �𝑢𝑢 + 𝑔𝑔(𝑧𝑧′ − 𝑧𝑧0′ ) − 𝑎𝑎s𝐿𝐿𝑦𝑦2 �∆𝑡𝑡 <  𝑥𝑥′ < 𝑥𝑥 + �𝑢𝑢 + 𝑔𝑔(𝑧𝑧′ − 𝑧𝑧0′ ) + 𝑎𝑎s𝐿𝐿𝑦𝑦2 �∆𝑡𝑡0  otherwise   (90) 87 is obtained. Let us now assume that the particle image intensity distribution is well approximated by eqs (37)-(40) and that the illumination varies only in z-direction (i.e. 𝐼𝐼o1�𝑥𝑥�⃗�  ≡𝐼𝐼o1(𝑧𝑧) and 𝐼𝐼o2�𝑥𝑥�⃗′�  ≡ 𝐼𝐼o2(𝑧𝑧′)), and the particle concentration is uniform. Substituting eq. (37)-(40) and (90) in eq. (88) and integration with respect to ?⃗?𝑥′ yields  〈𝑅𝑅𝐷𝐷(𝑠𝑠 −𝑀𝑀∆?⃗?𝑥)〉 = ∫ 〈𝑅𝑅𝐷𝐷(𝑠𝑠 − 𝑀𝑀∆?⃗?𝑥), 𝑧𝑧〉 𝑑𝑑𝑧𝑧  (91) with the local correlation function  〈𝑅𝑅𝐷𝐷(𝑠𝑠, 𝑧𝑧)〉 =𝑁𝑁𝐽𝐽p2𝐷𝐷a4𝛽𝛽364(𝜋𝜋)3/2𝑀𝑀𝑎𝑎s𝐿𝐿𝑦𝑦∆𝑡𝑡𝑠𝑠04 ∬𝑊𝑊I1�?⃗?𝑋 − ?⃗?𝑋1�𝑊𝑊I2�?⃗?𝑋 + 𝑠𝑠 − ?⃗?𝑋2�∬ 𝐼𝐼o1(𝑧𝑧)𝐼𝐼o2(𝑧𝑧)𝑑𝑑e3 exp �−4𝛽𝛽2𝑑𝑑e2 �(𝑋𝑋 −𝑀𝑀𝑥𝑥)2 + (𝑌𝑌 −𝑀𝑀𝑦𝑦)2 +�𝑌𝑌 −𝑀𝑀(𝑦𝑦 + ∆𝑦𝑦) + 𝑠𝑠𝑦𝑦�2�� �erf�2𝛽𝛽 𝑋𝑋+𝑠𝑠𝑥𝑥−𝑀𝑀𝑥𝑥−𝑀𝑀∆𝑡𝑡�𝑢𝑢+𝑔𝑔(𝑧𝑧−𝑧𝑧0)−𝜅𝜅s𝐿𝐿𝑦𝑦2 �𝑑𝑑𝑒𝑒 � −erf�2𝛽𝛽 𝑋𝑋+𝑠𝑠𝑥𝑥−𝑀𝑀𝑥𝑥−𝑀𝑀∆𝑡𝑡�𝑢𝑢+𝑔𝑔(𝑧𝑧−𝑧𝑧0)+𝜅𝜅s𝐿𝐿𝑦𝑦2 �𝑑𝑑𝑒𝑒�� 𝑑𝑑𝑦𝑦 𝑑𝑑𝑥𝑥 𝑑𝑑𝑋𝑋 𝑑𝑑𝑌𝑌,  (92) representing the contribution of particles at a certain depth z in the fluid to the overall correlation function.  Let us now integrate eq. (92) with respect to y to obtain  〈𝑅𝑅𝐷𝐷(𝑠𝑠, 𝑧𝑧)〉 =𝑁𝑁𝐽𝐽p2𝐷𝐷a4𝛽𝛽2√2256𝜋𝜋𝑀𝑀2𝑎𝑎s𝐿𝐿𝑦𝑦∆𝑡𝑡𝑠𝑠04 ∬𝑊𝑊I1�?⃗?𝑋 − ?⃗?𝑋1�𝑊𝑊I2�?⃗?𝑋 + 𝑠𝑠 − ?⃗?𝑋2� ∫ 𝐼𝐼o1(𝑧𝑧)𝐼𝐼o2(𝑧𝑧)𝑑𝑑e2 exp �−4𝛽𝛽2𝑑𝑑e2 �(𝑋𝑋 −𝑀𝑀𝑥𝑥)2 + 12�𝑠𝑠𝑦𝑦 −𝑀𝑀∆𝑦𝑦�2��𝜑𝜑�𝑌𝑌, 𝑠𝑠𝑦𝑦� �erf�2𝛽𝛽 𝑋𝑋+𝑠𝑠𝑥𝑥−𝑀𝑀𝑥𝑥−𝑀𝑀∆𝑡𝑡�𝑢𝑢+𝑔𝑔(𝑧𝑧−𝑧𝑧0)−𝜅𝜅s𝐿𝐿𝑦𝑦2 �𝑑𝑑𝑒𝑒 � −erf�2𝛽𝛽 𝑋𝑋+𝑠𝑠𝑥𝑥−𝑀𝑀𝑥𝑥−𝑀𝑀∆𝑡𝑡�𝑢𝑢+𝑔𝑔(𝑧𝑧−𝑧𝑧0)+𝜅𝜅s𝐿𝐿𝑦𝑦2 �𝑑𝑑𝑒𝑒�� 𝑑𝑑𝑥𝑥 𝑑𝑑𝑋𝑋 𝑑𝑑𝑌𝑌, (93) where   𝜑𝜑�𝑌𝑌, 𝑠𝑠𝑦𝑦� = lim𝑦𝑦→∞ �erf �√2 𝑠𝑠𝑦𝑦−𝑀𝑀∆𝑦𝑦−2(𝑀𝑀𝑦𝑦−𝑌𝑌)𝑑𝑑𝑒𝑒 � − erf �√2 𝑠𝑠𝑦𝑦−𝑀𝑀∆𝑦𝑦+2(𝑀𝑀𝑦𝑦+𝑌𝑌)𝑑𝑑𝑒𝑒 ��. (94) Physically reasonable ∆𝑦𝑦 are finite and 𝑊𝑊I1and 𝑊𝑊I2 put bounds on 𝑌𝑌 and 𝑌𝑌 + 𝑠𝑠𝑦𝑦, respectively, so that eq. (94) becomes 88  𝜑𝜑�𝑌𝑌, 𝑠𝑠𝑦𝑦� ≈ lim𝑦𝑦→∞�erf �− 2√2𝑀𝑀𝑦𝑦𝑑𝑑𝑒𝑒 � − erf �2√2𝑀𝑀𝑦𝑦𝑑𝑑𝑒𝑒 �� ≈ −2. (95) Although lim�𝑧𝑧−𝑧𝑧𝑓𝑓�→∞ 𝑑𝑑𝑒𝑒 = ∞, we believe that (95) is a good approximation as for physically reasonable local correlation functions lim�𝑧𝑧−𝑧𝑧𝑓𝑓�→∞〈𝑅𝑅𝐷𝐷(𝑠𝑠 − 𝑀𝑀∆?⃗?𝑥), 𝑧𝑧〉 = 0 regardless of the approximation eq. (95) (In other words, particles that are very far from the object plane will not contribute to the correlation function eq. (91)).  Following the same arguments for the integration of eq. (93) with respect to x gives  〈𝑅𝑅𝐷𝐷(𝑠𝑠, 𝑧𝑧)〉 =−𝑁𝑁𝐽𝐽p2𝐷𝐷a4𝛽𝛽256√2𝜋𝜋𝑀𝑀3𝑎𝑎s𝐿𝐿𝑦𝑦∆𝑡𝑡𝑠𝑠04 𝐹𝐹𝐼𝐼(𝑠𝑠) 𝐼𝐼o1(𝑧𝑧)𝐼𝐼o2(𝑧𝑧)𝑑𝑑e  exp �−2𝛽𝛽2𝑑𝑑e2 �𝑠𝑠𝑦𝑦 −𝑀𝑀∆𝑦𝑦�2� �erf�√2𝛽𝛽 𝑠𝑠𝑥𝑥−𝑀𝑀∆𝑡𝑡�𝑢𝑢+𝑔𝑔(𝑧𝑧−𝑧𝑧0)−𝜅𝜅s𝐿𝐿𝑦𝑦2 �𝑑𝑑𝑒𝑒� − erf�√2𝛽𝛽 𝑠𝑠𝑥𝑥−𝑀𝑀∆𝑡𝑡�𝑢𝑢+𝑔𝑔(𝑧𝑧−𝑧𝑧0)+𝜅𝜅s𝐿𝐿𝑦𝑦2 �𝑑𝑑𝑒𝑒�� ,  (96) with the in-plane loss of correlation   𝐹𝐹𝐼𝐼(𝑠𝑠) = �𝑊𝑊I1�?⃗?𝑋 − ?⃗?𝑋1�𝑊𝑊I2�?⃗?𝑋 + 𝑠𝑠 − ?⃗?𝑋2�𝑑𝑑𝑋𝑋𝑑𝑑𝑌𝑌 (97) accounting for the skewing of the cross correlation function due to the finite size of the interrogation windows [199].  4.2.3 Weighting functions  For the remainder of the current work, skewing of the correlation function due to in-plane correlation loss is neglected, i.e. 𝐹𝐹𝐼𝐼(𝑠𝑠) = 1. In practise the bias error of the measured velocity introduced by skewing can be eliminated by shifting the second interrogation [202] window and/or by using interrogation windows of appropriate different sizes [199], [73]. Since 𝐹𝐹𝐼𝐼(𝑠𝑠) is known the correlation function can be divided by 𝐹𝐹𝐼𝐼(𝑠𝑠) prior to peak detection in order to eliminate skewing completely [73].  Regardless of the specific flow field (and correlation function) at hand, an expression for the measured velocity 𝑢𝑢�⃗ 0 =  𝑠𝑠0𝑀𝑀∆𝑡𝑡  is determined from the correlation function eq. (91) by finding the location 𝑠𝑠0 = 𝑢𝑢�⃗ 0𝑀𝑀∆𝑡𝑡 of the signal peak where the partial derivatives of the overall correlation function are zero [79], [198], i.e. 89  𝜕𝜕〈𝑅𝑅𝐷𝐷〉𝜕𝜕𝑠𝑠𝑥𝑥�𝑠𝑠=𝑠𝑠0=𝑢𝑢�⃗ 0𝑀𝑀∆𝑡𝑡= ∫ 𝜕𝜕〈𝑅𝑅𝐷𝐷〉𝜕𝜕𝑠𝑠𝑥𝑥�𝑠𝑠=𝑠𝑠0=𝑢𝑢�⃗ 0𝑀𝑀∆𝑡𝑡𝑑𝑑𝑧𝑧 = 0, 𝜕𝜕〈𝑅𝑅𝐷𝐷〉𝜕𝜕𝑠𝑠𝑦𝑦�𝑠𝑠=𝑠𝑠0=𝑢𝑢�⃗ 0𝑀𝑀∆𝑡𝑡= ∫ 𝜕𝜕〈𝑅𝑅𝐷𝐷〉𝜕𝜕𝑠𝑠𝑦𝑦�𝑠𝑠=𝑠𝑠0=𝑢𝑢�⃗ 0𝑀𝑀∆𝑡𝑡𝑑𝑑𝑧𝑧 = 0. (98) In the landmark paper [79], it was shown that eq (98) in combination with eq (96) for the special case of vanishing gradients (i.e. as = 0 and g = 0) can be written as   𝑢𝑢0𝑥𝑥(𝑡𝑡) = ∫𝑊𝑊�𝑥𝑥�𝑢𝑢��⃗ ,𝑢𝑢�⃗ 0,𝑧𝑧�𝑢𝑢𝑥𝑥(𝑧𝑧,𝑡𝑡)𝑑𝑑𝑧𝑧∫𝑊𝑊�𝑥𝑥�𝑢𝑢��⃗ ,𝑢𝑢�⃗ 0,𝑧𝑧�𝑑𝑑𝑧𝑧 , 𝑢𝑢0𝑦𝑦(𝑡𝑡) = ∫𝑊𝑊�𝑦𝑦�𝑢𝑢��⃗ ,𝑢𝑢�⃗ 0,𝑧𝑧�𝑢𝑢𝑦𝑦(𝑧𝑧,𝑡𝑡)𝑑𝑑𝑧𝑧∫𝑊𝑊�𝑦𝑦�𝑢𝑢��⃗ ,𝑢𝑢�⃗ 0,𝑧𝑧�𝑑𝑑𝑧𝑧 , (99) where 𝑊𝑊�𝑥𝑥�𝑢𝑢��⃗ ,𝑢𝑢�⃗ 0, 𝑧𝑧� and 𝑊𝑊�𝑦𝑦�𝑢𝑢��⃗ ,𝑢𝑢�⃗ 0, 𝑧𝑧� are weighting functions that describe the relative influence of all particles in the interrogation window at any depth z in the fluid on the measured velocity. For vanishingly small flow velocity gradients it has been shown that the weighting functions in eqs (99) become independent of parameters describing the flow field [79]. However, in general the weighting functions depend on both, the vector 𝑢𝑢�⃗ = �𝑢𝑢𝑥𝑥(𝑧𝑧, 𝑡𝑡) 𝑢𝑢𝑦𝑦(𝑧𝑧, 𝑡𝑡)�𝑘𝑘 representing the (theoretically) measured velocity if only particles at a certain depth z in the fluid were present (i.e. 𝑢𝑢�⃗ = ∆𝑥𝑥∆𝑡𝑡 and ∆?⃗?𝑥 corresponding to the position of the peak of the local correlation function) as well as the measured flow velocity 𝑢𝑢�⃗ 0. Note that, in the special case of a flow field with zero in-plane gradients, 𝑢𝑢�⃗ (𝑧𝑧, 𝑡𝑡) = ∆?⃗?𝑥(𝑧𝑧, 𝑡𝑡) ∆𝑡𝑡⁄  is the actual particle velocity at a certain depth of the fluid.  The weighting functions were originally defined implicitly through eqs (98) and (99) with eq (96) for the special case of vanishing velocity gradients [79]. An attempt has been made to derive a more explicit expression for the weighting functions based on the curvature of the local correlation function [84]. More specifically, Bourdon et al. [84] proposed that the weighting function is equal to the Hessian matrix of the local correlation function evaluated at the local signal peak and showed that certain components of the Hessian matrix can indeed be equal to the weighting functions as defined through eqs (98) and (99) when the flow field is 90 approximately uniform4. More recent work derived expressions for weighting functions that account for flow velocity gradients from specific components of the Hessian matrix of the local correlation function [82], [83] even though it has never been verified if the original weighting functions as implicitly defined through eqs (98), (99) are related to the curvature of the local correlation function when the flow field is non-uniform. Thus, the present work will re-evaluate the link between the original micro PIV weighting functions as implicitly defined by (98) and (99) and the curvature of the local correlation function. As a first step towards this goal, explicit expressions of the weighting functions are sought. Therefore, it is crucial to note that eqs (99) is simply an algebraically manipulated version of eq (98). More specifically, eq (99) is obtained by solving eq (98) is for 𝑢𝑢�⃗ 0 [79], [198]. Hence, the original implicit equations eqs (98) and (99) defining the weighting functions suggest that more explicit expressions of the same weighting functions read   𝑊𝑊�𝑥𝑥�𝑢𝑢��⃗ ,𝑢𝑢�⃗ 0, 𝑧𝑧� = 𝑑𝑑〈𝑅𝑅𝐷𝐷(𝑠𝑠, 𝑧𝑧)〉𝑑𝑑𝑠𝑠𝑥𝑥 �𝑠𝑠=𝑠𝑠0=𝑢𝑢�⃗ 0𝑀𝑀∆𝑡𝑡𝑢𝑢𝑥𝑥(𝑧𝑧, 𝑡𝑡) − 𝑢𝑢0𝑥𝑥(𝑡𝑡)  𝑊𝑊�𝑦𝑦�𝑢𝑢��⃗ ,𝑢𝑢�⃗ 0, 𝑧𝑧� = 𝑑𝑑〈𝑅𝑅𝐷𝐷(𝑠𝑠, 𝑧𝑧)〉𝑑𝑑𝑠𝑠𝑦𝑦 �𝑠𝑠=𝑠𝑠0=𝑢𝑢�⃗ 0𝑀𝑀∆𝑡𝑡𝑢𝑢𝑦𝑦(𝑧𝑧, 𝑡𝑡) − 𝑢𝑢0𝑦𝑦(𝑡𝑡)  (100) It has to be emphasized, that the expressions eq (100) are not arbitrary definitions of new weighting functions. Instead, equation (100) represents the only valid expressions of the same weighting functions that were originally defined implicitly by equations (98) and (99). In contrast, weighting functions derived from the Hessian matrix of the local correlation function have never been shown to be related to eqs (98) and (99) when flow velocity gradients are present.  Equations (100) show that the weighting functions are, in general, directly related to the first derivative of the local correlation function evaluated at the position of the peak of the global correlation function rather than to the second derivative of the local correlation function 4 The explicit definition of the weighting function given in [81] (i.e. equation 16) is problematic, as a scalar weighting function cannot be equal to the Hessian matrix [81]. Furthermore, it remains unclear how to interpret the mixed derivatives of the Hessian matrix. 91                                                 evaluated at the position of the peak of the local correlation function as it was suggested by [84]. In the next step, it will be determined under which conditions the weighting functions are related to the second derivative of the local correlation function evaluated at the position of the peak of the local correlation function. Let us assume that our optical system behaves in such way that ?̃⃗?𝑠 implicitly defined by ∇�⃗𝑠𝑠 �∬ 𝐽𝐽o�?⃗?𝑋; 𝑧𝑧�𝐽𝐽o�?⃗?𝑋 + 𝑠𝑠; 𝑧𝑧�𝑑𝑑𝑋𝑋𝑑𝑑𝑌𝑌�𝑠𝑠=?̃⃗?𝑠= 0�⃗  does not depend on the distance of the particle from the object plane (𝑧𝑧 − 𝑧𝑧𝑓𝑓) (with ∇�⃗ = � 𝜕𝜕𝜕𝜕𝑠𝑠𝑥𝑥 𝜕𝜕𝜕𝜕𝑠𝑠𝑦𝑦�𝑘𝑘𝑠𝑠 ). Let us further assume that the flow field is constant in z-direction (zero out-of-plane gradients5). In that case, the position of the peak of the local correlation function is equal to the position of the peak of the overall correlation function so that l’Hospital’s rule applied to eqs (100) essentially yields  𝑊𝑊�𝑥𝑥�𝑢𝑢��⃗ ,𝑢𝑢�⃗ 0, 𝑧𝑧� = lim𝑢𝑢�⃗ 0−𝑢𝑢�⃗ →0�⃗ 𝜕𝜕𝜕𝜕(𝑢𝑢𝑥𝑥−𝑢𝑢0𝑥𝑥)�𝜕𝜕〈𝑅𝑅𝐷𝐷(𝑠𝑠,𝑧𝑧)〉𝜕𝜕𝑠𝑠𝑥𝑥 �𝑠𝑠=𝑠𝑠0=𝑢𝑢�⃗ 0𝑀𝑀∆𝑡𝑡� =𝑀𝑀∆𝑡𝑡𝜕𝜕2𝜕𝜕𝑠𝑠𝑥𝑥2 〈𝑅𝑅𝐷𝐷(𝑠𝑠, 𝑧𝑧)〉�𝑠𝑠=𝑀𝑀∆𝑥𝑥,  𝑊𝑊�𝑦𝑦�𝑢𝑢��⃗ ,𝑢𝑢�⃗ 0, 𝑧𝑧� = lim𝑢𝑢�⃗ 0−𝑢𝑢�⃗ →0�⃗ 𝜕𝜕𝜕𝜕(𝑢𝑢𝑦𝑦−𝑢𝑢0𝑦𝑦)�𝜕𝜕〈𝑅𝑅𝐷𝐷(𝑠𝑠,𝑧𝑧)〉𝜕𝜕𝑠𝑠𝑦𝑦 �𝑠𝑠=𝑠𝑠0=𝑢𝑢�⃗ 0𝑀𝑀∆𝑡𝑡=� =𝑀𝑀∆𝑡𝑡𝜕𝜕2𝜕𝜕𝑠𝑠𝑦𝑦2 〈𝑅𝑅𝐷𝐷(𝑠𝑠, 𝑧𝑧)〉�𝑠𝑠=𝑀𝑀∆𝑥𝑥.  (101) Equations (101) relate the weighting functions to specific components of the Hessian matrix of the local correlation function 〈𝑅𝑅𝐷𝐷(𝑠𝑠 − 𝑀𝑀∆?⃗?𝑥), 𝑧𝑧〉 evaluated at the local signal peak up to a constant factor (see Appendix A1 for clarification on eq (101)). In contrast to previous work [84], the present analysis clearly shows that the weighting functions can be determined without the mixed derivatives of the Hessian matrix of the local correlation function. Furthermore, the derivation of eq. (101) presented here clearly shows under which circumstances there is a connection between the weighting function and the curvature of the local correlation function: 5 With “out-of-plane gradients” we refer to flow velocity gradients in the direction of the optical axis z and “in-plane gradients” refer to gradients in the directions x and y. 92                                                 Eq. (101) is only valid if the positions of the peaks of all local correlation functions 〈𝑅𝑅𝐷𝐷〉 are identical to the position of the peak of the global correlation function6. As the weighting functions are a measure of the relative influence of particles on the measured velocity, eqs (100) can be divided by an arbitrary constant without changing the information content and often the weighting functions are made dimensionless by dividing by their value at the object plane 𝑧𝑧 = 𝑧𝑧𝑓𝑓 [84], [203], [80], [194]. Physically meaningful weighting functions (100) always approach 0 for 𝑧𝑧 − 𝑧𝑧𝑓𝑓 → + −⁄ ∞. Thus, they can be used to define the relevant depth of the measurement volume [79]; the depth of correlation 𝑧𝑧DoC is defined as the larger one of the width of the weighting functions where they fall below a certain threshold. In order to determine the DoC from the weighting functions one first has to solve for all the real roots 𝑧𝑧corrx,i of  𝑊𝑊�𝑥𝑥�𝑧𝑧 = 𝑧𝑧corrx,i� − 𝜀𝜀𝑊𝑊�𝑥𝑥�𝑧𝑧 =  𝑧𝑧𝑓𝑓� = 0, (102) where 0.01 is the typical value for ɛ [79], and then determine 𝑧𝑧DoCx = max�𝑧𝑧corrx,i� −min�𝑧𝑧corrx,i�. The same procedure needs to be repeated for 𝑊𝑊�𝑦𝑦 to determine 𝑧𝑧DoCy. Finally, the depth of correlation follows from 𝑧𝑧DoC = max (𝑧𝑧DoCx, 𝑧𝑧DoCy).  4.3 Weighting function and DoC for constant velocity gradients 4.3.1 Weighting functions Let us now focus on the specific flow field eq. (89) with   𝑔𝑔(𝑧𝑧 − 𝑧𝑧0) = 𝑏𝑏 ∙ (𝑧𝑧 − 𝑧𝑧0). (103) In order to determine the corresponding weighting functions eqs. (100), knowledge of the position of the peak of the local correlation function eq. (96), as well as the position of the peak 6 The author of [80] derived expressions for weighting functions accounting for out-of-plane shear from the curvature of the local correlation function. However, the present work suggests that there is no connection between the weighting functions and the curvature of the local correlation function as the position of the peak of the local correlation function can differ from the position of the peak of the global correlation function in this case. In the notation of [80], the approximation 𝑠𝑠𝑥𝑥 ≈ 𝑀𝑀𝑀𝑀∆𝑡𝑡 made in order to go from eq 23 to eq 24 is invalid: Regardless of whether the out-of-of plane gradient is small or large, the local signal peak will be exactly at 𝑠𝑠𝑥𝑥 ≈ 𝑀𝑀�𝑀𝑀 − 𝛼𝛼(𝑧𝑧 − 𝑧𝑧0)�∆𝑡𝑡. Thus, the weighting function eq 29 (in [80]) should not depend on the gradient parameter 𝛼𝛼 while the true weighting function that will be presented in the remainder of this paper does depend on 𝛼𝛼. 93                                                 of the overall correlation function eq. (91) is required. For uniform illumination (i.e. 𝐼𝐼o1 = 𝐼𝐼o2 = constant), the peak of the local correlation function eq. (96) with eq. (103) is located at  𝑠𝑠𝑥𝑥 = 𝑠𝑠𝑥𝑥0 = 𝑀𝑀∆𝑡𝑡𝑢𝑢𝑥𝑥(𝑧𝑧, 𝑡𝑡) = 𝑀𝑀∆𝑡𝑡�𝑢𝑢 + 𝑏𝑏 ∙ (𝑧𝑧 − 𝑧𝑧0)�, 𝑠𝑠𝑦𝑦 = 𝑠𝑠𝑦𝑦0 = 𝑀𝑀∆𝑡𝑡𝑢𝑢𝑦𝑦 = 𝑀𝑀∆𝑡𝑡𝑣𝑣. (104) The position of the peak of the overall correlation function eq. (91) is not as easily found by using analytical methods. However, we can simply guess that the position of the peak of the overall correlation function is identical to the position of the peak of the local correlation function evaluated at the object plane (at 𝑧𝑧 =  𝑧𝑧𝑓𝑓), i.e.   𝑠𝑠𝑥𝑥0 = 𝑀𝑀∆𝑡𝑡𝑢𝑢0𝑥𝑥(𝑡𝑡) = 𝑀𝑀∆𝑡𝑡 �𝑢𝑢 + 𝑏𝑏 ∙ �𝑧𝑧𝑓𝑓 − 𝑧𝑧0��, 𝑠𝑠𝑦𝑦0 = 𝑀𝑀∆𝑡𝑡𝑢𝑢0𝑦𝑦(𝑡𝑡) = 𝑀𝑀∆𝑡𝑡𝑣𝑣. (105) In other words, for the given flow field, the finite depth of the measurement volume does not lead to a bias error for the measured velocity (skewing neglected). This is plausible, because at every position y in the domain and for every position 𝑧𝑧𝑓𝑓 of the focus plane, the flow velocity eq. (89) with (103) can be decomposed into a uniform part and an odd part 𝑢𝑢�⃗ odd�𝑧𝑧 − 𝑧𝑧𝑓𝑓� =−𝑢𝑢�⃗ odd(−𝑧𝑧 − 𝑧𝑧𝑓𝑓). The uniform part of the velocity does not lead to a measurement error and the bias resulting from the odd part cancels out (given that the particle intensity distributions are symmetric with respect to 𝑧𝑧 = 𝑧𝑧𝑓𝑓).    94 Combining eq. (100) with (96) and eq. (104) and eq. (105) gives the weighting functions  𝑊𝑊�𝑥𝑥�𝑢𝑢��⃗ ,𝑢𝑢�⃗ 0, 𝑧𝑧� = 1𝑎𝑎s𝐿𝐿𝑦𝑦∆𝑡𝑡𝑑𝑑e2𝑏𝑏 ∙ �𝑧𝑧 − 𝑧𝑧𝑓𝑓� ⎣⎢⎢⎢⎡exp⎝⎜⎛−2𝛽𝛽2 �𝑀𝑀∆𝑡𝑡 �𝑏𝑏 ∙ �𝑧𝑧 − 𝑧𝑧𝑓𝑓� − 𝑎𝑎s𝐿𝐿𝑦𝑦2 ��2𝑑𝑑e2⎠⎟⎞− exp⎝⎜⎛−2𝛽𝛽2 �𝑀𝑀∆𝑡𝑡 �𝑏𝑏 ∙ �𝑧𝑧 − 𝑧𝑧𝑓𝑓� + 𝑎𝑎s𝐿𝐿𝑦𝑦2 ��2𝑑𝑑e2⎠⎟⎞⎦⎥⎥⎥⎤  𝑊𝑊�𝑦𝑦�𝑢𝑢��⃗ ,𝑢𝑢�⃗ 0, 𝑧𝑧� = 1𝑎𝑎s𝐿𝐿𝑦𝑦∆𝑡𝑡𝑑𝑑e3 �erf�√2𝛽𝛽𝑀𝑀∆𝑡𝑡 �𝑏𝑏 ∙ �𝑧𝑧 − 𝑧𝑧𝑓𝑓� +𝑎𝑎s𝐿𝐿𝑦𝑦2 �𝑑𝑑𝑒𝑒�− erf�√2𝛽𝛽𝑀𝑀∆𝑡𝑡 �𝑏𝑏 ∙ �𝑧𝑧 − 𝑧𝑧𝑓𝑓� − 𝑎𝑎s𝐿𝐿𝑦𝑦2 �𝑑𝑑𝑒𝑒�� (106) Note, that L’Hospital’s rule needs to be applied before the value  𝑊𝑊�𝑥𝑥(𝑧𝑧 = 𝑧𝑧𝑓𝑓)  of the weighting function can be computed.  In general, the weighting functions eqs. (106) differ significantly from previous results that were derived from the curvature of the local correlation function eqs. (101) [83]. For vanishing in-plane shear (as = 0), the weighting functions eqs. (106) become,  𝑊𝑊�𝑥𝑥�𝑢𝑢��⃗ ,𝑢𝑢�⃗ 0, 𝑧𝑧� = 𝑊𝑊�𝑦𝑦�𝑢𝑢��⃗ ,𝑢𝑢�⃗ 0, 𝑧𝑧� = 1𝑑𝑑e4 exp�−2𝛽𝛽2 �𝑀𝑀∆𝑡𝑡 �𝑏𝑏 ∙ �𝑧𝑧 − 𝑧𝑧𝑓𝑓���2𝑑𝑑e2� (107) This is identical to the expression derived by previous work [83] based on the curvature of the local correlation function eqs (101). However, evaluation of eqs (101), with local correlation function eq. (96) for as = 0 shows that eq. (107) is only equal to the curvature of the local correlation function for vanishing out-of-plane shear b = 0.    95 For vanishing out-of-plane shear (b = 0), the weighting functions eq. (106) become  𝑊𝑊�𝑥𝑥�𝑢𝑢��⃗ ,𝑢𝑢�⃗ 0, 𝑧𝑧� = 1𝑑𝑑e4 exp⎝⎜⎛−2𝛽𝛽2 �𝑀𝑀∆𝑡𝑡 𝑎𝑎s𝐿𝐿𝑦𝑦2 �2𝑑𝑑e2⎠⎟⎞ 𝑊𝑊�𝑦𝑦�𝑢𝑢��⃗ ,𝑢𝑢�⃗ 0, 𝑧𝑧� = 1𝑎𝑎𝐿𝐿𝑦𝑦∆𝑡𝑡𝑑𝑑e3 erf�√2𝛽𝛽𝑀𝑀∆𝑡𝑡 �𝑎𝑎s𝐿𝐿𝑦𝑦2 �𝑑𝑑𝑒𝑒� (108) Even though eq. (108) is identical to eq. (101) with local correlation function eq. (96), only the expression for 𝑊𝑊�𝑥𝑥 is identical to the result that was previously derived by [82] from the curvature of the local correlation function.  Finally, for vanishing gradients (as = 0 and b = 0), eqs (26-28) all reduce to the well-known result by [79]:  𝑊𝑊�𝑥𝑥�𝑢𝑢��⃗ ,𝑢𝑢�⃗ 0, 𝑧𝑧� = 𝑊𝑊�𝑦𝑦�𝑢𝑢��⃗ ,𝑢𝑢�⃗ 0, 𝑧𝑧� = 1𝑑𝑑e4 (109) Solving eq. (102) with the weighting function eq. (109) reveals two real roots and the corresponding expression for the depth of correlation [194]  𝑧𝑧DoC0 = 2�1 − √𝜀𝜀√𝜀𝜀𝐶𝐶1𝐶𝐶2= �1 − √𝜀𝜀√𝜀𝜀�𝑛𝑛02𝑁𝑁𝐴𝐴2− 1��𝑑𝑑p2 + 1.49𝜆𝜆2 � 𝑛𝑛02𝑁𝑁𝐴𝐴2 − 1�� (110) is valid if, in addition to the assumptions made for obtaining the weighting function eq. (109), 𝑧𝑧DoC0 ≪ 𝑠𝑠0 holds.  At this point it has to be mentioned that [194] derived expressions for the weighting functions based on the assumption of small gradients. The derivation presented here arrives at the identical expressions for the weighting functions eq (107) and 𝑊𝑊�𝑥𝑥 in eq (108) as previously derived by [82] and [83]. However, the current work shows that these expressions are exact solutions and hence, are also valid for large gradients. For small gradients, eq (107) and eq (108) all reduce to the well-known weighting function eq (109).   96 Equation (106) becomes independent of the optical parameters when expressed by the dimensionless parameters  𝐴𝐴 = 𝑎𝑎s𝐿𝐿𝑦𝑦𝑀𝑀∆𝑡𝑡𝑑𝑑e (𝑧𝑧=𝑧𝑧𝑓𝑓); 𝐵𝐵 = 𝑏𝑏𝑧𝑧DoC0𝑀𝑀∆𝑡𝑡𝑑𝑑e (𝑧𝑧=𝑧𝑧𝑓𝑓) ; 𝑍𝑍 = 𝑧𝑧𝑧𝑧DoC0; 𝑍𝑍𝑓𝑓 = 𝑧𝑧𝑓𝑓𝑧𝑧DoC0 (111) Figure 26 visualizes the effect of in-plane and out-of-plane shear on the weighting function eq. (106). For zero in-plane shear (A = 0 in Figure 26 a), the weighting functions become narrower with increasing B. The simple illustration in Figure 26 b explains this reduction of the depth of correlation. Here it can be seen that for zero out-of-plane shear, all particles within the depth of correlation contribute (‘significantly’) to the displacement peak. Out-of-plane shear separates the contributions of particles that are further away from the object plane from contributions of particles at the object plane to the displacement correlation peak. If an out-of-focus contribution is separated far enough from the in-focus contribution, then the out-of-focus contribution will have no effect on the position of the peak that determines the measured displacement (in both directions) and the depth of correlation is effectively reduced by out-of-plane shear.  As shown in the bottom right part of Figure 26 b, the dimensionless out-of-plane-shear parameter 𝐵𝐵 = 𝑀𝑀∆𝑥𝑥 𝑑𝑑e (𝑧𝑧 = 𝑧𝑧𝑓𝑓)⁄  represents the ratio of the separation of the out-of-focus contributions and the width of the in-focus contributions to the correlation function. As shown in Figure 26 c, for zero out-of-plane shear (B = 0), the widths of 𝑊𝑊𝑥𝑥 and 𝑊𝑊𝑦𝑦 increase with increasing in-plane shear A. The width of 𝑊𝑊𝑥𝑥increases more drastically with increasing A as compared to 𝑊𝑊𝑦𝑦, which is almost indistinguishable by eye for A = 2 and for A = 3. It is interesting to observe that for 𝐴𝐴 ≥ 1, the relative influence 𝑊𝑊𝑥𝑥(𝑧𝑧 = 𝑧𝑧𝑓𝑓) of particles at the object plane on the measured velocity decreases compared to the relative influence of particles that are further away from the object plane. For A > 2.5, particles at the object plane do not contribute significantly to the measured velocity 𝑢𝑢0𝑥𝑥 anymore.  Hence, it can be expected that the depth of correlation according to eq. (102) will diverge to infinity rapidly for A > 2.5 even though the actual thickness of the measurement domain stays finite. In order to overcome this discrepancy, we propose a slightly modified implicit definition of the depth of correlation  𝑊𝑊�𝑥𝑥�𝑧𝑧 = 𝑧𝑧corrx,i� − 𝜀𝜀 ∙ max (𝑊𝑊�𝑥𝑥) = 0,  (112) with 𝑧𝑧DoCx = max�𝑧𝑧corrx,i� − min�𝑧𝑧corrx,i�. For zero in-plane shear, max (𝑊𝑊�𝑥𝑥) = 𝑊𝑊�𝑥𝑥�𝑧𝑧 =  𝑧𝑧𝑓𝑓� so that equation (112) is consistent with previous results.  97 Figure 26d visualizes why the weighting functions become wider with increasing in-plane-shear A. For better clarity, it is assumed that a small out-of-plane shear is superimposed to the in-plane shear (according to eq. (89) and (103)). The left hand side of Figure 26d shows the reference case without in-plane-shear. The right hand side shows the effect of in-plane shear on the local correlation functions of particles at the object plane and at some distance from the object plane. As shown in the bottom right part of Figure 26 d, in-plane shear reduces the height of the peaks of the local correlation functions and (for the given flow field) stretches the peaks in 𝑠𝑠𝑥𝑥-direction. For a given in-plane shear, the reduction of the peak height as well as the stretching of the peak reduces with increasing particle size [201], [73]. In µPIV, particles appear larger the further they are away from the object plane. Thus, the effect of shear on the local correlation function (peak attenuation and stretching) is less significant the further the particles are from the focal plane (i.e., ℎ𝑓𝑓′ℎ𝑓𝑓< ℎ1′ℎ1.). As it can be seen in Figure 26 d, local correlation functions of out-of focus particles have a stronger influence on the position of the overall correlation peak (corresponding to the measured velocity) if in-plane shear is present as compared to the case when no in-plane-shear is present. The influence of the local correlation functions of out-of focus particles on the position of the overall peak is larger in the 𝑠𝑠𝑥𝑥 direction than in the  𝑠𝑠𝑦𝑦 direction since the local correlation functions of particles close to the object plane are stretched in the 𝑠𝑠𝑥𝑥 direction but not in the 𝑠𝑠𝑦𝑦 direction.  Finally we would like to emphasize, that both dimensionless shear parameters A and B represent ratios of displacements associated with the constant flow gradients and the particle image size. The importance of the ratio A on the displacement correlation peak in macroscopic PIV has already been identified by previous work [212], [201], [73].  The present work shows, that a similar ratio B is suitable to describe the effects of constant out-of-plane shear on the µPIV displacement correlation peak.   98  a)  b)    c)   d) Figure 26: a) Effect of constant out-of plane shear B on the weighting functions 𝑊𝑊�𝑥𝑥 and 𝑊𝑊�𝑦𝑦 for zero in-plane shear A = 0 and b) visual interpretation. c) Effect of in plane shear A on the weighting functions 𝑊𝑊�𝑥𝑥 and 𝑊𝑊�𝑦𝑦 and d visual interpretation for non-zero out-of-plane shear. The circles in b) and d) represent example particle patterns at t = t1 (lines) and at t = t2 (filled circles without lines). Solid lines represent in-focus particles and dashed lines out-of-focus particles. Out-of-focus particles are shown larger as they appear larger in an image. Also shown, are schematics of the contributions of in-focus (solid lines) and out-of-focus particles (dashed lines) to the overall displacement correlation peak 〈𝑅𝑅𝐷𝐷〉 . 99 4.3.2 Depth of correlation We were unable to solve eq. (102) with weighting function eq. (106) for the depth of correlation analytically. However, the depth of correlation 𝑧𝑧DoCx(𝐴𝐴,𝐵𝐵)𝑧𝑧DoC0, and   𝑧𝑧DoCy(𝐴𝐴,𝐵𝐵)𝑧𝑧DoC0 are also independent of optical parameters. We compute the latter numerically by evaluating the weighting functions eq. (106) on a uniform grid 𝑍𝑍𝑖𝑖 (i = 1,…,N) with spacing ∆𝑍𝑍 = 0.01 for each given pair of dimensionless shear rates A and B. Once the values of the weighting function are known on the uniform grid, linear interpolation is used to determine 𝑧𝑧DoCx(𝐴𝐴,𝐵𝐵)𝑧𝑧DoC0 and 𝑧𝑧DoCy(𝐴𝐴,𝐵𝐵)𝑧𝑧DoC0 from eq. (102) by interpolating the weighting function linearly. The accuracy of this method will not be worse than +/-0.01 but we believe it is much better than that as the piecewise linear interpolant approximates the weighting function quite well.  The results for the relative depth of correlation are shown in Figure 27. As shown by Olsen [82], [83], the depth of correlation in 𝑧𝑧DoCx decreases with increasing out-of-plane shear B and increases with increasing in-plane shear A. In contrast to previous results [82], 𝑧𝑧DoCy also increases with A, although less drastically as compared to 𝑧𝑧DoCx. For A > 3, 𝑧𝑧DoCy = 𝑐𝑐𝑐𝑐𝑛𝑛𝑠𝑠𝑡𝑡 =1.5𝑧𝑧DoC0. Olsen found that the reduction of the depth of correlation due to out-of-plane shear can be neglected over the increase of the depth of correlation due to in-plane-shear unless the out-of-plane shear is much larger than the in-plane shear [83]. However, the results in Figure 27 show that both, in- and out-of-plane shear are significant as long as A and B are in the same order of magnitude, i.e if 𝐴𝐴𝐵𝐵= 𝑎𝑎s𝐿𝐿𝑦𝑦𝑏𝑏𝑧𝑧DoC0≈ 1. Finally, it can be seen in Figure 27 that the slopes of 𝑧𝑧DoCx and of 𝑧𝑧DoCy at A = 0 and B = 0 are zero and thus, the effect of the corresponding velocity gradient on the depth of correlation can be neglected as long as A << 1 and for B << 1. Finally it is worth noting, the depth of correlation also depends on the inter frame time ∆𝑡𝑡 when gradients are present [82], [83].    100 a) b) Figure 27: Effect of dimensionless shear rates 𝐴𝐴 = 𝑎𝑎s𝐿𝐿𝑦𝑦𝑀𝑀∆𝑡𝑡𝑑𝑑e (𝑧𝑧=𝑧𝑧𝑓𝑓) and 𝐵𝐵 = 𝑏𝑏𝑧𝑧DoC0𝑀𝑀∆𝑡𝑡𝑑𝑑e (𝑧𝑧=𝑧𝑧𝑓𝑓)  on the relative depth of correlation: a) 𝑧𝑧DoCx(𝐴𝐴,𝐵𝐵)𝑧𝑧DoC0, b) 𝑧𝑧DoCy(𝐴𝐴,𝐵𝐵)𝑧𝑧DoC0. 4.4 Weighting function, DoC and velocity bias for common flow fields in micro channels Most microfluidic devices consist of flow channels with constant rectangular cross section for which analytical equations for the flow profile exist. Knowing the flow profile is useful as a single measurement of the maximum velocity at the center of the channel provides knowledge of the whole flow field, including important parameters such as the wall shear stress and flow rate. However, in this flow scenario commonly met in praxis, the velocity gradients in z-direction may not even be approximately constant within the depth of the measurement volume as it was assumed in the previous section. Furthermore, the measured velocity at the center of the microchannel can be biased as a result of the finite DoC ( [3]). The present work builds on the ideas of previous work ( [3]) and investigates how the bias error in the measured velocity at the center of a microchannel with rectangular cross section is related to the optical parameters, channel dimensions, flow intensity and frame rate. We then interpret these results in terms of the DoC. This work also derives dimensionless parameters that allow estimating the bias error of the measured center-velocity and correcting for them if necessary. We would like to emphasize that, although the theory presented next can be used to predict the measured velocity at other positions in the channel, the current work focuses on the measured velocity in the center of the channel. 101 4.4.1 Bias error In order to compute the bias error due to depth-averaging for the specific flow scenario at hand, we start at eq. (96). For the present study it is assumed that the interrogation windows are centered on the center of the channel and small enough for in-plane-gradients to be negligible (as = 0). Furthermore, we set u = 0, v = 0 and   𝑔𝑔(𝑧𝑧 − 𝑧𝑧0) = 𝑢𝑢max𝑞𝑞 �𝑧𝑧 − 𝑧𝑧0ℎd ,𝑤𝑤dℎd�  (113) where 𝑢𝑢max corresponds to the maximum velocity at the center of the channel, wd and hd are half the channel width and height (measured from the center of the channel), respectively, 𝑧𝑧0 is located at the center of the channel and   𝑞𝑞 �𝑧𝑧 − 𝑧𝑧0ℎd,𝑤𝑤dℎd� =  1 − �𝑧𝑧ℎd�2 + 4∑ (−1)𝑘𝑘�𝜋𝜋2 (2𝑘𝑘 − 1)�3cos �𝜋𝜋2 (2𝑘𝑘 − 1) (𝑧𝑧 − 𝑧𝑧0)ℎd �cosh �𝜋𝜋2 (2𝑘𝑘 − 1)𝑤𝑤dℎd�∞𝑘𝑘=11 + 4∑ (−1)𝑘𝑘�𝜋𝜋2 (2𝑘𝑘 − 1)�3 cosh−1 �𝜋𝜋2 (2𝑘𝑘 − 1)𝑤𝑤dℎd�∞𝑘𝑘=1. (114) Equations (89) and (113)-(114) describe the profile of a fully developed flow of a Newtonian fluid in the center of a channel with rectangular cross section [213]. Finally, we assume identical illumination pulses of the form   𝐼𝐼o1(𝑧𝑧) = 𝐼𝐼o2(𝑧𝑧) = 𝐼𝐼(𝑧𝑧) = 𝐼𝐼orect �𝑧𝑧−𝑧𝑧02ℎd �, (115) with the rectangular function rect, in order to take the finite channel height into account. With the modifications up to here, the local correlation function eq. (96) becomes  〈𝑅𝑅𝐷𝐷(𝑠𝑠, 𝑧𝑧)〉 =𝑁𝑁𝐽𝐽p2𝐷𝐷a4𝛽𝛽2512√𝜋𝜋𝑀𝑀2𝑠𝑠04𝐼𝐼2(𝑧𝑧)𝑑𝑑e2 exp �− 2𝛽𝛽2𝑑𝑑e2 �𝑠𝑠𝑦𝑦�2� exp�− 2𝛽𝛽2𝑑𝑑e2 �𝑠𝑠𝑥𝑥 − 𝑀𝑀∆𝑡𝑡𝑢𝑢max𝑞𝑞 �𝑧𝑧−𝑧𝑧0ℎd ,𝑤𝑤dℎd��2�. (116) After introducing the dimensionless parameters  𝑆𝑆𝑥𝑥 = 𝑠𝑠𝑥𝑥𝑑𝑑e (𝑧𝑧=𝑧𝑧𝑓𝑓); 𝑆𝑆𝑦𝑦 = 𝑠𝑠𝑦𝑦𝑑𝑑e (𝑧𝑧=𝑧𝑧𝑓𝑓);𝑋𝑋𝑚𝑚𝑎𝑎𝑥𝑥 = 𝑢𝑢max𝑀𝑀∆𝑡𝑡𝑑𝑑e (𝑧𝑧=𝑧𝑧𝑓𝑓); 𝑍𝑍 = 𝑧𝑧𝑧𝑧DoC0;𝑍𝑍𝑓𝑓 = 𝑧𝑧𝑓𝑓𝑧𝑧DoC0 ;  𝑍𝑍0 = 𝑧𝑧0𝑧𝑧DoC0 ;𝐻𝐻d =ℎd𝑧𝑧DoC0; 𝑊𝑊d = 𝑤𝑤d𝑧𝑧DoC0, (117) 102 the displacement correlation peak (eq. (91)) with (116) becomes  〈𝑅𝑅𝐷𝐷�𝑆𝑆,𝑍𝑍�〉 =𝐶𝐶3 ∫11+4𝐹𝐹(𝜀𝜀)�𝑍𝑍−𝑍𝑍𝑓𝑓� 2𝑍𝑍=+𝜕𝜕d𝑍𝑍=−𝜕𝜕d exp �− 2𝛽𝛽2𝑆𝑆𝑦𝑦21+4𝐹𝐹(𝜀𝜀)�𝑍𝑍−𝑍𝑍𝑓𝑓� 2� exp�− 2𝛽𝛽21+4𝐹𝐹(𝜀𝜀)�𝑍𝑍−𝑍𝑍𝑓𝑓� 2 �𝑆𝑆𝑥𝑥 −𝑋𝑋𝑚𝑚𝑎𝑎𝑥𝑥𝑞𝑞 �𝑍𝑍−𝑍𝑍0𝜕𝜕d,𝑊𝑊d𝜕𝜕d��2�𝑑𝑑𝑍𝑍, (118) with  𝐹𝐹(𝜀𝜀) = 1 − √𝜀𝜀√𝜀𝜀, (119) and with the constant 𝐶𝐶3 that depends on the optical parameters of the system. The goal is now to find the positions 𝑆𝑆0𝑥𝑥 = 𝑀𝑀∆𝑡𝑡𝑢𝑢0𝑥𝑥𝑑𝑑e (𝑧𝑧=𝑧𝑧𝑓𝑓) and 𝑆𝑆0𝑦𝑦 = 𝑀𝑀∆𝑡𝑡𝑢𝑢0𝑦𝑦𝑑𝑑e (𝑧𝑧=𝑧𝑧𝑓𝑓) of the displacement correlation peak eq. (118). Note, that 𝑆𝑆0𝑥𝑥 and 𝑆𝑆0𝑦𝑦 also correspond to the dimensionless measured particle displacements. The position of the peak 𝑆𝑆0𝑦𝑦 = 0 is easily found from eq. (118). To the best knowledge of the authors, no analytical solution for the position of the peak 𝑆𝑆0𝑥𝑥 of eq. (118) exists. However, 𝑆𝑆0𝑥𝑥 can be determined numerically. Therefore, 𝜀𝜀 = 0.01 and 𝛽𝛽2 = 3.67 are considered as constant, and  𝑍𝑍𝑓𝑓 = 𝑍𝑍0 = 0 in order to compute the weighting function at the center of the channel.  It can be seen from eqs (118) that the dimensionless measured velocity 𝑆𝑆0𝑥𝑥 now only depends on Hd, 𝑋𝑋max, and Wd/Hd but not explicitly on the optical parameters anymore. Hence, we conclude that Hd, 𝑋𝑋max, and Wd/Hd are a suitable and complete set of dimensionless parameters to describe the effect of velocity gradients on the bias error for the specific flow field under consideration (given that the particle images are well described by eq. (86)).  In order to determine 𝑆𝑆0𝑥𝑥 for a given set of Hd, 𝑋𝑋max, and Wd/Hd numerically, we first approximate the displacement correlation peak eq. (118) as   〈𝑅𝑅𝐷𝐷�𝑆𝑆𝑥𝑥,𝑗𝑗,𝑍𝑍𝑖𝑖�〉 = 𝐶𝐶4 ∑ 11+4𝐹𝐹(𝜀𝜀)(𝑍𝑍𝑖𝑖) 2 exp�− 2𝛽𝛽21+4𝐹𝐹(𝜀𝜀)(𝑍𝑍𝑖𝑖) 2 �𝑆𝑆𝑥𝑥,𝑗𝑗 − 𝑋𝑋𝑚𝑚𝑎𝑎𝑥𝑥𝑞𝑞 �𝑍𝑍𝑖𝑖𝜕𝜕d ,𝑊𝑊d𝜕𝜕d��2�𝑁𝑁𝑧𝑧𝑖𝑖=1 , (120) 𝐶𝐶4 is another constant and 𝑍𝑍𝑖𝑖 is a uniform grid with −𝐻𝐻d < 𝑍𝑍𝑖𝑖 < 𝐻𝐻d and with 𝑁𝑁𝑧𝑧 = 600 so that ∆𝑍𝑍 = 𝑍𝑍𝑖𝑖+1 − 𝑍𝑍𝑖𝑖 ranges from ∆𝑍𝑍 = 3.33 ∙ 10−4 for H = 0.1 to ∆𝑍𝑍 =  0.033 for 𝐻𝐻 = 10. It is worth noting that eq. (120) corresponds to the “Olsen and Adrian based reconstructed correlation 103 function” of [3] in good approximation if 𝑊𝑊d𝜕𝜕d> 10. The displacement correlation peak eq. (120) is computed on a uniform grid 0 < 𝑆𝑆𝑥𝑥,𝑗𝑗 < 1.1𝑋𝑋𝑚𝑚𝑎𝑎𝑥𝑥 with 𝑗𝑗 = 1, … ,𝑁𝑁𝑠𝑠 and 𝑁𝑁𝑠𝑠 = 3000 so that ∆𝑆𝑆𝑥𝑥 =𝑆𝑆𝑥𝑥,𝑗𝑗+1 − 𝑆𝑆𝑥𝑥,𝑗𝑗 ranges from ∆𝑆𝑆𝑥𝑥 = 3.67 ∙ 10−6 for 𝑋𝑋𝑚𝑚𝑎𝑎𝑥𝑥 = 0.01 to ∆𝑆𝑆𝑥𝑥 = 0.0055 for 𝑋𝑋𝑚𝑚𝑎𝑎𝑥𝑥 = 15. The displacement correlation peak has been computed by using a custom Matlab (Mathworks) script and the position of the maximum 𝑆𝑆0𝑥𝑥 is determined from the maximum value in the array of numbers representing 〈𝑅𝑅𝐷𝐷〉𝑗𝑗 in eq. (120). Consequently, the uncertainty of the position 𝑆𝑆0𝑥𝑥 is 2∆𝑆𝑆𝑥𝑥. Further increase of 𝑁𝑁𝑧𝑧 and 𝑁𝑁𝑠𝑠 did not result in a noticeable change of the results presented in the following. With the known dimensionless measured displacement 𝑆𝑆0𝑥𝑥, we define the relative measured channel center velocity    𝑆𝑆0𝑥𝑥𝑋𝑋𝑚𝑚𝜅𝜅𝑥𝑥= 𝑢𝑢0𝑥𝑥𝑢𝑢max, (121) as in [3]. If the relative measured channel center velocity eq. (121) is 𝑢𝑢0𝑥𝑥𝑢𝑢max= 1, then the measurement is free of error. The bias error increases with decreasing 𝑢𝑢0𝑥𝑥𝑢𝑢max< 1. Figure 28 shows the bias error for the special case of 𝑊𝑊d𝜕𝜕d= 10. In this case, which has already been considered by previous work [3], the velocity profile eq. (114) becomes  𝑞𝑞 � 𝑍𝑍𝜕𝜕d,𝑊𝑊d𝜕𝜕d> 10� ≈  1 − � 𝑧𝑧ℎd�2, (122) parabolic in good approximation. While previous work only recognized Hd as important parameter, the present work clearly shows that Xmax is another relevant parameter that is required to fully describe the effect of velocity gradients on the velocity measurement for the flow scenario at hand. In Figure 28a it can be seen, that for a given Xmax, the error due to velocity gradients increases with increasing ratio of DoC and channel height 1𝜕𝜕d= 𝑧𝑧𝐷𝐷𝐷𝐷𝐷𝐷0ℎd (1/Hd was chosen for better comparison to previous work). In Figure 28b it can be seen that for a fixed Hd, the velocity error decreases with increasing Xmax. Thus, for the given flow scenario and for fixed optical parameters and particle (image) size, it is desirable to maximize the inter-frame time Δt so that the particle displacements are large compared to the particle image size. Both figures show that for 1𝜕𝜕d< 1.7, the measured velocity is less than 5% lower than the actual velocity. Furthermore, it is worth noting, that there is a theoretical minimum for the relative 104 measured channel center velocity  � 𝑢𝑢0𝑥𝑥𝑢𝑢max�min for 𝐻𝐻d → 0 and 𝑋𝑋max → 0. In this extreme case, the measured velocity is simply the spatial average  � 𝑢𝑢0𝑥𝑥𝑢𝑢max�min= 1ℎd∫ 𝑞𝑞 �𝑧𝑧ℎd,𝑤𝑤dℎd> 10�ℎd0𝑑𝑑𝑧𝑧 = 23  (123) of the velocity. a)  b)  Figure 28: Relative measured channel center velocity for large channel aspect ratios 𝑤𝑤dℎd> 10 as a function of a 1𝜕𝜕d= 𝑧𝑧DoC0ℎd  (legend shows values of 𝑋𝑋𝑚𝑚𝑎𝑎𝑥𝑥 = 𝑢𝑢max𝑀𝑀∆𝑡𝑡𝑑𝑑e (𝑧𝑧=𝑧𝑧𝑓𝑓)) and b as a function of the ratio 𝑋𝑋max  between particle displacement to particle image size (legend shows values of Hd). Let us now focus on the case of smaller aspect ratios 𝑤𝑤dℎd that are also commonly met in praxis. Therefore we repeat the procedure described above while taking the first 30 terms of the infinite sum in eq. (121) into account. Figure 29 shows the underpredictions of the velocity for two different aspect ratios 𝑤𝑤dℎd. Here, the underprediction of the velocity follows the trend as before in Figure 28a for large aspect ratios. However, for a given Hd and Xmax, the error due to underprediction decreases with decreasing aspect ratio 𝑤𝑤dℎd. This is plausible, as the underprediction of the velocity results from out-of-plane velocity gradients and these become less significant in the center of the channel as the aspect ratio decreases.    105 a)  b)  Figure 29: Relative measured channel center velocity as a function for different aspect ratios: a) 𝑤𝑤dℎd= 1, b) 𝑤𝑤dℎd= 0.2. Legends show values of Xmax. It is important to keep in mind that the focus position was set to the center of the channel. If the focus position is instead set e.g. to the wall, the bias error will not follow the same trends as in Figure 28 and Figure 29. Although other focus positions could easily be investigated with the method presented here, for the sake of brevity, the current work focuses on the case where the focus is set to the center of the channel. Before we move on to the investigation of the weighting functions and the depth of correlation for the flow problem at hand, we would like to take a closer look at the relevant dimensionless parameters Xmax and Hd for a fixed aspect ratio 𝑤𝑤dℎd> 10. First of all we note, that two parameters are required instead of just one (B) for the case of constant out-of-plane gradients. However, the ratio  𝐵𝐵� = 𝑋𝑋𝑚𝑚𝜅𝜅𝑥𝑥𝜕𝜕d= 𝑏𝑏�𝑧𝑧𝐷𝐷𝐷𝐷𝐷𝐷0𝑀𝑀∆𝑡𝑡𝑑𝑑e(𝑧𝑧=𝑧𝑧𝑓𝑓) , (124) of the two relevant parameters 𝑋𝑋max and 𝐻𝐻d, with 𝑏𝑏� = 𝑢𝑢𝑚𝑚𝜅𝜅𝑥𝑥ℎd  representing a measure of the intensity of the out-of-plane gradients, is very similar to the dimensionless out-of-plane shear B in eq. (111). This should be kept in mind for the discussion of the depth of correlation and weighting function in the following sub section.  4.4.2 Weighting functions and Depth of correlation In order to gain a deeper understanding of the effects of velocity gradients on the underprediction of the velocity discussed in the previous sub-section, the weighting functions eq. (100) and the depth of correlation will be evaluated for the flow scenario at hand. Therefore, 106 the correlation function eq. (120), the (known) actual flow field eqs (89) and (113) as well as the numerically determined measured velocity (see previous sub section) are substituted into eq. (100). For the sake of brevity only weighting functions for large aspect ratios Wd/Hd are considered. At this point the reader has to be reminded that the bias error is directly governed by the parameters describing the flow field Hd, 𝑋𝑋max and Wd/Hd and not by the depth of correlation. The depth of correlation is a parameter that depends on Hd, 𝑋𝑋max and Wd/Hd and it allows to interpret the measurement error in terms of the depth of the effective measurement volume. Figure 30 shows weighting functions for three different scenarios, one where the ratio 𝐻𝐻d =ℎd𝑧𝑧𝐷𝐷𝐷𝐷𝐷𝐷0= 10 (Figure 30a) is relatively large, one where 𝐻𝐻d = 1 (Figure 30b) and one where Hd = 0.01 is small (Figure 30c). Also shown is the effective depth of correlation in the presence of shear as a function of 𝑋𝑋max for different values of Hd (Figure 30d).  If the ratio 𝐻𝐻d = ℎd𝑧𝑧𝐷𝐷𝐷𝐷𝐷𝐷0 is large (Figure 30a) then, for practically feasible values of 𝑋𝑋max, all particles visible in an image-pair are approximately displaced by the same (maximum) displacement. Hence, not only the velocity error is small in this case (see Figure 28 and Figure 29) but also the effect of the out-of-plane gradient on the depth averaging of the velocity can be neglected. In this case, by definition, the bell-shaped weighting functions take the value 𝑊𝑊𝑥𝑥 = 𝑊𝑊𝑦𝑦 = 0.01 at 𝑍𝑍𝜕𝜕d = 12𝜕𝜕d (in other words, at 𝑧𝑧 = 𝑧𝑧𝐷𝐷𝐿𝐿𝑁𝑁0/2) as it can be seen for all weighting functions in Figure 30a. Consequently, if gradients do not affect the depth averaging then the effective depth of correlation 𝑧𝑧𝐷𝐷𝐿𝐿𝑁𝑁 = 𝑧𝑧𝐷𝐷𝐿𝐿𝑁𝑁0 in Figure 30d.  If 𝑧𝑧𝐷𝐷𝐿𝐿𝑁𝑁0 is at the same order of magnitude as the channel height or larger (i.e. Hd ≤ 1 in Figure 30b and c) and 𝑋𝑋max ≠ 0, then not all particles visible in an image pair are displaced by the same amount and gradients affect the depth averaging. Since only particles within the channel can contribute to the cross-correlation, zDoC/ zDoC0 can never exceed 2Hd when Hd < 1 (see Figure 30d). According to Figure 30b and Figure 30c, for a given Hd < 1, the weighting functions become narrower as 𝑋𝑋max (or the dimensionless shear parameter 𝐵𝐵�  (eq. 44)) increases. Hence, the depth of correlation decreases with increasing 𝑋𝑋max (see Figure 30d), as does the measurement error discussed in the previous sub-section. With eq. (124) in mind, the trends seen in Figure 30d are also consistent with the results for constant out-of-plane gradients in the previous section (Figure 27).  107 Another interesting feature of the weighting functions in Figure 30c is that under certain circumstances, the particles at the object plane have a lower influence on the measured velocity than particles at a certain distance from the object plane. Finally, Figure 30c shows that the weighting function approaches the rect-function as Hd and 𝑋𝑋max approach zero. In other words, when all particles images are approximately in focus and the maximum particle displacement is small compared to the particle image diameter, then all particles will contribute evenly to the correlation function. This is in line with the observation in the previous sub-section, that the measured velocity is simply the spatial average of the velocity in z-direction in this limiting case. a)  b)  c)  d)  Figure 30: Weighting functions for large aspect ratios 𝑊𝑊d𝜕𝜕d> 10 for a) Hd = 10, b) Hd = 1, c) Hd = 0.01. d) Relative depth of correlation as a function of 𝑋𝑋max for different Hd. The legends in a), b) and c) show values of 𝑋𝑋max. The legend in d shows values of Hd. 4.5 Extension to different particle image models All results presented up to here are based on the most commonly used model for the particle image intensity distribution summarized in section 4.2.1. As demonstrated by [3], a model of the form eq. (1) with eq. (86) may not be suitable to describe experimental data if a specialized upright-microscope is used. In this section, we will investigate if the same dimensionless parameters (eq. (117)) presented in the previous section are meaningful when particle images are acquired through the specialized microscope used in [3]. Klostermann et al. [3] suggest using a more general form for the (Gaussian) particle image model eq (1), i.e,  108  𝐽𝐽0𝐺𝐺 = 𝐾𝐾 𝐽𝐽(𝑧𝑧−𝑧𝑧𝑓𝑓)𝐽𝐽(𝑧𝑧=𝑧𝑧𝑓𝑓) exp�− 4𝛽𝛽2�𝑋𝑋�⃗ �2𝑑𝑑�e2(𝑧𝑧−𝑧𝑧𝑓𝑓)�, (125) where 𝐽𝐽(𝑧𝑧) describes the z-dependency of the peak intensity distribution (per unit of illuminating intensity) and ?̃?𝑑e  is the particle image diameter. With the Gaussian particle image intensity distribution expressed as eq. (125), the equivalent of the local correlation function eq. (116) reads   〈𝑅𝑅𝐷𝐷(𝑠𝑠, 𝑧𝑧)〉 = 𝐾𝐾� �𝐽𝐽(𝑧𝑧−𝑧𝑧𝑓𝑓)𝐽𝐽(𝑧𝑧=𝑧𝑧𝑓𝑓)�2 exp �− 2𝛽𝛽2𝑑𝑑�e2 �𝑠𝑠𝑦𝑦�2� exp�− 2𝛽𝛽2𝑑𝑑�e2 �𝑠𝑠𝑥𝑥 − 𝑀𝑀∆𝑡𝑡𝑢𝑢max𝑞𝑞 �𝑧𝑧−𝑧𝑧𝐷𝐷ℎd , 𝑤𝑤dℎd��2�,  (126) where 𝐾𝐾� is another constant that is not relevant for the remainder of this work. The equivalent of weighting functions eq. (109) for the particle image model eq. (125) reads  𝑊𝑊�𝑥𝑥(𝑧𝑧) = 𝑊𝑊�𝑦𝑦(𝑧𝑧) = 𝐽𝐽2𝑑𝑑�e2. (127) The strategy pursued in this section is to find expressions for the zero-shear-DoC 𝑧𝑧𝐷𝐷𝐿𝐿𝑁𝑁0 for different particle image models J and ?̃?𝑑e . We will then determine whether the dimensionless parameters eq. (117) are still suitable to make the correlation function dimensionless. Finally, we will compare the predicted measurement error to the experimentally determined measurement error reported by [3]. We begin with the empirical models for the the z-dependency of the peak intensity distribution   𝐽𝐽(𝑧𝑧) =  𝐶𝐶3𝐶𝐶4𝐶𝐶5 + 𝐶𝐶6�𝑧𝑧 − 𝑧𝑧𝑓𝑓�2  (128) and for the  particle image diameter  ?̃?𝑑e = 𝐶𝐶7 �1 − 𝐶𝐶8�𝑧𝑧 − 𝑧𝑧𝑓𝑓�2�−1/2 (129) proposed by [3], where the constants 𝐶𝐶3 −  𝐶𝐶8 are to be determined from experiment. Figure 31 shows particle image data from [3] for M = 25× and NA = 0.45 (The data was taken from the figure in the original work by using the ‘GetData graph digitizer’ (http://getdata-graph-digitizer.com/index.php). Also shown as solid lines, are the models (128) and (129) with 𝐶𝐶3 = 2.65,  𝐶𝐶4𝐶𝐶5 = 2.8633, 𝐶𝐶6 =  0.0018µm−2, 𝐶𝐶7 = 19 and 𝐶𝐶8 = 0.0001µm−2. The models fit the data reasonably well. 109 We can now solve eq. (112) with the weighting function eq. (127) in order to determine expressions for the depth of correlation 𝑧𝑧𝐷𝐷𝐿𝐿𝑁𝑁0 in absence of shear for the model eqs (48-49). However, with the typical value for 𝜀𝜀 = 0.01, this procedure does not yield a physically meaningful expression for the depth of correlation, as the particle image diameter eq. (129) becomes complex before the corresponding intensity distribution drops below √𝜀𝜀 = 0.1. For the sake of brevity this (meaningless and rather cumbersome) expression for 𝑧𝑧𝐷𝐷𝐿𝐿𝑁𝑁0 will not be shown here. We would like to point out, that the dimensionless parameters eq. (117) with eq. (129) along with 𝑧𝑧𝐷𝐷𝐿𝐿𝑁𝑁0 corresponding to eq. (125)-(129) are not suitable to make the local correlation function eq. (126) dimensionless. Hence, the dimensionless parameters eq. (117) are not suitable to describe the effects of velocity gradients on µPIV cross-correlation when the particle model eq. (128)-(129) is used. In the following we propose two alternative models. Model 1:  ?̃?𝑑e = 𝐶𝐶5 + 𝐶𝐶6�𝑧𝑧 − 𝑧𝑧𝑓𝑓�2, (130)  𝐽𝐽(𝑧𝑧) =  𝑁𝑁3𝑁𝑁4𝑁𝑁5+𝑁𝑁6�𝑧𝑧−𝑧𝑧𝑓𝑓�2, (131) Model 2:   ?̃?𝑑e = 𝐶𝐶5 + 𝐶𝐶10�𝑧𝑧 − 𝑧𝑧𝑓𝑓�4, (132)  𝐽𝐽(𝑧𝑧) =  𝐶𝐶3𝐶𝐶5𝐶𝐶11 + 𝐶𝐶10�𝑧𝑧 − 𝑧𝑧𝑓𝑓�4 (133) the z-dependency of the particle image diameter and of the peak intensity distribution. The Model 1 is shown as solid line in Figure 31a (as before eq 48) and in as dashed line Figure 31b with 𝐶𝐶5 = 19 and 𝐶𝐶6 = 0.001751 1𝜇𝜇𝑚𝑚2. The second model is shown as dotted lines in Figure 31a and Gb with 𝐶𝐶10 = 4.609 ∙ 10−7 1𝜇𝜇𝑚𝑚4 and 𝐶𝐶5𝐶𝐶11 = 2.3617. Where the first model fits the intensity-data (Figure 31a) well, it deviates from the particle image diameter data (Figure 31b) for large separations of the particle from the object plane. The second model fits the particle image data reasonably well but has some shortcomings when compared to the particle intensity data in Figure 31a.  The corresponding expressions for the (zero-shear) depth of correlation for Model 1 110  𝑧𝑧𝐷𝐷𝐿𝐿𝑁𝑁0 = 2𝑓𝑓1(𝜀𝜀,𝐶𝐶4)�𝑁𝑁5𝑁𝑁6 = 177𝜇𝜇𝑛𝑛, (134)  and for Model 2   𝑧𝑧𝐷𝐷𝐿𝐿𝑁𝑁0 = 2𝑓𝑓2(𝜀𝜀,𝐶𝐶11)�𝐶𝐶5𝐶𝐶104 = 143𝜇𝜇𝑛𝑛 (135) with  𝑓𝑓1(𝜀𝜀,𝐶𝐶4) = �12��√𝜀𝜀𝑁𝑁42+4𝑁𝑁4−2𝑁𝑁4√𝜀𝜀+√𝜀𝜀�√𝜀𝜀 − 𝐶𝐶4√𝜀𝜀 − √𝜀𝜀, (136) and with  𝑓𝑓2(𝜀𝜀,𝐶𝐶11) = �𝑓𝑓1(𝜀𝜀,𝐶𝐶11). (137) are obtained by solving equation (112) with (127) for eq. (50-51) and for eq. (52-53), respectively. With the particle image models eqs (50-57) at hand, the dimensionless parameters eq. (117) make the displacement peak corresponding to eq. (126) dimensionless, i.e.   〈𝑅𝑅𝐷𝐷�𝑆𝑆,𝑍𝑍�〉𝑖𝑖 =𝐾𝐾� ∫1�𝑁𝑁?̅?𝑖+(2𝑓𝑓𝑖𝑖(𝜀𝜀)𝑍𝑍)2𝑖𝑖�2𝑍𝑍=+𝜕𝜕𝑍𝑍=−𝜕𝜕 exp �− 2𝛽𝛽2𝑆𝑆𝑦𝑦2�1+(2𝑓𝑓𝑖𝑖(𝜀𝜀)𝑍𝑍)2𝑖𝑖�2� exp�− 2𝛽𝛽2�1+(2𝑓𝑓𝑖𝑖(𝜀𝜀)𝑍𝑍)2𝑖𝑖�2 �𝑆𝑆𝑥𝑥 −𝑋𝑋𝑚𝑚𝑎𝑎𝑥𝑥𝑞𝑞 �𝑍𝑍𝜕𝜕d,𝑊𝑊d𝜕𝜕d��2�𝑑𝑑𝑍𝑍, (138) with i = 1 for Model 1 and i = 2 for Model 2, 𝐶𝐶1̅ = 𝐶𝐶4, 𝐶𝐶2̅ =  𝐶𝐶11 and with a another constant 𝐾𝐾� that is not relevant to the current work.  In order to determine the location of the peak of eq (138) corresponding to the measured velocity, the procedure presented in section 4.4.1 was repeated. The experimental conditions used by [3] were matched by setting the ratio of particle displacement and particle image size to Xmax = 10/𝐶𝐶5 = 0.53 and hd = 74 µm.  111 Figure 31c shows the predicted bias error corresponding to the two different particle models eqs (50-53). For hd = 74 µm, Model 1 (with 1/Hd = 2.39) predicts𝑢𝑢0𝑢𝑢max= 0.906 and model 2 (with 1/Hd =1.93) predicts 𝑢𝑢0𝑢𝑢max= 0.891. Note, that even though the different particle models give a rather significant difference for the zero-shear DoC in eqs (54-55), both models predict (nearly) the same measurement error in the presence of shear. These predictions are very close to the predictions 𝑢𝑢0𝑢𝑢max≈ 0.89 that [3] obtained from their “reconstructed correlation function” based on the model (48-49). These predictions of the measurement error all compare well to the measured error of 𝑢𝑢0𝑢𝑢max≈0.85 reported by (Kosterman et al. 2011) for the same conditions (symbols in Figure 31c). The model predictions deviate by about 4%-5% from the measured velocity error. A possible reason for the small discrepancy is the shortcoming of the particle image model to describe real particle image data. Furthermore, it seems likely that the measured velocity reported by [3]  𝑢𝑢0𝑢𝑢max≈ 0.85 contains an error resulting from the in-plane correlation loss FI, which has been neglected by the models at hand. With the (iterative) multipass interrogation strategy used by [3], the final shift of the second interrogation window will be less than the displacement of the particles at the object plane, which move with the velocity 𝑢𝑢max. Hence, some of the particles at the object plane that were present in the first interrogation window will not be found in the second window and hence, the measured velocity will be biased towards smaller velocities. This effect is not expected to be significant here, since the predicted bias error is small for the case considered. However, the effect of in-plane correlation loss will become more significant when the predicted bias error is large.    112 a)  b)  c)   Figure 31: a) Peak intensity and b) particle image diameter as a function of the separation 𝑧𝑧 − 𝑧𝑧𝑓𝑓 between particle and object plane with 𝑧𝑧𝑓𝑓 = 0. The symbols represent experimental data for 1.28 µm diameter particles seen through an upright combi microscope at M = 25× by [3]. The lines represent different empirical models (see text for explanation). c) Predicted bias error (lines) for two different particle image models and with Xmax = 0.53 and hd = 74 µm. Data-points show data by [3]. 4.6 Conclusions The present work clarifies the relationship between the µPIV weighting functions and the local correlation function. In contrast to assumptions made by previous work, we showed that the weighing functions are only equal to the peak-curvature of the local correlation function if, at each depth z in the fluid, the location of the peak of the local correlation function is equal to the location of the overall displacement correlation peak.  Subsequently, the effects of constant in-plane shear and out-of-plane shear on the depth averaging of the velocity were investigated. We showed that the relevant dimensionless parameters to describe the effect of gradients on the DoC are ratios between displacements associated with the gradients and the particle image size. The general trends found by previous works were confirmed, i.e. out-of-plane flow gradients reduce the DoC and in-plane-gradients increase the DoC. However, we point out that, even for equal in- and out-of-plane shear, neither the effect of in-plane shear, nor the effect of out-of-plane shear on the DoC can generally be neglected over the other as it has been concluded by previous work. Furthermore, 113 we showed that some expressions for weighting functions that were previously derived under the assumptions of small gradients are also valid for large gradients. For other previously published weighting functions, corrections were proposed. We then turned our attention to the non-uniform out-of-plane gradients encountered in straight microchannels with rectangular cross section. Although the ratio of the displacement associated with the gradients and the particle image size is still a relevant dimensionless parameter, two additional dimensionless parameters are required to fully describe the effect of flow gradients on the velocity bias error and on the DoC for this flow scenario. Furthermore, when optical parameters and flow parameters are fixed, both, the DoC and the bias error decrease with increasing inter-frame time for the considered flow fields and position of the focal plane. Finally we show that the dimensionless parameters derived in this work are not bound to the classical (and often criticised) model for particle images. Instead, we demonstrated that these parameters can also be meaningful when other particle image models have to be utilized. More specifically, we proposed two different empirical particle models and also provided expressions for the corresponding DoC (at zero shear).   114 5 Conclusions The key findings of this work have been listed in the conclusion sections of chapters 2 through 4. The following sections will discuss the limitations of this work and provide recommendations for future work. 5.1 Limitations This thesis investigated the transport and deposition of particles near the PDMS-water interface, as well as methods for the characterization of particle transport in microfluidic devices. As a result of research objective 1, important insights into the surface interaction forces in the polystyrene-water-PDMS system have been obtained. Most importantly, high-precision particle tracking provided evidence for lateral-surface interaction forces. Such forces have long been expected but manifestations of the same have not yet been observed within the framework of particle adsorption at the solid-liquid interface. However, the evidence provided was rather indirect and our interpretation of the displacement histograms in terms of lateral surface interaction forces is, at best, a hypothesis. It may be difficult to confirm the hypothesis until methods for the direct measurement of lateral surface interaction forces or improved methods for localized surface characterization are available. Nevertheless, we believe that the observations described in chapter 2 will be of interest to the community and we hope that this will trigger further research on this phenomenon (see recommendations for future work in the following sub-section). Another conclusion that was obtained as a result of research objective 1 is that the extended DLVO theory seems to provide a better description of the surface interactions in the polystyrene-water-PDMS system as compared to the DLVO theory. However, this conclusion is challenged by the large uncertainty of the input parameters required for these models. In the present example, the uncertainty in the Hamaker constant seems rather large and an accurate measurement of zeta-potentials by electrokinetic methods becomes increasingly difficult the higher the electrolyte concentration. Further note, that DLVO and extended DLVO are far from perfect in that they treat the different contributions to the total interaction energy as additive. More specifically, strong evidence suggests that the dispersive interactions are also affected by the presence of ions in solution, especially in highly polar media such as water (see [140]). Nevertheless, surface interaction forces are traditionally treated within the frameworks of the DLVO or xDLVO theory which have been shown to capture many aspects of experimentally observed surface interactions. As Hunter puts it: “It is not the first time (nor will it be the last) where the limited insights of an approximate theory are used until a more rigorous development becomes available” [140]. 115 Research objectives 2 and 3 led to models that describe the effect of commonly employed image band-pass filters and flow velocity gradients on the depth of correlation, respectively. The latter work was mainly motivated by discrepancies in previous literature on the topic. An attempt was made to present the framework of depth of correlation in micro PIV in the presence of flow velocity gradients in a more rigorous way than was done previously. In order to do so, explicit definitions of the weighting functions were proposed and the definition of the depth of correlation had to be modified to account for the fact that weighting functions are not always monotonically decreasing for z > zf. Overall, chapter 4 presented a general framework that allows determining weighting functions and the depth of correlation for practically relevant flow situations beyond the specific cases that were investigated in this thesis or by previous work. Most importantly, it was shown that the curvature of the local correlation function evaluated at its peak is not always related to the weighting function as it was previously assumed. We believe that these contributions will be of great interest to the field of µPIV and we hope that it will trigger some discussion on the concept of the depth of correlation in µPIV.  For example, one may raise the question of the practical relevance of the concept of the depth of correlation. The majority of the literature on µPIV refers to the original expression derived by Olsen and Adrian [79] for the case of negligible flow velocity gradients. For the case of negligible flow gradients, the depth of correlation may seem irrelevant as there will be no measurement error associated with finite depth of correlation in this case. However, in presence of out-of-plane flow velocity gradients it is desirable to minimize the depth of correlation in order to minimize the associated µPIV measurement error. As it was shown in section 4.3, the depth of correlation in presence of specific gradients is directly proportional to the depth of correlation for uniform flow. This suggests that the original expression for the DoC for uniform flow provides a useful tool for scaling the DoC in presence of flow velocity gradients.  One may also question the value of the expressions for weighting functions and DoC that account for flow gradients, if the flow field to be measured is typically unknown. In a very recent paper [214], it was demonstrated that the inverse problem corresponding to equation (99) can be solved for the true fluid velocity when the weighting function is known. Thus, in general, iterative schemes are conceivable, where the true weighting functions and flow velocity are determined in a step-wise fashion. Alternatively, the flow field may be estimated a priori by means of computational fluid dynamics.  Another question that may be raised is whether it is still meaningful to consider the weighting functions as a description of the relative influence of particles at a certain depth in the fluid on 116 the overall correlation function or the measured velocity. In section 4.3, a plausible interpretation of the shape of the weighting functions for constant in- and out-of-plane gradients was derived, which supports that the weighting functions really describe the relative influence of particles on the correlation function. The weighting functions for the more complicated flow scenario in section 4.4 were mostly in line with the predicted bias error, which again supports the present interpretation of the weighting functions. For example, the weighting functions approach a rectangular function for the case where the measured flow velocity is simply the spatial average of the true flow velocity field. However, one aspect of the weighting function in section 4.4 remains unclear. More specifically, an interpretation of the two maxima in the weighting functions in Figure 30 is not obvious. We would intuitively expect particles in regions of near zero out-of-plane gradient in the center of the channel to have the strongest influence on the measured flow velocity. 5.2 Recommendations for future work Particle deposition can currently not be predicted when conditions are unfavourable for deposition. This has typically been attributed to a combination of the factors surface charge heterogeneity, surface roughness and the related secondary minimum deposition. Particle adsorption experiments, where the surface charge (or potential) map in combination with surface roughness map of the substrate under investigation are known seem to be an ideal platform for rigorous testing of models that describe the effect of surface roughness, surface charge heterogeneity and the related secondary minimum deposition on particle adsorption. Thus, an effort to develop methods for localized surface charge characterization or improvement of the resolution of existing methods [173], seems worthwhile. Furthermore it would be interesting to see if manifestations of lateral surface interaction forces can be observed in adsorption experiments based on different particle and substrate materials. An investigation of the response of weakly immobilized particles to a sweep of the Peclet number may reveal the order of magnitude of the lateral forces. The investigation of the response of the trajectory of a weakly immobilized particle, or the displacement histograms, to transient chemical conditions may provide further insight into the nature of the lateral surface interactions forces. These studies could incorporate evanescent wave illumination into their setup, which allows tracking particles in close proximity of the substrate in 3D [215]. An investigation of the time evolution of the number of the weakly immobilized particles may provide additional clarification on whether these particles are located at a deep primary energy minimum (in which case the number of weakly immobilized particles can be expected to 117 increase linearly over time until surface blocking effects occur) or if they are in a shallow secondary minimum (in which case the number of weakly immobilized particles should reach a steady value). Furthermore, future experimental investigations of the clogging of microchannels due to particle deposition should be conducted at conditions that are favourable for the deposition of the particles on PDMS in order to avoid that the initial stages of clogging are governed by uncontrollable factors such as charge heterogeneity and surface roughness.  As for future work on the theory of µPIV cross correlation, several recent µPIV-based investigations [216], [217] of the evaporation driven flow of polymer solutions utilized an optical setup that produced asymmetrical particle image intensity distributions. This motivates an investigation of the implications of the asymmetry of the particle images on the µPIV measurement within the theoretical framework presented in this thesis. Furthermore, the results in chapter 3 suggest, that (reasonable) linear image processing methods may not be very effective in reducing the depth of correlation. We believe that some of the non-linear methods such as the base-clipping method or the image overlapping method may be more effective in reducing the depth of correlation in µPIV. Thus it is recommended to extend the theoretical framework presented in chapter 3 to account for the effect of non-linear image processing on weighting functions and the depth of correlation. 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Fluids, vol. 26, p. 033102, 2014.      140 Appendix A1 Simulation method and parameters: The boundary initial value problem described by eqs (4)and (6)-(7) (with the corresponding initial conditions) was solved for the logarithm of the dimensionless particle concentration by using the Matlab solver package pdepe. The lower and upper integration limits were set to x1min = ln(H1)-16 and 𝑥𝑥1max = 16. 5000 computational nodes were uniformly distributed between x1min < x1 < 2 and 30 additional nodes were uniformly distributed over the remainder of the integration domain. The method’s accuracy parameters were set to RelTol = 10-9 and AbsTol = 10-10. The transient solutions for the concentration fields were used in eq (11) together with the corresponding steady state concentration fields. All steady-state concentration fields and deposition rates were derived from solutions of the steady state version of eq (4) (i.e. with 𝜕𝜕𝑦𝑦1𝜕𝜕𝜕𝜕= 0). The steady state version of eq (4) was solved in two steps: First, eq (4) together with the perfect sink boundary condition in eq (6)(7) was solved for 𝜕𝜕𝑦𝑦1𝜕𝜕𝑥𝑥1 by using the Matlab solver ode15s. The method’s accuracy parameters were set to RelTol = 3∙1014 and AbsTol = 10-15. Next, with known 𝜕𝜕𝑦𝑦1𝜕𝜕𝑥𝑥1, the logarithm of the dimensionless particle concentration 𝑦𝑦1 was obtained by numerical integration based on the trapezoidal rule (Matlab function trapz) and the second boundary condition in eq (7) was enforced. In order to demonstrate that our implementation of the method was free of errors, Figure 4 of the original work [4] was reproduced as shown in Figure 32. Furthermore, all relevant simulation parameters used in this work, except for the zeta potentials and salt concentrations are listed in Table 5.  Figure 32: Figure 4 of [4] reproduced. jn is the dimensionless normal component of the particle flux as defined in [4]. 141 Table 5: Simulation Parameters Gravity constant G 9.81 m/s2 Temperature T 295 K Particle radius a 500 nm Primary minimum distance h1 0.158 nm Fluid density 𝜌𝜌f 997 kg/m3 Particle density 𝜌𝜌p 1053 kg/m3 Dynamic viscosity 𝜂𝜂 0.958 mPas Relative permittivity of water 𝜀𝜀𝑟𝑟 78.54 Reynolds number Re 0.75 Péclet number Pe 1.3∙10-4 Flow parameter 𝛼𝛼r 0.9975 Bulk diffusion coefficient D0 0.485 µm2/s Particle concentration nB 1.91∙107 cm-3 Chamber inlet radius Ri 1.1 mm Hamaker constant  A132 1.54∙10-21 J Lower integration limit  x1min; (hmin) ln(H1)-16; (1+exp(-16)) h1  Upper integration limit  x1max; (hmax)  (≈4 m)  142 A2 Implications of other choices for the Hamaker constant A132 Most of the discussion of the results depends to a strong degree on the selected value for the Hamaker constant. It seems mandatory to challenge the conclusions of this work by considering other values for the Hamaker constant in the range of the reported values. First of all, we rule out the negative Hamaker constants that can be obtained by combining the Hamaker constant for PDMS by [146] with any of the other reported values through equation (21). All other combinations indicate attractive van der Waals forces which are, in our opinion, more in line with the observed particle deposition events.  In the following discussion, we consider three choices for the Hamaker constants: 𝐴𝐴132(min) = 2.79 ∙ 10−22 𝐽𝐽 is the smallest (positive) constant that can be obtained by combining values listed in Table 1 through equation (21), 𝐴𝐴132 = 1.54 ∙ 10−21 𝐽𝐽 is the value that was chosen for this work and  𝐴𝐴132(max) = 5.2 ∙ 10−20𝐽𝐽 is the largest reported value that we found. Table 6 lists minimum separation distances 𝐻𝐻0 and expected number of mobile particles 𝑁𝑁pe in the field of view that are obtained by employing these Hamaker constants in combination with the DLVO theory and extended DLVO theory. Particle counts in the range 55 ≤ 𝑁𝑁pe ≤ 58 and the minimum separation distance 𝐻𝐻0 = 0.0003 indicate favourable conditions for deposition and 𝑁𝑁pe > 60 and 𝐻𝐻0 > 0.0003 indicate unfavourable conditions for deposition. All fields in Table 6 corresponding to favourable conditions for deposition are highlighted in gray. We believe, that the experimental conditions for deposition were favourable at the largest two salt concentrations (CNaCl = 0.1 M and 0.5 M) as the measured particle counts  𝑁𝑁pexp at these salt concentrations were similar and significantly lower than   𝑁𝑁pexp at lower salt concentrations CNaCl < 0.1 M and the measured deposition rates matched the predicted mass transfer rates reasonably well. We begin the discussion by considering 𝐴𝐴132(max). According to Table 6, both DLVO and extended DLVO theory seem to capture the transition from favourable to unfavourable deposition and additional measurements at salt concentrations between 10 mM and 100 mM would have to be conducted in order to favour one over the other. However, with 𝐴𝐴132(max) , both DLVO and xDLVO predict significant accumulation (𝑁𝑁pe = 498) of mobile particles due to deep secondary minima at CNaCl = 10 mM.  Comparison of the trends 𝑁𝑁pe and 𝑁𝑁pexp with respect to the salt concentration 𝐶𝐶NaCl suggests that there was no deep secondary minimum during the experiments as the measured particle counts for unfavourable conditions increase more gradually with the salt concentration as compared to the expected counts.  143 Let us now consider 𝐴𝐴132(min). From Table 6 it can be seen that the DLVO theory does not capture the observed transition from unfavourable to favourable conditions when 𝐴𝐴132(min) is chosen. The extended DLVO theory on the other hand does not allow us to favour the chosen 𝐴𝐴132 over 𝐴𝐴132(min) with the available experimental data as the corresponding trends of 𝐻𝐻0 and 𝑁𝑁pe with respect to 𝐶𝐶NaCl are nearly identical. However, in section 2.3.5 we suggest that the relatively larger fraction Υ of slow particles at CNaCl = 1mM and 10 mM may be explained by shallow secondary minima that make particle escape into the bulk less likely at these salt concentrations. With 𝐴𝐴132(min), the extended DLVO interaction potentials do not exhibit a secondary minimum at any of the investigated salt concentrations. Thus it can be concluded that overall, a Hamaker constant close to 𝐴𝐴132 = 1.54 ∙ 10−21𝐽𝐽 seem to allow the most plausible interpretation of the experiments performed in line with this work. Table 6: Minimum (dimensionless) separation distances 𝐻𝐻0 and expected particle counts 𝑁𝑁pe within the field of view obtained by employing the DLVO theory and the extended DLVO theory in combination with different Hamaker constants and for different salt concentrations 𝐶𝐶NaCl 𝐶𝐶NaCl  (mM)  DLVO Extended DLVO  𝑁𝑁pexp 𝐴𝐴132(min) 𝐴𝐴132 𝐴𝐴132(max) 𝐴𝐴132(min) 𝐴𝐴132 𝐴𝐴132(max) 𝐻𝐻0 𝑁𝑁pe 𝐻𝐻0 𝑁𝑁pe  𝐻𝐻0 𝑁𝑁pe  𝐻𝐻0 𝑁𝑁pe  𝐻𝐻0 𝑁𝑁pe  𝐻𝐻0 𝑁𝑁pe  0.1  0.3267 110 0.3267 110 0.3263 110 0.3267 110 0.3271 110 0.3262 111 164 1  0.1027 112 0.1027 112 0.1007 113 0.1027 112 0.1027 112 0.1007 113 220 10  0.0301 113 0.03 113 0.0253 498 0.0301 113 0.0299 113 0.0253 498 263 100  0.0065 113 0.0068 114 0.0003 55 0.0003 58 0.0003 58 0.0003 55 81 500  0.0003 60 0.0003 58 0.0003 55 0.0003 58 0.0003 58 0.0003 55 90  144 A3 Displacement distributions with standard deviation a) b)  c) d)  Figure 33: Average displacement distributions from Figure 13a) with associated standard deviation shown as error bars for different salt concentrations CNaCl: a) 0.1 mM, b) 1mM, c) 10 mM, d) 100 mM.  A4 Relationship between curvature of the local correlation function and the weighting function In this section additional clarification for the first equation in eq. (101) is given (clarification for second equation is analogous). We start at the expression for the weighting function eq (101)  𝑊𝑊�𝑥𝑥�𝑢𝑢��⃗ ,𝑢𝑢�⃗ 0, 𝑧𝑧� = 𝜕𝜕𝜕𝜕(𝑢𝑢𝑥𝑥 − 𝑢𝑢0𝑥𝑥)�𝜕𝜕〈𝑅𝑅𝐷𝐷�𝑠𝑠𝑥𝑥 − 𝑀𝑀𝑢𝑢𝑥𝑥∆𝑡𝑡, 𝑠𝑠𝑦𝑦 − 𝑀𝑀𝑢𝑢𝑦𝑦∆𝑡𝑡, 𝑧𝑧�〉𝜕𝜕𝑠𝑠𝑥𝑥 �𝑠𝑠=𝑠𝑠0=𝑢𝑢�⃗ 0𝑀𝑀∆𝑡𝑡�𝑢𝑢�⃗ 0−𝑢𝑢�⃗ −=0�⃗ , (139) and let ?⃗́?𝑥 = (?́?𝑥 ?́?𝑦)𝑘𝑘 with  ?́?𝑥 = 𝑠𝑠𝑥𝑥 − 𝑀𝑀𝑢𝑢𝑥𝑥∆𝑡𝑡 and ?́?𝑦 = 𝑠𝑠𝑦𝑦 − 𝑀𝑀𝑢𝑢𝑦𝑦∆𝑡𝑡 so that eq. (139) can be written as 145  𝑊𝑊�𝑥𝑥�𝑢𝑢��⃗ ,𝑢𝑢�⃗ 0, 𝑧𝑧� = 𝜕𝜕𝜕𝜕(𝑢𝑢𝑥𝑥−𝑢𝑢0𝑥𝑥) �𝜕𝜕〈𝑅𝑅𝐷𝐷(?́?𝑥,?́?𝑦,𝑧𝑧)〉𝜕𝜕?́?𝑥 �?⃗́?𝑥=𝑀𝑀∆𝑡𝑡(𝑢𝑢�⃗ −𝑢𝑢�⃗ 0)�𝑢𝑢�⃗ 0−𝑢𝑢�⃗ −=0�⃗ . (140) Now, let ?⃗́?𝑋 = (?́?𝑋 ?́?𝑌)𝑘𝑘 with  ?́?𝑋 = 𝑢𝑢𝑥𝑥 − 𝑢𝑢0𝑥𝑥 and ?́?𝑌 = 𝑢𝑢𝑦𝑦 − 𝑢𝑢0𝑦𝑦, so that eq. (140) becomes  𝑊𝑊�𝑥𝑥�𝑢𝑢��⃗ ,𝑢𝑢�⃗ 0, 𝑧𝑧� = 𝜕𝜕𝜕𝜕?́?𝑋 �𝜕𝜕〈𝑅𝑅𝐷𝐷(?́?𝑥,?́?𝑦,𝑧𝑧)〉𝜕𝜕?́?𝑥 �?⃗́?𝑥=𝑀𝑀∆𝑡𝑡?́?𝑋�⃗ �?́?𝑋�⃗ =0�⃗ = 𝑀𝑀∆𝑡𝑡 � 𝜕𝜕2𝜕𝜕𝑋𝑋�2 𝜕𝜕〈𝑅𝑅𝐷𝐷�𝑋𝑋�,𝑌𝑌� , 𝑧𝑧�〉�𝑋𝑋��⃗ =0�⃗ , (141) where the two first-order derivatives in the left hand side of eq. (141) were combined to a single second order derivative by making use of the Cartesian coordinates 𝑋𝑋�⃗ = (𝑋𝑋� 𝑌𝑌�)𝑘𝑘 . Finally, we may introduce 𝑋𝑋� = 𝑠𝑠𝑥𝑥 −  𝑀𝑀∆𝑥𝑥 and 𝑌𝑌� = 𝑠𝑠𝑦𝑦 −  𝑀𝑀∆𝑦𝑦 and interpret the weighting function eq. (141)   𝑊𝑊�𝑥𝑥�𝑢𝑢��⃗ ,𝑢𝑢�⃗ 0, 𝑧𝑧� = 𝑀𝑀∆𝑡𝑡 � 𝜕𝜕2𝜕𝜕𝑋𝑋�2 𝜕𝜕〈𝑅𝑅𝐷𝐷�𝑋𝑋�,𝑌𝑌� , 𝑧𝑧�〉�𝑋𝑋��⃗ =0�⃗ = 𝑀𝑀∆𝑡𝑡 𝜕𝜕2𝜕𝜕𝑠𝑠𝑥𝑥2 〈𝑅𝑅𝐷𝐷(𝑠𝑠 − 𝑀𝑀∆?⃗?𝑥), 𝑧𝑧〉�𝑠𝑠=𝑀𝑀∆𝑥𝑥 (142) as the curvature of the local correlation function evaluated at the local signal peak (times a constant).  146 

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