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Mechanics of polymer brush based soft active materials Manav 2019

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MECHANICS OF POLYMER BRUSH BASED SOFTACTIVE MATERIALSbyMANAVB.Tech., Indian Institute of Technology Kanpur, 2009M.A.Sc., The University of British Columbia, 2014A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Mechanical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)April 2019c© MANAV 2019The following individuals certify that they have read, and recommend tothe Faculty of Graduate and Postdoctoral Studies for acceptance, the dis-sertation entitled:Mechanics of polymer brush based soft active materialssubmitted by Manav in partial fulfillment of the requirements forthe degree of Doctor of Philosophyin Mechanical EngineeringExamining Committee:A. Srikantha Phani, Mechanical EngineeringSupervisorMauricio Ponga, Mechanical EngineeringCo-supervisorJoerg Rottler, Physics and AstronomySupervisory Committee MemberReza Vaziri, Civil EngineeringUniversity ExaminerIan Frigaard, Mechanical Engineering and Applied MathematicsUniversity ExaminerAdditional Supervisory Committee Members:Gwynn Elfring, Mechanical EngineeringSupervisory Committee MemberJayachandran N. Kizhakkedathu, Pathology and Laboratory MedicineSupervisory Committee MemberiiAbstractA brush-like structure emerges from the stretching of long polymer chains,densely grafted on to the surface of an impermeable substrate. This struc-ture is due to a competition between the conformational entropic elasticityof grafted polymer chains, and the intra and interchain excluded volumerepulsions. Polymer brushes occur in biology: neurofilaments, articulatecartilage, extra cellular biopolymers etc. Recently, engineered soft activematerials are developed to produce large controllable and reversible bend-ing and stretching deformations. These materials are the focus of this work.New theoretical models, molecular simulations to assess them, and exper-imental studies are presented in this work. Mechanical stress within a brushand its dependence on the molecular parameters of the brush and exter-nal stimulus (temperature) is studied for the first time. A continuum beammodel accounting for the Young-Laplace and the Steigman-Ogden curvatureelasticity corrections is developed first to understand the large deformationof a flexible substrate due to a brush grafted on it. This model yields ageneralized surface stress-curvature relation that enables one to determinestress from curvature measurements.Strong stretching theory (SST) from polymer physics is combined withcontinuum mechanics to obtain stress variation in a neutral brush with Gaus-sian chains. This theory predicts that the normal stress, parallel to the sub-strate, is a quartic function of the distance from the grafting surface witha maximum at the grafting surface. Idealizing the brush as a continuumelastic surface with residual stress, closed-form expressions for surface stressand surface elasticity as a function of molecular weight and graft density arederived. At a higher graft density, a more refined (semi) analytical SST withLangevin chain elasticity is advanced. Theoretical predictions are assessedby molecular dynamics simulation of a brush using bead-spring model.Experiments on a thermoresponsive brush grafted onto a soft beamshowed the surface stress is ∼ −10 N/m and its magnitude decreases grad-ually, and reversibly, on increasing solvent temperature. Molecular scaleparameters of the brush are estimated experimentally to enable qualitativecomparison with SST theories.iiiLay SummaryPolymer chains attached to the surface form a polymer brush structure, sim-ilar to the bristles on a tooth brush. This structure responds to changes inambient conditions, leading to large and controllable deformations, enablingseveral applications. For example, a catheter tip coated with a polymerbrush can assist the deployment of devices and delivery of drugs in a diffi-cult to reach branch in human circulation system. Such applications requirea fundamental understanding of the forces and deformations involved, whichthis work offers.Some degree of control over molecular scale parameters of the brush canbe exercised by controlling fabrication process parameters during the brushfabrication. However, the relation between the mechanical properties suchas the built-in stress of the brush, and its molecular scale parameters isnot well established. This thesis bridges this gap to allow better design ofactuators and sensors utilizing polymer brushes.ivPrefaceAll the work presented in this thesis was carried out as part of my PhDresearch. Parts of this thesis have appeared, is under review, or in prepara-tion for publication in refereed journals and conference proceedings, as listedbelow. I was responsible for all areas of concept formation, data collectionand analysis, as well as manuscript composition.• Parts of chapter 2 and 4 have been published in the Journal of theMechanics and Physics of Solids [1]. Parts of the same work were alsopresented at the following conferences: a) The 8th World Congress onBiomimetics, Artificial Muscles, and Nano-Bio, August 2015, Vancou-ver, Canada, and b) International Congress of Theoretical and AppliedMechanics, August 2016, Montreal, Canada. Dr. P. Anilkumar fabri-cated the polymer brush samples along with the measurement of themolecular weight of the polymers. Professor A. Srikantha Phani wasthe supervisory author on this project and was involved throughoutthe project in concept formation and manuscript composition.• Parts of chapter 3 has been published in the Journal of the Mechanicsand Physics of Solids [2]. Parts of the same work was also presented atThe Society of Engineering Sciences, 54th Annual Technical Meeting,July 2017, Boston, USA. Professor A. Srikantha Phani and ProfessorMauricio Ponga were the supervisory authors on this project and wereinvolved throughout the project in concept formation and manuscriptcomposition.• Parts of chapter 4 is in preparation to be submitted under the title‘Effect of temperature on stress in a thermoresponsive polymer brush’.Professor A. Srikantha Phani and Professor Mauricio Ponga were thesupervisory authors on this project and were involved throughout theproject in concept formation and manuscript composition.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . xvAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Soft material . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Polymer brush . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.1 Applications . . . . . . . . . . . . . . . . . . . . . . . 61.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Polymer theories . . . . . . . . . . . . . . . . . . . . . . . . . 91.4.1 Polymer Chain . . . . . . . . . . . . . . . . . . . . . . 101.4.2 Polymer brush theories . . . . . . . . . . . . . . . . . 151.4.3 Molecular simulation . . . . . . . . . . . . . . . . . . 211.5 Research objectives and methodology . . . . . . . . . . . . . 221.6 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.7 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . 23viTable of Contents2 Mechanics of a polymer brush . . . . . . . . . . . . . . . . . . 252.1 Stress in a polymer brush using SST . . . . . . . . . . . . . . 252.1.1 Good solvent . . . . . . . . . . . . . . . . . . . . . . . 272.1.2 θ-solvent . . . . . . . . . . . . . . . . . . . . . . . . . 322.1.3 Surface stress and surface modulus . . . . . . . . . . 332.1.4 Energetics of bending . . . . . . . . . . . . . . . . . . 342.2 Large deflection of a flexible beam grafted with a polymerbrush . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 Stress in a polymer brush . . . . . . . . . . . . . . . . . . . . 443.1 SST with Langevin chains . . . . . . . . . . . . . . . . . . . 443.2 Stress in a polymer brush using SST-L . . . . . . . . . . . . 453.2.1 Free energy density . . . . . . . . . . . . . . . . . . . 453.2.2 Calculation of the derivatives . . . . . . . . . . . . . . 503.3 Molecular dynamics simulation . . . . . . . . . . . . . . . . . 543.3.1 Calculation of stress . . . . . . . . . . . . . . . . . . . 583.4 Results: MD vs. SST-G and SST-L . . . . . . . . . . . . . . 603.4.1 Monomer density . . . . . . . . . . . . . . . . . . . . 603.4.2 End density . . . . . . . . . . . . . . . . . . . . . . . 643.4.3 Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734 Stimuli response of a brush: experiment and theory . . . . 754.1 Context for comparison . . . . . . . . . . . . . . . . . . . . . 764.2 Stimuli response . . . . . . . . . . . . . . . . . . . . . . . . . 774.2.1 Stimuli response of a polymer solution . . . . . . . . 774.2.2 Stimuli response of a polymer brush . . . . . . . . . . 774.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.3.1 Brush Fabrication . . . . . . . . . . . . . . . . . . . . 794.3.2 Measurement of surface stress . . . . . . . . . . . . . 794.3.3 Estimation of graft density and molecular weight . . 814.4 Theoretical modeling . . . . . . . . . . . . . . . . . . . . . . 844.4.1 SST-L for a thermoresponsive brush . . . . . . . . . . 864.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 884.5 Comparison between SST-L and experiment . . . . . . . . . 944.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97viiTable of Contents5 Conclusion and future work . . . . . . . . . . . . . . . . . . . 995.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.4 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . 105Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108AppendicesA A general force-extension scaling relation for real chains . 121A.1 Uni-axial force-extension of a single chain . . . . . . . . . . . 121A.1.1 Entropic spring . . . . . . . . . . . . . . . . . . . . . 122A.1.2 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . 122B Comparison of brush heights obtained from different theo-ries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126C Linearization of cantilever beam equation . . . . . . . . . . 128D SST-L calculations . . . . . . . . . . . . . . . . . . . . . . . . . 130D.1 Steps to calculate stress in a brush . . . . . . . . . . . . . . . 130D.2 Estimation of ∂zz∂xx from MD/experiment . . . . . . . . . . . 131E Generating initial configuration of a brush with approxi-mately parabolic density profile . . . . . . . . . . . . . . . . . 132viiiList of Tables1.1 The table below presents a comparison of scaling theory, SCFT,SST, and SCMF. . . . . . . . . . . . . . . . . . . . . . . . . . 201.2 A comparison of the assumptions and limitations of SST andMD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.1 Comparison of expressions for surface stress of a brush ingood and θ solvents obtained from different theories . . . . . 353.1 Stretching and extension parameters for brushes with differ-ent graft densities . . . . . . . . . . . . . . . . . . . . . . . . . 653.2 Comparison of expressions for the height, free energy and sur-face stress in a brush in a good solvent obtained from differenttheories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.1 The table compares the differences between experimentallystudied system and theoretically studied system. . . . . . . . 764.2 Experimentally measured brush parameters for brushes of dif-ferent polymerization hours. . . . . . . . . . . . . . . . . . . . 834.3 Number of monomers in a polymer chain and graft density ofbrushes with different polymerization hours . . . . . . . . . . 83B.1 Comparison of expressions for height of a brush in good, θand poor solvents obtained from mean field Flory theory . . . 127ixList of Figures1.1 SEM image and schematic of a polymer brush . . . . . . . . . 21.2 Mushroom are brush states are shown . . . . . . . . . . . . . 31.3 Mushroom to brush crossover for different solvent qualities . 41.4 Coil-globule transition in a polymer brush . . . . . . . . . . . 51.5 Different aspects of a polymer brush are shown . . . . . . . . 71.6 Force-extension relation for an isolated chain . . . . . . . . . 132.1 A schematic showing a side view of a planar polymer brush . 262.2 A brush layer can be construed as an elastic surface layer withresidual stress at the top of a beam. . . . . . . . . . . . . . . 332.3 Schematic of an elastic beam with surface stress . . . . . . . 363.1 Monomer density profile obtained from SST-L is comparedwith the prediction from the SST-G. . . . . . . . . . . . . . . 483.2 End density profiles obtained from SST-L is compared withthe prediction from SST-G. . . . . . . . . . . . . . . . . . . . 493.3 The figure shows ∂u¯∂xx and∂zz∂xxvs z¯ curves for three graftdensities along with ∂H¯∂xx . . . . . . . . . . . . . . . . . . . . . 523.4 Stress profiles obtained from SST-L is compared with the pre-dictions from SST-G. . . . . . . . . . . . . . . . . . . . . . . . 543.5 Front view of MD simulation box with bead spring chainsrepresenting polymers . . . . . . . . . . . . . . . . . . . . . . 553.6 Variation of the pair (ULJ) and bond (UFENE) potentialswith the distance between interacting beads. . . . . . . . . . 563.7 The effect of number of monomers in a chain (N) on thevariation of monomer density with distance from the graftingsurface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.8 Variation of monomer density in a polymer brush with thedistance from the grafting surface . . . . . . . . . . . . . . . . 623.9 SST-G prediction of monomer density compared with MDmonomer density. . . . . . . . . . . . . . . . . . . . . . . . . . 63xList of Figures3.10 Variation of the height of a brush with graft density . . . . . 633.11 Variation of end density in a polymer brush with distancefrom the grafting surface . . . . . . . . . . . . . . . . . . . . . 663.12 Effective stretching ratio (γ) is plotted vs stretching ratio (βs) 673.13 Typical stress profile . . . . . . . . . . . . . . . . . . . . . . . 683.14 Variation of stress component σXX in a polymer brush withdistance from the grafting surface . . . . . . . . . . . . . . . . 693.15 The plot shows variation of σxx with a quartic function of thedistance from the grafting surface . . . . . . . . . . . . . . . . 703.16 The plot shows the variation of normalized stress σxx withthe distance from the grafting surface. . . . . . . . . . . . . . 713.17 The plot shows variation of normalized stress σxx with a quar-tic function of the distance from the grafting surface . . . . . 723.18 Variation of surface stress with respect to graft density . . . . 724.1 Experimental set-up to measure curvature of a cantilever beamand images showing beam deflection at two temperatures . . 804.2 Variation of curvature of the brush coated beams and thesurface stress with temperature . . . . . . . . . . . . . . . . . 824.3 Schematic shows phase separation in polymer brush is favoureddue to the free energy minimization . . . . . . . . . . . . . . 874.4 Variation of interaction free energy density and chemical po-tential with changing volume fraction for different temperatures 894.5 Variation of monomer density with the distance from thegrafting surface at different temperatures . . . . . . . . . . . 904.6 Variation of brush height with temperature for different graftdensities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.7 Variation of end density with the distance from the graftingsurface at different temperatures . . . . . . . . . . . . . . . . 924.8 Variation of surface stress with graft density at two temper-atures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.9 The change in the resultant surface stress with increasing tem-perature for different graft densities . . . . . . . . . . . . . . 944.10 Surface stress variation with graft density obtained from theexperiment and theory . . . . . . . . . . . . . . . . . . . . . . 965.1 Contributions of this thesis . . . . . . . . . . . . . . . . . . . 100A.1 Force-extension curve for a bead spring polymer chain . . . . 125xiList of SymbolskB Boltzmann constantT TemperatureRg Radius of gyrationRF Flory radiusD Mean distance between grafting pointsν Flory exponent/Poisson’s ratio, depending on contexta Effective monomer lengthN Number of effective monomers in a chainlp Persistence lengthlk Kuhn lengthL Contour lengthρg Graft densityδ Deflection of a cantileverτs Surface stressB Bending modulusκsp Spontaneous curvatureR Chain end to end distancep Stretching force at chain endl Size of Pincus blobH Polymer brush heightVf Volume fractionφ Monomer densityg(z) End density at height z in a brushχ Flory Huggins parameterv Excluded volume or binary Virial coefficientw Ternary Virial coefficientζc Size of a correlation blobζT Size of a thermal blobF Helmholtz free energyFint Brush interaction free energy per unit surface areaFel Brush stretching free energy per unit surface area∆Fint Brush interaction free energy per unit volumexiiList of Symbols∆Fel Brush stretching free energy per unit volumef(z) Free energy density at height zfint Interaction free energy densityfel Stretching free energy densityE(z, ζ) Stretching in a chain with end at ζ at zE Young’s modulus of substrateE¯ Plane stress modulusEs Young’s modulus of surfaceC Steigmann-Ogden constantI Area momentθ Rotationh Thickness of substratehs Thickness of surfaceκ Curvature of a beamV (z) Mean field potential at height zµ Chemical potentialΠ Osmotic pressured Effective monomer diameterpl Asymmetry ratioσij Stressij StrainΩ Volume of a solvent moleculeu z-displacement of a thin layerλ Mid plane stretchingE Young’s modulus of substrate·¯ Normalized parametere Local chain stretchingmi Mass of the ith beadri Position of the ith beadria ath component of position of the ith beadvia ath component of velocity of the ith beadU Total potential energy of the systemΓ Bead frictionδij Kronecker delta functionδ(·) Dirac delta functionNtot Total number of unconstrained beadUFENE FENE potentialULJ Lennard Jones potentialUSoft Soft interaction potentialxiiiList of Symbolsσ Length scale in LJ potentialR0 Maximum stretching of a bond Energy scale in LJ potentialVbin Volume of a binNbin Number of beads in a binβs Stretching parameterβe Extension parameterγ Effective stretching ratioMn Number averaged molecular weightMw Weight averaged molecular weighthdb Thickness of dry brushρ DensityNA Avogadro’s numberxivList of AbbreviationsCS Carnahan-StarlingFENE Finitely extensible nonlinear elasticFH Flory-HugginsFJC Freely jointed chainGPC Gel permeation chromatographyLAMMPS Large-scale Atomic/Molecular Massively Parallel SimulatorLCST Lower critical solution temperatureLJ Lennard-JonesMC Monte CarloMD Molecular dynamicsPDI Polydispersity indexPNIPAm Poly(N- isopropylacrylamide)PNIPAm-co-PDMA Poly(N- isopropylacrylamide)-co-Poly(N,N-dimethylacrylamide)pPVC plasticized poly(vinyl chloride)SAM Soft active materialsSCFT Self consistent field theorySCMF Single chain mean field theorySEM Scanning electron microscopexvList of AbbreviationsSI-ATRP Surface-initiated atom transfer radical polymerizationSST Strong stretching theorySST-G Gaussian chain SSTSST-L Langevin chain SSTUCST Upper critical solution temperatureWCA Weeks-Chandler-AndersenWLC Worm like chainxviAcknowledgementsFirst and foremost, I would like to express my gratitude to my supervi-sor Professor Srikantha Phani for his guidance and support throughout theproject, and not limited to research alone.I want to thank my co-supervisor Professor Mauricio Ponga for his guid-ance particularly in the computational part of the work, and for giving meconfidence to modify bits of the molecular dynamics simulation code. Hiscareer advices have been very practical and valuable.I would like to thank Dr. Parambath Anilkumar for fabricating polymerbrush samples and providing molecular weight data. The experimental workreported in the thesis used those polymer brush samples. I would like to alsothank Dr. Madhab Bajgai for his contribution in the development of proto-col for fabrication of polymer brush, and Madeshwaran Selvaraj and DianaNino for their help with experimental set-up. I am grateful to Professor Jay-achandran Kizhakkedathu for allowing use of his lab for brush fabrication,and for the related discussions and advices.I am thankful to Professor Joerg Rottler for his kind help whenever Iasked for his advice and clarifications, and for pointing me to some recentuseful works.I would like to acknowledge the generous funding support from differentsources that I have received through the course of my PhD: the award offour year fellowship (4YF) from UBC, the support and training from theNatural Sciences and Engineering Research Council of Canada (NSERC)NanoMat CREATE program at UBC, and other NSERC funding supportsthrough Discovery grant and through the Collaborative Health Researchproject jointly with the Canadian Institute of Health Research (CIHR). Iam thankful to WestGrid and Compute Canada for providing computationalresources. I would also like to thank Professor Eliot Fried and OkinawaInstitute of Science and Technology (OIST) for supporting my internship atOIST.The course of PhD has been one of ebbs and flows, highs and lows,personally as well as professionally, and I am thankful to have friends whohelped me come ashore. I would like to thank them all.xviiTo my familyxviiiChapter 1IntroductionThis chapter introduces soft materials, particularly polymer brushes. Aftera brief introduction to soft materials in Section 1.1, polymer brushes arediscussed in Section 1.2. After that, motivation for this work is describedin Section 1.3. Since the thesis makes use of the theories of polymer physicsextensively, an overview of the theories of an isolated polymer chain aswell as the theories of polymer brushes are presented in Section 1.4. This isfollowed by the research objectives (Section 1.5) and the scope (Section 1.6),ending with an outline of this thesis (Section 1.7).1.1 Soft materialSoft materials are a class of materials which show large and nonlinear re-sponse to weak external forces. Examples include colloids, liquid crystals,surfactants, materials composed of polymers, among others. Their shearmodulus is orders of magnitude smaller than conventional hard materials [3],giving them softness. The physical origin of their softness lies in the funda-mental length and energy scales associated with their constituent structuralunits. The structural units (size ∼ 1 nm−1 µm) are at mesoscopic scale [4],sandwiched between atomic length scale (∼ 1 A˚) and macro scale. The en-ergy associated with interactions between molecules in these structural unitsis on the order of thermal energy, kBT (≈ 1/40 eV ≈ 4 pN · nm at roomtemperature). This is very small in comparison to the energy of a covalentbond (∼ 10 eV ). Hence, it is easy to make them respond to a stimulus. Thelarge size of structural units also results in their slow response to an appliedstrain: their response time can be as large as ∼ 1 − 104 s in comparisonto ∼ 10−9 s of a simple fluid [5]. Because of the large response time, theyexhibit viscoelasticity.Soft active materials (SAM) are an important category of soft materialswith growing technological applications in areas ranging from biology torobotics. They undergo large reversible deformation in response to externalstimuli such as temperature [6], hydration [6], pH [7], light [8] etc., offeringa facile route to achieve stimulus response. Examples of SAMs include11.2. Polymer brushstimuli-responsive gels [9, 10], electroactive polymers [11, 12], liquid crystalelastomers [13, 14], shape memory polymers [15, 16] and polymer brushes [6],among others. Their applications [17] in the areas of drug delivery [18], selfassembly [19], biomedical engineering [20], soft robotics [21] etc. have ignitedscientific inquiry into their mechanical response [22–24]. Typically, stimuliresponse in most SAMs originates from their bulk properties. However, inthe case of a polymer brush, stimulus response originates from the surfacemodification of a substrate. This allows one to achieve stimuli responsewithout sacrificing bulk mechanical properties.1.2 Polymer brush11   Figure 7S A) SEM image of partial PDMA grafting on one side of pPVC; B) two wet specimen with partial PDMA grafting on single side was prepared from the same batch of SI-ATRP. Similar curvature was observed verifying the uniform coating and polymerizaiton.  The coated and uncoated lengths are denoted as L1 and L2, respectively in Figure 3A (main text). Every cross section within the coated/grafted region is subjected to a constant, internal bending moment,  , of magnitude        about the mid-plane. Here,   and   denote the width and thickness of the pPVC substrate, respectively. Outside the coated region there is no bending moment. The deformed elastic curve can be obtained by solving the moment-curvature relation for a thin Euler-Bernoulli beam 6c together with the boundary conditions of zero displacement and rotation at the fixed end:                                                             where    is bi-axial Young’s modulus related to the uni-axial Young’s modulus   and Poisson ratio   via the relation         ;   is the second moment of area of cross section about the mid-plane. For a specimen with rectangular cross-section,        . The solution of the above equations (1a) and (1b) gives the elastic deflection curve      of the substrate, from which the deflection,    at the free end        , is given by 500 μmUncoated PVCPDMA brushA B WetSingle-side CoatingNon-coatedNon-coated500 µmlamide (DMA) for 2, 6, 12, and 24 h. As shown in Figure 1B,the pPVC substrates were deformed with increased polymer-ization time. All pPVC substrates curved in the direction ofthe side onto which the PDMA chains were grafted. Longerpolymerization times resulted in a gradual decrease in theradius of the curvature of the substrates (Figure 1C1). Themolecular weight (Mn: number-average molecular weight,Mw: weight-average molecular weight) of the PDMA chainsformed in solution along with the surface-grafted chainsremained constant after 2 h (Mn andMw/Mn values were 1.5!106, 1.8! 106, 2.1 ! 106, 2.0! 106 and 1.68, 1.75, 1.87, 2.00,respectively for 2, 6, 12, 24 h of SI-ATRP), suggesting that theincrease in bending deformation can be attributed to theincrease in graft density of the polymer chains (i.e. increase inchain–chain repulsion) on the surface. Although the directmeasurement of the molecular weight is desirable,[6] the Mnvalues of the grafted PDMA chains were estimated from thesolution polymers in the current study because the amidelinkage between the polymer and the surface was cleavedincompletely. The presence of amide bonds in the PDMA alsocomplicated the cleavage process. The gradual increase in thePDMA graft density and the ultrahigh molecular weight ofthe chains are consistent with our previous observation of SI-ATRP of DMA from unplasticized PVC in aqueous solu-tion.[7]A high degree of reversibility is essential for an idealactuator design. Figure 2A shows the effect of dehydration–rehydration under atmospheric conditions on the bending–flattening of the PDMA-grafted pPVC substrate (12 h SI-ATRP). Under these conditions (45% relative humidity,22 8C), the PDMA-grafted pPVC gradually dehydrated,flattened, and finally reached an equilibrium state after nineminutes. The dry sample reverted to its original shape uponrehydration within eight seconds (see video 1 in the Support-ing Information). The flattening and bending of the PDMA-brush-grafted pPVC is due to conformational changes of thePDMA chains on the surface during the drying–wettingprocess. The chain dimensions of the grafted PDMAdecreased during the drying process which resulted in reducedchain–chain interactions. During the rehydration, the poly-mer chains regained their original dimensions and thesubstrate reverted to its original shape. A control nontreatedpPVC substrate did not show noticeable shape changes duringthe wetting–drying process (see video 2 in the SupportingInformation).To obtain quantitative information on the reversibility ofthe bending–flattening process, a wet PDMA-grafted pPVCsubstrate was dried by two approaches: vacuum drying (22 8C.0.1 Pa, 15 minutes) and hot-gun drying (180 8C, 10 seconds).As shown in Figure 2B1, B2, and B5, the vacuum dryingafforded a highly reversible bending–flattening process indi-cated by the minimal variation in the bending angles when theprocess was repeated (Figure 2B5). In contrast, hot-gundrying (complete drying) led to a gradual decrease in thebending angle (Figure 2B1, B3, B6) with repeated wetting–drying cycles, suggesting a more pronounced initial irrever-sibility. In the attenuated total reflectance Fourier transforminfrared (ATR-FTIR) spectrum the water peak (3400 cm!1)observed for the vacuum-dried substrate disappeared afterhot-gun drying (Figure 2B4). The differences in the behaviorof the substrate subjected to the two drying methods reflectthe importance of the residual water for the reversibility ofthe bending–flattening pro ess. On possible reason for theirreversible deformation in the case of hot-gun drying couldbe entanglement of the polymer chain. The residual water inthe specimen after vacuum drying might have prevented suchentanglement, resulting in the high reversibility of theprocess.To verify that the covalently grafted PDMA chains areresponsible for the bending, we examined the bending–flattening process of a pPVC substrate spin-coated withPDMA on one side. TheMn value of the spin-coated PDMAwas comparable to that of the grafted chains and the thicknessof the dry coating was approximately 31 mm. There was nobending observed for the spin-coated sample in the hydratedstate and it bended slightly to the side of the PDMA coatingupon drying, which is presumably due to the contraction ofthe PDMA layer. The structures of unmodified pPVC, thePDMA-grafted pPVC substrate (24 h SI-ATRP), and thepPVC spin-coated with PDMA were compared by scanningelectron microscopy (SEM). The topography of the pPVCand the PDMA-grafted pPVC substrates is rougher than thatof the spin-coated PDMA (see Figure 6S in the SupportingInformation). Cross-sectional images of the samples areshown in Figure 2C. A sparsely spaced interface filled withvertically aligned fiberlike structures was observed forFigure 2. A) Effect of dehydration–rehydration on the bending of apPVC substrate with PDMA chains grafted on one side (12 h SI-ATRP).B) Effect of different drying processes on the reversibility of thebending. B1) Photograph of a wet PDMA-grafted pPVC substrat ,B2) photograph of the substrate dried under vacuum, B3) photographof the substrate dried by a hot gun, B4) ATR-FTIR spectra of a PDMA-brush-grafted pPVC substrate dried in vacuum (dot-dashed line) andwith a hot gun (solid line), B5 and B6) relationship of the bendingangles to different wetting–dryi g cycles under ifferent drying con-ditions (B5: vacuum drying, B6: hot-gun drying). C) Cross-sectionalback-scattered electron SEM images of C1: unmodified pPVC, C2:pPVC grafted with PDMA (24 h SI-ATRP), and C3: spin-coated PDMAon pPVC. The scale bars in C1–C3 represent 5 mm.5117Angew. Chem. Int. Ed. 2011, 50, 5116 –5119 ! 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.angewandte.org1    𝐷 𝐷 Flory Radius (𝑅𝐹) Mushroom: 𝐷 > 𝑅𝐹 (∝ 𝑁) Brush: 𝐷 < 𝑅𝐹  Graft density: Σ = 𝐷−2 Number of monomers: 𝑁 Monomer  size: 𝑎 Polymer chain Flexible substrate Grafting point 𝑡 𝑍 𝑋 𝐻 Substrate 𝑎 𝑁𝑎 𝑑 2Figure 1.1: SEM image of a polymer brush grafted on a plasticized poly(vinylchloride) (pPVC) substrate showing brush like structure (left) (adaptedfrom [6]), and an SEM image of a cross-section of a pPVC substrate withdried polymer brush along with a schematic of polymer chains grafted ontothe surface of a substrate (right).Polymer chains densely grafted onto a planar or curved impermeablesubstrate, give rise to a polymer brush [25]. Figure 1.1 shows the Scan-ning electron microscope (SEM) image as well as a schematic of a polymerbrush. Depending on the substrate geometry, polymer brushes are catego-rized as planar brush, cylindrical brush or spherical brush. In a cylindri-21.2. Polymer brushcal or spherical brush, radius of curvature of the substrate is smaller thanthe thickness of the brush layer on top of the substrate. Polymer brushesinvolving electrostatic charge fall into the categories of polyelectrolyte orZwitterionic brushes. Otherwise, they are called neutral polymer brush andhenceforth will be referred to as polymer brush in this thesis. Examples innature include extracellular biopolymers [26], neurofilaments [27], articulatecartilage [28] etc.    𝐷 𝐷 ∼ 𝑅𝐹 Mushroom: 𝐷 > 𝑅𝐹 Brush: 𝐷 < 𝑅𝐹 Polymer chain Flat substrate Grafting point 𝑡 𝑍 𝑋 𝐻 Substrate Figure 1.2: The schematic shows mushroom and brush states of polymerchains grafted to a substrate.To understand the structural characteristics of a polymer brush, we startwith an isolated chain grafted onto a rigid substrate. Such an isolated chainhas a gyration radius Rg ∼ RF = Nνa, where RF is called Flory radius (endto end distance of the chain), N is the number of effective monomers (see Sec-tion 1.4.1 for a definition) in the chain, a is the size of an effective monomer,and ν is the Flory exponent. ν depends on the solvent quality, which can bedivided in three categories: a) good solvent, in which monomers have highaffinity for solvent molecules (ν = 3/5) and polymer chains are swollen, b)θ-solvent, in which excluded volume1 v of a monomer vanishes (ν = 1/2) andpolymer chains behave like ideal chains (see Section 1.4.1), and c) poor sol-vent, in which monomers minimize exposure to solvent molecules (ν = 1/3)and form globules. So, the size of an isolated chain depends on the numberof effective monomers in a chain, the size of an effective monomer, and thesolvent quality. Now, a distinguishing feature of a polymer brush is thepresence of a new length scale, the mean distance between grafting points.When the distance between grafting points of polymer chains on a substrateis more than the gyration radius (1/√ρg > Rg, where ρg is graft density2),they do not interact and form the so called ‘mushroom configuration’ (see1It equals the second Virial coefficient of a polymer solution [30].2Number of grafting points per unit surface area of the substrate31.2. Polymer brushSCFTGood  solventMushroomBrushMarginal  solventScaling theoryIncreasing  solvent strengthlog(ρgR2g)log(vN2 /R3 g)H∼RgH ∼ Rg(ρgR2g) (1 − 2ν)2νH ∼ Rg(ρgR2g)1/3(v N2R3g)3 − 5ν3(2 − 3ν)H ∼ Rg (ρgR2g)(v N2R3g)1/3(vN 2/R 3g ) ∼ (ρg R 2g ) −1(vN2 /R3g) ∼(ρgR2g)2 − 3ν2νFigure 1.3: The schematic (adapted from [29]) shows different regimes ofgraft density (ρg) and excluded volume (ν) in which polymers grafted toa flat surface are in mushroom or brush configuration. The division of theregimes depends on the number of monomers in each grafted chain (N), andgyration radius of a chain (Rg). The shaded region on the left is mushroomregion. In a good solvent condition (above the red dashed line), excluded vol-ume is constant and equals a3. Hence we get a vertical line separating mush-room and brush region. In a marginally good solvent (marginal solvent), theexcluded volume varies with the solvent quality, resulting in a slanting lineseparating mushroom and brush regions. The schematic also shows the re-gions of applicability of scaling theory (discussed in Section 1.4.2) and Selfconsistent field theory (SCFT) (discussed in Section 1.4.2) for brushes. Itmust be noted though that the separating lines do not indicate sharp bound-aries, but represent smooth crossovers.Figure 1.2). However, when 1/√ρg < Rg, the chains adapt a stretchedconfiguration and form brush structure [25, 31, 32] (see Figure 1.3). Thisbrush-like structure, reminiscent of bristles on a tooth brush, is governedby a combination of repulsion between monomers in a crowded monolayer,entropic stretching of polymer chains and the constraints set by end graft-ing. The entropic spring force strives to bring the two ends of a polymerchain together as a closed chain configuration has the maximum entropy.41.2. Polymer brushExcluded volume repulsion between monomers of the same as well as theneighbouring chains, tend to extend the chain in a good solvent. Theircompetition leads to the extension of the chains away from the substrate,forming a brush structure. Significantly, these interactions make a brushlayer grafted on a substrate behave like an elastic surface layer with resid-ual surface stress3. These stresses deform the underlying elastic substrate[6, 23]. Further, in the presence of an external stimulus, the interactionbetween monomers and solvent is altered, such as through hydrogen bondbreaking between the polymer and the solvent. This modifies the qualityof solvent, making the solvent a poor solvent and eventually leading to acollapsed chain structure (see Figure 1.4). In a stimuli-responsive polymerbrush, residual surface stress as well as surface elasticity can be reversiblymodified by switching between stretched (good solvent) and collapsed (poorsolvent) states.  Stimulus Add Coil state Globule state Remove 𝐻 𝑧 Substrate Brush Figure 1.4: Coil-globule transition in a polymer brush grafted to a flexi-ble substrate. A brush with swollen chains deforms a flexible substrate.Transition to the globule state makes the substrate flat.Polymer brushes have attracted considerable scientific interest recently.One of the primary reasons for this interest is the advancement in synthesistechniques (for example Surface-initiated atom transfer radical polymeriza-tion (SI-ATRP)) which allow grafting of polymer chains on a variety ofsubstrate materials (organic, inorganic or polymeric) and shapes (micro andmacro planar substrates, spherical or cylindrical nanoparticles etc.) [33].To fabricate a brush, either polymer chains with reactive end groups aretethered to a surface or the chains are synthesized starting from the poly-merization initiators on the substrate surface. The former method is called‘grafting to’ method [34] and the latter ‘grafting from’ method [6, 35]. In3Note that the surface stress in this thesis refers to the stress resultant due to thedistributed stress within a brush and not to the stress at the interface between the brushand the solvent.51.2. Polymer brushboth these methods, chains are anchored to the grafting surface preferablyby a covalent bond as it is stable in varying solvent qualities [33]. The graft-ing to technique usually produces low graft density brushes. The graftingfrom method provides greater control over the synthesis and helps fabricatehigh density brushes with long polymer chains.A diagram summarizing various aspects of a polymer brush discussed inthis chapter is shown in Figure 1.5. The aspects of polymer brush utilizedin the thesis are shown in black text in colored circles.1.2.1 ApplicationsA wide range of applications exist for polymer brushes [33, 36, 37]. Sensingand actuation rely on the stress in a brush and its modification due to achange in the brush itself or the solvent [38, 39]. Application in lubricationrelies on its resistance to compressive forces and low friction between twocompressed polymer brush layers [40, 41]. The hydrophilicity of a brush andof its free surface controls its adhesion to foreign objects [37], enabling itsapplication in anti-fouling surfaces [42] and for cell culture [43]. We discusspertinent applications of a polymer brush below.Sensing and actuationStimulus sensitivity of a polymer brush facilitates its use in sensing and ac-tuation [38, 39]. Polymer brush grafted on one surface of a cantilever beaminduces deflection in the beam. If an analyte, present in the solution, bindswith the chains in the brush, then the deformation state of the cantileverbeam changes. Microcantilever coated with polymer brush has been usedin glucose sensing [44] and selective metallic ion sensing [45]. Similarly, bychanging the solvent quality in a controlled fashion, a cantilever with brushcoating can be used to perform mechanical work. Brush coated microcan-tilever has been utilized in actuation [46, 47]. Macroscale bending/stretchingactuators exhibiting large elastic deformations have also been reported [6],which hold promise for biomedical application, for example in catheters withactively deforming tip to access difficult to reach vessels inside human body.Surface lubricationA polymer brush bilayer – two surfaces, grafted with polymer brushes,pressed against each other – displays a very low friction and the osmoticpressure in the brushes oppose compressive forces on polymer brush bilay-ers [40, 41]. Nature uses this mechanism in synovial joint lubrication [28].61.2. Polymer brushPolymerbrushChargedPoly-electrolyteZwitterionicNeutralPolymerchainsHomo-polymerHetero-polymerRegularBlockRandomRealchainIdealchainFJCGaussianLangevinWLCPlanarbrushCurvedbrushCylindricalbrushSphericalbrushMono-dispersePoly-disperseSolventqualityLCSTpolymerUCSTpolymerScalingtheorySCFTSSTMD MCSensing/actuationSurfacelubricationOtherapplicationsChargeGeometryDispersityTypeModelsConstituentStimuliTheories SimulationApplicationsFigure 1.5: Principal aspects of a polymer brush are illustrated above. Top-ics of interest to this work are shown as black text inside coloured circles.For abbreviated terms, see list of abbreviations.71.3. MotivationNeutral polymer brush layers provide efficient lubrication (friction coeffi-cient < 10−2) at moderate velocity (∼ 500 nm/s) and loads (< 1 MPa)[40]. Small interpenetration between the brushes grafted on opposing sur-faces lead to low energy dissipation, resulting in low friction coefficient. Incomparison, a polyelectrolyte and a polyzwitterionic brush shows excellentwater-lubrication (friction coefficient ∼ 10−4). Osmotic pressure of counte-rion and the strong hydration of charged groups are thought to be responsi-ble for the lubrication [48, 49]. Dense neutral brushes have also been foundto show excellent lubrication (friction coefficient ∼ 10−4) owing to strongresistance of brushes present in opposing layers to mix with each other [50].Other chemical and biological applicationsPolymer brushes have also been used as a programmable material [51]. Theyfacilitate control of surface forces, and hence find use in the stabilization ofcolloidal dispersions [52, 53] and in self-assembly of brush grafted particles[54, 55]. They help control the adhesion and wetting properties of surfaces[37, 43]. They have also found use in smart drug delivery [56]. Increasingly,they are being used in biotechnology and nanotechnology [57].1.3 MotivationThe application of polymer brush in actuation and sensing crucially dependson chemomechanical stress in the brush, elasticity of the brush layer, andhow they are modified by a stimulus. So, enabling these applications requiresan understanding of the mechanical properties of the brush and how theydepend on the molecular parameters of a brush. Deflection of a cantileverbeam due to a polymer brush on one surface was shown to scale as δ ∼ρ13/6g [58] using the Daoud-Cotton model of curved brushes [59]. A scalingapproach to evaluate surface stress (τs) due to a polymer brush in differentsolvent conditions was presented in [23]. In a good solvent, τs ∼ Nρ11/6g .Surface modulus follows the same scaling as surface stress. In the case of apoor solvent, surface stress was found to be negligible. The scaling approach(details in Section 1.4.2) however, is based on incorrect assumptions aboutbrush structure, and does not give quantitative results. It does not providethe details of stress in a brush either. Furthermore, the results do not holdfor high graft density brushes.Spontaneous curvature and bending rigidity of polymer grafted mem-branes have been investigated extensively, both theoretically [60–63] as well81.4. Polymer theoriesas experimentally [64]. Strong stretching theory (SST) (see Section 1.4.2)was used to find bending modulus (B) of brush layer: B = 964(12pi2)1/3N3ρ7/3g[65]. This is the same scaling as obtained from the scaling arguments [60].Using Daoud-Cotton blob model [59], spontaneous curvature, κsp, was shownto scale as κsp ∼ N2ρ13/6g for 1/κsp >> h, where h is membrane thick-ness [62]. The bending modulus of the brush layer was shown to scale asB ∼ N3ρ5/2g . There is a small difference in scaling of bending modulus withrespect to ρg in [65] and [62] due to negligence of correlation effects in SST[62]. It was pointed out in [63] that Daoud-Cotton blob model does notcorrespond to a minimum in free energy of curved brush. But, no explicitscaling law was obtained. Numerical SCFT (see Section 1.4.2) has beenused to obtain dependence of bending rigidity of a membrane with denselygrafted brush (ρg = (0.1 − 0.9)/a2) in a good, θ and poor solvent recently[66]. Like previous analyses of a brush coated membrane, bending modulusis extracted from curvature dependent free energy expression. It has beenshown that B ∼ N3. However, no simple power law for dependence of B onρg was found. The work [66] does not show results corresponding to spon-taneous curvature (resulting from stress in a brush) of a polymer graftedmembrane.To conclude, scaling theory predicts slightly different scaling of bendingrigidity than SCFT. Recently, SCFT has been used to study bending rigidityof high density brushes. However, there is a lack of a systematic study ofstress and its variation in a brush and how it is modified by a stimulus.Understanding this aspect of polymer brush is one of the main goals of thisthesis. Our work borrows heavily from the theories of polymer physics.Hence, a brief review of relevant polymer theories is presented first, beforeformulating the research objectives. This is done to serve as a background tomechanics researchers entering into the field. Section 1.4.1 may be skippedby an expert.1.4 Polymer theoriesPolymer brushes are made of polymer chains. Polymer chains are a string ofconnected repeating units (slightly different from a monomer)4 of the sameor different kinds. At a small length scale, chemical properties of a polymerchain are determined by the chemical composition of monomers. However,at large length scales, its physical properties are determined by its topology4Repeating units differ from monomers as monomers are molecules, while repeatingunits are part of a polymer molecule.91.4. Polymer theoriesand only weakly affected by the chemical composition of monomers, givinguniversality to physical properties of soft matter composed of polymers [4,30]. Physical properties include mechanical properties, flow properties etc.They are of interest to polymer physicists and different models of polymerchains have been developed over the past decades. In the following section,a brief description of different types of polymer chains, models of linearpolymer chains, and theories of a polymer brush are presented.1.4.1 Polymer ChainPolymers are ubiquitous in our daily lives, present in plastics, fibers, food,medicine and in our own bodies (biomolecules such as DNA, proteins, mi-crotubules etc.). Polymer molecules are composed of ∼ 10 − 104 repeatingunits, each of a fraction of a nm size and 10− 106 Dalton5 mass. Contourlength of a polymer chain is 10 nm− 1 m (length of the largest known mi-crotubule [67]). They have a large number of internal degrees of freedomenabling us to design polymers with various properties and functionalities.However, this also leads to a wide range of length and time scales associatedwith their behavior, making it challenging to study them.From Figure 1.5, based on their composition, polymers are divided as ho-mopolymers (made of the same repeating units) and hetropolymers (madeof more than one type of repeating units). Depending upon the arrange-ment of different types of repeating units, heteropolymers can be a blockcopolymer (repeating units appear in blocks), regular copolymer(repeatingunits alternate regularly), or a random copolymer (random arrangement ofrepeating units). Based on topology, polymer chains can be categorized aslinear, comb, star, random, ring etc. The theoretical analysis in this thesis islimited to a linear homopolymer. Experiments involve random copolymers.Implications of this difference is discussed in Chapter 4.Based on the response of a solution of a polymer to an increase in tem-perature, polymers can be classified as polymers with Upper critical solutiontemperature (UCST polymers) or polymers with Lower critical solution tem-perature (LCST polymers) (see Figure 1.5). UCST polymers in a solutiontransition from a globule or collapsed phase to a coil or swollen phase onincreasing the temperature to a critical value. LCST polymers in a solutionare in swollen state at a lower temperature and forms a globule on increas-ing the temperature to a critical value. This transition is called coil-globuletransition. In the study of the effect of temperature on a brush, a LCST51 Dalton ≡ 1 g/mol101.4. Polymer theoriespolymer has been used, both in theory and experiment.Repeating units in a polymer chain are connected by covalent bonds,forming the backbone of a polymer chain. Since energy of a covalent bond islarge, stretching (or breaking) of a bond is difficult. However, polymer chainsare very flexible in bending. The flexibility arises from two main molecularmechanisms. In some polymers, torsional angle between consecutive bondscan vary as torsional rotation is allowed (free or hindered). This results in anoverall bending of a chain. Other polymers like DNA are relatively stiff andundergo continuous bending instead of bond rotation. Bending as well as noncovalent bonds such as Van der Waals bond, hydrogen bond etc. with whichnonbonded repeating units typically interact, compete with kBT . Hence,polymer chains at a finite temperature wiggle due to thermal fluctuations,and the correlation between tangents on two locations on a polymer chaindecrease exponentially with the distance between the locations. There existsa length scale, much smaller than the contour length of a polymer chain,over which tangents to a polymer chain are practically uncorrelated. Thislength is called Kuhn length of a polymer chain. Kuhn length of a flexiblepolymer chain is typically ∼ 1 nm. For semiflexible chains such as DNA, itis ∼ 50−100 nm. In the simplest model of a polymer chain, Kuhn segments(a segment of a polymer of Kuhn length) are assumed to be one dimensionalwith only nonzero length and noninteracting. This allows one to neglect self-avoidance between Kuhn segments. This model is known as the ideal chainmodel, taking inspiration from the ideal gas model in which gas particles areassumed to have zero volume and no interaction. This assumption facilitatesthe development of simple models to describe polymers. In a realistic model,each Kuhn segment is apportioned an excluded volume (v) and the model isknown as a real chain model. This model accounts for self-avoidance betweenKuhn segments. In the following, we briefly review both these models.Ideal ChainThough ideal chain model seems unrealistic, it describes a polymer chainin a θ-solvent, and in a linear polymer melt or concentrated solution6 verywell [67]. There are two primary models of an ideal chain: Freely jointedchain (FJC) model and Worm like chain (WLC) model. FJC is a discretemodel in that each Kuhn segment is considered as a rigid link and thedirection of any two links are uncorrelated. Typically, in polymer physics, aKuhn segment is called an effective monomer or simply a monomer. From6Interaction between two nonbonded Kuhn segments of a chain is screened by sur-rounding chains [4].111.4. Polymer theorieshereafter, we will follow this convention. FJC model describes the physicsof a flexible polymer chain. On the other hand, WLC is a continuous model,and describes a semi-flexible polymer. The two models are described below.Freely Jointed Chain (FJC)The simplest model to describe a polymer chain is the freely jointed chain(FJC) model (see Figure 1.6) [67]. The statistics of FJC is described by arandom walk model of constant step size. If we have a chain of N links,each of length a, then the mean end to end distance 〈R〉 = 〈∑Ni=1 ri〉 = 0,where ri is a vector of magnitude a along the ith link, R is chain end toend vector, and 〈·〉 represents phase average. But mean square end to enddistance is 〈R ·R〉 = 〈∑Ni=1 ri ·∑Nj=1 rj〉 = Na2. So, for a fixed a, the rootmean square end to end distance is not proportional to contour length (Na)of a chain, but to the square root of the contour length! Furthermore, theprobability P (R) of finding chain ends at a distance R = |R| (| · | denotesmagnitude of a vector) is:P (R) ∝ exp(− 3R22Na2). (1.1)This is a Gaussian distribution with zero mean and Na2/3 standard de-viation. Owing to this, this model is called Gaussian chain model. Thefree energy associated with a chain with ends fixed at a distance R isF (R) = 3R22Na2kBT . This free energy is of entropic origin as stretching endsof a chain leads to a decrease in the number of possible chain conformationsand hence a decrease in configuration entropy of a chain. The elasticityof rubber results from this entropic elasticity. The free energy reminds ofHooke’s law for a spring of zero length. If one performs a displacementcontrolled experiment by fixing the ends of a chain, then the mean forceexperienced at the fixed ends 〈p〉 = 3RNa2kBT .Gaussian chain model is adequate to describe force-extension relation foran FJC, if stretching is small (R . Na/3). However, it gives linear force-extension relation for any magnitude of extension. This is clearly incorrectas the length of a chain is finite. So, a force controlled model respectingfinite extensibility of a chain is used in which alignment of each link of achain accrues an energetic penalty if the link is not aligned with the directionof the applied force (P (ri) ∝ exp(p · ri/(kBT )), where p is force vector with|p| = p). Using the fact that alignment of each link is independent of theothers, the partition function is obtained which in turn is used to find the121.4. Polymer theoriesGibbs free energy and subsequently the force-extension relation given below.RNa= L(pakBT), L(·) := coth(·)− 1(·) , (1.2)where R is the mean end-to-end distance. Due to the Langevin functionpresent in the force-extension relation, the model is called a Langevin chainmodel. Force-extension relations of a Gaussian chain and a Langevin chainare compared in Figure 1.6. The relation for a Langevin chain shows diver-gence at higher extensions resulting in a “hardening” characteristic causedby finite extensibility of a chain. Note that in the small extension limit(R/(Na) . 1/3), the force-extension curve for Langevin chain and Gaus-sian chain are indistinguishable due to the fact that free energies in bothcases are the same in the long chain limit.RapRpContour length = NappContour length = Lt(s )1t(s )2sri0 0.2 0.4 0.6 0.8 1R/(Na)0246810pa/(kBT)GaussianLangevinWLC (lp =a/2, L =Na)Figure 1.6: FJC and WLC being stretched with a force p (left) and force-extension curves for an isolated Langevin chain, the corresponding isolatedGaussian chain, and WLC with lp = a/2 and L = Na (right). Note the force-extension divergence at higher stretching giving a “hardening” spring (WLCis the stiffest). For small extensions, the three curves are indistinguishable.Worm Like Chain (WLC)WLC, also called Kratky-Porod model, models a chain whose contour con-tinuously deviates from a straight line due to thermal fluctuations. So, thecorrelation between tangents at two points located at a distance s1 ands2 from a reference point along the chain contour decreases exponentially:〈t(s1) · t(s2)〉 = exp(−|s2 − s1|/lp), where lp is the persistence length of achain, and t(s1) and t(s2) are unit tangent vectors at s1 and s2. For a DNA131.4. Polymer theoriesmolecule lp ≈ 50 nm. Energy of a chain conformation with ends stretchedby a force p is∫ L0 (B(∂t(s)∂s )2 − p · t)ds, where the first term is the bendingenergy, and the second, the energetic penalty for a segment of a chain notaligned with the force direction. L is chain contour length. B is bendingrigidity and is ≈ 50 nm · kBT for a DNA molecule. By integrating over allthe conformations, partition function can be obtained which can be used tofind the Gibbs free energy and subsequently force-extension relation. How-ever, there is no analytical solution and Marco and Siggia’s approximation[68] is frequently used as the force-extension relation.plpkBT=RL+14 (1−R/L)2 −14(1.3)Real ChainIdeal chain model assumes that segments of a polymer chain do not interact.However, in reality two segments of a chain far apart along the length ofa chain may interact. Using, Flory mean field arguments, the monomernumber density in a volume occupied by a polymer chain with N effectivemonomers is ∼ N/R3, where R is end-to-end distance [30]. If the excludedvolume of a monomer is v, then the probability of having another monomerwithin the excluded volume of a given monomer is ∼ vN/R3. Energy cost ofeach interaction is kBT . So, interaction energy per chain (N monomers) is∼ kBTvN2/R3. Since chain ends are separated by a distance R, the entropicfree energy, assuming Gaussian chain statistics is ∼ kBTR2/(Na2). In Floryapproximation, the total free energy is the summation of the interaction andelastic free energies. By minimizing the total free energy with respect to R,one finds R ∼ v1/5a2/5N3/5 [30]. This simple model from Flory predicts theexponent of N accurately as confirmed by modern calculations such as usingrenormalization group theory (R ∼ N0.588). But that is merely a coincidenceas the overprediction of interaction free energy balances with overpredictionin elastic free energy to give accurate scaling of height [30]. Notice that in aGaussian chain R ∼ N1/2. So, a general universal relation R ∼ Nν can beused to relate the size of a chain with the number of monomers in a chain. νis called Flory exponent. For random walk conformation of an unperturbedchain (isolated chain), ν = 1/2 and for self avoiding random walk, ν ≈ 3/5in three-dimensions.To obtain the force-extension relation for a real chain, we consider areal chain stretched by a force p at its ends. Below a length scale l =kBT/p, chain feels no stretching and hence l ∼ ag3/5, where g is number141.4. Polymer theoriesof links. In a blobological description (details in Section 1.4.2), the blobassociated with this length scale is called tension blob or Pincus blob [69].Pincus blobs align along the force direction. Hence, the end-to-end distanceR ∼ l(N/g) where N/g is total number of Pincus blobs. Free energy ofstretching of a chain is of the order of kBT per blob. So, the total stretchingenergy is kBT (N/g) ∼ kBT (R/RF )5/2, where RF is the unstretched size ofa real chain RF ∼ aN3/5. From the free energy of stretching, we obtainthe force-extension relation p ∼ kBTRF(RRF)3/2. This force extension relationis nonlinear and is valid for RF << R << Na. Also, this scaling relationholds only for very long chains (N & 105) [70]. The force-extension relationfor different chains can be combined into the following form:p ∼ kBTRF(RRF) ν(1−ν). (1.4)See Appendix A for details.In the above, different chain models have been described with a focuson the size of polymer chains and the corresponding force-extension rela-tion. In the theoretical development in this thesis, FJC model for linearhomopolymers will be used.1.4.2 Polymer brush theoriesFor several decades, polymer brushes have been a system of immense in-terest to polymer physicists. Multiple, often complementary, theoreticalapproaches have emerged to relate the macroscopic brush properties such asbrush height (H) to molecular scale parameters such as effective monomersize (a), number of monomers in a chain (N), graft density (ρg) etc. of abrush. A detailed comparative review can be found in [71–73]. Here, a briefreview of the theories is presented to serve as a useful background to thisstudy.Scaling theoryWe first look at the scaling picture for a brush developed by Alexander [31]using Flory like mean field arguments. This model assumes that all thechains in a brush are uniformly stretched with their ends reaching the freeboundary of the brush. Also, monomer density profile in a brush is step-like with a constant value of the monomer density throughout the brush.Furthermore, it assumes polymer chains to be ideal chains. On assuming151.4. Polymer theoriesbrush height to be H, monomer density equals φ = Nρg/H. We can writethe free energy of a polymer chain in a brush as a sum of the contributionsfrom the chain stretching (the first term in the equation below) and thepolymer solvent interaction (the remaining terms) as follows:Fstep =(3H22Na2+(12vφ2 +16wφ3)Hρg)kbT, (1.5)where H/ρg is the volume occupied by one polymer chain, and v and ware binary and ternary interaction parameters, respectively. By minimizingFstep with respect to the height of the brush (H) (dFstep/dH = 0), one ob-tains the expression for the height of the brush. In a good solvent condition,binary interaction (the term with φ2 in (1.5)) dominates and contributionfrom ternary interactions (the term with φ3 in (1.5)) can be ignored [74].Under this condition,H =(16)1/3v1/3ρ1/3g a2/3N. (1.6)Now, let us consider the blob model. The powerful and yet simple andintuitive concept of blobs was introduced by Daoud and deGennes [30, 75]using the analogy between polymers and critical phenomena [76, 77]. Inblobology [71], different interaction energies or different boundary conditionsin a polymer system define corresponding blobs of different sizes. The blobsize equals the size of a chain segment for which the interaction energy equalskBT . A polymer chain can be viewed as a string of blobs. Chain segmentswithin a blob behave as a single unperturbed (isolated) chain performing arandom walk or a self avoiding random walk.In a polymer brush, there are two blobs of interest. One is correlationblob (same as screening blob), whose size is determined by the mean dis-tance between the grafting points and is given by ζc = ρ−1/2g . Screeningblob determines that at a length scale larger than the size of this blob,self avoidance between segments of a single chain is screened by the neigh-bouring chains and the chain behaves as an ideal chain [71]. The other isthermal blob, whose size is dependent on the solvent quality (ζT =a|2χ−1| ,where χ is Flory-Huggins parameter, defining solvent quality) [23, 67]. Inan athermal solvent (χ = 0), ζT = a. Links of a chain within a thermal blobfollow Gaussian statistics (random walk), thermal blobs within a screeningblob perform self avoided random walk and screening blobs again performrandom walk [23]. Correlation blob per polymer chain is N/(gcgT ), where161.4. Polymer theoriesgc (= (ζc/ζT )1/ν) and gT (= (ζT /a)2) are number of thermal blobs per cor-relation blob and number of links per thermal blob, respectively. Brushheight H = NgcgT ζc = Na1/νρ(1−ν)/(2ν)g |2χ − 1|2−1/ν . In a good solvent,H = Na5/3ρ1/3g . Scaling of height with respect to graft density is the sameas in Alexander’s scaling theory. See Appendix B for a comparison of brushheight obtained from different theories.Using the height expression obtained above, we now calculate free energyof a brush. Stretching is assumed to happen only at the level of correla-tion blobs. Below this size thermal blobs and links within a thermal blobare unstretched. The free energy of stretching of a chain in the brush is32(HRF)2kBT , where RF = (N/(gT gc))1/2ζc is unstretched size of an idealchain of correlation blobs. Interaction energy per chain equalsN/(gT gc)kBT .Adding them together gives F = 52Nρ1/(2νb4/ν−6)g v2−1/ν . In a good solvent,F = 5/2ρ5/6g a2/3v1/3. Notice that in comparison to Alexander’s scaling,scaling of free energy with respect to graft density is different. In Florylike mean field calculation, stretching as well as interaction free energies areoverestimated as the effect of self avoidance is neglected [30].The drawback of blobology is that it lacks a systematic approach todefine blobs and relies on intuition, and only provides asymptotic scalingrelations without prefactors [71]. In the case of brushes there are furtherassumptions that all the chains are equally stretched, ends of all the chainsstraddle the free end of brush and monomer density is uniform through thebrush.Self consistent field theory (SCFT)SCFT [78], also called mean field theory, is one of the most successful poly-mer theories describing the equilibrium behavior of polymeric systems [79].In SCFT, a given polymer chain (typically modeled as a random walk) is as-sumed to lie in an effective mean potential field which is dependent on localmonomer density, thus accounting for the influence of neighboring chains.Fluctuations in the interaction field of a chain with the surrounding chainsis ignored. Conformations of the given chain are affected by the mean field.Since conformations of the given chain affect the monomer density and con-sequently the mean field, a set of self consistent equations are obtained.Solving the SCF equations numerically yields the local minimum free en-ergy. Both, continuum [80] as well as lattice [81–83] formulations exist tosolve SCFT equations numerically.A departure from numerical calculation is the recognition that polymer171.4. Polymer theorieschains follow lowest energy path, also called classical paths7 [84], providedthe brush is strongly stretched. In this strong stretching regime, the clas-sical paths of polymer chains dominate the partition function of the brushand fluctuations from these paths can be ignored [84]. This crucial insightallowed the development of the so called strong stretching theory (SST) forbrushes [65, 74, 85]. SST allows analytical calculation of brush properties.Strong stretching theory (SST)SST is an analytical theory and can be formulated as an energy minimizationproblem [74]. Free energy of a polymer brush covering unit surface area ofa substrate can be written as:F = Fint + Fel=∫ H0fint(φ(z))dz+∫ H0fel(z)dz (1.7)where Fint is free energy contribution from interaction among monomersand Fel is contribution from extension of chains in a brush. fint(φ(z)) is freeenergy density per unit volume at height z. fel(z) is free energy density dueto elastic stretching of chains at height z.Based on the Flory-Huggins (FH) solution theory8, the most encounteredtheory of polymer solutions, the mixing free energy density is given by:fint(Vf ) ≈ kBTa3[(1− Vf ) log(1− Vf ) + χVf (1− Vf )], for N →∞, (1.8)where Vf is the volume fraction of the polymer in a solution, χ is Flory-Huggins parameter, and a3 is the volume of an effective monomer. Monomerdensity φ = Vf/a3. It is often expanded and approximated to the followingform:fint(φ) ≈ kBT[12vφ2 +16wφ3]. (1.9)7“The classical polymer path is defined as the path which minimizes the free energy, fora given start and end positions, and thus corresponds to the most likely path a polymercan take. The name follows from the analogy with quantum mechanics, where the classicalmotion of a particle is given by the quantum path with maximal probability. Since forstrongly stretched polymers the fluctuations around the classical path are weak, it isexpected that a theory that takes into account only classical paths, is a good approximationin the strong-stretching limit.” [72]8Note that Flory-Rehner theory, often encountered in hydrogel literature, is used forpolymers with crosslinking, which is not the case here.181.4. Polymer theoriesIn a moderately dense brush (ρga2 . 0.1) in a good solvent, binary interac-tion dominates and the term with φ3 can be neglected, and in a θ-solvent,v = 0. Also,fel(z) = kBT∫ Hzg(ζ)E(z, ζ)dζ, (1.10)where g(ζ) is density of chain ends (∫ H0 g(ζ)dζ = ρg), and E(z, ζ) is localstretching in a chain at height z if the chain end is located at ζ. E(z, ζ) = dzdn ,the ratio of the end extension (dz) of a chain with dn monomers. To findthe equilibrium structure of the brush, functional F (see (1.7)) needs to beminimized and the expressions for functions φ(z), g(z) and E(z, ζ) needs tobe found using the following self consistency condition [74]:φ(z) =∫ Hzg(ζ)E(z, ζ)dζ, (1.11)and the normalization conditions,∫ ζ01E(z, ζ)dz = N,∫ H0φ(z)dz = ρgN. (1.12)If the mean potential field is V (z), exchange chemical potential µ(φ) satisfiesthe following relation:V (H)− V (z) = µ(φ(z))− µ(φ(H)). (1.13)This is an important equation relating mean field with chemical potentialwithin a brush and is used in the subsequent chapters to find monomer den-sity. In a brush with Gaussian chains, V (z) = 3pi28N2a2z2. In a good solvent,when the binary interaction is prominent, µ(φ) = vφ. Then the above rela-tion leads to a parabolic monomer density profile [65, 74]. SST and scalingtheory differ in their prediction of the structure of the polymer brush, par-ticularly in terms of the chain end distribution throughout the height of thebrush and the monomer density profile. However, SCFT calculations havebeen shown to match with SST for graft densities ρg . 0.1/a2 [83]. Also,the parabolic monomer density profile is confirmed to be correct by rigorousMolecular dynamics (MD) simulation studies [86–88] and by experiments[89, 90]. Local deviations are recognized at the grafted and free ends of abrush due to a depletion layer and a tail, respectively. Furthermore, thechain ends are assumed to be stretch free in SST, but they have been ob-served to undergo different end-stretching depending on their location fromthe grafting surface [91], When the free ends of the chain are far away from191.4. Polymer theoriesthe grafting surface, they point away from the surface, while those closeto the surface point toward the surface, as observed in Monte Carlo (MC)and MD simulations, however mean end stretching is found to be zero asassumed in SST [91, 92]. Also, since SST does not capture the excludedvolume correlations that occur in the limit of strong excluded volume inter-actions, its applicability is limited to brushes with weak excluded volumeinteractions (v2 << ρga8 << 1) [65, 93].A further refinement of SST theory is made by accounting for the force-extension divergence [94] using Langevin chain elasticity in [95]. A (semi)analytical procedure emerges which can predict monomer density profilesover a range of graft densities, which smoothly bridge the parabolic andnear-step profiles.The blob theory accounts for self-avoiding random walks of individualpolymer chains, but it still has the step profile ansatz. SCFT provides detailsof brush structure, however self-avoidance is neglected and polymer chainsare assumed to be ideal. But at length scales smaller than the screeninglength, this is incorrect, particularly in a good solvent condition. An attemptto improve SCFT has been made in Single chain mean field theory (SCMF)[96–98]. In SCMF, interactions of a single chain are fully accounted, butinteraction of the chain with the neighboring chains and surrounding solventmolecules is accounted within a mean field approximation [73]. The resultsfrom SCMF are found to be in good agreement with molecular dynamicssimulation [96]. However, typically, SCMF has been used to study smallerchains (N ≤ 100) [73]. See Table 1.1 for a comparison of theories.Table 1.1: The table below presents a comparison of scaling theory, SCFT,SST, and SCMF.Feature Scaling theory SCFT SST SCMFChain elasticity Real chain Ideal chain Ideal chain Real chainSelf-avoidance Considered Neglected Neglected ConsideredChain configurations All All Classical AllCalculation Analytical Numerical Analytical NumericalThe methods described earlier involve multiple approximations: Scal-ing theory accounts for self-avoidance but assumes step-profile for monomerdensity variation; SCFT assumes ideal chain conformation and neglects fluc-tuations in the interaction field; SST goes a step further and neglects all thechain conformations except the one with the minimum energy. SCMF alsoneglects fluctuation in the interaction field of a chain. In contrast, molecu-201.4. Polymer theorieslar simulations model complex multi-body interactions in full detail. Thisallows one to check the validity of the assumptions as well obtain guidancefor systematic development of better models. In the following section, webriefly review molecular simulations of polymer brushes.1.4.3 Molecular simulationMolecular simulations [92], MD [99–101] simulation [86–88, 102] as well asMC simulations [103–106] have been used extensively to study the polymerbrushes. They have validated predictions from SCFT and SST as well aspointed to their limitations [86, 87]. However, it remains a challenge toobtain quantitative predictions from molecular simulation of polymeric sys-tems due to large spread of length and time scales associated with polymersystems, and a dearth of reliable models of interaction among constituents(interaction potentials). Also, a detailed model requires much more com-putational resources. So, often coarse-grained models are used to studymesoscopic properties of brush.A bead-spring model of a polymer chain was pioneered in [107]. In thismodel, each polymer chain is represented by a series of connected beads.The beads represent effective monomers. The interaction between bondedbeads, and nonbonded beads are governed by different potentials in the MDsimulation. Typically, Lennard-Jones (LJ) potential with appropriate cut-off is used for nonbonded interaction. Weeks-Chandler-Andersen (WCA)potential, which combines Finitely extensible nonlinear elastic (FENE) po-tential with LJ potential, is used for bond interaction. The parameters of thepotentials are chosen to ensure that essential physics, such as connectivityalong a chain, finite extensibility of a bond, noncrossability of chains etc. isnot violated by the model. The model was used to simulate polymer brushesin different solvent conditions [87, 88]. MD studies of moderate and highgraft density brushes have been reported in [102]. A comparison betweenstatic properties of a brush in a good solvent obtained from different modelscan be found in [93]. In this work, bead-spring model for polymer chains isused to obtain stresses. The behavior of polymer brushes of different graftdensities is investigated, and the simulation results are used to assess theo-ries. A limitation of MD simulations is that to achieve strong stretching ofchains in order to be able to compare MD with SST, one requires a largenumber of beads per chain. Also, this number increases considerably forlow graft densities. Unfortunately, this makes it exceedingly expensive toequilibrate the system, and one seeks a reasonable trade off between accu-racy and computational efficiency. We will see in Chapter 3 that this has211.5. Research objectives and methodologyimplications in the prediction of mechanical stress. For completeness, Ta-ble 1.2 compares the main assumptions and features of Gaussian chain SST(SST-G), Langevin chain SST (SST-L) and MD.Table 1.2: A comparison of the assumptions and limitations of SST andMD.Feature SST-G SST-L MDChain elasticity Gaussian chain Langevin chain No limitationStretching Infinite Infinite FiniteSelf-avoidance Ideal chain Ideal chain ConsideredVirial truncation Binary No truncation Not applicableChain paths Classical Classical All pathsChain end Force free Force free UnconstrainedComputational cost None Small Very high1.5 Research objectives and methodologyThis thesis aims to understand stress in a polymer brush, how it is affectedby stimulus, and the deflection of a thin flexible substrate with one surfacegrafted with polymer brush. The objectives of the thesis are outlined below:• To understand the stress and its distribution in polymer brushes of awide range of graft density,• To study the effect of stimuli on resultant surface stress due to a poly-mer brush, and• To study the deflection of a flexible beam due to polymer brush graftedto its surface.To understand the stress in a brush, SST, with Gaussian chain assump-tion (SST-G) as well as Langevin chain assumption (SST-L), has been em-ployed. The effect of solvent has also been considered within SST formalism.Semi-analytical SST calculations were performed in MATLAB. Results fromSST in athermal solvent case have been validated using MD simulation of apolymer brush with bead-spring chains. MD simulation was performed us-ing Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS)program [108]. To study the mechanics of deflection of a beam due to poly-mer brush grafting, an analytical model is developed.221.6. ScopeSurface stress (and its variation with temperature) due to a thermore-sponsive brush grafted to a flexible substrate is also estimated experimen-tally. Experiments reported in the thesis were performed using a ther-moresponsive random copolymer brush, Poly(N- isopropylacrylamide)-co-Poly(N,N-dimethylacrylamide) (PNIPAm-co-PDMA) brush, grafted on oneside of a plasticized poly(vinyl chloride) (pPVC) thin beam.1.6 ScopeThis thesis is limited to neutral flexible homopolymer brushes grafted to aflat substrate and in solvents of varying quality. Spherical and cylindricalbrushes, or brushes with charge have not been considered. The polymerchains in the brush are assumed to be flexible homopolymers in the the-oretical development as well as in MD simulation. Furthermore, polymerbrushes are assumed to be monodisperse. The experimental results thoughare obtained from brushes of random copolymers composed of polydispersechains. Also, we only consider equilibrium state of the brush and dynamicsdoes not form part of this work.We have used SST [65, 74, 95, 109] for planar brushes in the derivationof stress in a brush. The substrate on which a brush is grafted has beenassumed to be rigid allowing the use of SST results for brushes. For thepolymer chains in the brush, freely jointed chain model has been used. Forsmall graft densities, Gaussian chain model is appropriate, whereas for highdensity brushes, Langevin chain model is employed.Molecular simulation is based on bead-spring model of polymer chainsin the brush. We perform an NVT simulation with Langevin thermostat inwhich each bead is assumed to be connected to a thermal bath. The MDstudy is confined to athermal solvent condition.In the analytical model of deflection of a beam due to polymer brushcoating, it is assumed that thickness of the beam as well as the radius ofcurvature of the deflected beam is more than two orders of magnitude higherthan the brush height. This allows us to use the results for a planar polymerbrush in the case of deflected beam as well.1.7 Thesis outlineThe remainder of the thesis is organized as follows:In chapter 2, SST is combined with a continuum mechanics model toobtain the residual stress variation in a brush. A closed form expression for231.7. Thesis outlinesurface stress and surface elasticity of the brush layer will be derived. Inthe second part of the chapter, an analytical model of a beam with polymerbrush grafted on top surface will be developed. The generalized continuumbeam model accounts for the Young-Laplace and Ogden-Steigman curvatureelasticity correction terms, and yields a surface stress-curvature relation,which contains existing relations in the literature as special cases.In chapter 3, stress in a polymer brush with Langevin chain will bediscussed. The Langevin chain assumption allows calculation of stress in abrush over a wide range of graft density and matches the results from thefirst chapter based on Gaussian chain for low graft density. Also, in thederivation, modified Carnahan-Starling equation of state is used instead ofFlory-Huggins solution theory to account for interaction among monomers.In the second part of the chapter, MD model will be discussed and monomernumber density, end density and stress in a brush of varying graft densityobtained from SST and MD will be compared. Quartic variation of stressobtained in chapter 2 will be validated by MD simulation.Chapter 4 contains SST results for the effect of stimuli on the structure ofa brush and surface stress due to a brush. The derivations assume Langevinchain model for the chains in a brush. Effect of vertical phase separationin an LCST polymer on surface stress will be studied. Finally, experimen-tal measurement of surface stress due to a thermo-responsive random co-polymer brush grafted to a flexible substrate will be presented. Molecularparameters such as graft density, molecular weight of the chains in the brushand polydispersity index has also been estimated. The experimental findingsin light of the theoretical results will be discussed.In the last chapter, conclusions of the thesis are presented. The limita-tions of the work and the future directions are discussed.24Chapter 2Mechanics of a polymerbrushAs mentioned in introduction, brushes grafted onto a substrate have residualstress which can be used to produce controllable and reversible bending de-formation of the flexible host substrates. This allows use of polymer brush insensing and actuation. To understand such systems and improve their func-tional properties, we study the stress distribution in a brush, and developsurface stress-curvature relation for an elastic beam of a soft material graftedwith a polymer brush, in this chapter. Stress in a polymer brush graftedto a rigid substrate is derived by combining SST-G for brushes [65, 74] ingood and θ- solvents with a continuum mechanics model. It is shown thatthe residual stress variation in a brush is a quartic function of the distancefrom the grafting surface, with the maximum stress occurring at the graftedsurface. By idealizing the brush as a continuum elastic surface layer witha residual stress, we derive a closed form expression for surface stress andthe surface elasticity of the layer as a function of brush parameters, suchas graft density and molecular weight in Section 2.1. From free energycomparisons, it is shown that a polymer brush can produce large bendingdeformation in a flexible substrate. Then, an analytical model for a thinflexible beam coated with a polymer brush layer on its top surface is devel-oped in Section 2.2, using virtual work principle. The model incorporatesYoung-Laplace and Steigmann-Ogden curvature elasticity corrections. Thesurface stress-curvature relation developed in Section 2.2 is used to estimatesurface stress experimentally in Chapter 4. The chapter ends with conclud-ing remarks in Section 2.3.2.1 Stress in a polymer brush using SSTConsider the schematic of a planar polymer brush shown in Figure 2.1. Thesubstrate is assumed to be rigid. This allows use of SST-G results fromthe literature as they also assume rigid substrate. This is justified, since252.1. Stress in a polymer brush using SSTSubstratedzdDxzHFigure 2.1: A schematic showing a side view of a planar polymer brush ofheight H, effective monomer diameter d, effective length of a monomer aand contour length of a polymer chain Na. Typically, d = a, as assumed inthis chapter as well. Inverse square root of graft density equals the averagedistance between grafting points in the brush (〈D〉 = ρ−1/2g ). A thin layerat height z is also shown which we frequently refer to in this as well as thenext chapters.using this approach, we can calculate the stress in a brush as a functionof the molecular properties of the brush. If a brush is grafted to a flexiblesubstrate, the substrate deforms due to the stress in the brush. The de-formation changes only the mean spacing (〈D〉) and hence graft density ofthe brush. This is based on the assumption that the brush remains planarafter the deformation and any other substrate-brush coupling such as inter-action between substrate surface and monomers in the brush is negligible.Now, since only graft density is changing, we can obtain the brush proper-ties for the new graft density using the relations we will obtain. The brush,as mentioned in introduction, is neutral and monodisperse, made of linearhomopolymers. Also, SST-G assumes the polymer chains in the brush to beGaussian chains.To derive the stress in a brush, we start by writing the free energy ofthe brush. It is the sum of interaction free energy Fint and chain stretchingfree energy Fel. Interaction free energy is a function of monomer numberdensity φ(z) and two functions are required to evaluate it: g(ζ), quantifyingthe number of chains per unit substrate area ending at height ζ (∫ H0 g(ζ)dζ =ρg), and E(z, ζ), local stretching at z in polymer chains with ends at height262.1. Stress in a polymer brush using SSTζ (z ≤ ζ ≤ H), as stretching in a chain is not uniform along the length. Freeenergy of the brush can then be expressed as:F = Fint + Fel=12kBT∫ H0vφ2(z)dz+16kBT∫ H0wφ3(z)dz+12kBTβ∫ H0∫ Hzg(ζ)E(z, ζ)dζdz, β =3pla2, pl =lka, (2.1)where v and w, respectively, are the binary and ternary interaction param-eters. The form of interaction free energy is based on FH solution theory,and the effect of solvent is accounted for implicitly in this theory. Here, lk isKuhn length of polymer and pl is asymmetry parameter, denoting numberof monomers in a Kuhn segment. pl = 1 for a flexible polymer chain andpl > 1 for a semiflexible chain. β has no special physical meaning and isused only to simplify the free energy expression.Monomer number density φ(z) is related to g(ζ) and E(z, ζ) by theself consistency condition (1.11). Also, E(z, ζ) and φ(z) have to satisfynormalization conditions (1.12). Minimization of free energy in (2.1) withrespect to g(ζ) and E(z, ζ), under the constraints (1.11) and (1.12) yields theequilibrium properties of the brush [74]. Considerable simplification arisesunder good solvent conditions for moderate graft density brushes (ρg . 0.1)(the second term containing w in (2.1) is negligble) and θ-solvent (v =0) conditions. Quality of the solvent can be varied by varying v. In thefollowing, we derive stress expressions for these two cases.2.1.1 Good solventIn a moderately dense brush (ρga2 . 0.1) in a good solvent, binary inter-action dominates, and contribution of higher order interactions to the freeenergy density (w term in (2.1)) can be ignored, leading to the followingsimplified expression:F = Fint + Fel≈ 12kBT∫ H0vφ2(z)dz+12kBTβ∫ H0∫ Hzg(ζ)E(z, ζ)dζdz. (2.2)Minimization of free energy in this case yields analytical expressions for theunknown functions φ(z), g(ζ) and E(z, ζ), as well as H [65, 74] as given272.1. Stress in a polymer brush using SSTbelow:φ(z) =pi2β8N2v(H2 − z2) , (2.3)H =(12pi2)1/3 v1/3ρ1/3g Nβ1/3, (2.4)g(ζ) = γζ√H2 − ζ2, γ = pi2β4N3v, (2.5)E(z, ζ) =pi2N√ζ2 − z2. (2.6)Free energy distribution obtained from SST for a planar brush is alsononuniform and it shows a variation through height of the brush. By cal-culating the change in the free energy density in a brush due to a uniformuniaxial strain xx applied to the brush, we can obtain distribution of normalstress σxx within the brush.In order to calculate variation of σxx in z-direction, we consider an in-finitesimally thin layer of brush of unit horizontal area and of small thicknessdz at height z above the grafted surface (see Figure 2.1). Free energy of thisinfinitesimal layer is given by:∆F = ∆Fint + ∆Fel = f(z)dz,f(z) =[(12vφ2(z) +12β∫ Hzg(ζ)E(z, ζ)dζ)kBT], (2.7)where f(z) is free energy density at height z. Brush has a residual stressσij, i, j=x, y, z. For an infinitesimal strain ij experienced by the thinlayer when an infinitesimal strain xx is applied to the substrate,∆(f(z)V0) = σijijV0 + µ∆N, (2.8)where V0 is the initial volume of the thin layer; µ is the chemical potential ofsolvent in the layer and ∆N is the change in the number of solvent moleculesin the thin layer due to the application of strain.We assume that σzz = 0 and σxz = σxz = 0. This assumption is reason-able since no normal or shear force along z-direction is applied on the brush.The assumption is also tested using MD simulation in the next chapter. As-suming plane strain condition in y-direction, (2.8) yields:∆(f(z)V0)V0= σxxxx +µ∆NV0. (2.9)282.1. Stress in a polymer brush using SSTAs the polymer molecules and the solvent are incompressible,∆VV0= xx + zz =Ω∆NV0, (2.10)∆NV0=xx + zzΩ, (2.11)where Ω is the volume of a solvent molecule. Substituting (2.11) in (2.9)and diving by xx yields,1V0∆(f(z)V0)xx=(σxx +µΩ)+µΩzzxx. (2.12)On expanding the left hand side of the above equation, we obtain:∆f(z)xx+ f(z)1xx(∆VV0)=(σxx +µΩ)+µΩzzxx. (2.13)By using (2.10) in the above, and for xx approaching 0, we get,∂f(z)∂xx+ f(z)(1 +∂zz∂xx)=(σxx +µΩ)+µΩ∂zz∂xx. (2.14)As the solvent molecules within and outside the brush are in chemical equi-librium, the movement of solvent particles does not contribute to a changein the free energy (µ = 0). Substituting this in the above, we obtain theexpression for residual stress in x-direction.σxx =∂f(z)∂xx+ f(z)(1 +∂zz∂xx). (2.15)The above equation is valid for any form of f(z) and can be used for SSTwith interactions of order higher than binary. Note that the expression ∂zz∂xxrepresents the ratio of the change in thickness of the thin layer at z due tothe applied strain xx. Though it appears like negative of local poisson’sratio, it should not be interpreted as such. The reason is that the materialin the thin layer is not the same throughout as fluid molecules are allowedto move in and out of the thin layer when strain xx is applied.To evaluate the above expression, we need to find ∂f(z)∂xx and∂zz∂xx. Toevaluate ∂zz∂xx , we assume that the monomers in the layer of thickness dz inthe initial configuration remain in the layer after strain xx is applied on thesubstrate. However, the layer displaces by u in z-direction and volume ofthe layer also changes due to the movement of solvent molecules in and out292.1. Stress in a polymer brush using SSTof the layer (polymer and solvent are incompressible however) as the brushreaches a new equilibrium. So, volume occupied by monomers in the layerremains unchanged but monomer density (φ(z)) changes due to a change inthe volume of the layer. Hence,∆(φ(z)V0) =1V0∆(φ(z)V0)xx= 0. (2.16)On expanding the above as in (2.13) and using (2.10), and for xx approach-ing 0, we get,∂φ(z)∂xx+ φ(z)(1 +∂zz∂xx)= 0. (2.17)Using (2.3) in the above, we obtain:∂φ(z)∂xx=pi2β4N2v(H∂H∂xx− z ∂u∂xx). (2.18)We also notice that∂zz∂xx=∂∂xx(∂u∂z)=∂∂z(∂u∂xx). (2.19)Using (2.18) and (2.19) in (2.17):pi2β4N2v(H∂H∂xx− z ∂u∂xx)+ φ(z)(1 +∂∂z(∂u∂xx))= 0. (2.20)We can use (2.4) to evaluate ∂H∂xx . Note that xx changes H by changing graftdensity ρg. Due to strain xx, surface area changes (Adeformed = A(1+xx)).Since graft density is the number of chains grafted to a unit surface area,the change in surface area results in a change in graft density, ρdeformedg ≈ρg(1− xx). Hence,∂ρg∂xx= −ρg,∂H∂xx= −13H. (2.21)Substituting the above in (2.20) yields:pi2β4N2v(−13H2 − z ∂u∂xx)+ φ(z)(1 +∂∂z(∂u∂xx))= 0. (2.22)302.1. Stress in a polymer brush using SSTAs φ(z) is a quadratic function in z (see (2.3)), we assume that ∂u∂xx is apolynomial in z, ∂u∂xx =∑i=0anzn. Substituting this in the above equationand recognizing that(∂u∂xx)z=H= ∂H∂xx = −13H, we obtain,∂u∂xx= −13z. (2.23)Substituting the above in (2.18) and (2.19) gives:∂φ(z)∂xx= −23φ(z), (2.24)∂zz∂xx= −13. (2.25)Now, we evaluate ∂f(z)∂xx using the expression in (2.7).∂f(z)∂xx=vφ(z)∂φ(z)∂xxkBT+12kBTβ(g(H)E(z, H)∂H∂xx− g(z)E(z, z) ∂u∂xx)+12kBTβ(∫ Hzg(ζ)∂E(z, ζ)∂xxdζ +∫ Hz∂g(ζ)∂xxE(z, ζ)dζ). (2.26)From (2.5) and (2.6), g(H) = 0 and E(z, z) = 0, and∂g(ζ)∂xx=γζH√H2 − ζ2∂H∂xx= − γζH23√H2 − ζ2 ,∂E(z, ζ)∂xx= − pi2Nz√ζ2 − z2∂u∂xx=pi6Nz2√ζ2 − z2 . (2.27)g(H) = 0 means that the density of free ends at H is zero. E(z, z) = 0means that the stretch in a polymer chain at its free end is zero.The expression for σxx in (2.15) transforms to the following on substi-tuting the relations in (2.24), (2.25) and (2.26).σxx =− 13vφ2(z)kBT+12kBTβ∫ Hzg(ζ)E(z, ζ)dζ+12kBTβ(∫ Hzg(ζ)∂E(z, ζ)∂xxdζ +∫ Hz∂g(ζ)∂xxE(z, ζ)dζ). (2.28)312.1. Stress in a polymer brush using SSTIn the above expression, the first term involving v results from nonbondedinteraction. The remaining terms result from polymer chain stretching. Sub-stituting (2.5), (2.6) and (2.27) in the above and carrying out the integrationyields:σxx = −pi4β2kBT192vN4(H2 − z2)2︸ ︷︷ ︸Nonbonded interaction−pi4β2kBT384vN4(H2 − z2)2︸ ︷︷ ︸Chain stretching= −pi4β2kBT128vN4(H2 − z2)2 = −98(pi212)2/3v1/3ρ4/3g β2/3(1−( zH)2)2kBT.(2.29)We observe that the nonbonded interactions contribute twice as much asthe chain stretching in the expression for stress in a brush layer in a goodsolvent. Stress shows strong dependence on graft density. Its dependence onchain length is through brush height H. Also, stress goes to zero at z = Hsmoothly (with zero slope and curvature) compared to monomer densityφ(z) (see (2.3)).It should be observed that since a plane shear strain (xy) at the surfaceof a substrate does not cause any change in surface area and consequentlyin graft density and free energy of a brush, plane residual shear stress (σxy)and the associated shear modulus of the brush equal zero.2.1.2 θ-solventIn a θ-solvent v = 0, and monomer-monomer interaction is governed byternary interaction parameter w. Free energy density in a brush at a heightz is given by [74]:f(z) =(16wφ3(z) +12β∫ Hzg(ζ)E(z, ζ)dζ)kBT, (2.30)andφ(z) =pi2N√βw√H2 − z2, (2.31)H =2pi(4wβ)1/4ρ1/2g N, (2.32)g(ζ) =pi24N2√βwζ, (2.33)E(z, ζ) =pi2N√ζ2 − z2. (2.34)322.1. Stress in a polymer brush using SSTNotice that monomer density profile is not parabolic. Also, height of thebrush varies as square root of graft density. The number density of chainswith end at height ζ is an increasing function and it assumes the maximumvalue at height H. The local stretch in a chain (E(z, ζ)) remains the same asin a chain in a good solvent. Following the same method as in Section 2.1.1,stress distribution in brush can be shown as:σxx =43/43β3/4w1/4ρ3/2g(1−( zH)2)3/2kBT, (2.35)wherein nonbonded interaction and chain stretching contribute equally inthe expression for stress. Note that stress variation is no longer quarticwith respect to the distance from the grafting surface and the dependenceof stress on graft density becomes stronger. Dependence of stress on chainlength is again through the brush height H.2.1.3 Surface stress and surface modulusSubstrate Polymer chains Elastic surface layer with residual stress Spring: FENE potential Truncated and Shifted 𝑉𝐿𝐽 Figure 2.2: A brush layer can be construed as an elastic surface layer withresidual stress at the top of a beam.To understand the mechanics of a beam grafted with polymer brush, thebrush layer can be idealized as an elastic surface layer with residual stress.We call the resultant residual stress in the brush layer as surface stress inthis thesis. Note that this surface stress is not the stress at the interface ofthe brush and solvent. The surface stress due to the brush layer is obtainedby the summation of stress in all the layers in a brush. In a good solvent,τs =∫ H0σxxdZ = −(35(pi212)1/3v2/3ρ5/3g β1/3N)kBT. (2.36)We observe that τs has the same value for normal surface stress in anydirection along the surface. Notice that τs has stronger dependence on graftdensity compared to number of monomers in a polymer chain.332.1. Stress in a polymer brush using SSTTo obtain surface elastic modulus, we need to determine the change insurface stress as a brush coated substrate is stretched. Stretching of sub-strate causes change in graft density, which leads to a change in surfacestress. Tangent surface elastic modulus is obtained using the following rela-tion:Es =∂τs∂xx=((pi212)1/3v2/3ρ5/3g β1/3N)kBT, (2.37)where we have used ρdeformedg = ρg(1− xx). We observe that the scaling ofthe surface stress and the surface modulus are the same. Their magnitudesare also of the same order.We can repeat the above calculation for the case of a θ-solvent to obtain:τs = −(12piw1/2ρ2gβ1/2N)kBT, (2.38)Es =(1piw1/2ρ2gβ1/2N)kBT. (2.39)Table 2.1 compares surface stress expressions obtained from differenttheories. Note that difference in scaling of stress in a good solvent betweenSST and scaling theory originates from the difference in scaling of the freeenergy predicted by the two theories, even though brush height predictionsagree (see Table B.1). The difference in scaling originates from the fact thatthe mean field theory and the SST do not take into account the effect ofself-avoidance that occur in the case of strong excluded volume interactions.Also, for a θ-solvent, expression from scaling theory has an additional termcoming from the term in the virial expansion of mean field free energy ofmixing (1.8) proportional to φ [23]. For large N , this term is typicallyignored in the mean field theory and the SST.2.1.4 Energetics of bendingWe now qualitatively compare the change in free energy of a polymer brushdue to a strain in substrate with the strain energy of bending of an Eulerbeam and obtain Young’s modulus of a beam that can bend substantially dueto a brush layer. This comparison is meaningful only when the brush remainsplanar, that is the length scale associated with the substrate curvature islarge compared to brush height.For bending of an Euler beam of thickness h due to a brush grafted toits top surface,|τss| ∼ 12Eh312κ2, (2.40)342.1. Stress in a polymer brush using SSTTable 2.1: Comparison of expressions for surface stress (in kBT units) ofa brush in good and θ solvents obtained from mean field Flory theory, SSTand scaling theory [23]. Scaling of surface stress obtained from Flory argu-ment are the same as in SST, but the prefactors are different. Also, scalingobtained from the scaling theory is different. Note that the expressions fromSST shown below are obtained by assuming pl = 1 and hence substitutingβ = 3/a2 in (2.36) and (2.38) for good and θ solvents, respectively.Good solvent θ solventFlory argument −3 (16)2/3 v2/3ρ10/6g a−2/3N −w1/2ρ2ga−1NSST (this work) −35(pi24)1/3v2/3ρ10/6g a−2/3N −√32pi w1/2ρ2ga−1NScaling theory ∼ −13v1/3ρ11/6g a2/3N ∼ −12(ρg + wρ2ga−4N)where the term on the left is the change in the free energy of brush in a unitarea because of the strain due to bending in the substrate, and the term onthe right is bending strain energy per unit length of a beam with unit width.s is strain at the top surface of the beam due to bending and κ is curvatureof bending. E is Young’s modulus of substrate. On neglecting mid planestretching in the beam, s = κh/2. Substituting this in the above relationyields:E ∼ 12|τs|h(κh). (2.41)Assuming that s ∼ 10−3, κh ∼ 10−3. In this case, deflection of the tip of acantilever beam of length 50h is ∼ h. Now, we need to obtain τs to be ableto estimate Young’s modulus of a beam that will show this level of strainand deflection due to bending.We use (2.36) to estimate surface stress due to a brush. Excluded vol-ume parameter v = a3(1 − 2χ), where χ is Flory-Huggins interaction pa-rameter [30], which characterizes interaction between solvent molecules andmonomers. In a good solvent, v ≈ a3 as χ approaches 0 [30]. We assumegraft density ρg ∼ 0.1/a2. Note that max(ρga2) ≤ 1. For brushes withhigh molecular weight polymer chains (fabricated using SI-ATRP method),N ∼ 104. Taking a = 1 nm and pl = 1 (flexible polymer), and at roomtemperature, τ0s ∼ −1 N/m and Es ∼ 1 N/m. Height of the brush is∼ 1 µm. For a beam with 100 µm thickness, bending is considerable ifE ∼ 100 MPa. Also, in this case radius of curvature is ∼ 10 cm. Since theradius of curvature is much larger than the height of the brush, the brush is352.2. Large deflection of a flexible beam grafted with a polymer brushplanar.In summary, energetics of bending of a beam due to polymer brushsuggests that polymer brush can produce large deflection in thin beams ofmaterials with low elastic modulus.2.2 Large deflection of a flexible beam graftedwith a polymer brushIn this section, we develop an analytical model for the deformation of athin flexible beam (Young’s modulus E, Poisson’s ratio ν) with an elasticsurface layer of nonzero thickness (elastic modulus Es, Steigmann-Ogdenconstant9 C) with residual stress due to polymer brush at its top surface,using the principle of virtual work. Surface stress in the undeformed state ofthe substrate is τ0s , and in the deformed beam, it is τs. Figure 2.3 shows theconfiguration with the undeformed substrate along with the deformed con-figuration. The model allows large deformation of the beam, however, strainshould remain small (valid for thin beams). We assume that plane sectionsremain plane and perpendicular to the centreline of the beam because strainenergy due to shear stresses is very small and can be neglected.dxdX SubstrateBrush layerXZMid planehUndeformed configurationDeformed configuration�θ(X)Figure 2.3: Schematic of an elastic beam with surface stress due to poly-mer brush layer in undeformed (top) and deformed states (bottom) of thesubstrate.9This will equal tangent bending rigidity of a brush in the small strain case. In a goodsolvent, C = 964(12pi2)1/3N3ρ7/3g [61].362.2. Large deflection of a flexible beam grafted with a polymer brushBending of a classical Euler-Bernoulli beam is governed by:EIdθdx−M = 0, (2.42)where I is the second moment of area of cross-section of the beam and Mis bending moment at the cross-section. In the presence of a surface layer,effective Young’s modulus is modified [110–113]. Assuming an identicalsurface layer at the top and bottom surfaces of a beam and accountingfor the mechanical equilibrium of surface led to the inclusion of Young-Laplace correction term10 [112]. Accounting for nonzero thickness of surfacenecessitated the introduction of curvature elasticity of surface [113] in theeffective modulus. Below, we review the conditions when the correctionsbecome considerable and then formulate the governing equation for a beamwith a surface layer at its top surface by including the Young-Laplace andcurvature elasticity corrections.For very thin substrates, mechanical equilibrium of the surface layerintroduces correction in stress in the substrate through the Young-Laplacerelation. For a rectangular cross section with thickness h, the correction isconsiderable if τs ∼ Ehν . From our experiment to be discussed in Section 4.3,τs ∼ −10 N/m. Hence, correction due to the Young-Laplace relation isconsiderable if Eh ∼ 1 N/m (ν ∼ 10−1). For E ∼ 106 Pa, correction due tothe Young-Laplace relation is considerable for a micron thick beam.Furthermore, the thickness of a polymer brush layer is ∼ 1 µm [6].Hence, for very thin substrates, the surface layer can not be assumed to be ofzero thickness and curvature dependence of surface elasticity [113, 114] mayintroduce considerable correction to the effective modulus. For a rectangularbeam with thickness h, effective elastic modulus [113] is given as:Eeff = E(1 +3EsEh+12CEh3). (2.43)The second term in the above relates to the elasticity of the surface layer.The ratio Es/E is a material length scale [115]. The third term has its originin curvature dependence of surface energy. Also, the effective thickness of asurface, hs, has been defined as hs =√CE−1s [113]. So, for a given thicknessof a surface layer, C = Esh2s. By substituting this relation in (2.43), we can10A typical example of Young-Laplace relation appears in a bubble or an inflated balloon.Mechanical equilibrium of a curved membrane with stress such as a bubble or an inflatedballoon, requires a difference in pressure inside and outside of the membrane. Young-Laplace relation relates the stress in the membrane with the difference in pressure.372.2. Large deflection of a flexible beam grafted with a polymer brushconclude that the correction to effective modulus (Eeff ) due to Young’smodulus of the surface and the curvature elasticity are of the same order ifhs ∼ h. So for a polymer brush layer, the two corrections will be of similarorder when the beam is a micron thick.Now, we develop governing equations for deformation of the beam, shownin Fig. 2.3. Lagrangian strain at the mid plane of the beam is give by:xx(x) =(dx)2 − (dx)22(dx)2≈ dxdx− 1 (2.44)The approximation holds since strain is assumed to be small. Lagrangianstrain in the beam is given byxx(x,z) = (λ(x)− 1)− dθ(x)dxz, λ(x) =dxdx, (2.45)where z is the distance of a point from the centreline of the beam. Weassume that the deformation of the beam does not change this distance.λ(x) is mid plane stretch.The axial stress (second Piola-Kirchhoff stress, conjugate of Lagrangianstrain) in the beam is obtained from strain in the beam by using the followingconstitutive relation:σxx(x,z) = E¯xx +ν1− ν σzz, (2.46)where E¯ = E/(1 − ν2) (plane stress modulus). The relation is obtainedfrom the generalized Hooke’s law by assuming yy = 0. Note that the normalstress in the bulk just underneath the surface layer in the direction normal tothe centreline of the beam (σ+zz) appears to maintain mechanical equilibriumof the surface layer, akin to the pressure inside an inflated balloon. Itsrelation to the deformation of the surface layer is obtained by using Young-Laplace theory [112].σ+zz(x) = λdθdxτs ≈ dθdxτs, (2.47)whereτs(x) = τ0s + Es((λ− 1)− dθdxh2). (2.48)Also, since there is no brush on the bottom face of the beam σ−zz(x) =0. Since the beam is very thin, we assume that σzz varies linearly in z-direction [116], and is given by the following expression:σzz(x, z) =12dθdxτs +zhdθdxτs. (2.49)382.2. Large deflection of a flexible beam grafted with a polymer brushSubstituting the above in (2.46) yields the expression for stress in the bulk.σxx = E¯xx +ν1− ν(12dθdxτs +zhdθdxτs). (2.50)Now, to obtain governing equation, we utilize the principle of virtualwork. In equilibrium configuration,δWint + δWext = 0, (2.51)where δWint and δWext are internal and external virtual works respectively.As there is no external force in our case δWext = 0. Hence, (2.51) reducesto δWint = 0. Internal virtual work has contribution from the bulk and thesurface (δWint = δWbulk + δWsurf ). The two contributions are given by:δWbulk =∫ L0[∫ h/2−h/2σxxδxxdz]dx, (2.52)δWsurf =∫ L0(τs [δxx]z=h2+ Cdθdxδ(dθdx))dx, (2.53)where C dθdx is surface moment stress (curvature κ =dθ˜(x)dx ≈ dθdx for smallstrain, where θ˜(x) = θ(x(x))) [113, 114]. Note that we can use the linearizedform of the surface moment stress [113], because in our coated beam system,strain is small even though rotation need not be. From (2.45), variation instrain, δxx, can be written as:δxx = δλ− zδ(dθdx). (2.54)On substituting the expressions corresponding to σxx (2.50) and δxx (2.54)in (2.52), and integrating with respect to z, the expression becomesδWbulk =∫ L0[h(E¯(λ− 1) + ν1− ντs2dθdx)δλ]dx+∫ L0[h312(E¯dθdx− ν1− ντshdθdx)δ(dθdx)]dx. (2.55)The expression for the virtual work contribution from the surface layer,δWsurf is given by:δWsurf =∫ L0(τsδλ− τsh2δ(dθdx)+ Cdθdxδ(dθdx))dx. (2.56)392.2. Large deflection of a flexible beam grafted with a polymer brushUsing (2.55) and (2.56), the expression for total internal virtual work δWintis obtained as:δWint = 0 =∫ L0[(h(E¯(λ− 1) + ν1− ντs2dθdx)+ τs)δλ]dx+∫ L0[(h312(E¯dθdx− ν1− ντshdθdx)− τsh2+ Cdθdx)δ(dθdx)]dx. (2.57)At the boundary, constraints can be applied on displacement or rotation,but not on stretch. So, we use the following relation to convert variationin stretch (δλ) in terms of variation in displacement and rotation (see Fig-ure 2.3).duxdx= λ cos θ − 1, (2.58)δλ = sec θ δ(dudx)+ λ tan θ δθ, (2.59)where ux is displacement in x-direction. Replacing δλ with expression in(2.59), simplifying the terms in (2.57) using integration by parts and apply-ing the principle of variation, we obtain the following governing equationsfor a beam with surface stress.ddx[(h(E¯(λ− 1) + ν1− ντs2dθdx)+ τs)sec θ]= 0, (2.60)ddx[(h312(E¯dθdx− ν1− ντshdθdx)− τsh2+ Cdθdx)]−(h(E¯(λ− 1) + ν1− ντs2dθdx)+ τs)λ tan θ = 0. (2.61)The boundary conditions (BC) are given by the following equations:[(h(E¯(λ− 1) + ν1− ντs2dθdx)+ τs)sec θδux]0,L= 0, (2.62)[(h312(E¯dθdx− ν1− ντshdθdx)− τsh2+ Cdθdx)δθ]0,L= 0. (2.63)It should be noted that the above equations are nonlinear. Furthermore, τsitself depends on λ and dθdx (see (2.48)).The general equations obtained above can be used to determine surfacestress on a cantilever beam by measuring curvature in the beam. For a402.2. Large deflection of a flexible beam grafted with a polymer brushcantilever beam, boundary conditions (2.62) and (2.63) at the free end atx = L give: [(h(E¯(λ− 1) + ν1− ντs2dθdx)+ τs)sec θ]L= 0, (2.64)[(h312(E¯dθdx− ν1− ντshdθdx)− τsh2+ Cdθdx)]L= 0. (2.65)Using the above, governing equations (2.60) and (2.61) turn out to be:h(E¯(λ− 1) + ν1− ντs2dθdx)+ τs = 0, (2.66)h312(E¯dθdx− ν1− ντshdθdx)− τsh2+ Cdθdx= 0. (2.67)Using (2.67), relation between curvature (κ) and effective surface stress (τs)is obtained.κ ≈ dθdx=6τsE¯h2 − ν1−ν τsh+ 12Ch. (2.68)Using (2.66), axial stretch in the beam can be obtained.λ− 1 = −(1 +12ν1− νdθdxh)τsE¯h. (2.69)The term 12ν1−νdθdxh appears because of Young-Laplace correction.To obtain κ for a known initial surface stress τ0s , we invoke small strainassumption and linearize (2.68) using κh << 1 (Appendix C) to obtain:κh ≈ 6τ0sE¯h+ 4Es +12Ch2+ 12CEsE¯h3− ν1−ν τ0s. (2.70)This is the general form of equation relating the curvature of a cantileverbeam with the surface stress in the undeformed substrate configuration. Byneglecting the Young-Laplace correction term(ν1−ν τ0s)in the denominator,we obtain the same expression for curvature as in [113] for no surface stressat the bottom face of the beam.11κh ≈ 6τ0sE¯h+ 4Es +12Ch2+ 12CEsE¯h3. (2.71)11The coefficient of Es in the denominator of (60) in [113] should be 4 (instead of 2) asin (2.71)).412.3. ConclusionsIn this case, the neutral axis is independent of surface stress and Young’smodulus of the surface layer and is given by:xn = −h6− 2CE¯h2. (2.72)On discarding the curvature elasticity contribution as well by setting C = 0,we obtain a curvature relation that can also be obtained from equations in[117] for a cantilever beam.κh ≈ 6τ0sE¯h+ 4Es≈ 6τ0sE¯h1 + 4 EsE¯h. (2.73)The above relation suggests a pertinent length scale as a ratio of elasticmodulus of surface and that of the substrate at which effect of surface elasticmodulus becomes significant. In [113], molecular dynamics simulation ofsilver nanowire gives Es ∼ 10 N/m. For silver, E¯ ∼ 1011 Pa. Hence, surfaceelastic modulus becomes significant for substrate thickness ∼ 10−10 m.It is shown in Section 2.1.3 that Es ∼ |τ0s | for polymer brush. Now,κh << 1 implies τ0s /(E¯h) << 1 and hence Es/(E¯h) << 1. So the termEs/(E¯h) in denominator can be neglected to obtain the famous Stoney’srelation.κh ≈ 6τ0sE¯h. (2.74)2.3 ConclusionsThis chapter is aimed at understanding two aspects of polymer brush me-chanics: a) stress in a polymer brush with Gaussian chains, and b) themechanics of a flexible beam with polymer brush grafted on one of its sur-faces. Main conclusions are as follows.1. We have derived the distribution of lateral stress in polymer brushgrafted to a rigid substrate using mean field theory and shown thatthe stress variation is quartic along height direction of brush in a goodsolvent as given by (2.29). Maximum stress occurs near the graftingsurface and the stress smoothly goes to zero at the free surface of thebrush.2. The expression for stress in θ-solvent is also derived and it shows avariation different from the good solvent case, and a stronger depen-dence on graft density as seen in (2.35).422.3. Conclusions3. In the expression for stress in brush, nonbonded interaction contributestwice as much as the chain stretching in a good solvent as observed in(2.29). Their contributions are equal in a θ-solvent.4. Surface stress due to polymer brush has stronger dependence on graftdensity than on molecular weight of polymer chains in the brush assuggested by (2.36).5. Surface stress and the surface elastic modulus for a brush layer are ofthe same order of magnitude. Hence, the effect of surface elasticity insmall strain deformation is negligible (see (2.36), (2.37)).6. Governing equations for finite deformation (but small strain) of a beamwith a coating of polymer brush on its top surface have been derived(2.70). The Young-Laplace term, needed to satisfy equilibrium of sur-face layer, and effect of curvature on elasticity of surface layer havealso been included in the equations. On neglecting the contributionsof the Young-Laplace effect and the curvature elasticity of surface, theexpression for curvature of a beam with surface layer reduces to theresults reported in literature.In this chapter, the derivation of stress was based on a brush with Gaus-sian chains. Also, only binary interaction in good solvent condition (andternary interaction in θ-solvent condition) was considered. This limits thevalidity of the results to only moderate graft density brushes. In the nextchapter, stress will be obtained using SST-L which is valid for high graftdensity brushes. Also, the prediction of stress from this chapter as well asfrom the next will be validated using MD simulation. Assumptions made inthe derivations will also be verified.43Chapter 3Stress in a polymer brushIn the previous chapter, stress in a polymer brush with Gaussian chains isderived. However, at high grafting densities, end to end distance of a chainin a good solvent is large, and hence the force-extension divergence, absentin Gaussian elasticity, must be considered in elastic free energy calculation.Here, we employ SST with Langevin chains to account for force-extensiondivergence and study stress in high graft density brushes.This chapter builds on the previous chapter in two respects: extensionof theory and MD simulations. An overview of SST for brushes with non-Gaussian chains [94, 95, 109] is given in Section 3.1. Then, SST for brusheswith non-Gaussian chains is extended to calculate stress in the brush. UsingLangevin chain elasticity and a modified Carnahan-Starling (CS) equationof state [109], stress distribution in a densely grafted polymer brush in agood solvent is calculated in Section 3.2. This particular choice enables thederivation of semi-analytical expression for stress, free of fitting-parameters,to cover a wide range of graft densities studied in our MD simulations. MDsimulations of brush in an athermal solvent aimed at understanding thestress variation within a brush as a function of its molecular parameters isperformed. Section 3.3 presents the computational details of our MD model.Predictions by various theories are compared with MD simulation results anddiscussed in Section 3.4, ending with concluding remarks in Section 3.5.3.1 SST with Langevin chainsGaussian chain elasticity is valid for small chain extensions and shows finitestretching for any finite extension. However, the finite chain length leadsto a divergence in the force-extension relation at high extensions (see Fig-ure 1.6). Using a dimensionless extension parameter, βe = H/(Na), one cancharacterize different regimes where the divergence of the force-extensionneeds to be accounted for. For instance, when βe > 1/3, divergence inforce-extension due to finite extensibility cannot be ignored.A semi-analytical framework to account for finite extensibility effectsin SST was proposed in [94]. However, the form of stretching free energy443.2. Stress in a polymer brush using SST-Lwas chosen based on mathematical convenience. Semi-analytical SST us-ing stretching free energy of a Langevin chain (SST-L), which accuratelydescribes large stretching of a freely jointed chain, was developed in [95].It predicts that, with increasing density, a brush approaches a step profilefor monomer density as suggested by scaling theory and the chain free endsincreasingly straddle the free surface of the brush. However, the mean fieldpotential in this work was obtained as a series solution. A rational polyno-mial approximation for the series solution is proposed in [109]. Furthermore,and conventionally, interaction free energy in SST is calculated based on FHtheory. A modified CS equation of state from liquid-state theory for hard-sphere mixtures with correction for connection between beads in a polymerchain to account for interaction is employed in [109, 118]. Remarkably, pre-diction of brush structure in an athermal solvent12 from this SST showsclose match with the bead-spring MD simulation results without any needfor a fitting parameter [109, 118]. Here, we extend this theory to calculatestress in a brush.3.2 Stress in a polymer brush using SST-LTo evaluate stress within a brush, we use (2.15) derived in the previouschapter. First we need to find free energy density f(Z) in the brush andits derivative with applied strain xx along with∂zz∂xx. In the section below,free energy density is derived first, followed by monomer density and enddensity calculations. Subsequently, the required derivative terms ( ∂f∂xx and∂zz∂xx) are evaluated.3.2.1 Free energy densityFree energy density in a brush with Langevin chains has two contributors:(a) Langevin chain elasticity, fel and, (b) interactions among the monomers,fint. Intuitively, the area under force-extension curve (p − z curve) of aLangevin chain (see Figure 1.6) furnishes the free energy of elastic stretchingof a single chain, Fchain, as:Fchain =∫ z0p′(z′)dz′ = pz−∫ p0z′(p′)dp′=(p¯z¯− log(sinh(p¯)p¯))kBTN, p¯ =pakBT, z¯ =zNa, (3.1)12A Flory-Huggins equation of state or an enthalpic correction to CS equation of statecan be considered to include different solvent qualities [109].453.2. Stress in a polymer brush using SST-Lwhere z is end to end distance of a Langevin chain, p is stretching forcealigned along z direction, and we use force extension relation in (1.2). Also,the complementary energy (second term) is evaluated first by carrying outthe integration and then subtracted from the total energy (first term, pz).Also notice that the stretching force and height have been normalized in theabove.A polymer chain in a brush is like a chain in a one dimensional externalfield. This results in a stretching force p¯ in the chain which varies alongits length. To find stretching free energy in this case, we consider chainsegments in a slit of width dz in the brush, as shown in Figure 2.1. Assumingthat there are dn monomers of a chain segment within this slit, the freeenergy is obtained from (3.1) by replacing N with dn:dFchain =kBTae(p¯)(p¯e(p¯)− log(sinh(p¯)p¯))dz, e(p¯) =1adzdn= L(p¯), (3.2)where e(p¯) is local stretching in a chain due to the stretching force p¯. Thestretching force (p¯) in the above depends on height z¯ of the segment abovethe grafting surface as well as the height of the chain end ζ¯ (= ζ/(Na)). Fora given mean potential field (V (z) = V¯ (z¯)), p¯ at height z¯ for a chain withend at ζ¯ is obtained by the following relation [95]:log(sinh(p¯)p¯)= V¯ (ζ¯)− V¯ (z¯), (3.3)where V¯ (z¯) is given by [109]:V¯ (z¯) = 2z¯2a(2− 45 z¯21− z¯2). (3.4)So, we can conclude that end density, g(ζ), plays a significant role in deter-mining the stretching free energy density at z. Now, the total elastic freeenergy density of the brush at height z can be written in terms of end-densitydistribution function, g(ζ), as:fel =∫ HzdFchaindzg(ζ)dζ =∫ H¯z¯kBTae(p¯)(p¯e(p¯)− log(sinh(p¯)p¯))g¯(ζ¯)dζ¯,H¯ =HNa, g¯(ζ¯) = Nag(ζ). (3.5)Note that the above form of V¯ (z¯) is an empirical rational fraction approx-imation to the accurate power series in z¯ reported in [95]. Also, calcula-tion of g¯(ζ¯) follows the description in [109] and is summerized in 3.2.1. A463.2. Stress in a polymer brush using SST-L(semi)analytical procedure to find fel at a given z¯, then is to find V¯ in (3.4)first, followed by solving for p¯ in (3.3) and g¯(ζ¯) in (3.10), and finally usingp¯ and g¯(ζ¯) in (3.5).To calculate fint, we make use of CS equation of state for hard spheremixtures with a correction for connection between monomers in a polymerchain [109]. Here, polymer chains are viewed as a series of beads of volumeA0a, where A0 =pi6d2, d is the size of a bead and a is the length of a bead.Using the following relation between the chemical potential and free energyfrom [94],µd =1kBT∂fint∂φ=A0akBT∂fint∂Vf, (3.6)where φ is monomer density, and Vf is volume fraction (Vf = A0aφ), theinteraction free energy density follows:fint =kBTA0a∫ Vf0dµ˜(Vf )dVf , (3.7)where µ = µ(φ) = µ˜(Vf ). Chemical potential per unit chain length (µ˜(Vf ))from the modified CS equation of state is [109]:µ˜(Vf ) =1d(Vf7− 7Vf + 2V 2f(1− Vf )3 + log(1− Vf )). (3.8)In summary, to numerically calculate fint, Vf is calculated first for a brushof a given height (see 3.2.1), followed by (3.8) to find µ˜(Vf ), which in turnis used in (3.7) to determine fint.Calculation of Vf and ρgTo find Vf at z¯ in a brush of given height H¯, first we evaluate the left handside of (1.13) using the expression for V¯ (z¯) in (3.4). On the right hand sideof (1.13), we substitute (3.8), as µ(φ) = µ˜(Vf ) and solve for Vf numerically.In the SST-L approach, in contrast with SST-G, height is input and notthe graft density. Hence, we need to find ρg for a given height using thefollowing:ρg =1A0∫ H¯0Vfdz¯. (3.9)Now we can compare the monomer density profiles predicted from SST-Land SST-G for a brush with a given ρg, and the results are shown in Fig-ure 3.1. To generate the plot, we make use of the relation Vf = A0aφ to473.2. Stress in a polymer brush using SST-Lfind φ. Note that, the normalized brush height (H¯) for both the theoriesare prescribed to be the same for a graft density when comparing the twotheories. Based on height and graft density, binary interaction parameter(v) in SST-G is obtained using (2.4). Using this v, monomer density profile(and latter end density as well as stress profile) is obtained using (2.3). Atlow graft density, the prediction for density profile from the two theories areparabolic and match closely, as expected. However, with increasing graftdensity, unlike SST-G, density profile predicted by SST-L, approaches stepprofile.0 0.2 0.4 0.6 0.8 1Z/H00.511.5(Z)/g2/3g=0.005g=0.05g=0.5SST-GFigure 3.1: Monomer density profile obtained from SST-L is compared withthe prediction from the SST-G. Notice that SST-L (solid lines) predicts thatthe profile for low graft density is parabolic and matches closely with theSST-G (dashed lines) and it approaches a step profile with increasing graftdensity.Calculation of end density and its derivativeNormalized end density g¯(z¯) in a brush is obtained using the following rela-tion (see [95, 109] for details):g¯(z¯) =1A0dV¯ (z¯)dz¯∫ z¯∗0dV ′fdνdz¯′, (3.10)where z¯∗ is found from the following implicit relation:V¯ (z¯∗) = V¯ (H¯)− V¯ (z¯), (3.11)483.2. Stress in a polymer brush using SST-Land ν = V¯ (H¯)− V¯ (z¯)− V¯ (z¯′). Finding dV′fdν directly is difficult. So, we firstfind dνdV ′fusing (3.8) as suggested in [109]:dνdV ′f=1d6 + 3V ′f − 4V ′2f + V′3f(1− V ′f )4, (3.12)and obtaindV ′fdν usingdV ′fdν = 1/(dνdV ′f). Note that V ′f 6= Vf , and is obtained bysolving the following equation for given z¯ and z¯′:µ˜(V ′f ) = V¯ (H¯)− V¯ (z¯)− V¯ (z¯′). (3.13)Figure 3.2 compares end density profiles for a few graft densities as predictedby SST-G and SST-L. Again, for low graft densities, the two predictionsmatch very well. For high graft densities, however, most of the chain endsapproach the free end of the brush.0 0.2 0.4 0.6 0.8 1Z/H0123456Nag(Z)/g2/3g=0.005g=0.05g=0.5SST-GFigure 3.2: End density profiles obtained from SST-L is compared with theprediction from SST-G. At high graft density, chain ends are predicted tolie near the free end of the brush, unlike the prediction from SST-G.The calculation of stress in a brush requires evaluation of[∂g¯(ζ¯)∂xx]∂ζ¯∂xx=0(ζ¯ is integration parameter). It is obtained by taking derivative of (3.10):∂g¯(ζ¯)∂xx=1A0dV¯ (ζ¯)dζ¯ ∂ζ¯∗∂xx[dV ′fdν]ζ¯=ζ¯∗+∫ ζ¯∗0∂∂xx(dV ′fdν)dz¯′ , (3.14)493.2. Stress in a polymer brush using SST-LdV¯ (ζ¯)dζ¯is obtained using (3.4). To find ∂ζ¯∗∂xx, we take derivative of (3.11).∂ζ¯∗∂xx=H¯ζ¯∗2H¯4 − 4H¯2 + 52ζ¯∗4 − 4ζ¯∗2 + 5(1− ζ¯∗21− H¯2)2∂H¯∂xx. (3.15)dV ′fdν is obtained from (3.12) and∂∂xx(dV ′fdν)is obtained by taking derivativeof (3.12):∂∂xx(dV ′fdν)=∂∂xx 1dνdV ′f= −d((1− V ′f )3(V ′3f − 5V ′2f + V′f + 27)(6 + 3V ′f − 4V ′2f + V′3f )2)∂V ′f∂xx, (3.16)where∂V ′f∂xx=85da(H¯(2H¯4 − 4H¯2 + 5)(1− H¯2)2)((1− V ′f )46 + 3V ′f − 4V ′2f + V′3f)∂H¯∂xx. (3.17)Since we already know H¯, ∂H¯∂xx , Vf and we can numerically calculate ζ¯∗,all the expressions above can be numerically calculated to finally obtain[∂g¯(ζ¯)∂xx]∂ζ¯∂xx=0.3.2.2 Calculation of the derivativesIn this section, we first calculate ∂zz∂xx and∂f∂xx, and subsequently numericallycalculate the stress variation σxx(z¯). The derivative of zz with respect tothe applied strain, xx, can be expressed as:∂zz∂xx=∂∂xx(∂u¯∂z¯)=∂∂z¯(∂u¯∂xx), (3.18)where u¯ = u/(Na) and u is the displacement of a thin layer at z¯ (see Fig-ure 2.1) in the z-direction due to the applied strain. Finding the abovederivative requires us to first find ∂u¯∂xx . To this end, we assume that the num-ber of monomers within a thin layer of volume V0 at height z¯ (see Figure 2.1),φV0, does not change due to the applied strain (∆(φV0) = ∆(VfV0)/(A0a) =0), which yields [1]:∂∂z¯(∂u¯∂xx)= − 1Vf∂Vf∂xx− 1, (3.19)503.2. Stress in a polymer brush using SST-Lwith the boundary condition:[∂u¯∂xx]z¯=H¯=∂H¯∂xx. (3.20)The derivative of volume fraction with respect to the applied strain,∂Vf∂xx, isobtained by taking the derivative on both sides in (1.13).∂Vf∂xx=85da(H¯(2H¯4 − 4H¯2 + 5)(1− H¯2)2∂H¯∂xx− z¯(2z¯4 − 4z¯2 + 5)(1− z¯2)2∂u¯∂xx)× (1− Vf )46 + 3Vf − 4V 2f + V 3f, (3.21)where the first term within bracket results from the change in the brushheight due to the applied strain and the second from the displacement ofthe thin layer at height z¯. The derivative of H¯ with respect to the appliedstrain is evaluated numerically.∂H¯∂xx=∂ρg∂xx∂H¯∂ρg= −ρg ∂H¯∂ρg, (3.22)and ∂H¯∂ρg is obtained by finding H¯ = H¯(ρg) using (3.9).After substituting (3.21) in (3.19), (3.19) with the boundary condition(3.20) is solved numerically to obtain ∂u¯∂xx and subsequently∂Vf∂xx. By takingthe derivative of ∂u¯∂xx with respect to z¯, we obtain∂zz∂xx. Figure 3.3 shows thevariation of ∂u¯∂xx and∂zz∂xxwith z¯ for three graft densities and compares thenumerically obtained curves from SST-L with analytical relation ∂u¯∂xx = −13 z¯(and ∂zz∂xx = −13) obtained from SST-G [1]. For the lowest graft density,curves obtained from SST-L and SST-G agree well, as expected. This canbe seen in the insets of Figure 3.3 where the values of ∂u¯∂xx are almost on top ofthe SST-G prediction while for ∂zz∂xx only small deviations are seen. However,as graft density is increased, deviations from the SST-G theory become moreapparent as depicted in the plot for ρg = 0.05 and 0.5. A method to estimatethis ratio experimentally (or from simulation) is suggested in Appendix D.2.We also observed another interesting feature predicted by the SST-L. Forρg ≥ 0.5, the predicted values of ∂zz∂xx for z¯ > 0.73 become positive indicatingthat the layers above this z¯ undergo an expansion when the brush is stretchedin x-direction. The critical point at which this occurs, is highlighted in theplot with a blue asterisk. This change in the sign of strain is observed onlyfor very high graft density brushes and is only captured by SST-L. However,513.2. Stress in a polymer brush using SST-Laverage value of ∂zz∂xx is always negative and is ≥ −1/3. It should be notedthat −∂zz(z)∂xx need not be construed as local Poisson’s ratio. The reasonis that in the calculation of ∂zz(z)∂xx , the thin layer (shown in Figure 2.1) isassumed incompressible. The local volume of the layer may change uponthe application of strain xx to ensure that the monomers within the volumeremain the same. Solvent molecules, however, are allowed to move in or outof the layer.Z/(Na)-0.3-0.2-0.10u/XXH/ XX0 0.2 0.4 0.6 0.8Z/(Na)-0.500.5ZZ/XX0 0.1 0.2-0.0500 0.1 0.2-0.4-0.35-0.3SST-Gg=0.5g=0.05g=0.005g=0.5SST-Gg=0.005g=0.05 (0.73,0)Figure 3.3: The figure shows ∂u¯∂xx and∂zz∂xxvs z¯ curves for three graft densitiesalong with ∂H¯∂xx . Analytical relations obtained from SST-G are also plotted.As ∂zz∂xx is obtained by numerical differentiation of∂u¯∂xx(see (3.18)), thereare jumps at the end of curves in the lower plot but they are not shown. Inthe inset, numerical curve for the lowest graft density is compared with theanalytical curve and a close agreement is observed, specifically in ∂u¯∂xx plot.However, considerable deviation is observed for high graft densities. Also,for very high graft density, ∂zz∂xx becomes positive for z¯ > 0.73, which is notcaptured by SST-G. The average value of ∂zz∂xx , however, is always negativeand is ≥ −1/3, suggesting a reduction in the brush height for xx > 0.To evaluate derivative of free energy density at height z¯ with respectto the applied strain, we find the derivative of interaction part(∂fint∂xx)and523.2. Stress in a polymer brush using SST-Lstretching part(∂fel∂xx)independently, and then sum them up. ∂fint∂xx is ob-tained by taking derivative of (3.7):∂fint∂xx=kBTA0a(Vf7− 7Vf + 2V 2f(1− Vf )3 + log(1− Vf ))∂Vf∂xx. (3.23)Finding ∂fel∂xx is more involved. Taking derivative of the expression for felin (3.5) gives:∂fel∂xx=kBTa∫ H¯z¯log(sinh(p¯)p¯)(1− (coth(p¯))2 + 1p¯2(e(p¯))2)g¯(ζ¯)∂p¯∂xxdζ¯+kBTa∫ H¯z¯(p¯− 1e(p¯)log(sinh(p¯)p¯))∂g¯(ζ¯)∂xxdζ¯, (3.24)where the first integral in the expression results from the change in stretchingof parts of chains within the thin layer and the second integral from thechange in end density. To evaluate the above relation, we need to obtain thederivative of local stretching force ( ∂p¯∂xx ) and normalized end density (∂g¯(ζ¯)∂xx).By making use of the implicit relation involving p¯ in (3.3) and recognizingthat ∂V¯ (ζ¯)∂xx = 0 as ζ¯ is the integration variable in (3.24), we obtain the desiredderivative:∂p¯∂xx= − 1e(p¯)∂V¯ (z¯)∂xx= − 1e(p¯)85z¯a2z¯4 − 4z¯2 + 5(1− z¯2)2∂u¯∂xx. (3.25)See 3.2.1 for calculation of ∂g¯(ζ¯)∂xx . On solving the above numerically andsubstituting the values of f , ∂f∂xx and∂zz∂xxin (2.15), we obtain the stressprofile σxx(z¯). See Appendix D.1 for a summary of the steps involved incalculation of the stress. SST-G and SST-L stress profiles are compared inFigure 3.4. Note that for an accurate comparison between the two theo-ries, we prescribe the same brush height for SST-G as given by SST-L forany given graft density. This allows the determination of excluded volumeparameter v in SST-G for each graft density, and subsequent calculation ofσxx using (2.15). Based on (2.15), we expect σxx/ρ4/3g vs z/H curves fordifferent graft densities obtained from SST-G to fall on a master curve. Thesmall deviations observed in Figure 3.4 are due to a very small differencein excluded volume parameters for different graft densities. Turning ourattention to the values predicted by SST-L, we observe that for small graftdensities the predictions are close to SST-G. However, for large values of533.3. Molecular dynamics simulationgraft density, we observe that the prediction of stress distribution changessignificantly with changes in the shape of the distribution. Note that thejump in stress profile near the top of a brush in Figure 3.4 is a numericalartefact. It occurs due to the fact that the end density shows sharp descentnear the top of the brush (see Figure 3.2) and numerical evaluation of thederivative of the end density (in (3.14)) near the top requires much smallerstep size than the step size in the rest of the brush. The jump is observedat the z¯ where step size changes.0 0.2 0.4 0.6 0.8 1Z/H-8-6-4-20XX/g4/3g=0.005g=0.05g=0.5SST-GFigure 3.4: Stress profiles obtained from SST-L is compared with the predic-tions from SST-G. Curves obtained from SST-G for different graft densitiesfall on top of each other. SST-G and SST-L curves corresponding to thelowest graft density are very close. However, at high graft density, SST-Lpredicts much higher stress. The jump in stress profile near the top of abrush is a numerical artefact.3.3 Molecular dynamics simulationThe purpose of MD simulations is to verify the predictions of SST-G andSST-L without placing any restrictions a priori on (a) virial truncation,(b) Langevin or Gaussian assumptions for chain elasticity, and (c) classicalpaths restriction on chain conformations.We use the LAMMPS [108] code, to simulate a neutral polymer brushgrafted to a rigid substrate. A cartoon of our model is illustrated in Fig-ure 3.5. Let us now consider a system of Ng chains with each chain madeof N + 1 beads. The first bead of each chain is fixed to the substrate. Thetotal number of unconstrained beads in the system is Ntot = NgN . Here, we543.3. Molecular dynamics simulationperform a Langevin dynamics simulation wherein temperature is controlledby attaching a heat bath to each of the unconstrained beads. Consequentcoupling results in a random force on each bead along with a viscous forcegoverned by the fluctuation-dissipation theorem. The governing Langevinstochastic differential equation to be solved then is:mid2ridt2= −∂U∂ri− Γdridt+ Fi(t), i = 1 . . . Ntot, (3.26)where mi and ri are mass and position, respectively, of the ith unconstrainedbead, U is the total potential energy of the system, and Γ is bead friction.In the simulations, Γ = 2.0τ−1, where τ is unit of time in LJ units. Notethat LJ units are used throughout the MD simulation section. Fi(t) is aGaussian white noise satisfying the following relation:〈Fi(t) · Fj(t′)〉 = δijδ(t− t′)6kBTΓ, i, j = 1 . . . Ntot, (3.27)in accordance with the fluctuation-dissipation theorem [119]. Note that δijis Kronecker delta function and δ(·) is Dirac delta function.XZUFENEULJBeadFixed BCPeriodic BCUwallFigure 3.5: Front view of MD simulation box with bead spring chains repre-senting polymers, along with the interactions involved in the brush and theboundary conditions. Red coloured beads are fixed to the rigid substrate.The total potential energy of the system has three-main contributions:(i) bond potential UFENE, (ii) non-bonded pair potential ULJ , and (iii) po-553.3. Molecular dynamics simulation0.8 0.9 1 1.1 1.2 1.3 1.4 1.5r (in )020406080100U (in ) UFENEUFENE    without U LJCut and    shifted U LJFigure 3.6: Variation of the pair (ULJ cut at its minimum and shifted up)and bond (UFENE) potentials with the distance between interacting beads.Observe the short range repulsion in UFENE is due to the LJ term presentin it (see (3.29)).tential governing interaction of the beads with the grafting surface Uwall.U =12Ntot∑i=1Ntot∑j=1UFENE(rij) +Ng∑k=1UFENE(rikg )+12Ntot∑i=1Ntot∑j=1j 6=i±1ULJ(rij) +Ntot∑i=1Uwall(zi), (3.28)where rij = |ri−rj | is the distance between beads i and j, and rikg = |ri−rgk|is the distance between bead i and constrained (grafted) bead k. zi is theperpendicular distance of a bead from the grafting surface. Also, the beadsfixed to a substrate interact only with the unconstrained bead bonded toit. Beads of unit mass are connected by FENE spring representing a bondbetween effective monomers as done in earlier MD studies on brushes [86, 87].The potential associated with FENE springs is given as (see Figure 3.6):UFENE(rij) = bij[−0.5KR20 log(1−(rijR0)2)+ 4pc(( σrij)12 − ( σrij)6+14)],(3.29)where bij is a bond order parameter that is 1 for adjacent beads in a polymerchain, and 0 otherwise, K is a constant determining stiffness, R0 is the563.3. Molecular dynamics simulationmaximum extension in the spring.  and σ are the energy and length scalesassociated with the second term which is LJ potential. pc is a piecewisecontinuous function used to truncate the LJ potential to only account forrepulsion forces. Thus, pc = 1 for rij ≤ rc = 21/6σ and is 0 otherwise. Notethat this σ is different from the symbol for stress tensor (σij), which alwayshas a subscript in this work.The first term in the expression above is attractive and is balanced bythe repulsive second term at equilibrium bond length. In the simulation,R0 = 1.5σ. In an athermal simulation at reduced temperature T = 1.2/kBand for K = 30/σ2, average bond length is equal to 0.97σ. So, whilecomparing MD simulation results with SST-L, we take a = 0.97σ and d = σin interpreting MD results. Figure 3.6 shows the two terms of the FENEpotential, and the total interaction potential as previously described.The interaction between nonbonded beads is governed by LJ potentialwith appropriate cut-off (see Figure 3.6).ULJ(rij) =4((σrij)12 − ( σrij)6 − ( σrc)12 + ( σrc)6) rij ≤ rc,0 rij > rc,(3.30)where rij is the distance between a pair of interacting monomers, and rc isthe cut-off distance. To simulate brush in a good solvent condition, rc =21/6σ such that pair interaction is purely repulsive. This is often referred toas athermal simulation since the potential is close to a hard sphere potential[87].The polymer chains have their one end fixed to a rigid wall. To ensurethat the polymer chains do not cross the wall, the bead-wall interaction isrepulsive, and governed by the following potential:Uwall(zi) =4((σzi)12 − ( σzi)6 − ( σzc)12 + ( σzc)6) zi ≤ zc = 21/6σ,0 zi > zc = 21/6σ.(3.31)Athermal simulations at T = 1.2/kB are performed. Length and widthof the simulation box and hence, of the grafting surface, is chosen to be thesame and slightly larger than the brush height except for ρg ≥ 0.2, whereinto limit the total number of beads (Ntot) at ∼ 500, 000, the box size wassmaller than the brush height. The first monomer of each of the chainsis fixed to one of the uniformly spaced grid points on the grafting surface.A random walk conformation of a chain starting at each of the grafting573.3. Molecular dynamics simulationpoints is obtained and used as the starting brush configuration. An efficientway to generate initial configuration, particularly for low graft graft densitybrushes, is described in Appendix E. In the directions along the length andwidth of the box (x and y), periodic boundary conditions are specified,see Figure 3.5. In z-direction, fixed boundary is specified and height of thesimulation box is chosen sufficiently large so that no particle goes out of thebox during a simulation.Particle velocities are randomly assigned to ensure a reduced tempera-ture of T = 1.2/kB. Note that the initial brush configuration may havean overlap between monomers. Because LJ potential is unstable when thedistance between interacting particles approaches zero, we initially run thesystem with the following soft pair potential instead of LJ pair potential for∼ 30, 000 time steps, before switching to the LJ potential.Usoft(rij) ={A(1 + cos(pirijrc))rij ≤ rc,0 rij > rc,(3.32)where rij is the distance between a pair of interacting monomers. rc waschosen to be σ, and A was increased from 0 to 30 in 10 steps to ensurethat the configuration becomes stable upon switching to LJ potential. Afterswitching to LJ pair potential, the system is run for ∼ 107 steps to equili-brate. Once the monomer density profile becomes stable, we run the systemfor another ∼ 5×106 steps to obtain data to calculate property values. Notehowever that for very small graft densities, where length of each chain (Na)is large, equilibration took ∼ 10 times more steps.To obtain the variation of the brush properties, for example numberdensity, end density, stress etc., with distance from the grafting surface, wedivide the simulation volume in bins of thickness σ, and length and widthalong the grafting surface the same as that of the simulation box. Value ofany of the above properties at the center of a bin is calculated by averagingthe property values over the bin, and over the length of the simulation.3.3.1 Calculation of stressWe take virial stress as the stress measure. At each time step, we computethe following quantity for the ith bead:Siab = −miviavib + 12Np∑j=1(riaPijb + rjaPjib)+12Nb∑k=1(riapikb + rkapkib)(3.33)583.3. Molecular dynamics simulationwhere mi is the mass of the ith bead, ria and via are the ath component ofposition vector and velocity of the ith bead, Np and Nb are the number ofpair neighbors and bonds of the ith bead, respectively. P ijb is bth componentof force on the ith bead due to pair interaction with the jth bead, and pikb isbth component of force on the ith bead due to bond interaction with the kthbead. Now, let us consider the nth bin with volume Vbin. It has Nbin beadsat the lth time step. The instantaneous virial stress in the nth bin, definedat the lth time step, is:[σnab]l =Nbin∑i=1SiabVbin, (3.34)which accounts for the behavior of multiple beads in the bin. In order toreport statistically meaningful quantities, we computed the averaged stresstensor per bin at the kth time step as follows:〈σnab〉k =Nk∑i=1[σnab](k+i)Nk, (3.35)where Nk is a number of consecutive time steps of the simulation. We ob-serve that the number of time steps used to average the stress componentsneeds to be large enough to reduce fluctuations and spurious measures thatcould appear during entropic oscillations of the polymer brush. Our simu-lations showed that for Nk ≥ 1000 the results are insensitive to the choiceof Nk. Thus, we took Nk = 1000. We systematically do this for multi-ple time instances (NI ∼ 5000) and report the phase-averaged virial stresscomponents in the nth bin as:σnab =NI∑k=1〈σnab〉kNI. (3.36)We remark that the reported values of the components of virial stressreflect the stress state of a collection of beads, and not a point wise measureof stress in the system. To obtain a point wise measure, other stress metric[120] should be employed.593.4. Results: MD vs. SST-G and SST-L3.4 Results: MD vs. SST-G and SST-LWe compare the results from SST calculation and MD simulation in thissection. Since we have used generic potentials in MD simulation, the resultsare qualitative and quantitative mapping to a physical system requires oneto determine  and σ for the system first.We studied brushes with graft densities ranging from 0.005 to 0.5. Tostart with, we ensure that chains are strongly stretched so that brush heightis proportional to number of beads in a chain N . This allows an accuratecomparison between MD and SST. To achieve this, we performed a conver-gence test wherein N was increased to ensure that φ(z) vs z/(Na) curvesfor different N converge to a single curve as shown in Figure 3.7. We noticethat the curves converge towards a single curve as N is increased. Alsonotice the depletion layer and tail in the monomer density profile whichare not present in SST predictions. They naturally appear in simulations,and shrink with an increasing N as expected from numerical SCFT [121].Guided by this convergence test, we specify a minimum stretching parameterβs = 3/2(H2/(Na2)) > 30 as the convergence criterion for all the graft den-sities simulated. Note that it is shown in [121] that for large βs, numericalSCFT monomer density profiles agree well with SST profiles. We make useof this result by choosing N according to this criterion. This ensures thatpolymer chains in a brush are stretched to a size at least ' 4.5 times theend to end distance of corresponding ideal chain with no interacting chainsnearby. We could not choose a higher threshold for βs because that wouldhave required an exceedingly large N for low graft densities, incurring muchhigher computational cost to reach equilibration.3.4.1 Monomer densityMonomer density in a brush varies with distance from the grafting surfaceas shown in Figure 3.8. We divided the range of graft densities simulatedinto three regimes: low graft density (ρg < 0.02), intermediate graft den-sity 0.02 ≤ ρg < 0.1, and high graft density (ρg ≥ 0.1), and show differentplots accordingly. Predictions from SST-G are expected to be valid onlyin low graft density regime. Notice that SST-L [109] closely approximatesthe monomer density profile for all graft densities simulated, and the agree-ment improves with increasing graft density. Interestingly, the simulationsnaturally predict a smooth transition from a parabolic profile to a step likeprofile as graft density is increased.To validate monomer density profile predicted by SST-G in the low graft603.4. Results: MD vs. SST-G and SST-L0 0.1 0.2 0.3 0.4 0.5Z/(Na)00.020.040.060.080.10.120.14Monomer density, (Z)N=50N=100N=150N=200N TailDepletion layerFigure 3.7: The effect of number of monomers in a chain (N) on the variationof monomer density with distance from the grafting surface in a polymerbrush of graft density 0.03. Monomer density curves converge to a singlecurve as N is increased. Stretching parameter βs = 14.66, 23.43, 32.29 and40.98 for N = 50, 100, 150 and 200, respectively. From this, N for eachgraft density is chosen such that βs > 30 in all the athermal simulationscarried out in this work.density regime, we plot scaled monomer density (φ(z)/ρ2/3g ) with scaleddistance from the grafting surface ((z/H)2) in Figure 3.9. This plot validatestwo predictions, first, that φ(z) ∼ ρ2/3g , and second, that the the monomerdensity has a parabolic profile (shows quadratic variation with distance fromthe grafting surface). This is clearly highlighted in the plot, where MD pointsfall on a line in the middle region of the polymer brush. However, we noticethat the profile deviates from a parabola at the grafted as well as free enddue to the effect of depletion layer and tail. The profile increasingly deviatesfrom these predictions as graft density is increased.Due to the presence of a tail in monomer density profile, height is difficultto identify clearly. So, average height is defined as the first moment ofmonomer density [86]:H :=83∫∞0 zφ(z)dz∫∞0 φ(z)dz. (3.37)The normalizing pre-factor 8/3 ensures that the height predicted by SST-G matches with the calculation above if parabolic monomer density profileobtained from SST-G is used in the above formula.613.4. Results: MD vs. SST-G and SST-L0 0.05 0.1 0.15 0.2 0.25 0.3Z/(Na)00.020.040.060.08Monomer density, (Z)g=0.005g=0.006g=0.008g=0.01SST-LgTailDepletion layer0 0.1 0.2 0.3 0.4 0.5 0.6Z/(Na)00.050.10.150.20.25Monomer density, (Z) g=0.02g=0.03g=0.05g=0.06g=0.08gSST-L0 0.2 0.4 0.6 0.8 1Z/(Na)00.20.40.60.8Monomer density, (Z)g=0.1g=0.2g=0.4g=0.5SST-LgFigure 3.8: Variation of monomer density in a polymer brush with thedistance from the grafting surface. Parabolic profile at low graft density(top) smoothly transitions to a step-like profile with increasing graft density(bottom). Note that density profiles obtained from SST-L agree well withMD predictions without the need for a fitting parameter.623.4. Results: MD vs. SST-G and SST-L0 0.2 0.4 0.6 0.8 1 1.2 1.4(Z/H)200.511.5(Z)/g2/3g=0.005g=0.006g=0.008g=0.01Linear fitTailDepletion layerFigure 3.9: SST-G predicts the φ(z)/ρ2/3g vs (z/H)2 plot to be independentof ρg and a straight line. The simulation shows good match with the theoryin the bulk of the brush for small graft densities.10 -2 10 -1g0.20.40.60.81H/(Na)MDFitSST-LFigure 3.10: Variation of the height of a brush with graft density. A line ofslope 0.33 is drawn to show the match between MD prediction and SST-Gand scaling theory. Height obtained from SST-L is also shown. Notice thedeviation from linear fit at high graft densities.The dependence of brush height on graft density is shown in Figure 3.10.For the graft densities studied in this work, the scaling of height with re-spect to graft density matches closely with the theoretical prediction forρg ≤ 0.2. However, a deviation can be observed on increasing graft densityfurther. The plot also shows the height predicted by SST-L which showsclose agreement with the MD values. Interestingly, we observe an increase633.4. Results: MD vs. SST-G and SST-Lin slope of a curve obtained by joining MD points in Figure 3.10, pointing toan increase in scaling exponent of ρg in the expression for brush height fromtheoretically predicted 1/3. However, height obtained from SST-L showsexactly the opposite trend. This discrepancy is an artefact of the way theheight is calculated in (3.37). Note that for a step profile without a depletionlayer or tail at the ends of a brush, (3.37) predicts a height greater than theactual height of the brush. An unwanted consequence of this is that heightpredicted may be higher than the contour length of a chain (H/(Na) > 1),as observed for the last point in Figure 3.10 corresponding to ρg = 0.5.We use the the height obtained from MD to calculate βs and βe. Table 3.1lists N , βs, and βe for different graft densities. Note that βs > 30 for all graftdensities. Extension in a chain βe helps determine the validity of Gaussianchain assumption. Based on βe values in Table 3.1, Gaussian elasticityis not valid for ρg > 0.02. To determine its validity at ρg = 0.02, weneed to consider nonuniform chain extension predicted by SST. Hence, wecheck the value of local stretching (E(z, ζ)) to determine validity of Gaussianassumption. As monomer density is highest close to the grafting surface, wefind local stretching at z ≈ 0 using (2.6) from SST-G.E(0, ζ) =piζ2N=pia2ζ¯. (3.38)E(0, ζ) is lower than 0.33 for ζ¯ < ζ¯0 = 0.22. The proportion of chainswith ζ¯ < ζ¯0, P (ζ¯ < ζ¯0), can be obtained using (2.5) on recognizing thatg¯(ζ¯) = Nag(ζ), as follows:P (ζ¯ < ζ¯0) =∫ ζ¯00g¯(ζ¯)ρgdζ¯ = 1−(1−(ζ¯0H¯)2)3/2= 0.62 (3.39)For ρg = 0.02, H¯ = βe = 0.32, hence P (ζ¯ < ζ¯0) = 0.62. Only 62% of chainssatisfy the condition for ρg = 0.02. For ρg = 0.01, this fraction is 85%. So,Gaussian chain assumption is not valid for ρg = 0.02 and only lower graftdensities may follow the assumption. Based on Figure 3.9, we can concludethat it is valid for ρg ≤ 0.01.3.4.2 End densityWe plot the variation of the scaled end density of monomers with scaleddistance from the grafting surface in Figure 3.11, obtained from the MDsimulations and SST-L. We observe SST-L prediction deviates considerablyfrom the MD prediction for low graft density brushes. Generally, the curves643.4. Results: MD vs. SST-G and SST-LTable 3.1: Stretching and extension parameters for brushes with differ-ent graft densities are listed. Large value of stretching parameter suggestsstrong stretching, and hence SST is applicable. However, the gaussian chainassumption is acceptable only if the extension in chains is less 1/3, limitingvalidity of SST to graft densities less than 0.03. Note that strong stretchingcan be achieved by increasing N , however extension is not affected by achange in N in a strongly stretched brush.ρg N Stretching Extension(βs =32H2Na2) (βe =HNa)0.005 500 31.22 0.200.006 500 34.25 0.210.008 500 40.58 0.230.01 300 30.34 0.260.02 300 45.63 0.320.03 200 40.87 0.370.05 200 57.05 0.440.06 200 64.48 0.460.08 200 78.57 0.510.1 200 91.95 0.550.2 200 153.52 0.720.4 100 133.63 0.940.5 100 158.42 1.03from MD show sharper peaks and a smooth transition to zero at the brushend than those predicted by SST-L. However, with increasing graft density,we obtain a better agreement as is depicted in the last plot in Figure 3.11.The difference at lower graft densities is related to the small value of βs (seeTable 3.1), which results in large depletion layer and tail. Also, brush freeends increasingly concentrate to the end of the brush, as assumed in scalingtheory, when graft density is very high.We also plot effective stretching ratio γ, defined as 〈ζ〉 / 〈ζ〉0, where 〈ζ〉is mean chain end height in a brush and 〈ζ〉0 is mean end height of a singlepolymer chain with no neighbouring chains as a function of βs in Figure 3.12.We observe that it follows the pattern suggested in [91], however since βs islarge in our plot, we do not see the lower end of the plot as in [91]. Notethat in our calculations, to find 〈ζ〉0, we assume polymer chain to be ideal,in which case 〈ζ〉0 =√2/3Na [91].653.4. Results: MD vs. SST-G and SST-L0 0.05 0.1 0.15 0.2 0.25 0.3Z/(Na)00.010.020.030.040.050.060.07Nag(Z)g =0.005g =0.006g =0.008g =0.01SST-Lg0 0.1 0.2 0.3 0.4 0.5 0.6Z/(Na)00.10.20.30.4Nag(Z)g =0.02g =0.03g =0.05g =0.06g =0.08SST-L g0 0.2 0.4 0.6 0.8 1Z/(Na)01234Nag(Z)g=0.1g=0.2g=0.4g=0.5SST-LgFigure 3.11: Variation of end density in a polymer brush with distance fromthe grafting surface. Prediction of end density profile from SST-L for lowergraft densities (top) is very different from MD result, however we observe abetter match at high graft densities (bottom).663.4. Results: MD vs. SST-G and SST-L40 60 80 100 120 140s3456789Figure 3.12: Effective stretching ratio (γ) is plotted vs stretching ratio (βs)and it follows the increasing pattern observed in [91]. However, unlike [91],we do not have points for lower βs in the plot.3.4.3 StressAfter validating our simulations and presenting a detailed study of the struc-tural properties of a brush in the previous sections, we now consider thevariation of stress in the polymer brush using MD and theory. The typicalvariation of the components of the virial stress in a brush obtained as de-scribed in Section 3.3.1 are shown in Figure 3.13 (the plots are for ρg = 0.03).Notice that the normal stresses in x and y directions are the same as ex-pected from symmetry among the two directions. Also, shear stresses are anorder of magnitude smaller compared to normal stress in x and y directionsand thus, they are neglected. Normal stress in z direction is found to be upto one third of normal stress in the x direction for the lowest graft densityand the fraction decreases with increasing graft density to become less than1% for the highest graft density. This likely results from the fact that brushis not very strongly stretched at low graft densities.Figure 3.14 shows the stress (σxx) variation in a brush as graft density isvaried. Again, we distinguish low, intermediate and high graft densities. Tocheck the validity of quartic variation of stress in low graft density brushes,as predicted in [1], we plot σxx as a quartic function of z in Figure 3.15.The stress profile indeed shows quartic variation within the bulk of thebrush for graft densities up to ρg = 0.03. At the grafted and the freeends of the brush, variation from the quartic profile is observed due to adepletion layer and a tail, respectively. Furthermore, even though we find673.4. Results: MD vs. SST-G and SST-L0 0.1 0.2 0.3 0.4Z/(Na)-0.015-0.01-0.0050IJXXYY0 0.1 0.2 0.3 0.4Z/(Na)-4-20246810IJ10-4ZZXYYZXZFigure 3.13: Typical stress profile in a polymer brush. The magnitudes ofshear stresses are an order of magnitude smaller than the magnitude of σxxand σyy and hence is neglected. σzz is up to ∼ 1/3 of σxx in the lowest graftdensity brush and decreases to less than 1% for the highest graft densitysimulated. Also, note that near the grafting surface, σzz has a very largemagnitude (0.28 /σ3, not shown in the plot) due to wall repulsion.that monomer density profile shows parabolic profile, as predicted by SST-G, for ρg ≤ 0.01, the quartic stress profile (also predicted by SST-G) persistsup to ρg = 0.03. This numerical evidence, obtained with MD simulations,validates the previous theoretical results about stress variation obtainedusing SST-G [1].For higher graft densities, SST-G theory eventually breaks down andhence, we have to rely on SST-L to find stress profile. Figure 3.16 comparesstress profile obtained from MD with the SST-L prediction. We find a good683.4. Results: MD vs. SST-G and SST-L0 0.05 0.1 0.15 0.2 0.25 0.3Z/(Na)-2-1.5-1-0.500.51XX10-3g=0.005g=0.006g=0.008g=0.01g0 0.1 0.2 0.3 0.4 0.5Z/(Na)-0.08-0.06-0.04-0.020XX g =0.02g =0.03g =0.05g =0.06g =0.08g0 0.2 0.4 0.6 0.8Z/(Na)-2.5-2-1.5-1-0.50XXg =0.1g =0.2g =0.4g =0.5gFigure 3.14: Variation of stress component σXX in a polymer brush withdistance from the grafting surface. Comparisons with SST-G and SST-L areshown in Figure 3.15 and in Figure 3.16, respectively, for appropriate rangesof graft densities.693.4. Results: MD vs. SST-G and SST-L0 0.2 0.4 0.6 0.8 1(1-(Z/H)2)2-2.5-2-1.5-1-0.50XX10-3g=0.005g=0.006g=0.008g=0.01Linear fitg0 0.2 0.4 0.6 0.8 1(1-(Z/H)2)2-0.015-0.01-0.0050XXg=0.02g=0.03Linear fitgFigure 3.15: The plot shows variation of σxx with a quartic function ofthe distance from the grafting surface. Simulation results show good agree-ment with the quartic variation prediction from SST-G for small graftingdensities. Note that even though finite extensibility effects cause deviationfrom parabolic monomer density profile for ρg = 0.02, 0.03, stress still showsquartic variation.agreement between them. The agreement improves with increasing graftdensity, as monomer density and end densities are closely predicted by SST-L at high graft densities. Note that y-axis in the plots is stress divided byτs/(Na), where τs =∫ H0 σxxdz is surface stress, the resultant of stress in abrush. This normalization helps separate the magnitude part of the stressfrom the stress variation profile and we find that stress variation profile iswell predicted by SST-L. Also, the stress variation curve (obtained fromSST-L) near the free end of the brush has points where the curve is not703.4. Results: MD vs. SST-G and SST-Lsmooth. As explained in Section 3.2, this is a numerical issue due to a sharpfall in the end density profile near the top of a brush.0 0.1 0.2 0.3 0.4 0.5Z/(Na)01234XXNa/sg =0.05g =0.06g =0.08SST-L0 0.2 0.4 0.6 0.8Z/(Na)00.511.522.533.5XXNa/sg=0.1g=0.2g=0.4g=0.5SST-LgFigure 3.16: The plot shows the variation of normalized stress σxx with thedistance from the grafting surface. Simulation results show good match withthe prediction from the SST-L for high graft densities.Remarkably, for very high graft densities, the stress obtained with MDsimulations suggests a bilinear profile when plotted against (1− (z/H)2)2 asseen in Figure 3.17. This suggest two regions where the polymer chain hasdifferent local stretching, which ultimately has impact on the free energydensity and therefore, on stress.Finally, we plot the dependence of the resultant surface stress (τs) withgraft density (ρg) in Figure 3.18. Considering the fact that the SST-L doesnot use any fitting parameter to predict the surface stress, we find that itclosely predicts the magnitude of the surface stress. For high graft densities,the SST-L values match well with the MD data, while for low graft densities,713.4. Results: MD vs. SST-G and SST-L0 0.2 0.4 0.6 0.8 1(1-(Z/H)2)200.511.522.533.5XXNa/sg=0.1g=0.2g=0.4g=0.5gFigure 3.17: The plot shows variation of normalized stress σxx with a quar-tic function of the distance from the grafting surface for very high graftdensities. Notice that the stress profile appears to have a bilinear profile.10 -2 10 -1g10 -410 -210 0-s/(Na)MDFitSST-LFitFigure 3.18: Variation of surface stress with respect to graft density. For thelinear fit, data only up to ρg = 0.08 has been used. The scaling exponent forMD data is 2.1350±0.0320 as opposed to 1.7430±0.0160 for SST-L. SST-Gpredicts this exponent to be 1.6667, whereas scaling theory predicts it to be1.8333. Also, MD and SST-L both predict increasing scaling exponent forρg > 0.1.more deviations are observed. This is attributed to the difference in pre-diction of monomer density (see Figure 3.8) as well as end density at lowergraft densities (Figure 3.11). The scaling of surface stress with respect tograft density are, however, very different. Scaling exponent of surface stress723.5. Conclusionswith respect to graft density is 2.1350 ± 0.0320 from MD simulations and1.7430 ± 0.0160 from SST-L on fitting the surface stress up to ρg = 0.08.Below, we discuss the deficiencies of different methods to understand thisdiscrepancy.Table 3.2 compares scaling of height, free energy and stress obtained fromdifferent theories and from our simulations. Scaling of height with respectto graft density closely matches in all the theories as well in computation.In contrast, scaling of free energy and stress with respect to graft densityare different in different theories. As discussed in [1], mean field Flory the-ory inaccurately assumes that chain ends are concentrated to the free endof the brush and hence all the chains are equally and uniformly stretched.Also, it does not accounts for excluded volume correlations which occur inthe limit of strong excluded volume interactions. These shortcomings leadto higher free energy predictions for a brush. Scaling theory correctly ac-counts for excluded volume correlations, although it also assumes that chainends are concentrated to the free end of the brush and hence all the chainsare equally stretched. Additionally, neither of the two theories account forfinite extensibility of chains. SST-G does not assume equal stretching ofchains and hence chain ends are distributed throughout the brush, leadingto parabolic monomer density profile. But it does not account for finite ex-tensibility of chains which is present in SST-L. However, neither SST-G norSST-L account for excluded volume correlations, leading to over predictionof free energy as well as stress. MD simulations do not have these restric-tions in principle (see Table 1.2), and give a higher scaling exponent of graftdensity in the expression for stress. However, it should be noted that MDsimulation results are sensitive to stretching parameter βs. For very smallgraft densities, βs is not very high, which may affect the scaling exponent.3.5 ConclusionsStresses in a polymer brush is studied in this work using SST-L and MDsimulations. The conclusions are as follows.1. Molecular dynamics simulations verify the quartic stress profile pre-diction of SST-G from our earlier work [1], in the low graft densityregime. Gaussian elasticity assumption is valid in this range due tosmall extensions, as quantified by βe. The agreement between sim-ulations and SST-G prediction is within the bulk of the brush andaway from depletion layer and tail. Our simulations also confirm theparabolic monomer density profile.733.5. ConclusionsTable 3.2: Comparison of expressions for the height, free energy and surfacestress in a brush in a good solvent obtained from scaling theory, mean fieldFlory theory, SST-G, SST-L, and MD. Free energy and stress are in kBT .Note that the scaling exponent of ρg in the expressions for height is the samefrom all the theories and closely matched by semi-analytical calculation andMD on fitting the data up to ρg = 0.08. The same is not true for freeenergy. Scaling of surface stress with respect to graft density is also differentin different theories.Method Height Free energy Surface stressScaling theory ∼ ρ1/3g a5/3N ∼ 52v1/3ρ11/6g a2/3N ∼ −13v1/3ρ11/6g a2/3NMean field Flory theory(16)1/3v1/3ρ1/3g a2/3N92(16)2/3v2/3ρ10/6g a−2/3N −3(16)2/3v2/3ρ10/6g a−2/3NSST-G(4pi2)1/3v1/3ρ1/3g a2/3N910(pi24)1/3v2/3ρ10/6g a−2/3N −35(pi24)1/3v2/3ρ10/6g a−2/3NSST-L ∼ ρ1.02/3g ∼ ρ10.24/6g ∼ ρ(10.46±0.1)/6gMD ∼ ρ1/3g - ∼ ρ(12.81±0.2)/6g2. Gaussian elasticity of chains breaks down at higher graft densitiesand has lead to discrepancies between the SST-G and MD results.This motivated the advancement of a semi-analytical parameter freetheory (SST-L) based on Langevin elasticity of polymer chains, whichaccounts for the divergence in force-extension relation. Further, SST-L does not restrict itself to binary interactions among monomers asdoes SST-G. These two features are found to explain the MD resultssatisfactorily.3. Prediction from SST-L for monomer density (see Figure 3.8) end den-sity profile (see Figure 3.11), brush height (see Figure 3.10), and stressprofile (see Figure 3.16) agree well with MD simulations at high graftdensities. We also note that SST-L predictions for these parameterssmoothly transition from SST-G at low graft densities to those of step-profile used in scaling theories at high graft densities.4. Surface stress predicted by SST-L matches closely with MD resultsfor high graft density. For lower graft densities, SST-L over-predictsthe surface stress. Also, scaling exponents of surface stress with graftdensity obtained from the two theories for ρg ≤ 0.08 are different. Theprecise reason for this is yet to be understood, though one can spec-ulate about the validity of stress measures, differential end-stretchingwith distance from the grafting surface, MD potentials and low valuesof N .74Chapter 4Stimuli response of a brush:experiment and theoryMultiple applications of polymer brushes depend on the response of a brushto an external stimuli, specifically deformation magnitude and the responsetime. The previous chapters were limited to the structure of a brush and thestress in it, under a good solvent condition. However, a stimulus changesthe quality of the solvent, leading to changes in the structural as well asthe mechanical properties of a brush. This chapter presents the effect oftemperature on the structure of a thermoresponsive brush and on the surfacestress due to the brush. We experimentally estimate the surface stress dueto a thermoresponsive brush and then compare the experimental findingswith the theoretical results.Section 4.1 sets the context. Then, stimuli response of a polymer solutionand a polymer brush is discussed in Section 4.2. After that, experimentalestimation of surface stress due to PNIPAm-co-PDMA random copolymerbrush grafted on to a pPVC thin beam is presented in Section 4.3. We makeuse of the curvature-surface stress relation, developed in Section 2.2, in thissection. Fabrication of the brush is also discussed briefly along with themeasurement of the molecular parameters of the brush.In the second part of the chapter, we extend SST-L to study thermore-sponse of a Poly(N- isopropylacrylamide) (PNIPAm) (an LCST polymer)brush. The choice of the polymer brush is guided by the brush system stud-ied experimentally. Interaction free energy is modelled using FH theory inwhich temperature dependent Flory-Huggins interaction parameter (χ) ac-counts for the effect of temperature on the brush. A qualitative comparisonbetween experimental results and the theoretical predictions is presentedbefore concluding the chapter.754.1. Context for comparison4.1 Context for comparisonPolymer brush material systems are notoriously difficult to fully characterizeexperimentally. For example, graft density is inferred from dry thickness of abrush. Invariably, theoretical support is relied upon to estimate parameters.Even the dependence of Flory Huggins parameter χ on volume fraction andtemperature is challenging to investigate. There is no consensus on χ forPNIPAm, see [122] for a review on the progress made over past fifty years!On the other hand, theories have assumptions. For example, while thetheories for homopolymer brushes are tractable, random copolymer brushesstill pose a challenge. Hence, a context for comparison between theory andexperiments is necessary.We first experimentally estimate the surface stress caused by a polymerbrush. This informs us of the magnitude of the surface stress due to apolymer brush. We also obtain the effect of temperature on the surfacestress experimentally. The molecular scale parameters of the brush are alsoestimated which helps us gouge the dependence of the surface stress onmolecular parameters of a brush qualitatively. Then, we develop an SST-Lmodel to study the structure of thermoresponsive brush and calculate thesurface stress due to the brush. Since, in the end we attempt to comparethe results from the experiment with the result from theory, it is imperativeto spell out the differences between the systems studied experimentally andtheoretically (see Table 4.1).Table 4.1: The table compares the differences between experimentally stud-ied system and theoretically studied system.Theory ExperimentHomopolymer Random heteropolymerMonodisperse PolydispersePlanar Planar with local roughnessUniform brush grafting Nonuniform brush graftingApart from the differences in the system, there are multiple approxima-tions in the choice of parameters in modeling. These approximations areresult of lack of accurate information on molecular scale parameters of aPNIPAM chain [123] and phase diagram of PNIPAm solution [122]. Thesedifferences and approximations makes it very difficult to make a one to onecomparison between experiment and theory.764.2. Stimuli response4.2 Stimuli responseThis section starts with a description of stimuli response of polymer solu-tions, followed by the description of the stimuli response of brushes. We willsee that these two systems respond differently to a stimulus, primarily dueto the end constraint on the polymer chains in a brush.4.2.1 Stimuli response of a polymer solutionStimuli such as temperature, pH, light etc. can modify the solution prop-erties of a polymer solution. They do so by modifying the interaction be-tween a polymer and the solvent molecules around it, thereby modifyingthe strength of a solvent. In a polymer solution, at a critical value of astimulus, polymer chains exhibit a phase transition from an expanded coil(or swelled) conformation to a globule (or collapsed) conformation, or viceversa. This transition is called coil-globule transition. In the case of a ther-moresponsive polymer, the critical temperature is called UCST or LCST.In a UCST polymer, the coil to globule transition happens upon decreasingthe solution temperature below a critical value. At a high temperature, ahigh entropy of mixing (Smix) decreases free energy (E − TSmix, where Eis energy and T is temperature) considerably. Hence, coil state, the statewith high entropy of mixing, is favored at high temperatures. In an LCSTpolymer, the transition shows the exact opposite trend. This is becauseof some other chemical effects that overcome the obvious entropy gain athigher temperatures. Breaking of hydrogen bond between polymer and sol-vent with increasing temperature is among the reasons for favoring globulestate at higher temperatures.4.2.2 Stimuli response of a polymer brushA polymer brush responds to a stimulus differently from a polymer solutiondue to the end constraint on polymer chains in a brush. The coil-globuletransition described for a polymer solution in the previous section, may notbe sharp in a polymer brush. In fact, it is suggested that the transition ina classical planar polymer brush (a UCST polymer brush) is a cooperativetransition and not a phase transition [74]. Also, phase separation (vertical aswell as lateral) in a polymer brush has been reported based on experiments[124–126] as well as theory [123, 127] and simulations [87].To study the effect of temperature on thermoresponsive polymer brushes,theoretical models [123, 127, 128], as well as simulation studies [129] have774.3. Experimentsbeen employed. In theoretical models, the effect of a change in temperatureis accounted by a temperature dependent enthalpic term in the interactionfree energy [30, 128]. The SST model developed in [123, 128] captures thevertical phase separation in a brush which has been observed experimen-tally in thermoresponsive brushes [124–126]. In MD simulation, the effect oftemperature on a thermoresponsive brush can be captured by an all atomsimulation. However, considering that multiple interacting polymer chainsneed to be considered to simulate a brush, computational cost will be pro-hibitively large. So, a coarse grained model can be employed. Unfortunately,computationally efficient and reliable coarse grained models for PNIPAm islacking. A united atom model to study temperature response of PNIPAmis presented in [130], but simulating polymer brush with large chains will bevery costly. A coarse grained model of PNIPAm brush using Virial coeffi-cients (up to third order) from the equation of state of a PNIAPAm solutionto model soft nonbonded interaction was presented in [129]. Solvent wasmodeled implicitly. This model, being computationally less expensive canbe used to simulate brushes with large chains. Polymer brush with bead-spring model for polymer chains have also been employed to study the effectof temperature on a thermoresponsive brush. The effect of temperature isstudied by allowing beads to have attractive interaction and varying simu-lation temperature [87]. However, the model cannot simulate brushes withLCST polymers.In the following, we study the effect of temperature on the surface stressdue to a brush experimentally. This is followed by SST-L model to studythe effect of temperature on the structure of a thermoresponsive brush andthe surface stress due to it.4.3 ExperimentsWe conducted experiments to estimate surface stress due to a PNIPAm-co-PDMA brush grafted on to a pPVC thin beam. This informs us of themagnitude of surface stress caused by a polymer brush. The molecular scaleparameters of the brush are also estimated which helps us gouge the depen-dence of the surface stress on molecular parameters of a brush qualitatively.Below, a brief description of the fabrication process is given in Section 4.3.1.Surface stress due to polymer brush was estimated experimentally by themethod of substrate curvature measurement and using the results in Sec-tion 2.2. The protocol and experimental set-up along with the experimentalresults are described in Section 4.3.2. Molecular weight of the polymer chains784.3. Experimentsin a brush and graft density of the fabricated brushes were also estimated.They are described in 4.3.3.4.3.1 Brush FabricationA temperature sensitive random copolymer brush, PNIPAm-co-PDMA brush,was grafted on one side of a pPVC film using SI-ATRP method [6, 131]. Tostart with, one surface of a cleaned and dried pPVC substrate of the desiredshape and size was sealed by attaching it to a sticky surface and coveringthe edges. The exposed surface was treated with allylamine plasma for 10minutes for surface activation. This process attaches amine groups to thesurface. After this, the substrate is detached from the sticky surface, dippedin deionized water and sonicated for 30 minutes to remove unreacted ally-lamine. Then, the substrate is kept in an ATRP initiator solution for 24hours. This allows initiator molecules to get attached to the plasma treatedsurface and they act as the grafting points from which polymerization starts.After this, the substrate is taken out of the initiator solution, dipped in wa-ter and sonicated for 3 minutes to remove free initiators. This is followed bykeeping the initiator tethered substrate in ATRP polymerization solution.This final process is carried out in a glove box filled with argon at roomtemperature. In polymerization solution, 90% of monomer by weight is NI-PAm as against 10% DMA. The polymerization time controls graft densityand molecular weight of the polymer chains in the brush. This method iscapable of producing high graft density brushes with large molecular weightsof the polymer chains in the brush leading to a large residual surface stress.Samples for the experiment were prepared with polymerization times of 16,14, 12 and 9 hours, which gave brushes with decreasing graft density. Also,some free initiator molecules are left in the polymerization solution for thegrowth of free polymer chains in the solution. The solution is purified afterpolymerization is done, and the purified solution is dried and then used toestimate molecular weight.4.3.2 Measurement of surface stressWe experimentally estimated the surface stress due to a polymer brush bymeasuring the curvature of a cantilever beam grafted with a polymer brushon one surface. Also, since the brush is sensitive to temperature, the changein surface stress with temperature is also estimated. Here, the analyticalresults from Section 2.2 are used.The pPVC beams used in the experiment are 2 cm long (apart from794.3. Experiments0.5 cm length clamped between glass slides), 5 mm wide, and 254 µm thick.Young’s modulus for the pPVC film is measured to be 11.13 MPa. We takea typical value of Poisson’s ratio for pPVC of 0.4. A brush coated beamassumes a curved shape in water. The curvature of the beam changes as12345(a)55  Co(b)Figure 4.1: (a) Experimental set-up to measure curvature of a cantileverbeam (1. camera, 2. brush coated beam, 3,4. temperature sensor anddisplay, and 5. temperature bath), and (b) brush coated beam at 15 ◦C (top)and 55 ◦C (bottom). Experimentally obtained image of a cantilever beamwith edge (black dashed line) traced using image processing in MATLABand the arc of the circle fitted to the traced edge (green line).temperature is varied, due to a change in surface stress (see Fig. 4.1(b)).To measure curvature of a beam coated with the brush at different tempera-tures, a water bath (see Fig. 4.1(a)) was brought to the desired temperatureand then the beam was placed in it. Temperature of the bath is varied from15 ◦C to 55 ◦C by adding hot water to the bath. Note that the same levelof water in the bath is maintained throughout the experiment by takingout requisite amount of water after mixing added hot water. After placingthe beam in the bath at the desired temperature, we waited for two min-utes for the brush to reach a steady state before taking a photograph of thebeam. During this waiting time, small amount of hot water was added tothe bath to ensure that temperature stays within ±0.5 ◦C of the desiredtemperature level. The beam was photographed at different temperaturesby a fixed camera. Using image processing in MATLAB, the edge of thecantilever beam was traced, and using circle fitting, curvature of the coated804.3. Experimentsbeam was determined (see Figure 4.1(b)). Note that some inaccuracy inmeasurement is introduced by the fact that the traced edge is not the sameedge of the cantilever throughout its length. Near the fixed end of the beam,bottom edge of the beam is traced but near the free end, top edge of thebeam is traced. Figure 4.2 shows the variation of curvature of the beamwith temperature.From Figure 4.2, magnitude of curvature is (|κ|) ∼ 102 m−1 and thethickness of the substrates is h ∼ 102 µm. Hence the magnitude of strains(∼ |κh|) are ∼ 10−2. Therefore the assumption of small strain in the beamsin the experiment is valid. So, we can use the measured curvature, κ, tofind surface stress τs in the beam using expression in (2.68) with the addedassumption that C ≈ 0, since substrate thickness is much higher than theheight of brush (∼ 1 µm).τs =E¯h2κ6 + κh+ ν1−νκh. (4.1)Note that we have discarded curvature elasticity contribution in (4.1) be-cause our beam is much thicker than the polymer brush layer. The maximumcorrection in the effective surface stress due to the Young-Laplace term inthe denominator ( ν1−νhκ) in (4.1)) is < 1%, suggesting that the term can bediscarded.The surface stress variation with temperature is as shown in Figure 4.2.Observe that the surface stress at a given temperature decreases with de-creasing polymerization hours. Also, increasing temperature causes a de-crease in the magnitude of the stress. Note that a caveat in the abovecalculation is that the Young’s modulus of a pPVC substrate is independentof temperature. However, this may not be true.4.3.3 Estimation of graft density and molecular weightWe also experimentally estimated molecular weight of polymer chains ina brush and graft density of the brushes. The number averaged molecu-lar weight (Mn) and Polydispersity index (PDI)13 of the polymer chainswere estimated from Gel permeation chromatography (GPC) of the puri-fied and dried free polymer chains in the polymerization solution [6]. Table4.2 shows molecular weight as well as PDI for brushes grown with different13It equals the ratio of weight averaged molecular weight (Mw) and number averagedmolecular weight (Mn) and characterizes presence of chains of varying lengths. In themean field theory based calculation presented here, PDI is ignored.814.3. Experiments10 20 30 40 50 60−250−200−150−100−500Temperature (o C)Curvature (m−1 )  16h14h12h9h10 20 30 40 50 60−30−25−20−15−10−50Temperature (o C)Surface stress (N/m)  16h14h12h9hFigure 4.2: Variation of curvature of brush coated beams (left) and thesurface stress (right) with temperature for polymer brushes with differentpolymerization times. Increasing temperature leads to decrease in mag-nitude of surface stress. Also, magnitude of surface stress decreases withdecreasing polymerization time.polymerization hours. Degree of polymerization (number of repeating unitsin a polymer chain) in the brush equals Mn/113.16 as molecular weight ofNIPAm is 113.16 g/mol. Note that since only 10% by weight of DMA is824.3. Experimentsused in polymerization, we ignore it in degree of polymerization calculation.Table 4.3 shows the estimated degree of polymerization.Table 4.2: Experimentally measured brush parameters for brushes of dif-ferent polymerization hours.Polymerization Molecular Polydispersity Dry brushtime weight (Mn) index (PDI) thickness (hdb)(hours) (g/mol) (Mw/Mn) (µm)16 1.27× 106 1.19 1.31± 0.4914 1.03× 106 1.17 0.68± 0.2312 1.02× 106 1.19 0.59± 0.259 1.03× 106 1.14 -Table 4.3: Number of monomers in a polymer chain and graft density ofbrushes with different polymerization hours estimated from experimentallyobtained parameters in Table 4.2.Polymerization Degree of Graft density (ρg)time (hour) polymerization (chains/nm2)16 11239 0.74± 0.2814 9115 0.48± 0.1612 9026 0.42± 0.189 9115 -To estimate graft density of brushes, dry thickness of the brush layer(hdb) needs to be measured. A polymer brush grafted substrate was freezedried. A cross section of the dried brush grafted substrate (see Figure 1.1,top right image) was imaged using SEM after coating (≈ 5 nm thick) thesubstrate with gold-Pelladium (Au-Pd). The SEM image was used to mea-sure average dry thickness of the brush by image processing in MATLAB.Dry thickness (hdb) of brushes with different polymerization hours is listedin Table 4.2. It should be noted that the measured values have high uncer-tainty primarily due to three reasons: a) nonuniform brush grafting, b) lackof a clean cut at the cross section resulting in the absence of a planar section,c) roughness of the brush grafted pPVC surface making the measurement ofthe thickness of the dry brush inaccurate, and d) difficulty in discerning theboundary between the substrate and the brush. Also, dry thickness of thesample with nine hour polymerization time could not be measured due tothe above mentioned reasons. Measured dry thickness was used to estimate834.4. Theoretical modelinggraft density (ρg) using the following relation [6]:ρg =hdbρNAMn, (4.2)where ρ and NA are density of polymer (taken as 1.20 g/cm3 and convertedto appropriate units) and Avogadro’s number (6.022 × 1023), respectively.Estimated graft densities are given in Table 4.3.To conclude, degree of polymerization does not vary significantly withpolymerization time for the brushes fabricated using the process describedin Section 4.3.1. However, significant change in graft density is observedon varying polymerization time. Also, significant variability in the esti-mated graft density is observed. The surface stress at a given temperatureis ∼ −10 N/m and its magnitude decreases with decreasing graft density.Increasing temperature leads to a monotonic decrease in the magnitude ofthe surface stress. The total change in the surface stress on varying tem-perature from 15◦C to 55◦C also decreases with decreasing polymerizationtime.For the purposes of recollection, we list the limitations and uncertaintiesin the experiment:1. Estimation of surface stress assumes that the Young’s modulus ofpPVC is independent of temperature in the temperature range ofstudy.2. Measurement of the molecular weight assumed that the molecularweight of chains grown on free initiators in the polymerization solutionis the same as the molecular weight of the chains in a brush.3. There is a large variability in the dry thickness of the brush due toinhomogeneous brush growth and roughness of the grafted substratesurface. This leads to large variability in graft density. Also, estima-tion of graft density assumes that the density of the dry brush is thesame as the density of PNIPAm in solid form.4.4 Theoretical modelingIn this section, theoretical modeling of stimuli response of a brush is de-scribed. We extend the SST model of the effect of temperature on a ther-moresponsive brush with Gaussian chains [123, 127] to brushes with Langevinchains. Furthermore, in a theoretical study of polymer solution, the effect844.4. Theoretical modelingof a stimulus (or solvent quality) is taken into account by introducing anenthalpic term in the interaction free energy. In FH theory, solvent qualityis specified by the Flory-Huggins interaction parameter χ [30].For a brush, interaction free energy density (fint) is given by the followingexpression [30, 123]:fint = [(1− Vf ) log(1− Vf ) + χVf (1− Vf )] kBTa3, (4.3)where Vf is the volume fraction of the polymer in a solution. Here, weconsider volume of an effective monomer to be a3. So, monomer den-sity φ = Vf/a3. For a classical polymer, χ = χ(T ) = Θ2T , where Θ isθ-temperature of the solution, the temperature at which excluded volumeinteraction becomes zero [30]. Notice that χ for a classical polymer is depen-dent on temperature but independent of Vf and thus such an expression forχ cannot describe LCST behavior of polymers. So, FH theory is generalizedto include χeff = χ(T, Vf ) which is assumed to have the following form:χeff =∑ni=0 χi(T )(Vf )i, where χi are functions of T , and n is an integer[127, 128, 132]. The dependence of χ on Vf leads to a more complicatedbrush response to stimuli such as the possibility of vertical phase separa-tion into a polymer rich and a solvent rich phases within a brush [123, 127].Phase separation happens when separation into polymer rich and solventrich phases results in an overall decrease in the free energy of the polymersolution as compared to a homogeneous solution (see Figure 4.3).The monomer volume fractions in the two phases after the phase sep-aration are obtained from the conditions that the chemical potentials andosmotic pressure in the two phases should be equal: µ˜(V +f ) = µ˜(V−f ) andΠ(V +f ) = Π(V−f ), respectively, where V+f and V−f are volume fractions inpolymer rich and solvent rich phases, respectively. In a poor solvent, in aclassical brush as well as a PNIPAm brush, monomer density profile mayalso show a jump at the free end of the brush, that is Vf (H¯) 6= 0. Thevalue of Vf (H¯) is obtained from the condition that the osmotic pressure atz¯ = H¯ should be zero (Π(Vf ) = 0). Note that for the given interaction freeenergy density (4.3), chemical potential and osmotic pressure are given by854.4. Theoretical modeling[123, 127]:µ˜(Vf ) = a3∂fint∂Vf=[−1− log(1− Vf ) + (1− 2Vf )χ+ Vf (1− Vf ) ∂χ∂Vf]kBT, (4.4)Π(Vf ) = Vf∂fint∂Vf− fint= −[Vf + log(1− Vf ) + V 2f χ− V 2f (1− Vf )∂χ∂Vf]kBTa3. (4.5)At a given temperature, one checks the condition for phase separation. Ifthe conditions are satisfied then one obtains V −f and V+f . A volume frac-tion value in between these limits (V −f < Vf < V+f ) cannot occur. Thismay result in vertical phase separation within a brush (see Figure 4.3(a)).Similarly, if the condition for a jump in the monomer density profile atthe boundary is satisfied, one obtains Vf (H¯) = V+f (see Figure 4.3(b)). Avolume fraction value smaller than this (0 < Vf < V+f ) cannot occur.4.4.1 SST-L for a thermoresponsive brushWe employ SST with Langevin chains to study the effect of temperature ona thermoresponsive brush. Then, the mean field potential in a brush at z¯is the same as given in (3.4). Chemical potential within a brush is given by(4.4). So, volume fraction variation within a brush can be calculated using(1.13) by replacing V (z) with V¯ (z¯), and µ(φ) with µ˜(Vf ). However, solvingthe equation numerically may turn out to be inconvenient, since µ˜(Vf (H¯))is not necessarily zero. So, we use the following relation to obtain volumefraction profile within a brush [123, 127]:µ˜(Vf (0))− µ˜(Vf (z¯)) = V¯ (z¯), (4.6)which is obtained by taking difference of the two following expressions, thefirst of which is obtained from (1.13) by substituting z¯ = 0.µ˜(Vf (0))− µ˜(Vf (H¯)) = V¯ (H¯)− V¯ (0), (4.7)µ˜(Vf (z¯))− µ˜(Vf (H¯)) = V¯ (H¯)− V¯ (z¯). (4.8)The procedure to obtain Vf (z¯) is to choose Vf (0) and use (4.6) to obtainVf (z¯) and subsequently φ(z) = φ¯(z¯). As pointed out earlier, if the rela-tions in the previous section suggest that phase separation will happen at a864.4. Theoretical modeling(a)f intVf VfV fVf- Vf+ Vf-V f-Vf+V f+Zμ~-(b)f intVfV fVf+V f+Z-0Figure 4.3: The schematic shows variation of free energy with volume frac-tion at a given temperature. Notice that in (a), for V −f < Vf < V+f , freeenergy is minimized if the solution separates into two states with V −f and V+fvolume fractions. This results from the concavity in the free energy vs vol-ume fraction plot. Also shown is the variation of the chemical potential withvolume fraction (middle). In case of phase separation, µ˜(V +f ) = µ˜(V−f ). Vol-ume fraction variation in a polymer brush with vertical phase separation isshown in the right schematic in (a). In (b), globule formation with Vf ≥ V +fis energetically favored. This results in jump in volume fraction profile atthe brush free end.particular temperature, then V −f < Vf < V+f cannot occur within a brush.Then, a jump in the volume fraction profile is obtained (see Figure 4.3(a)).Similarly, a jump in the volume fraction profile at the boundary is obtainedwhen Vf < V+f is not allowed. Then Vf (H¯) = V+f (see Figure 4.3(b)).Calculation of the end density profile is more involved. We start withthe self-consistency equation (1.11).φ¯(z¯) =∫ H¯z¯g¯(ζ¯)E(z¯, ζ¯)dζ¯, (4.9)where φ¯(z¯) = φ(z), g¯(z¯) = Nag(z), and E(z¯, ζ¯) is local stretching at z ina polymer chain with end at height ζ¯. E(z¯, ζ¯) = e(p¯), where p¯ is localstretching force (see (3.2)). Local stretching is finite for finite z¯ and ζ¯. Now,874.4. Theoretical modelingif g¯(ζ¯) is also finite, then as z¯ approaches H¯, φ¯(z¯) should approach zero.However, as shown in the schematic in Figure 4.3(b), this is not always thecase. In such a scenario, end density g¯(ζ¯) diverges. In fact, at any locationin the brush, if monomer density does not change smoothly, end densitydiverges. Also, a sharp change in the monomer density leads to a sharppeak in the end density profile. Here, end density is calculated numericallyafter obtaining the monomer density profile.After obtaining monomer density and end density profiles, the elastic freeenergy density within a brush is calculated using (3.5) and the interactionfree energy density is calculated using (4.3). By numerically integrating thefree energy density over the brush thickness, the total brush free energy(F¯tot = Ftot/(Na)) is obtained. The resultant surface stress due to thepolymer brush is calculated using the following relation [23]:τ¯s =∂F¯tot∂xx= −ρg ∂F¯tot∂ρg(4.10)where τ¯s = τs/(Na).∂F¯tot∂ρg, is also obtained numerically.4.4.2 Resultsχ is crucial to the study of stimuli response of a brush. The results presentedin this section are based on empirically obtained effective χ of PNIPAm-water solution. The choice is motivated by the presence of PNIPAm inthe brushes used in the experiment. However, note that χ for a randomcopolymer PNIPAm-co-PDMA is not available.Decades of research on phase diagram of PNIPAm has not yielded adefinitive χ for PNIPAm [122]. Here, we use χ of PNIPAm-water solutionreported in [133]. The choice is guided by the fact that the vertical phaseseparation in a PNIPAm brush predicted by this form of χ has been observedexperimentally [124–126].χ = χ(T, Vf ) =− 12.947 + 0.044959T + 17.920Vf − 0.056944VfT+ 14.814V 2f − 0.051419V 2f T. (4.11)This empirical result is based on fitting the experimentally obtained demix-ing temperature of PNIPAm solution on the χ = χ(T, Vf ) model proposedin [132]. Solutions up to very high PNIPAm mass concentration (/ 0.7)were used in the experiment. The number averaged molecular weights ofPNIPAm chains used in the measurement were < 105 g/mol. However, it884.4. Theoretical modeling0 0.2 0.4 0.6 0.8 1Vf-0.4-0.3-0.2-0.100.1a3fint(kBT)T=20T=30T=39T=500 0.2 0.4 0.6 0.8 1Vf-1-0.500.5 (kBT)T=20T=30T=39T=50Figure 4.4: Variation of interaction free energy (top) and chemical potential(bottom) in a PNIPAm brush with changing volume fraction at differenttemperatures. It can be observed that the free energy plot is convex at20 ◦C and phase separation does not happen. Observe concavity in the freeenergy plot at 39 ◦C and 50 ◦C. They lead to a jump in the monomerdensity profile at the brush free end. Vertical phase separation occurs at30 ◦C, which can be inferred from the chemical potential plot correspondingto 30 ◦C.is pointed that demixing temperature is chain length independent in highconcentration solution.We start by calculating the variation of interaction free energy densitywith volume fraction at each temperature of interest (see Figure 4.4). Then,the conditions for phase separation or jump in the volume fraction profile894.4. Theoretical modelingare checked. Observe from Figure 4.4 that the volume fraction profile of abrush will be smooth at temperatures ≈ 20 ◦C. At ≈ 30 ◦C, vertical phaseseparation will occur. At temperatures ' 30 ◦C, a jump in the profile atthe free end of the brush will be observed.Based on the above information, monomer density and brush height, enddensity and surface stress due to a brush are calculated and are reported inthe following sections.Monomer density0 0.05 0.1 0.15 0.2 0.25 0.3 0.35Z/(Na)00.20.40.60.8(Z)T=20T=30T=39T=500 0.1 0.2 0.3 0.4 0.5 0.6Z/(Na)00.20.40.60.81(Z) T=20T=30T=39T=50Figure 4.5: Variation of number density with the distance from the graftingsurface for two graft densities: ρg = 0.05 (top) and ρg = 0.5 (bottom).Monomer density profile for brushes of two graft densities are shown inFigure 4.5. Good solvent condition is observed at lower temperatures. So,904.4. Theoretical modelingmonomer density profile is continuous and reaches zero at the brush freeend. Also, the profile is close to parabolic for the lower graft density and isstep like for the higher graft density. The monomer density profile changeswith increasing temperature. It becomes step like at higher temperatures.Notice the drastic change in the profile for the lower graft density in Fig-ure 4.5. But the effect of temperature on the profile for higher graft densityis minimal. At 30 ◦C vertical phase separation in the brush is observed. Athigher temperatures, notice the jump in the monomer density profile at thebrush free end. Also, temperature affects brush height at low graft densities,specifically below 33 ◦C (see Figure 4.6). However, its effect on the hight ofhigh graft density brushes is very small.20 25 30 35 40 45 50Temperature (°C)00.10.20.30.40.50.6H/(Na)IncreasinggFigure 4.6: Variation of brush height with temperature for different graftdensities (ρg = 0.01, 0.02, 0.03, 0.05, 0.08, 0.1, 0.2, 0.4, 0.5). Notice thattemperature has little effect on the brush height at very high graft densities.End densityEnd density profile and its variation with temperature for two graft densitiesis shown in Figure 4.7. In the good solvent case (T = 20 ◦C), end density isfinite everywhere and reaches zero at the free end. However, at higher tem-peratures at which jump in monomer density profile (either vertical phaseseparation or jump at the brush free end) occurs, end density diverges andapproaches infinity. At (T = 30 ◦C), end density profile has a discontinuitywithin the brush. For higher temperatures, end density diverges near thefree end of the brush. Also, notice that the end density curve for ρg = 0.5914.4. Theoretical modeling0.05 0.1 0.15 0.2 0.25 0.3Z/(Na)00.51Nag(Z)T=20T=30T=39T=50Verticalasymptotes0 0.1 0.2 0.3 0.4 0.5 0.6Z/(Na)0246810Nag(Z)T=20T=30T=39T=50VerticalasymptotesFigure 4.7: Variation of end density with the distance from the graftingsurface for two graft densities: ρg = 0.05 (top) and ρg = 0.5 (bottom).at 20 ◦C has a sharp peak. This results from the sharp drop observed inmonomer density profile in Figure 4.5.Surface stressThe variation of the surface stress (comparative) with graft density is plottedin Figure 4.8. Notice a sharp increase in the surface stress with increasinggraft density. Also, the plot suggests that for ρg & 0.25, surface stress willstay compressive even at 50 ◦C.The resultant surface stress for brushes with different graft densities isplotted with increasing temperature in Figure 4.9. For low graft densities,the resultant surface stress is positive and has a minimum near 30 ◦C. For924.4. Theoretical modeling0.1 0.2 0.3 0.4 0.5g-0.8-0.6-0.4-0.200.2s/(Na) (kBT 0/a3 )T=20T=50Figure 4.8: Variation of surface stress with graft density at two tempera-tures. Magnitude of the surface stress increases with increasing graft density.(T0 = 293.15 K)intermediate graft densities (ρg = 0.08 − 0.2), surface stress increases withtemperature and is compressive below 33 ◦C for ρg = 0.2. In fact near ρg =0.2, a maximum increase in surface stress due to the change in temperatureis observed. At high graft densities, it is compressive at all temperatures.Near ρg = 0.4, the change is the surface stress due to changing temperatureis very small. Surface stress shows decreasing pattern with temperature atρg = 0.5 (except near 30◦C). This is likely a consequence of the fact thatbrush height increases and chain ends concentrate towards the tip of thebrush with increasing temperature in a brush with ρg = 0.5 (see Figure 4.7).This leads to an increase in the stretching free energy of the brush withincreasing temperature, like in a classical brush. Also, contribution of elasticstretching of chains in surface stress is much larger than the contributionfrom interaction of monomers for high graft densities. So, like in a classicalbrush, the surface stress increases with temperature.To summarize, the effect of temperature on the structure of a PNIPAmbrush and the surface stress is studied using SST-L. Among the temperaturesat which calculations have been performed, vertical phase separation in thebrush occurs at 30 ◦C. On increasing the temperature further, a jump inmonomer density profile occurs at the free end of a brush. Surface stressis tensile at low graft densities and has a minimum near 30 ◦C. Surfacestress is compressive at high graft densities, and its magnitude increaseswith increasing graft density. The effect of temperature on the surface stress934.5. Comparison between SST-L and experiment20 25 30 35 40 45 50Temperature (°C)-0.02-0.0100.010.020.030.04s/(Na) (kBT0/a3)g=0.01g=0.02g=0.03g=0.05g=0.08g=0.1g=0.220 25 30 35 40 45 50Temperature (°C)-0.7-0.6-0.5-0.4-0.3-0.2s/(Na) (kBT0/a3)g=0.4g=0.5Figure 4.9: The change in the resultant surface stress with increasing tem-perature for different graft densities. Notice that for low graft densities,the resultant surface stress is positive. It is compressive below 33 ◦C forρg = 0.2. At high graft densities, it is compressive at all temperatures.varies considerably with graft density. For low graft density brushes, surfacestress is minimum near 30 ◦C. For intermediate graft densities, it increaseswith increasing temperature. At high graft densities, surface stress decreases(increases in magnitude) with increasing temperature.4.5 Comparison between SST-L and experimentIn this section, we attempt to estimate the surface stress due to a brush bysubstituting the molecular scale parameters of the brush measured in the944.5. Comparison between SST-L and experimentprevious section into SST. This estimation requires us to know:1. graft density,2. length of an effective monomer,3. number of effective monomers in each chain, and4. Flory-Huggins parameter χ.We have an estimate of graft density. However, it is important to to pointout that the measured values have a large variability. Length of an effectivemonomer is the Kuhn length of polymer chains in the brush. However, thereis no consensus on Kuhn length of a PNIPAm polymer chain in literature[123]. In fact, it is known to depend on temperature as well as concentration.So, we use a very rough estimate for this, as described latter on. Since wehave an estimate of the degree of polymerization of chains in the brush,number of effective monomers can be calculated if we know the length of arepeating unit in the polymer chains. Unfortunately, there is no accuratevalue available. Finally, χ depends on the polymers in the brush. In theexperiment, PNIPAm-co-PDMA random copolymer brush is used. PDMAis hydrophilic in 20− 50 ◦C temperature range. Empirical relation for χ ofPNIPAm exists, but χ of PNIPAm copolymerized with PDMA is not known.So, with uncertainties abound, we attempt to calculate the surface stress ina brush using SST and compare the results with experimental results fromthe previous section.First, we estimate the effective monomer length. Ideally, a should beobtained experimentally for the given brush density [134] and at the tem-perature of interest. However, in the absence of experimental data, we as-sume p = 1, since PNIPAm is a flexible polymer, and we estimate effectivemonomer size a from radius of gyration (Rg) of PNIPAm in a solution at20 ◦C given in [135].Rg = 0.022×M0.54w (nm) =Na26, (4.12)where Mw = Mn×PDI. Note that Rg depends on mass density of solutionand needs to be corrected to account for the difference in mass density inour polymer brushes and the polymer solution in which Rg was measured[134]. However, in the absence of the measured mass density data, we usethe uncorrected values. Also, since we do not know N either, we assumeN = degree of polymerization, obtained in the previous section. Then, a is954.5. Comparison between SST-L and experimentestimated to be ≈ 1 nm using the above relation. Furthermore, even thougha ≈ 1 nm is based on the measurement at 20 ◦C, we assume the same valueof a at other temperatures as well.Based on the above assumptions, variation of the surface stress withgraft density is shown in Figure 4.10. The surface stresses are of the sameorder of magnitude as obtained experimentally. The surface stress increasesin magnitude with increasing graft density. Also, they do not go to zero evenat 50 ◦C, as observed in the experiment. Note that the large variability inexperimentally estimated graft density is not shown in the figure.0.4 0.5 0.6 0.7 0.8g (chains/nm2)-30-25-20-15-10-5s (N/m)T=20T=500.4 0.5 0.6 0.7 0.8g (chains/nm2)-120-100-80-60-40-200s (N/m)T=20T=50Figure 4.10: Surface stress variation with graft density obtained from ex-periment (left) and from SST calculation by using experimentally estimatedmolecular scale parameters of the brushes (right). At both the temperatures,surface stress increases in magnitude with increasing graft density, both inthe experiment and in theory. Large variation in experimentally estimatedgraft density is not shown.The discrepancies may be arising from multiple reasons as listed below.1. The foremost reason is that the brush in the experiment consists ofPNIPAm-co-PDMA polymer chains. The semianalytical calculation isbased on PNIPAm.2. Crucially, SST-L uses an empirical relation for χ of PNIPAm pro-posed in [133]. But no consensus exists on the phase diagram andconsequently χ of PNIPAm [122].3. Even though the brush is polydisperse, we neglect the effect of poly-964.6. Conclusionsdispersity at the outset.4. The grafting of brush on a substrate surface is not uniform, result-ing in large fluctuation from mean graft density in Table 4.3. In thesemianalytical calculation, we neglect the effect of variation in graftdensity on a substrate surface.5. The length of an effective monomer is taken to be 1 nm in the calcu-lation. However, this is an approximation. Also, it is known that itvaries with temperature.6. We assume the substrate to be planar and smooth, neglecting the effectof roughness. However, roughness of the substrate surface, speciallywhen peak to valley roughness is comparable to the brush height, cansignificantly alter the behavior a brush.However, even with these limitations and assumptions, we are able to esti-mate the magnitude of the surface stress due to the brush from the molecularscale parameters of the brush closely.4.6 ConclusionsIn this chapter, the effect of temperature on a polymer brush is studied,experimentally as well as using SST-L. The main conclusions are as follows.• PNIPAm-co-PDMA brushes were fabricated by grafting them on ap-PVC thin beam, and surface stress due to them was estimated bymeasuring curvature of a brush grafted beam. The curvature - surfacestress relation (2.68) obtained in Section 2.2 was used to estimatesurface stress. The surface stress due to the brushes is found to be ofthe order of −10 N/m and it decreased in magnitude by ≈ 10 N/mon increasing temperature from 15 ◦C to 55 ◦C.• SST-L was used to study PNIPAm brushes. Vertical phase separationis observed in a brush at 30 ◦C. This leads to a jump in monomer den-sity profile which causes end density profile to diverge at the locationof the jump. At higher temperatures, a jump in monomer density pro-file at the free end of a brush is observed. The monomer density profileof low graft density brushes are considerably affected by a change intemperature. For high graft density brushes, the effect of tempera-ture on monomer density is very small. Finally, surface stress due toa brush increases sharply with increasing graft density. For brushes974.6. Conclusionswith ρg . 0.4, the surface stress in general increases with increasingtemperature. For ρg = 0.5, an opposite trend is observed.• Graft density and molecular weight of the brush have also been esti-mated experimentally and used to estimate surface stress using SST-L.The estimated surface stress is of the same order of magnitude. Also,the surface stress does not reach zero even at 50 ◦C, as observed inthe experiment. However no quantitative conclusions can be drawndue to large number of approximations involved in the estimation ofparameters such as ρg and χ.98Chapter 5Conclusion and future workThis thesis was motivated by the application of a polymer brush in actuationand sensing in particular soft materials such as pPVC film grafted withpolymer brush on its top surface. The aims are to understand a) the stressin a neutral planar polymer brush made of linear homopolymers, b) theeffect of temperature on the surface stress due to a thermoresponsive brush,and c) the mechanics of a thin beam grafted with a polymer brush. Prior tothis work, in the literature, structure of a brush and its other macroscopicproperties have been extensively studied using scaling theory, SCFT, SST,MC, MD etc. These theories establish the relation between molecular scaleparameters of a brush with its structure and other properties. However,stress in a brush remained unexplored. This thesis attempts to address thisgap in fundamental knowledge. Figure 5.1 summarizes the study of stressin a brush and the contributions of this thesis.Conclusions of each chapter were provided at the end of the chapter.Here, we first provide an overview of the work done in Section 5.1. Thenthe main conclusions of the thesis are delineated in Section 5.2. The limita-tions of the work are discussed in Section 5.3. Finally, future directions areoutlined in Section 5.4.5.1 SummaryIn this thesis, (semi) analytical SST and MD simulations were employed inthe study of stress in a brush. We first used SST-G which provides analyticalrelations between stress in a brush and its molecular scale properties for lowgraft densities. We started by deriving analytical expressions for stress in apolymer brush in a marginally good solvent and a θ-solvent by combiningSST for brushes of Gaussian chains with continuum mechanics. To modelthe free energy of interaction, it was assumed that the binary interactionsdominate and hence the higher order interactions can be ignored. The keypoints in the derivation were to obtain a) the free energy density distribu-tion within a brush, and b) the change in the free energy density due to asmall uniaxial strain applied to the brush along a direction parallel to the995.1. SummaryρgScaling  theoryResultant  surface stress SST-GSST-L Stress distribution Utz and Begley,  2008MD Stress distribution Resultant  surface stress SST-L (PNIPAm  brush)This thesisStress in a polymer brushResultant  surface stress Experiment (Macroscale)Graft density,Stress distribution MushroomFigure 5.1: The figure shows the available literature on stress in a polymerbrush and the contributions of this thesis. Red color indicates poor solventcondition and green color indicates good solvent condition. The colors inbetween indicate solvent qualities in between poor and good.substrate. This gave us an analytical expression for the distribution of stresswithin a brush and the dependence of stress on molecular scale propertiesof the brush.Since the Gaussian chain model of a polymer chain is valid only in thelimit of small extension of a chain, the results obtained in the Chapter 2 werelimited to low graft density brushes only. This motivated the advancement ofa semi-analytical theory based on Langevin elasticity of polymer chains. InChapter 3, SST for brushes with Langevin chains (SST-L) was used to studybrush structure and the stress in a brush, albeit in an athermal solvent. Weobtained semi-analytical expressions for the structural properties of a brushand the stress in a brush. Also, to be able to make a quantitative comparisonwith MD simulation results, a modified Carnahan-Starling equation of statewas used to model the interaction free energy.MD simulations of brushes with graft densities varying over two ordersof magnitude were carried out to validate SST results and to point out theirshortcomings. We used bead-spring model to represent polymer chains ina brush and LJ pair potential to model monomer-monomer interaction. Tosimulate athermal condition, LJ potential was cut-off and shifted to ensure1005.1. Summarythat the interaction between a pair of beads was only repulsive. In thesimulation of different graft densities, we have typically used large enoughchains such that the assumption of strong stretching is satisfied. However,for the lowest graft densities, we had to limit the number of monomers in achain due to the difficulty in equilibrating the simulation system.Stimuli response of a brush was studied experimentally as well as usingSST in Chapter 4. Surface stress and its variation with temperature dueto thermoresponsive brushes grafted to flexible thin beams was estimatedexperimentally from the measurement of the curvature of the flexible beams.To obtain surface stress from the curvature of a beam, a curvature-surfacestress relation was used. The relation was obtained from the analyticalmodel for finite deformation (but small strain) of a beam with a coatingof polymer brush on its top surface. The analytical model also accountedfor the Young-Laplace and the Steigman-Ogden curvature elasticity correc-tions. On neglecting the contributions of the Young-Laplace effect and thecurvature elasticity of surface, the expression for curvature of a beam withsurface layer reduces to the results reported in literature.To study the effect of temperature on a thermoresponsive brush, SSTwith Langevin chains was also employed. FH theory with χ = χ(Vf , T )was used to model the free energy of interaction. We studied the structuralproperties of a brush along with the surface stress due to a brush. Weused the empirical expression for χ(Vf , T ) of PNIPAm in the numericalcomputation.To relate the surface stress with the molecular scale properties of a brushexperimentally, graft density and molecular weight of the polymer chains inthe thermoresponsive brush were estimated. These values were inserted intothe SST model to obtain an estimation of the surface stress in a brush.Finally, an attempt was made to relate the surface stress due to a poly-mer brush and the effect of temperature on the surface stress due to athermoresponsive brush with the molecular scale parameters of the brush,experimentally. This requires us to know a) graft density, b) length of aneffective monomer, c) number of effective monomers in polymer chain, andd) the effective Flory-Huggins parameter for the polymer in consideration.Graft density and molecular weight of the chains in the fabricated brusheswere estimated experimentally. There is no consensus on the Kuhn length ofPNIAPam in the scientific community. Hence, by using the number of effec-tive monomers in the chain the same as the number of repeating units in apolymer chain, length of an effective monomer was estimated to be ∼ 1 nm,after ignoring presence of PDMA in the polymer chains. Flory-Huggins pa-rameter of PNIPAm-co-PDMA random copolymer is also not know. So,1015.2. Conclusionsignoring PDMA, and assuming the expression of χ(T, Vf ) for PNIPAm pro-posed in [133], we evaluate surface stress variation with temperature andgraft density. Note that there is no consensus on the χ(T, Vf ) either, but itleads to vertical phase separation in brushes in some temperature range, aphenomena observed in some experiments.5.2 ConclusionsUsing SST-G, we found that the stress variation is quartic along heightdirection of a brush in a marginally good solvent (see (2.29)) with a maxi-mum near the grafting surface. The stress smoothly goes to zero at the freesurface of the brush. The distribution of stress is different in a θ-solvent,with a stronger dependence on graft density (see (2.35)). In the expressionfor stress in brush, nonbonded interaction contributes twice as much as thechain stretching in a good solvent as observed in (2.29). Their contributionsare equal in a θ-solvent. Surface stress due to polymer brush has strongerdependence on graft density than on molecular weight of polymer chainsin the brush as suggested by (2.36). Surface stress and the surface elasticmodulus for a brush layer are of the same order of magnitude. Hence, the ef-fect of surface elasticity in small strain deformation is negligible (see (2.36),(2.37)).Molecular dynamics simulations carried out in Chapter 3 verified thequartic stress profile prediction of SST-G in the low graft density regime ρg ≤0.03 and within the bulk of the brush and away from depletion layer and tail.Gaussian elasticity assumption is valid in this range due to small extensions,as quantified by βe. Our simulations also showed the parabolic monomerdensity profile within the bulk of the brush for low graft density brushes,which we consider as a validation of our simulation. Scaling of height withgraft density matches with the prediction from SST-G and scaling theoryfor ρg < 0.2. For higher graft densities, however, it shows some deviations(see Figure 3.10).Gaussian elasticity of chains breaks down at higher graft densities andleads to discrepancies between the SST-G and MD results. However, pre-dictions from SST-L for monomer density (Figure 3.8), end density (Fig-ure 3.11), brush height (Figure 3.10), and stress (Figure 3.16) agree wellwith MD simulations at higher graft densities (ρg ≥ 0.05). Also, predictionsof monomer density from SST-L show a smooth transition from SST-G atlow graft densities to those of step-profile used in scaling theories at highgraft densities. Surface stress predicted by SST-L matches closely with MD1025.3. Limitationsresults for high graft density. For lower graft densities, SST-L over-predictsthe surface stress. Also, scaling exponents of surface stress with graft den-sity obtained from the two theories for ρg ≤ 0.08 are different. The precisereason for this is yet to be understood, though one can speculate aboutthe effect of excluded volume correlations ignored in SST-L, the validity ofstress measures, differential end-stretching with distance from the graftingsurface, MD potentials and low values of N .Surface stress due to PNIPAm-co-PDMA brushes were estimated bymeasuring curvature of brush grafted beams. The surface stress due tothe brushes is found to be of the order of −10 N/m and it decreased inmagnitude by ≈ 10 N/m on increasing temperature from 15 ◦C to 55 ◦C.SST-L was used to study effect of temperature on PNIPAm brushes.Vertical phase separation is observed in a brush at 30 ◦C. This leads to ajump in monomer density profile which causes end density profile to divergeat the location of the jump. At higher temperatures, a jump in monomerdensity profile at the free end of a brush is observed. The monomer densityprofiles of low graft density brushes are considerably affected by a changein temperature. For high graft density brushes, the effect of temperatureon monomer density is very small. Finally, surface stress due to a brushincreases sharply with increasing graft density. For brushes with ρg . 0.4,surface stress in general decreases in magnitude with increasing temperature.For ρg = 0.5, an opposite trend is observed.An attempt to estimate the surface stress due to a brush by pluggingin the estimated (or approximated) molecular scale parameters of the brushin SST-L was also made. Even with a large number of approximations in-volved and uncertainties in the measured parameters, the estimated surfacestresses are of the same order of magnitude as estimated from the curvaturemeasurement. Also, the surface stress does not reach zero even at 50 ◦C, asobserved in the experiment. However, no further conclusions can be drawndue to large number of approximations involved in the estimation.5.3 LimitationsThe (semi) analytical models (SST-G and SST-L), MD simulations and ex-perimental estimations employed in the study have multiple assumptions,which limits their applicability. The limitations of the work are outlinedbelow.1035.3. LimitationsLimitations of modelingThe analytical SST-G as well as semi-analytical SST-L assumes polymerchains to be freely jointed chains. However, real chains have excluded vol-ume, which is ignored in these theories. This leads to inaccuracy in theenergy scaling obtained from these theories. Also, the study is limited tobrushes made of neutral linear homopolymer chains.Strong stretching theories, employed extensively in this thesis, consideronly classical paths of chains in a brush ignoring the contribution of fluctu-ations of a chain away from the classical paths to the partition function of abrush. Specifically, the effect of depletion layer and tail cannot be accountedfor in this model. In the bulk of the brush, this approximation works wellif the chains in a brush are strongly stretched. However, for brushes withshort chains, the results have only limited validity. Furthermore, the theorypresented in this thesis ignores polydispersity entirely. But a real polymerbrush is always polydisperse.The derivation of stress in a polymer brush is based on the propertiesof a brush grafted to a rigid substrate. The effect of substrate flexibilityon these results is not studied. Also, a substrate is assumed to be flat andsmooth. Though, surface of a real substrate, for example the surface ofa pPVC substrate used in the experimental work presented in this thesis,is rough. The effect of the roughness of a substrate may be considerablespecifically when the peak to valley distance is of the order of brush height.However, it has been neglected.Limitations of MD simulationMD simulations reported in the thesis are generic with no chemical speci-ficity. In the simulation, polymer chains are represented as bead-springs andinteraction among the beads is governed by the ubiquitous LJ pair potential.In short, they contain minimal physics in simulating polymer chains. Thismakes it difficult to capture polymer specific properties such as behaviourof a PNIPAm brush. Also, stimuli response of a brush of LCST polymeris difficult to simulate using this model. Furthermore, the substrate wasassumed to be rigid in MD simulation as well.Limitations of experimentWe attempted to compare the experimentally obtained surface stress dueto a thermoresponsive polymer brush and its variation with temperature1045.4. Future workwith the surface stress obtained from SST-L. This was aimed at experimen-tally validating SST-L calculations. However, an uncertainty in parametersprevented a clear one to one validation as discussed below.Molecular weight measurement was based on polymer chains grown fromfree initiators in the polymerization solution. However, growth of the freechains are not affected by the presence of nearby chains as is the case in abrush. This may lead to differences in actual molecular weight of chains ina brush and the measured molecular weight.Graft density estimation was based on the dry thickness of the polymerbrush and the assumption that the density of the dry brush is the sameas the density of PNIPAm in solid form. Also, the estimation requires themolecular weight. Furthermore, considering the brush samples used in theexperiment were macroscale, there is considerable variability in the brushdry thickness through the grafted surface and hence in the estimated graftdensity.The length of an effective monomer of the random copolymer brush(PNIPAm-co-PDMA) is not known. In fact, there is no consensus on theKuhn length of a PNIPAm polymer chain either. This hampered the calcu-lation of the total number of effective monomers in a chain as well.The χ parameter for the random copolymer is not known. There is noconsensus on χ of PNIPAm either [122]. So, we used one of the expressionsfor χ of PNIPAm available in the literature in SST calculation.The pPVC substrate has roughness which affects the measurement ofdry brush in SEM imaging and consequently introduces inaccuracy in thegraft density estimation. Also, roughness may invalidate the planar brushassumption in SST.5.4 Future workLimitations of the work and the problems which could not be studied ad-equately point to the future directions. Also, the study of stress can beexpanded to new systems. A few avenues for future exploration are outlinedbelow.Polymer brush model with real polymer chainsSST-G models can be improved by using real chains instead of FJC in thestress calculation. The work reported in [118], valid in the limit of smallextension of real chains, can form the basis for such a development. However,1055.4. Future workwe are not aware of any (semi) analytical models for large stretching of realchains.Inclusion of a depletion layer and a tail in the modelSince SST ignores fluctuations in chain paths, it is unable to capture deple-tion layer near the grafting surface in a brush and the tail at the free end ofthe brush. However, these effects can be included in the theory presented inthis thesis following the estimation of the monomer density and end densityin depletion layer and tail as presented in [136].Polymer brush model with random copolymer chainsBrushes in our experiment were random copolymer brushes made of anLCST polymer and a hydrophilic polymer. There is no brush model availablein literature describing the stimuli-response of such a brush. Such a modelcan be developed in future.MD simulation with realistic potentialsCoarse grained models of specific polymers can be used in MD simulation ofa brush to obtain results which can be directly compared with experimentalfindings. Also, this will help understand the stimulus response of a brush.Improvement in experimentsOne can use microscale cantilevers in the measurement of surface stress dueto a brush instead of macro-cantilevers used in the experiments reported inthis thesis. This will decrease nonuniformity in brush grafting. Also, brushstructure measurement using methods such as ellipsometery or small angleneutron scattering will help validate the predictions from SST.Effect of the substrate roughnessAll surfaces have some roughness. Peak to valley roughness of the order ofbrush height may play a considerable role in determining the brush prop-erties. But the literature on the effect of roughness on brush properties isscant. Also, roughness introduces inaccuracy in the measurement of drythickness of a brush which is used to estimate graft density. So, this is avery challenging problem but extremely useful to study from the point ofview of applications.1065.4. Future workSubstrate flexibilityThe thesis assumes substrate is planar and rigid, and uses results from SSTof brushes grafted to a planar rigid substrate when deriving stress in abrush. However, a very flexible substrate such as a lipid bilayer graftedwith polymer brush, will deflect considerably, invalidating the planar brushassumption. In such a case, a brush may become spherical or cylindrical orbe in a transition region between planar and cylindrical or spherical brush.The stress in a brush in such a system is yet to be studied.Polymer brush with semiflexible chainsThe SST model used here is confined to FJC model of polymer chains. Itcan be extended to semiflexible chains by using WLC model to account forfree energy of stretching of polymer chains.Polyelectrolyte brushPolyelectrolyte brushes are used extensively, for example, beams graftedwith single stranded or double stranded DNA are used in sensing. Theyalso generate large surface stress due to presence of charge. So, consideringtheir potential applicability, extending the theory to polyelectrolyte brushesis an obvious future direction.Effect of solvent flowThis thesis uses brush models with implicit solvent to study stress in abrush. In these models, solvent is assumed to be stationary. However, inmany applications, brush is immersed in a flowing solvent. 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Matsen, “Finite-stretching corrections to themilner-witten-cates theory for polymer brushes,” The European Phys-ical Journal E, vol. 23, no. 2, pp. 135–144, 2007.120Appendix AA general force-extensionscaling relation for realchainsA.1 Uni-axial force-extension of a single chainDescribing the force-extension relation of a single polymer chain is a canon-ical problem of fundamental theoretical interest with wide ranging practicalapplications: from rubber elasticity to DNA elasticity. The central questionis to relate the magnitude p of force p = pe applied to a polymer chain andextension R along the direction of application of force e. Two cases must bedistinguished: extension under imposed end forces (force controlled) and ex-tension under imposed displacements (displacement controlled) across theends of a polymer chain. These two distinct physical situations lead todifferent boundary conditions, and different ensembles in the statistical de-scription of the conformations of a polymer chain. For a very long polymerchain (thermodynamic limit) however the two cases lead to the same force-extension relation. In addition to the above distinction, the polymer chainitself has several idealized representations: freely jointed chain (FJC), freelyrotating chain (FRC), worm like chain (WLC) among others.Regardless of the idealization adopted and the manner in which a poly-mer is extended, we seek a functional relation between p and R. It is wellknown, for over 200 years now, that entropic elasticity plays a significant rolein this relation. Experiments reveal that increasing temperature contractsa polymer (such as an elastic band) instead of the usual expansion exhib-ited by metals. There are two principal methods to model this mechanicalbehavior:1. following a macrosopic and phenomenological approach inspired byexperiments. This phenomenological continuum approach has beenpioneered by Rivlin and Mooney.121A.1. Uni-axial force-extension of a single chain2. following a microscopic andstatistical mechanics based approach.A.1.1 Entropic springThe entropic spring constant can be derived from many approaches such as:1. Scaling arguments pioneered by Flory and de Gennes.2. Statistical mechanics applied to chain idealizations.3. Fluctuation-Dissipation theorem, again from statistical mechanics.4. Phenomenological continuum descriptions of Rivlin and Mooney.Here we present analysis based on scaling arguments.A.1.2 ScalingNo exlcuded volume effectsThink of a polymer chain conformation as that of a random walk in a d-dimensional space. When the chain is not subjected to any external stretch-ing each monomer is located by a random walk: for example a monomer munits away from a reference monomer is reached by m random walks. Asthe chain is stretched random walks are directed by the externally imposedstretch. Over length scales (l) much shorter than the root-mean-square end-to-end distance of an unstretched chain, RF = N12a, that is l << N12a,random walks are unperturbed by the stretching. Here, a is the effectivemonomer size and N is the number of such monomers making up the poly-mer chain. Thus monomers within a blob of size l are oblivious to externalstretch and fluctuate randomly. However, on larger length scales the se-quence of blobs are oriented along the stretch direction, that is they arenot randomly oriented. If a blob contains g number of monomers then theblob size is given by l2 ≈ a2g. In the polymer chain comprising N Kuhnsegments, there are Ng blobs and hence the chain extension isR ≈ lNg≈ N a2l→ l ≈ Na2Rg ≈ N2a2R2. (A.1)Since each blob is forced to go along the stretching direction, one degree offreedom is lost per blob. Hence from equipartition theorem, the free energy122A.1. Uni-axial force-extension of a single chainincrease is kBT per blob. The free energy increase in the extended chain isthenF ≈ kBT Ng≈ kBT R2Na2≈ kBT(RRF)2(A.2)The force extension relation is given by:p =∂F∂R≈ kBT RR2F≈ kBTlHooke’s law (A.3)The length scale l of the blob thus corresponds to kBT of stored energy.Some limitations of the above scaling argument must be noted.1. Scaling relation is not exact, arbitrary constants need to be determined(from experiment, presumably) in the force-extension relation in (A.3).2. The random walks of a real chain can be self avoiding. It arisesfrom the repulsion between monomers (within a chain and from thesurrounding chains), the so-called excluded volume effects. Flory,de Gennes and Pincus have elaborated on this aspect. In particu-lar de Gennes’ renormalization of the above linear Hooke’s law, inwhich RF in (A.3) assumes the expression for a self-avoiding chain,does not account for the reduction in excluded volume effects whenRF << R << Na. It turns out from the scaling analysis of Pin-cus that the force-extension is nonlinear for R >> RF , as we shall seebelow.With excluded volume effectsFlory argument: Consider a polymer of N monomers swollen to sizeRF > N12a. Flory assumes that monomers are distributed in a volumeR3F . Probability of a second monomer being within the exclude volume v ofany given monomer is the product of v and number density of monomers inthe pervaded volume NR3F. Taking an energy cost of kBT per exclusion, theenergy cost per monomer is NR3FvKBT and since there are N monomers, theenergy cost for exlcuded volume for the entire chain is:Fexc ≈ kBTvN2R3F. (A.4)To the above we add the entropic contributionFent ≈ kBT R2FNa2. (A.5)123A.1. Uni-axial force-extension of a single chainto obtain the total free energyF ≈ kBT R2FNa2+ kBTvN2R3F. (A.6)The minimum free energy configuration is given by the size where RF sat-isfies ∂RFF = 0 and is obtained asRF ≈ v 15a 25N 35 (A.7)An important point to recognize is that Flory’s theory leads to a universalpower law for the size of a polymerRF ∼ Nν . (A.8)ν = 35 for a swollen linear polymer and ν =12 for an ideal linear chain.Including the case with the exlcuded volume interactions, our earlier scalinganalysis in a general form can be expressed as:size of blob: l ≈ agνend-to-end distance: R ≈ Ngagνfree energy: F ≈ kBT Ng≈ kBTN[RNa] 1(1−ν)≈ kBT(RRF) 1(1−ν)(A.9)Setting ∂RF = p we obtainp ≈ kBTRF(RRF) ν(1−ν). (A.10)For an ideal chain, this relation simplifies to (A.3). For a chain with excludedvolume interactions, the above relation suggests nonlinear force-extension re-lation. See Figure A.1 for force-extension relation of a chain with excludedvolume effect obtained from MD simulation of a bead-spring chain describedin Chapter 3. The curve shows deviation from linearity as expected. How-ever, we do not observe a region with slope 3/2 due to small number ofbeads in the chain [70].We can make the following remarks on the above improved scaling anal-ysis:1. For an ideal chain without exlcuded volume interactions, we set ν = 12in the above to obtain the Hooke’s law p ≈ kBTR/R2F .124A.1. Uni-axial force-extension of a single chain2. For a good solvent, taking ν = 35 , we obtain p ≈ kBT/RF (R/RF )32 .This suggests that excluded volume effects produce nonlinearity forRF << R << Na!3. Appropriate choices for ν can be made depending on the solvent qual-ity.4. de Gennes suggested the following linear force-extension relation byreplacing the Flory radius of an ideal chain (N1/2a) with the generalexpression for the Flory Radius RF ≈ Nνa: f = 3kBT RR2F . This istrue only for weak stretching defined as R << RF .5. Pincus developed a scaling analysis that accounts for reduced ex-cluded volume interactions in strong stretching, defined as the regionof extension RF << R << Na, which validates the nonlinear force-extension relation above. Further a more general relation of the formR = RFφ(RF , f) is derived (see Eq (II.5) in Pincus’ 1976 paper).6. One of the draw back of our scaling analysis is that the force-extensionrelation is unbounded. This is not physical. The divergence shown byFJC and WLC models of polymer chains when R is large is not shownby this model.10 1 10 2<z>10 -110 0pRFFigure A.1: Force-extension curve for a bead spring polymer chain with 200beads. Notice that unlike a Gaussian chain, the curve is not linear and itsincreasing. We do not see a region with constant slope of 3/2 however. Toobserve the region with constant slope of 3/2, one needs number of beadsto be & 105 [70].125Appendix BComparison of brush heightsobtained from differenttheoriesHere we obtain the variation in height of a polymer brush with a changein solvent quality using mean field Flory theory, and compare it with SSTprediction in [74] and scaling theory based prediction in [23].To start with, we assume a step profile for the brush. Then, monomerdensity in the brush is a constant and is given by Nρg/H. Employing Florylike arguments, we can write the free energy of a polymer chain in a brushas a sum of the contributions from the chain stretching and the polymersolvent interaction, accounting for ternary interaction as follows:Fstep =(3H22Na2+12v(NρgH)2 Hρg+16w(NρgH)3 Hρg)kbT, (B.1)where H/ρg is the volume occupied by a polymer chain. By minimizingFstep with respect to the height of the brush (H) (dFstep/dH = 0), we canobtain the expression for the height of the brush. Below we discuss heightapproximation in good, θ and poor solvent regime.In a good solvent condition, binary interaction dominates and contri-bution from ternary interaction can be ignored [74]. In θ solvent, v = 0and hence contribution from binary interaction is zero. In a poor solventwith large magnitude of binary interaction parameter but with a negativesign, energy contribution from the chain stretching can be ignored [74]. Theabove assumptions allow derivation of asymptotic expressions for the heightof a brush in solvents of different solvent qualities. The tables below showthe heights obtained from the above approximation with predictions fromSST and scaling theory.Scaling of brush height with N and ρg obtained from each of the theoriesis the same. Also, for a classical polymer, v = a3(1 − 2χ) and w = a6 [30].So, we can see that scaling with respect to monomer size is also consistent.126Appendix B. Comparison of brush heights obtained from different theoriesTable B.1: Comparison of expressions for height of a brush in good (thoughapplicability is limited to solvent conditions leading to weak excluded volumeinteraction), θ and poor solvents obtained from mean field Flory theory [31]and SST [74] and scaling theory [23, 32].Flory arguments SST ScalingGood solvent(16)1/3v1/3ρ1/3g a2/3N(4pi2)1/3v1/3ρ1/3g a2/3N ∼ ρ1/3g a5/3Nθ solvent(19)1/4w1/4ρ1/2g a1/2N4pi(112)1/4w1/4ρ1/2g a1/2N ∼ ρ1/2g a2NPoor solvent 23w|v|ρgN23w|v|ρgN ∼ a3ρg|1− 2χ|−1NFor the good solvent case, SST gives 34% larger brush height comparedto the height derived from Flory arguments, but it is 26% lower than theprediction from the blob model.127Appendix CLinearization of cantileverbeam equationLet us first define a nondimensionalization scheme.τ˜s =τsE¯h, τ˜0s =τ0sE¯h, C˜ =CE¯h3, E˜s =EsE¯hκ˜ = κh ≈ dβdXh. (C.1)Under this scheme, (2.66) and (2.67) transform to:(λ− 1) + 12ν1− ν τ˜sκ˜+ τ˜s = 0 (C.2)112(κ˜− ν1− ν τ˜sκ˜)− 12τ˜s + C˜κ˜ = 0. (C.3)(C.2) is used to obtain stretch in the mid plane.(λ− 1) = −(1 +12ν1− ν κ˜)τ˜s (C.4)Also, using (2.48), the nondimensionalized effective surface stress τ˜s can beexpressed as:τ˜s = τ˜0s + E˜s((λ− 1)− κ˜2). (C.5)Substituting expression for λ in (C.4) into the above equation and solvingfor τ˜s gives:τ˜s =τ˜0s − E˜s κ˜21 + E˜s(ν1−νκ˜2 + 1) . (C.6)128Appendix C. Linearization of cantilever beam equationSubstituting the above expression for τ˜s in (C.3), and expressing the equa-tion as a polynomial equation in κ gives:12ν1− ν E˜s(2 + 12C˜)κ˜2 +(1 + 4E˜s + 12C˜(1 + E˜s)− ν1− ν τ˜0s)κ˜− 6τ˜0s = 0. (C.7)It can be noticed that the coefficients of κ˜2 is of the same order or of ordersmaller than the coefficient of κ˜. So, for κ˜ << 1, quadratic term in κ˜ canbe ignored to obtain the following expression for κ˜:κ˜ =6τ˜0s1 + 4E˜s + 12C˜(1 + E˜s)− ν1−ν τ˜0s. (C.8)Shifting back to the dimensional form, we obtain:κh ≈ 6τ0sE¯h+ 4Es +12Ch2+ 12CEsE¯h3− τ0s ν1−ν. (C.9)129Appendix DSST-L calculationsD.1 Steps to calculate stress in a brushFirst, choose models for chemical potential (µ(φ) = µ˜(Vf )), and mean po-tential field (V (z¯)) (depends on choice of chain model, for example Gaussianor Langevin chain). Then, follow the steps below to obtain stress. Note thatequations specific to a brush with Langevin chains and chemical potentialfollowing modified CS equation of state are referred to within brackets.1. Choose a brush height H¯.2. Obtain Vf by solving (1.13).3. Obtain graft density for the chosen brush height using (3.9).4. Obtain end density using (3.10).5. Obtain density of the free energy of interaction (fint(z¯)) using (3.7).6. Obtain density of the free energy of stretching (fel(z¯)) (using (3.5)).7. Obtain the free energy density (f(z¯)) by the adding the interactionand stretching contributions obtained above.8. Obtain the derivative of the end density with respect to the appliedstrain (∂g¯(ζ¯)∂xx ) using (3.14).9. Obtain the derivates ∂u¯∂xx and∂Vf∂xx(by simultaneous solving (3.19) and(3.21) with boundary condition (3.20)).10. By fitting a high order polynomial, make ∂u¯∂xx curve smooth. This isneeded for the next step.11. Obtain ∂zz∂xx using (3.18).12. Obtain(∂fint∂xx)(using (3.23)).13. Obtain ∂fel∂xx (using (3.24)).14. Obtain ∂f∂xx by adding the derivatives corresponding to interaction andstretching free energy densities obtained in the last two steps.130D.2. Estimation of ∂zz∂xx from MD/experiment15. Obtain stress variation within a brush using (2.15).D.2 Estimation of ∂zz∂xx from MD/experimentThe assumption that the monomers within a thin layer of a brush remainin the layer as the brush is stretched with a strain xx, gives the followingexpression for ∂zz∂xx :∂zz∂xx= − 1φ∂φ∂zz− 1 (D.1)If one knows the solvent conditions, then monomer density is function of thegraft density and location z: φ = φ(ρg, z). Then its derivative with respectto xx can be written as:∂φ∂zz=∂φ∂ρg∂ρg∂zz+∂φ∂z∂z∂zz(D.2)= −ρg ∂φ∂ρg+∂φ∂z∂u∂zz(D.3)where we have used the relations ρg = ρ0g(1− xx) (ρ0g is graft density beforedeformation) and the displacement of the layer at height z due to the appliedstrain xx is u. Substituting this relation in (D.1) and multiplying all theterms with φ yields:φ∂zz∂xx= ρg∂φ∂ρg− ∂φ∂z∂u∂zz− φ (D.4)Using the relation (3.18) in the left hand side term in the above, bringingthe second term in the right hand side to the left hand side of the equationand combining the left hand side terms yields:∂∂z(φ∂u∂zz)= ρg∂φ∂ρg− φ (D.5)Notice that the right hand side terms are graft density, monomer density andderivative of monomer density with respect to graft density, all the termsmeasurable in an MD simulation or experiment!Using (D.5), ∂u∂zz can be obtained by integrating the right hand sideterm:∂u∂zz=1φ∫ z0(ρg∂φ∂ρg− φ)dz (D.6)Using (3.18), ∂zz∂xx can be obtained from∂u∂zz.131Appendix EGenerating initialconfiguration of a brush withapproximately parabolicdensity profileFor low graft density brushes, achieving hight βs to be able to make com-parisons with SST requires large number of beads per chain (N). However,equilibrating a brush with large N is very difficult due to the fact that re-laxation time for a chain increases very fast with increasing N . So, startingfrom a good initial configuration is imperative. As brushes at low graft den-sity show parabolic profile, SST-G results, summarized in Section 2.1.1, canbe used to generate initial brush configuration. Below we describe the stepsinvolved.1. Determine total number of chains, Ng, in the brush. Define a surfaceand decide locations of Ng grafting points. I chose equispaced gridpoints as grafting points.2. Calculate a tentative brush height, Ht, using (2.4). Divide the regionbetween z = 0 − Ht into nbins. Choose an optimal value of nbins sothat g(ζ) vs ζ is close to the curve predicted by (2.5).3. Using end probability PE(ζ) = g(ζ)/ρg, where g(ζ) is found using(2.5), calculate number of chain ends in the ith bin as NCE(i) =round(Ng ×∫ zup(i)zlow(i)PE(ζ)dζ). Here zlow(i) and zup(i) are the lowerand upper boundaries of ith bin. Ng −∑iNCE chains are added tothe bin with the maximum NCE .4. Starting from the first bin, randomly assign NCE(i) grafting points toeach bin. By doing this, we ensure that a chain starting from a givengrafting point ends in a particular bin.132Appendix E. Generating initial configuration of a brush with approximately parabolic density profile5. Now we start defining chains originating from each grafting locationone by one. For each grafting location we already know the chainend ζ. Also, we know dzdn = E(z, ζ) from (2.6). Using ∆n = 1,we find ∆z = E(z, ζ). Hence, z-coordinate of (i + 1)th bead in agiven chain is given by, zi+1 = zi + E(zi, ζ). ∆x and ∆y, such thatxi+1 = xi + ∆x and yi+1 = yi + ∆y, are randomly chosen (with theconstraint that beads do not go outside the simulation box) to ensurethat√∆x2 + ∆y2 + ∆y2=bond length. If zk ≥ ζ, we constrain zi = ζfor all i ≥ k.133

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