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Computer simulation of infrared spectra and structures of molecular nanoparticles Firanescu, George 2009

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Computer Simulation of Infrared Spectra and Structures of Molecular Nanoparticles by George Firanescu A THESIS SUBMITTED IN PARTIAL FULLFILLMENT OF THE REQUIREMENT FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Chemistry) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) August 2009 © George Firanescu 2009 Abstract Molecular nanoparticles, i.e. molecular aggregates held together by weak intermolecular interactions, are ubiquitous in planetary atmospheres and the interstellar space. Although they play a crucial role for radiative energy transfer and chemical processes, the understanding of their properties — which can differ significantly from those of the bulk — is still in its infancy. The present thesis is devoted to a better understanding of the influence of intrinsic properties of these particles on their infrared spectra. The influence of shape, size, architecture and phase on infrared spectra is modeled at a molecular level and propensity rules are established. The high complexity of these huge aggregates, which are composed of up to tens of thousands of molecules, makes a straightforward interpretation of their infrared spectra difficult or even impossible. The present thesis makes use of a combination of a quantum mechanical model for the calculation of the vibrational spectra — the extended vibrational exciton model — and a molecular dynamics approach for the generation of the particle structures. Calculations are performed for pure and mixed aggregates containing NH3, SF6, C02, CO, and CHF3.With a microscopic model at hand, it becomes even possible to go beyond system specific effects to uncover general underlying trends. 11 Table of contents Abstract.ii Table of contents iii List of tables vii List of figures ix Acknowledgements xviii Co-authorship statement xix 1 Introduction 1 1.1 Background and motivation 1 1.1.1 The relevance of molecular nanoparticles 1 1.1.2 Classical models and their limitations 2 1.1.3 Quantum mechanical models 5 1.1.4 Thesis overview 7 1.2 Theory and remarks on experiments 9 1.2.1 The vibrational exciton model 10 1.2.1.1 Induced dipole effects 12 1.2.1.2 Local environment effects 15 1.2.2 Numerical implementation 17 1.2.3 Particle model 21 1.2.3.1 Shape 21 1.2.3.2 Phase 22 111 1.2.3.3 Architecture .24 1.2.4 Correlating spectral and structural properties — state and excitation densities 26 1.2.4.1 State density 26 1.2.4.2 Excitation density 28 1.2.5 Remarks on the experiments 31 1.3 References 32 2 Size effects in the infrared spectra of NH3 ice nanoparticles studied by a combined molecular dynamics and vibrational exciton approach 37 2.1 Introduction 37 2.2 Experiment 39 2.3 Theory and simulation details 40 2.3.1 Model potential 41 2.3.2 Structural models 48 2.3.3 The extended vibrational exciton model 50 2.4 Results 55 2.4.1 Experimental spectra 55 2.4.2 Simulated spectra 60 2.5 Summary 68 2.6 References 72 3 Phase, shape and architecture of SF6 and SF6/C02aerosol particles: infrared spectra and modeling of vibrational excitons 75 3.1 Introduction 75 3.2 Theory and numerical approach 77 3.3 Pure SF6 aerosol particles 83 iv 3.3.1 Phase and shape of freshly prepared particles 85 3.3.2 Time evolution 91 3.4 Two-component SFdCO2aerosol particles 93 3.4.1 Core-shell particles 93 3.4.2 Statistically mixed particles 96 3.5 Summary 98 3.6 References 100 4 Vibrational exciton coupling as a probe for phase transitions and shape changes of fluoroform aerosol particles 103 4.1 Introduction 103 4.2 Experiment and calculations 104 4.2.1 Experiment 104 4.2.2 Calculations 105 4.3 Results and discussion 109 4.3.1 Spectral features of crystallization and shape change 109 4.3.2 Analysis of the particle shape by exciton calculations 118 4.4 Summary 125 4.5 References 127 5 Predicting the influence of shape, size and internal structure of CO aerosol particles on their infrared spectra 130 5.1 Introduction 130 5.2 Computational approach 131 5.2.1 Exciton model 131 5.2.2 Potential model 135 V 5.2.3 Particle model and molecular dynamics simulations 142 5.3 Results and discussion 146 5.3.1 Size, shape and surface effects in nanosized particles (<10 nm) 146 5.3.1.1 Crystalline particles 146 5.3.1.2 Amorphous particles and partially amorphous core-shell particles 151 5.3.2 Phase and shape effects in large particles (>10 nm) 155 5.3.2.1 Phase of the particles 156 5.3.2.2 Shape effects 159 5.3.3 Refractive index data 162 5.4 Summary 164 5.5 References 166 6 Conclusions 169 6.1 Summary and discussion 169 6.1.1 Differentiation between shape and phase effects 169 6.1.2 General shape effects 170 6.1.3 Time evolution of the particle shape 171 6.1.4 General phase effects: crystalline and amorphous particles 172 6.1.5 Size and surface effects 172 6.1.6 General comments on the computational approaches used 173 6.2 Outlook 175 6.3 References 177 vi List of tables Table 1.1 A list of the substances and vibrational modes studied in this thesis 31 Table 2.1 Potential parameters for the intramolecular (see Eq. (2.1)) and the intermolecular (see Eqs. (2.2) to (2.7)) contributions 45 Table 2.2 Comparison of monomer, dimer, and crystalline bulk properties derived from the model potential with experimental and theoretical reference values: dissociation energy De, hydrogen bond length RN..H, nitrogen distance RN..N, v2 fundamental wavenumber frequencies, and zero wave vector phonon wavenumbers cüph. The experimental monomer transition wavenumbers and dipole moments are averaged over the two tunneling components 46 Table 2.3 Spectral decomposition (broad component b in %) of the experimental particle spectra (Figure 2.1). r is the estimated amorphous shell thickness assuming core/shell particles with a crystalline core and an amorphous shell. T is the temperature, [NH3] the sample gas concentration, and R is the estimated particle radius 57 Table 2.4 Spectral decomposition (broad component b in %) of the simulated core/shell particle spectra after annealing. N is the number of molecules in a particle and R is the particle radius. The amorphous shell thickness r is derived from the broad component b. The amorphous volume fractionfs and the core/shell boundary at r (Figure 2.7 dashed line) are defined by construction before annealing 67 vii Table 3.1 Spectroscopic parameters used in the vibrational exciton calculations of SF6 and CO2 particles: transition wavenumbers i7, transition dipoles p, and polarizabilities a. s is the vacuum permittivity 82 Table 4.1 Spectroscopic parameters used in the vibrational exciton calculations of the CHF3 particles in the region of the v5 I v stretching vibrations: transition wavenumbers iY, transition dipoles p. and polarizabilities a. s is the vacuum permittivity 107 Table 5.1 Adapted Nutt & Mewly (ANM) potential parameters: harmonic frequency P, Lennard Jones parameters, and coefficients for the variable partial charges placed on the C and 0 atoms. 137 Table 5.2 Dimer geometry and dissociation energy (De) for the slipped anti-parallel configurations (Si and S2, see Figure 5.2a). R is the vector connecting the molecular centers of mass, 8A and 6B are the angles between R and the respective molecular axes. eB 0 corresponds to trans (like atoms on opposite sides) and 8B 0 to cis configurations (like atoms on the same side). The C-O bond length is i.i28323A 139 Table 5.3 Crystal minimum energy configuration (lattice constant, molecular center of mass shift from the fcc symmetry positions; fractional coordinates: (0,0,0), (0,Y2,4), (0,Y2,Y) (0,Y,Y2)) [46], corresponding lattice energy contributions and phonon frequencies as assigned from experiment and reassigned from theoretical work 140 viii List of figures Figure 1.1 Calculated absorption spectrum at constant resolution, using different time steps, illustrating the onset of numerical instability on the example of a small (CHF3)2 cube (2x2x2 unit cells). Accurate spectrum (solid), At = 0.02 ps, N = 1000; spectrum to demonstrate the effect of a slight numerical error (dashed), At = 0.08 ps, N = 250. The line indicates the mean energy E/hc=1141.7cnf’ 21 Figure 2.1 Experimental infrared spectra in the region of the umbrella vibration of ammonia ice particles. Particles were generated in a collisional cooling cell at the He bath gas temperatures indicated using a sample gas concentration of 400 ppm N}{3 in He. The spectra are normalized to equal maximum extinction and offset relative to each other. Dotted line: Particle spectrum at 20 K with sample gas concentrations adjusted (1600 ppm) to generate particles of comparable size to 54K/400ppm 40 Figure 2.2 Comparison between extinction spectra calculated for spherical particles with radii as indicated, using a) the Extended Vibrational Exciton (EVE) model and b) a complete normal mode analysis (NMA). The particles were generated as spheres cut from the perfectly crystalline bulk and then annealed 52 Figure 2.3 Spectral decomposition. a) Components derived from experimental spectra. b) Components derived independently for two particle sizes from spectra calculated for particles with different amorphous shell volume fractions. Solid line: (NH3)1015 , R = 2.0 nm. Dashed line: (NH3)1812, R = 2.4 nm. The dotted line represents the decomposition of a (NH3)4075 particle ix spectrum into its core (34%) and shell (66%) contributions using Eq. (2.16). See also Table 2.4. 56 Figure 2.4 Extinction spectra calculated for amorphous (NH3)32 particles of different shape as indicated. The spectra were calculated after annealing the particles 58 Figure 2.5 The v2 band of ammonia ice particles with different architectures: a) crystalline, b) amorphous core with crystalline shell, c) crystalline core with amorphous shell, d) amorphous particle. The spectra were calculated with the EVE model for spherical (NH3)1015 particles before (lhs) and after (rhs) annealing. The shell volume fraction before annealing was 50% corresponding to a thickness of 0.4 nm 61 Figure 2.6 The influence of particle architecture and annealing on calculated excitation densities for (NH3)1015 particles (R 2.0 nm). Rows: a) Extinction spectra. b) Radial excitation densities R2u(i7,R) (Eq. (2.16)). c) Radial densities of state R2p(i,R) (Eq. (2.12)). Columns: (1,2) Crystalline particle. (3,4) Crystalline core / amorphous shell particle (50% shell volume fraction). Columns (1,3) refer to particles before annealing. Columns (2,4) refer to those after annealing. The dashed line indicates the core/shell boundary. The integral of row b) over R yields row a). .63 Figure 2.7 Deviation from crystallinity A (Eq. (2.8)) for a (NH3)1015 particle before (dashed lines) and after (solid lines) annealing. The maximum value for ciA lies between 1.3 and 1.4 A for completely amorphous particles 65 x Figure 2.8 Calculated extinction spectra for an ensemble of (NH3)1812 particles (after annealing) with a crystalline core and an amorphous shell of varying thicimess. The volume fraction of the amorphous phase is defined before annealing. The typical change in the amorphous to crystalline ratio after annealing is shown in Figure 2.7b 66 Figure 2.9 Comparison between a) experimental (see also Figure 2.1) and b) simulated extinction spectra. The particle radii and the contributions of the broad spectral component are indicated for each observed spectrum. The simulations were obtained by linear combination of the spectral components derived from calculated extinction spectra, with the ratios adjusted for best agreement as indicated. The broad contribution was blue-shifted by 19 cm’ as explained in the text 70 Figure 3.1 Vibrational exciton calculations for an SF6 particle with an axis ratio of 1:1:6 and a cubic crystal structure. Thin line: Only dipole coupling (DD) is taken in to account (see Eq. (3.1)). Thick line: Dipole coupling (HDD) and dipole-induced dipole coupling (HDJD) are taken into account (see Eq. (3.3)). i7s the wavenumber in cm 81 Figure 3.2 Infrared spectra of SF6 aerosol particles. The temperature during particle formation in the cooling cell was 78 K. a) Region of the threefold degenerate v3 band. b) Region of the threefold degenerate v4 band. The bottom traces were recorded directly after particle formation at time to = 0 s. The middle and top traces show the spectral evolution of the same ensemble of particles after t1 = 33 s and t2 = 750 s, respectively 84 xi Figure 3.3 Traces a 1) and b 1) show the experimental infrared spectrum of freshly prepared SF6 particles (to = 0 s in Figure 3.2). Traces a2) to a4) show simulated spectra for particles with a cubic crystal structure for three different particle shapes (cube, quasi-octahedron, sphere). Traces b2) to b4) show the same for particles with a monoclinic crystal structure 86 Figure 3.4 Upper panels: Infrared spectra. Lower panels: Contour plots and single shell cuts of the normalized excitation densities as defined in Eqs. (3.11-13) in section 3.2. The shell index refers to concentric spherical shells around the center of mass of the particle. Trace a): for a particle with a cubic shape and a cubic crystal structure. Trace b): for a spherical particle with a cubic crystal structure 90 Figure 3.5 Trace a): Experimental infrared spectrum of SF6 aerosol particles recoded at t2 750 s after particle formation (Figure 3 .2a). Trace b): Calculated infrared spectrum. This spectrum is a linear combination of a spectrum of a cubic particle (46%) and spectra of elongated particles with different axis ratios (18% with an axis ratio of 1:1:3, 18% with an axis ratio of 1:1:6, 18% with an axis ratio of 1:1:12). The two side bands marked by arrows arise from the elongated particles. .. 92 Figure 3.6 Experimental infrared spectra of a SF6-C02core-shell particle. Traces a) and b) show two different spectral regions 94 Figure 3.7 Upper panel: Calculated infrared spectrum of a SF6-C02 core-shell particle. The corresponding experimental spectrum is depicted in Figure 3.6a. Lower panel: Excitation densities as defined in Eqs. (3.11-13) in section 3.2 95 xii Figure 3.8 a) Experimental infrared spectrum for statistically-mixed particles. b) Exciton calculation for statistically-mixed particles. For the simulations, we assumed a quasi-octahedral shape of the particle with a 36% overall cut-off 97 Figure 4.1 Time-dependent infrared spectrum of CHF3 aerosol in the region of the v5 / v2 bands. (a) Temporal evolution during the crystallization of the particles in the presence of trace amounts of water ice nuclei. (b) Temporal evolution during the change of the particles’ shape from cube- like to elongated particles. t is the time after particle formation (t = 0 s) 112 Figure 4.2 The same as in Figure 4.1, but in the region of the v4 band 113 Figure 4.3 The same as in Figure 4.1, but in the region of the v3 band 114 Figure 4.4 Experimental infrared spectrum of pure fluoroform particles without ice nuclei. The spectrum was recorded 1165 s after particle formation. a) Region of the v4 band. b) Region of the v5 / v2 band. The long tails towards lower wavenumbers are due to elastic scattering of the light by the particles, which have in this case sizes in the upper nanometer range 117 Figure 4.5 Infrared spectra in the region of the v5 / v2 band. a) Vibrational exciton calculation for a crystalline spherical particle. b) Vibrational exciton calculation for a crystalline cube-like particle. c) Experimental infrared spectrum (see t = 38 s in Figure 4.1) 120 xiii Figure 4.6 Left panels: crystalline spherical particles. Right panels: crystalline cube-like particles. Upper panels: calculated infrared spectra in the region of the v5 / v2 band. Lower panels: normalized excitation densities as defined in Eq. (4.10) 122 Figure 4.7 Infrared spectra of crystalline particles in the region of the v5 / v2 band. a) Vibrational exciton calculations. Thick line: a 1:1 mixture of cube-like and elongated particles. Thin dashed line: pure cube-like particles (see Figure 4.5). b) Experimental spectra. Thick line: a mixture of cube-like and elongated particles (estimated ratio 1:1) after 960 s of particle growth. Thin dotted line: cube-like particles immediately after crystallization is complete (see Figure 4.5) 124 Figure 5.1 EVE spectra of a (CO)748 crystalline sphere (r = 2.0 nm): with (solid) and without (dashed) polarization effects. The spectra are convoluted with a 0.2 FWHM Gaussian line shape. 132 Figure 5.2 a: Dimer structures corresponding to local minima on the ab initio potential surface of Vissers et al., see text. b: Potential energy surface for planar CO dimer as a function of molecular orientation. &A and 8B are the angles between the molecular axes of the monomers and the axis that connects their respective centers of mass. 0B 0 correspond to trans (like atoms on opposite sides) and °B 0 to cis configurations (like atoms on the same side). At each point the energy is minimized with respect to the intermolecular distance R. hi: Original Nutt & Mewly surface. b2: Adapted Nutt & Mewly surface, this work 138 xiv Figure 5.3 Standard deviation of the molecular centers of mass (upper trace) and of the molecular orientation (lower trace) from the crystalline structure (see Eq. (5.14)) for an ensemble of 10 spheres with a radius of 4 nm and a 40 vol% amorphous shell after 50 fs of simulated annealing 145 Figure 5.4 EVE spectra of crystalline spheres as a function of size. Absorbances are scaled to the same value (normalized to the maximum). a) Individual particles with increasing radii of r = 2, 4 and 6 nm from top to bottom. b) Lognormal distributions of particles with increasing mean radii of i = 2, 4 and 6 nm from top to bottom. o = 1.3 for all distributions. The spectra are convoluted with a 0.2 cm1 FWHM Gaussian line shape 147 Figure 5.5 Same as Figure 5.4 for crystalline cubic particles, a) Single size particles with the same volumes as in Figure 5 .4a. b) Lognormal size distributions with the same average volumes as in Figure 5.4b 148 Figure 5.6 Comparison of crystalline NH3 (1) and CO (2) spheres in (r = 2 nm). a) Calculated EVE JR spectra without polarization effects. b) Corresponding excitation density (Eq. (5.7)). c) Corresponding local density of states (Eq. (5.9)). The CO data were convoluted with a 0.2 cm1 FWHM Gaussian line shape to resolve the details of the very narrow spectral region, whereas a 6 cm1 FWHM was used for the broad N}13 band 150 xv Figure 5.7 EVE spectra for ensembles of spheres with radii of 4 nm. All spectra are normalized to max. absorbance. Each ensemble is composed of 10 spheres with a crystalline core-amorphous shell architecture with a shell volume of 20% (thick line), 40% (dashed), 60% (thin line) and 80% (dashed-dotted). The spectra are convoluted with a 0.2 FWFIM Gaussian line shape 153 Figure 5.8 EVE spectrum and excitation density (Eq. (5.8)) for an ensemble of 10 spheres with radii of 4 nm and crystalline core-amorphous shell architecture. The asymmetry in the spectrum and the long tails on both sides of the main peak are clearly associated with the 40 vol% amorphous shell extending from 3.4 to 4 nm. The spectra are convoluted with a 0.2 FWHM Gaussian line shape 154 Figure 5.9 Infrared spectra for different large CO aerosol particles: a) Calculated spectrum of a crystalline spherical particle. b) Calculated spectrum of a crystalline cubic particle. c) Experimental spectrum measured in a cooling cell at a temperature of 12 K (this work). d) Calculated spectrum of a 100% amorphous spherical particle. The calculated spectra are convoluted with a 0.5 FWHM Gaussian line shape 157 Figure 5.10 Calculated EVE spectra for mixtures of crystalline particles with different shapes. Spheres, cubes, and cuboids with an axis ratio of 1:1:3 and 1:1:9 are included. The ratio of the different shapes are: solid thick line: 21:34:22:23%, dash-dotted line: 14:34:24:26%, solid thin line: 7:32:28:33% and dashed line: 7:21:31:41%. The spectra are convoluted with a 0.5 FWHM Gaussian line shape 160 xvi Figure 5.11 Refractive index data derived from the calculated EVE spectra of large spherical CO particles, a) Completely amorphous CO ice. b) CO ice with an 80% amorphous contribution. The spectrum is derived from a particle with a crystalline core/amorphous shell architecture with an 80 vol% amorphous shell. c) CO ice with 40% amorphous contribution. The spectrum is derived from a particle with a crystalline core/amorphous shell architecture with a 40 vol% amorphous shell. d) Completely crystalline CO ice 163 Figure 5.12 Refractive index data from the literature derived from experimental spectra of thin CO films at temperatures of 10-16K: a) ref. [2], b) ref. [3], c) ref. [4], d) ref. [5] 164 xvii Acknowledgements Foremost, I would like to thank Prof. Dr. Ruth Signorell for her steady support throughout my entire PhD. I could not have wished for a better supervisor. Special thanks go to Dr. PD. David Luckhaus for an open door policy regarding my never ending questions on theory. I am especially grateful for his patience with my often vague questions to which he provided clear answers nonetheless. I am grateful to Dr. Martin Jetzki and MSc. Omar Freyr Sigurbomsson who conducted the cooling cell experiments, providing most of the experimental data used in this thesis. The collaboration with Omar helped me to a better understanding of the experimental aspects. I thank all my colleagues in the Signorell research group for their friendship and support, providing a fun working environment. xviii Co-authorship statement Chapters 2 through 5 are co-authored published articles (2-4) or submitted for publication (5). My contribution to each chapter is as follows: Chapter 2 (first author): • Identified and designed research program with my supervisor • Extended the vibrational exciton model to account for local structural variations in the particles by deriving individual transition dipoles and transition frequencies for each molecule within the force field of all others with the help of the fellow authors • Adjusted and enhanced a potential model for NH3 together with fellow authors to give a good and consistent description of structural and vibrational properties of NH3 particles • Wrote the software for the calculations and data analysis • Performed software validation • Obtained all theoretical data used in the analysis • Prepared the manuscript figures • Shared manuscript text preparation with my supervisor xix Chapter 3 (first author): • Identified and designed research program with my supervisor • Changed the numerical implementation of the vibrational exciton model to increase the maximum tractable particle size • Extended the vibrational exciton model to treat dipole-induced dipole interactions. • Derived the calculation of the excitation density from time correlation functions. • Obtained all theoretical data used in the analysis • Shared manuscript text preparation with my supervisor Chapter 4 (second author) • Wrote all software for exciton calculations and related data analysis • Obtained all theoretical data related to exciton calculations • Performed data analysis and prepared figures related to exciton analysis • Wrote the theoretical part of the manuscript Chapter 5 (first author) • Identified and designed research program with my supervisor • Adjusted potential model to meet simulation requirements • Wrote all software for the calculations and data analysis • Obtained all theoretical data used in the analysis • Performed data analysis with my supervisor • Wrote first version of the manuscript xx 1 Introduction 1.1 Background and motivation 1.1.1 The relevance of molecular nanoparticles Molecular nanoparticles are molecular aggregates held together by weak molecular interactions such as van der Waals forces and hydrogen bonds. Ranging from the sub-nanometer region up to several hundreds of microns in size, they play a crucial role both in planetary atmospheres and in interstellar space. Despite the ubiquitous presence of these aggregates, the understanding of their properties, which can vary significantly from those of the bulk, is still in its infancy. In planetary atmospheres, suspended molecular aggregates are termed aerosols. They have a major impact on the radiative transfer and chemical processes [1]. Indirect effects also arise, e.g. due to the smallest particles which serve as cloud condensation nuclei and affect, through their intrinsic properties, the physical properties of clouds. While water plays the dominant role on Earth, other substances have been found to fulfill a similar role in the atmospheres of other planets such as ammonia clouds on Jupiter and Saturn [2,3] or methane rain on Titan [4]. In interstellar space, the particle density and correspondingly the probability of molecular collisions are very low. The surfaces of emerging molecular icy grains serve as support for molecular accretion and chemical reactions, playing a key role in interstellar chemistry [5-7]. Furthermore the formation and destruction of these icy grains can provide information on the conditions and processes in interstellar clouds. The first important step towards understanding molecular nanoparticles is the investigation of their intrinsic properties, such as shape, size, phase and architecture. Since remote detection is crucial for molecular aggregates, spectroscopic methods play a central role. In 1 this category, infrared (IR) spectroscopy is particularly well suited [8-16]. Vibrational spectra contain a wealth of information on molecular nanoparticles. This includes information on local structure within the aggregates as well as on intrinsic particle properties. In the laboratory, combining JR spectroscopy with particle generation methods such as collisional cooling [8- 14,17,18], molecular nanoparticles can be studied under atmospherically or astrophysically relevant conditions at thermal equilibrium over periods of hours. In principle, information on the particles’ shape, size and internal structure is contained in these vibrational spectra. Given the high complexity of these systems with up to billions of molecules, however, a straightforward analysis is often difficult or even impossible. The development of theoretical models is essential to access this information and in particular uncover the underlying mechanisms responsible for the observed spectral features. 1.1.2 Classical models and their limitations Both classical and quantum mechanical models are used to analyze infrared spectra and thus to gain a deeper insight into the properties of small molecular aggregates. Due to the many degrees of freedom of molecular nanoparticles classical continuum models clearly dominate [19,20]. They calculate the scattering and absorption of infrared light by small molecular aggregates using classical scattering theory. The particles’ optical properties are treated as continuum properties. These classical models involve solving Maxwell’s equations for the particle’s boundary conditions. They require as input optical data, either in the form of the complex index of refraction n + 1k or the complex dielectric function 8 = 6 + 182. n and e are the real and k and 82 the imaginary parts. These two sets of data are equivalent representations of the optical properties of the system. The optical data must be obtained from independent 2 experiments. The scattering and absorption problem has an exact solution only for spherical particles, known as Mie theory [19]. If the particle boundaries deviate from a spherical shape, approximate solutions can be employed [20,2 1] such as the direct dipole approximation (DDA) [22] in which the particle is replaced by a lattice of dipoles. The major difficulties with all classical methods are the availability and accuracy of optical data for the studied systems. The crux lies in the fact that they have to be determined from independent experiments on either thin films or aerosols. Optical data are thus simply not available for many systems. So far databases [23-27 and refs. therein] are limited to pure substances and a few mixed systems. Since optical data vary for mixed particles continuously with the constituents’ ratio and the particle’s architecture (e.g. core-shell or statistical mixing), extensive databases would be required, which are not available nowadays. Aside from the availability, accuracy is a problem. The optical data are not directly measurable quantities — they are derived from infrared spectra using a Kramers-Kronig inversion [19,28,29] or a Lorenz model [19]. For both models, the accuracy is limited by the following factors. In thin film measurements, the thickness of the film must be determined and the error associated with this value is typically between 5-50% [29-31]. In aerosol measurements, one deals with an ensemble of particles of different shapes with a certain size distribution. Exact knowledge of the shapes and size distribution is required for accurate results, but in practice these quantities are rarely well defined. In addition it is often not clear what the internal structure of the thin film or aerosols is, e.g. it is not clear whether the refractive index data are for an amorphous or partially crystalline substance. The Kramers-Kronig method performs an inversion of the spectrum without making any initial assumptions in contrast to the Lorenz model discussed below. The complex index of 3 refraction is calculated, starting from an initial guess for the imaginary part, using the Kramers Kronig relation between the real and imaginary parts 1 V2 a(v’) ,2 2dv (1.1)2r ‘v—v where n is the real and k the imaginary part of the index of refraction, a = (4r / 2)k, and v is the frequency. In principle, the integral extends over the whole frequency region to infinity. However, the complex index of refraction varies strongly only in the vicinity of strong absorptions. Thus the well separated electronic contribution can be replaced to a very good approximation [31] by a constant value o. For many substances, however, the value for flo is not available and further inaccuracies are introduced through approximations [29,30]. For example, if one substance is dominant in a mixed particle, the o value of the respective pure compound is used. Within the Lorenz model [19], the optical data are fitted using a set of isotropic damped harmonic oscillators coupled to the electromagnetic radiation by their effective charges. The complex dielectric function can then be obtained as 2 2 . (1.2)a)—a) —17/0 j 2 2 ivewith w, =—, m60 and s is the permittivity of vacuum, o is the resonance frequency and is the damping factor of oscillatorj, which is inversely proportional to the width at half maximum. N is the number of oscillators, e their effective charge and m their mass. The parameters of the oscillators are not 4 based on the molecular properties of the system — they are simply fitted to the measured spectrum. For very simple systems, oscillators can be assigned to vibrations, providing an acceptable description of the optical data [19]. For systems such as aerosol particles that are composed of hundreds to tens of thousands of oscillators such an approach only leads to a very approximate description of the system and lacks the necessary accuracy. Furthermore the oscillators used in this case have no microscopic physical meaning and thus do not provide an understanding of the infrared spectra at a molecular level. In addition to availability and accuracy of optical data, classical scattering theory has another severe deficiency. It cannot properly describe the infrared spectroscopic properties of very small aerosol particles [16]. Very small means in this context molecular aggregates with sizes below about 10 nm. The reason lies in the fact that classical continuum models cannot reflect the modulation of inter- and intramolecular interactions by the finite size of the nanoparticles. 1.1.3 Quantum mechanical models In light of the failings of the classical models, a molecular model which does not rely on continuum approaches is desirable. A quantum mechanical model is necessary to understand the features observed in the vibrational spectra of molecular nanoparticles as well as to understand the underlying mechanisms at a molecular level. A full quantum mechanical treatment becomes computationally prohibitive beyond a few molecules. Thus the key to access larger systems lies in focusing on the dominant interactions. The present thesis shows that certain strong intermolecular interactions need to be considered in order to correctly simulate the intrinsic particle properties in infrared spectra. Several research groups have applied customized 5 molecular models to investigate particles of NH3 [32,33], CO2 [8,34,35] and H20 [12,36], consisting of up to a thousand molecules. The system sizes accessible through these models, however, are insufficient to fully characterize size effects in small aggregates (chapter 2 and section 5.3.1) or to study shape effects of large particles (chapters 3-5). A quantum mechanical model that has been shown to be successful in describing infrared spectra of systems composed of several tens of thousands of molecules [16,37-41, chapters 2-5], is the vibrational exciton model presented in detail in chapter 1.2. The foundation of the model is resonant transition dipole coupling, which is the interaction that gives rise to the appearance of vibrational bands with strong molecular transition dipoles (>0.1-0.2 D). This dipole coupling lifts the degeneracy of the uncoupled molecular states and thus leads to vibrational eigenflinctions of the nanoparticles that are delocalized over the whole particle. This makes an excellent probe for intrinsic particle properties. The calculated spectra presented here are not fits to experimental spectra — they are predictions that allow us to identify shape, size or structural effects on the particles’ infrared spectra and to determine whether the origins of these phenomena are system specific or more general mechanisms. For example, in particles with a high surface to volume ratio the delocalized vibrational modes are modulated by the particle boundary leading to unique, size specific spectra. This is an effect that the classical models do not describe, but that is captured by the exciton model. Similarly the model unravels other effects of intrinsic particle properties on infrared spectra, such as the influence of shape, phase or architecture. 6 1.1.4 Thesis overview The aim of this work is to investigate the effect of intrinsic properties on the infrared spectra of molecular nanoparticles by modeling their infrared spectra and structures. For that purpose an extended vibrational exciton model is developed and combined with molecular dynamics approaches for structure determination. Of particular interest is to go beyond system specific effects and to uncover general trends. For this it is important to establish propensity rules for the effects of shape, size, phase and architecture on infrared spectra of aerosols. Another aspect is to clarif’ which interactions apart from transition dipole coupling have a strong influence on spectroscopic features. As discussed in section 1.2. 1.1, the inclusion of the molecular polarizability turns out to be important and is thus included in the vibrational exciton model. Small molecular aggregates can be divided into three categories based on the spectroscopic effects found in their vibrational spectra. The smallest particles, up to approximately 10 nm (10 molecules) in size, have a high surface to volume ratio which leads to pronounced size and surface effects. For particles in the 10-100 nm region (1O-lO molecules) the influence of the surface becomes negligible. The resonant transition dipole coupling (scaling as r3 with distance) reaches convergence as a function of size and is modulated only by the shape of the particle boundary and the internal structure. Consequently the spectra in this region are dominated by shape and structural effects. Beyond the 100 nm limit, the size of the particles becomes comparable to the wavelength of the probing infrared light and elastic scattering of the light by the particles becomes important in addition to the phenomena already mentioned. Elastic scattering cannot be calculated with the vibrational exciton model directly. However, it is possible to derive optical data directly from the exciton calculations and use these data as input 7 for scattering calculations (section 5.3.3). This approach leads to much more accurate results than the use of experimentally determined optical data. The model systems studied in this thesis (chapters 2-5) are molecular aggregates of ammonia, sulfur hexafluoride, fluoroform and carbon monoxide with sizes ranging from 1 nm to about 100 nm. Molecules, clusters, bulk phase, as well as aerosols of these substances have been investigated extensively in the literature and in our own research group allowing for thorough checks of the theoretical model. These simple molecules possess vibrational transitions with strong molecular transition dipoles (>0.1 D) which lead to pronounced exciton coupling in the vibrational bands of these aggregates. Molecular aggregates of such simple systems are ubiquitous in planetary atmospheres and interstellar space which in combination with their simplicity make them excellent probes for those environments [42]. NH3 aerosols have also been detected within planetary atmospheres in our own solar system [2-4]. Chapter 2 investigates surface and size effects for small hydrogen bonded ammonia particles. In NH3 aggregates, resonant transition dipole coupling is strongly influenced by the highly anisotropic hydrogen bonds, which leads to structural variations in the particles. The exciton model is extended to account for this by including local variations of transition dipoles and transition frequencies. The effects of phase, shape and architecture on degenerate vibrations are studied in chapter 3 on the example of the triply degenerate v3 stretching mode of SF6 aerosols. A time dependent numerical implementation of the exciton model is introduced to treat the large number of oscillators (up to 30000) in these systems. Bulk phase SF6 shows a phase transition from a cubic [43] to a monoclinic [44] phase at 96 K. The lack of vibrational spectroscopic evidence for the phase transition of SF6 aerosols at 96 K is explained using this microscopic model. Furthermore, the experimentally observed time evolution of pure SF6 aerosol infrared spectra is 8 explored based on the information on phase and shape. The effects of core-shell architectures and statistical mixing on vibrational spectra are studied for the example ofSF6i’C02particles. In chapter 4, the effects of shape and phase in CHF3 aerosols are examined. CHF3 is used as a supercritical solvent due to its low critical data (T0 = 26.1°C, P = 48 bar [45]). The properties of CHF3 aerosols are important for the understanding of particle formation by Rapid Expansion of Supercritical Solutions, which is an attractive method for the micronization of pharmaceuticals [46]. The same time dependent approach as used in chapter 3 is necessary in the exciton calculations to describe the coupling between the doubly degenerate v5 and the non- degenerate v, mode of these CHF3 aggregates. Finally, the vibrational exciton model is applied to CO aggregates for the whole size range between 1-100 nm. Predictions on shape, size, surface and ensemble effects are made for small particles (1-10 nm, i.e. several hundred to 310 molecules) where no experimental infrared data exist. For large particles (10-100 nm, i.e. 310-l0 molecules), the available infrared spectra [13,47,48] are analyzed in terms of phase and shape effects with the help of modeling. The work within this thesis was covered in four publications [49-52] and contributed to two review articles [16,53]. 1.2 Theory and remarks on experiments The subsequent chapters demonstrate the application of the vibrational exciton model to predict and analyze the effects of intrinsic aerosol properties, such as shape, size and internal structure on their vibrational spectra. This is a manuscript based thesis: the core chapters (3-6) are manuscripts published or submitted for publication, required to be implemented with minimal changes. As a consequence each of the following chapters contains only a brief description of the 9 model itself tailored to the specifics of the studied system. Therefore, a comprehensive description of the vibrational exciton model is given in this section. 1.2.1 The vibrational exciton model The vibrational exciton model is a quantum mechanical model used to predict the vibrational infrared spectra of large molecular systems built of many weakly bound subunits. The model was initially applied in the pioneering work of Fox and Hexter [54] to describe the vibrational dynamics of molecular crystals. A comprehensive discussion on this subject can be found in ‘Molecular vibrations in crystals’ by Decius and Hexter [70]. The first attempt to apply the vibrational exciton model to finite particles was made by Cardini et al. [34] who tried to reproduce the experimental infrared spectra of CO2 clusters measured by Barnes and Gough [55]. Without definite information on the experimental conditions to allow a thorough comparison system size and temperature were only estimated in ref. [55] — the suitability of the model in describing molecular aggregates remained uncertain. Later attempts to use the exciton model were made by Ewing and coworkers [8,35,56], but were only partially successful due to computational limitations regarding particle size. The best agreement with experimental results was obtained for thin molecular slabs, where the reduced dimensionality of the problem led to a convergence of the exciton spectra as a function of system size already for 450-1800 molecules [8]. The exciton model is designed to describe the spectra of systems that are dominated by resonant transition dipole coupling. Within a molecular aggregate, this interaction lifts the degeneracy of the molecular vibrational eigenstates leading to modes that are delocalized over the entire particle, called vibrational excitons. Their delocalized nature makes exciton coupling an excellent probe for intrinsic particle properties. Research on various model systems by Signorell 10 et al. [37-40,53,57] revealed that for molecular vibrations with transition dipoles above 0.1 D, resonant transition dipole coupling dominates the appearance of infrared extinction spectra of molecularly structured particles. These vibrations are therefore ideally suited to study intrinsic particle properties. With transition dipole coupling as the dominant interaction, the Hamiltonian is greatly simplified, making the calculation of infrared spectra for particles with ten of thousands of molecules tractable. In its simplest form the Hamiltonian is limited to dipole-dipole interactions: H = H0 + HDD, with (1.3) IIDD = (1.4) I[ is the vibrational Hamiltonian of the uncoupled molecules. The sum in HDD extends over pairs of molecules, p is the dipole moment vector of molecule i ( denotes the conjugate transpose) and is a scaled projection matrix: (1— s..) = (3ee; —i) (1.5) where R is the distance between molecules i and j, and e the unit vector pointing from the center of mass of molecule ito that of moleculej. I is diagonalized in a direct product basis of molecular oscillator eigenfunctions. For the purposes of this work, the basis is restricted to near resonant single oscillator, single quantum excitations im), with ‘m) representing the product function with vibrational mode m excited on molecule I and all other oscillators in the ground state. I 0) denotes the overall ground state. We further approximate matrix elements by assuming 11 harmonic oscillators and linear dipole functions. Within this double harmonic approximation the Hamiltonian is completely defined by transition wavenumbers and molecular transition dipole moments Pim = (op1 im). These input parameters are derived independently, e.g. from gas phase data. The resulting calculated spectra are thus true predictions, not fits to experimental spectra. 1.2.1.1 Induced dipole effects An extension to the vibrational exciton model is the inclusion of dipole-induced dipole (DID) coupling. Research on the SF6 dimer [58,59] revealed that induced dipole effects amounted to approximately 10% of the frequency splitting in the JR spectrum of the v3 vibrational mode. While not dominant, it is still a significant molecular interaction and its inclusion leads to an improved description of the calculated exciton spectra. Furthermore, the apparent lack of a systematic investigation on the effect of DID coupling on the vibrational spectra of large molecular aggregates in the literature makes it an interesting point of study (chapters 3-5). The starting point for the induced dipole contribution(1IDJD) to the Hamiltonian is the definition of the energy (I]) of a dipole (Pd) induced by an electric field E: H=—1pEfldE=—(aE)E (1.6) where a is the polarizability tensor. Given a particle composed of N molecules, a dipole moment is induced on each molecule i by the electric field E1 arising from all other molecular dipoles, at the location of molecule i: 12 E. = 1 1 [3(1ue)e — = (1.7) U j Substitution into Eq. (1.6) and summation over all molecules yields [60] IZIDID = (1.8) This expression differs from the one used in refs. [58,59] where cross-terms of the form (a2p )÷2jkfik’ j k were erroneously omitted. The overall Hamiltonian is then given by I = + + “DID = “0 + (1.9) with = — - + ikk2J Inserting the series expansion of the dipole moment operator up to first order in the m-th normal coordinate of thej-th molecule, = + p7qjm , with p7 = (1.10) m q1 q.=O and rearranging the factors by normal modes q1, gives: = ‘0 +[(fiAP +qjm(pAp + im(Pi4uP;in] (1.11) 13 The 0th order terms in q contribute the same constant value to the diagonal terms (Jm Vl fm) and (0 i 0) and can therefore be disregarded. Furthermore, the first order terms do not couple (near) resonant levels. They do not contribute to resonant transition dipole coupling, the dominant intermolecular interaction to which our exciton model is restricted. Neglecting 0th and order terms in the dipole interaction yields — + qjm(p)Ap;qjn (1.12) i,m,j,n (p” ) Ap is a scalar and the matrix elements can be written as ( ijç) = SrtSsuhCiYrs + (p)Ap;(i qjmqjntu) (1.13)i,m,j,n where 5 represents Kronecker’s function, (Srf = 5(r — t)), h is Planck’s constant, c is the speed of light and i7 are the molecular transition wavenumbers. Within the basis of single harmonic excitations only the following terms are non-zero (Oim qjm1jm) = =, (Ojm q0jm) = , (urn q1jm) = (1.14) Oim) and 1,,,) denote the ground and first excited state of vibrational mode m of the single molecule 1. Applying Eqs. (1.14) to Eq. (1.13) leaves only the following terms jtu)=5rt5suhCi7rs+L (1.15) i,m,j,n with z = Szj8mn(1 + 2SjrSrns) + +5t3rnujrns 14 This gives for diagonal elements (r i) = hCVrs + (4)Arrp + (1.16) and for non-diagonal elements (tu)=(p:)Artii +(pAfrp: =(IArt. (1.17) The sum in Eq. (1.16) is the same contribution as to the ground state (0 H 0). Thus, shifting the zero energy arbitrarily but without any loss of generality to the uncoupled ground state energy and using ,uim = (0gm pj1jm) = -=j4”, the matrix elements of the Hamiltonian are cast into the following simple form: (r3 ji?jç) = 6rtSsuhCi7rs + 2jLtAr/J (1.18) The overall computational effort in setting up this matrix scales to the third power with respect to the number of oscillators in the system. The treatment of induced dipole effects is the most expensive part of the calculations. Considering resonant dipole coupling alone, the calculations scale only quadratically with the system size. 1.2.1.2 Local environment effects In its simplest form the vibrational exciton model uses as input the same transition dipole and transition frequency for all molecules. This assumption, however, is only correct in highly symmetric environments, such as some crystalline phases, where all molecules can be considered 15 equivalent. Variations in the structure of a particle, such as the disorder in the amorphous phase or the different environment experienced by molecules in the particle surface, lead to different local potentials for individual molecules with corresponding changes of transition dipoles and frequencies. These local variations are accounted for by deriving individual transition frequencies and transition dipoles for each molecule within the field of all others using an explicit potential and dipole function. The electrostatic interactions in the intermolecular potential function are closely related to the molecular transition dipoles. Therefore the dipole function should ideally be derived directly from the potential’s definition, not using an independent function. In doing so, the calculated intensities provide an independent check on the quality of the intermolecular potential. The transition frequencies and transition dipoles are obtained in the double harmonic approximation by performing a local normal mode analysis, i.e. for each molecule the corresponding block F, in the overall Hessian matrix F is calculated and diagonalized to provide the local transition frequency. 82v 02v 0x118x,, F = , with F, = . (1.19) a2v 1N1 1NN ... axnioxli axni and where V denotes the potential and (x11 . . . x,) are the Cartesian atomic coordinates of molecule i. This approach considers, within the harmonic approximation, all interactions between each molecule and the rest of the particle, save for the coupling between the normal modes of different molecules, represented by the off-diagonal F blocks. Then the derivatives of the molecule’s dipole moment are calculated with respect to all atomic Cartesian coordinates and converted to the local normal mode representation, which yields the local transition dipoles. 16 1.2.2 Numerical implementation The particles of interest in this work cover a size range of approximately 1-100 nm, i.e. from several hundred to billions of molecules. The computational problem is simplified by the fact that the absorption spectra of particles converge as a function of size at diameters around 10 nm, i.e. the band positions and shapes do not change if the size of the particle is increased further. It is thus sufficient to exceed the size convergence threshold in the simulations. This still leaves tens of thousands of degrees of freedom to be dealt with, however. The direct diagonalization of the Hamiltonian for such systems becomes increasingly impractical. For N = 15000 oscillators, this approach requires about 2 GB of memory and 1-2 days of computer time on a Pentium 4 processor at 2 GHz. This is barely sufficient to reach spectral convergence as a function of size in the simplest scenario, when simulating a single non- degenerate vibrational mode. The memory requirements scale with N2 while CPU time scales with N3. At some point the stability with respect to numerical round-off also becomes a problem. When the complexity of the system increases, an efficient numerical implementation of the exciton model becomes essential to calculate particle spectra in a reasonable timeframe. A few examples where direct diagonalization is no longer a viable method are when larger systems are required to achieve spectral convergence or when dealing with molecules with several degenerate or near resonant vibrational modes. The time-dependent numerical approach used throughout this work calculates the absorption spectra directly from the electric dipole autocorrelation function. If {E1 } is the set of eigenvalues of ft with corresponding eigenvectors I) and transition moments M1 = (0 ,u I) (overall dipole function p = p1 ) then the absorbance spectrum is proportional to: 17 cT(E)=IMj2*f(E_E1) (1.20) where * denotes a convolution and f(E) is an appropriate line shape, e.g., a Gauss or Lorenz function. Calculating the Fourier transform of (E) gives: ö(t) = g(t)M12je_iEt5(E — E1 )dE (1.21) = g(t) (0 Iu I)e_ht (I p 0) = g(t0 pehf1tp o) (1.22) = g(t) (OIpIm)Qm e_iHtl j)(j pjo) (1.23) i,m,j,n = g(t) (1m pjme’,ujnjn) = gQ)C(t) (1.24) i,m,j,n where gQ) is the Fourier transform of f(E) and cQ) is the dipole correlation function. The time propagation of the dipole weighted wavefunction in Eq. (1.24) is performed using a second order scheme, noting that c(— t) = C (t). Inserting the Taylor expansion of the time evolution operator e_t = 1— iIEt/h + ... (1.25) where & is the propagation time step, into — lt’) = (e_t — (1.26) and rearranging Eq. (1.26) gives —i-Ijyi,) + (1.27) 18 Within this numerical implementation of the exciton model, based on Eqs. (1.18), (1.24) and (1.27), the calculation of particle spectra at a resolution of -1 cm1 requires approximately 3 hours of computer time on a 2 GHz Pentium 4 processor for the largest systems (30000 degrees of freedom), once the Hamilton matrix is set up. This is about a factor of 100 faster than full diagonalization. The time propagation depends quadratically on the system size and is scalable as a function of two parameters: the propagation step At and the total number of time steps, N. Together they define the resolution of the calculated spectra: At is related to the Nyquist frequency and thus defines the frequency interval, while N defines the number of grid points within the interval. The Nyquist frequency °Nyquist = (1.28) is the largest frequency value that can still be represented correctly by a Fourier transform — higher frequencies are folded back into the (0, aNyqujstl interval leading to artifacts. Ideally At should be chosen such that the largest frequency of interest is just below or equal to cüNyquisf, minimizing the required number of propagation steps N, i.e. grid points, to the desired spectral resolution. Vibrational transitions however, lie between several hundred and a few thousand wavenumbers in frequency, imposing a rather strict limitation on At. Since in our case the ground state is uncoupled from excited states, it is possible to circumvent this problem by subtracting the average value E of the diagonal elements of the Hamiltonian from its diagonal. This roughly corresponds to the contribution of the uncoupled molecular vibrations (H0). This constant shift is added back to the spectrum after time evolution and Fourier transformation. Thus the Nyquist frequency needs only to be comparable to the width of the vibrational band, in most 19 cases less than 100 cm’, reducing the number of propagation steps at equal resolution by up to two orders of magnitude. The time step is also limited by the numerical accuracy of the propagation scheme. The first sign of numerical error is an increasing broadening of the spectrum, nearly negligible at and more pronounced further away. To illustrate this effect, the spectrum of a small CHF3 cube (2x2x2 unit cells) was first calculated with a time step of 0.02 Ps (Ny 833 cm’) over N = 1000 steps (full line) and a second time with a time step of 0.08 Ps (ONY 208 cm’) over N = 250 steps (dashed line) to introduce a small numerical error as shown in Figure 1.1. In both cases exceeds the width of the band in question. The number of time steps was changed simultaneously to keep the simulation time constant, i.e. maintain the same frequency grid. Comparing the slightly erroneous calculation (At 0.08 ps) against the accurate result (At = 0.02 ps), shows that around E 1141.7 cm’ the spectrum is almost unchanged. The further the spectral features are situated from E the stronger their broadening, clearly visible for the shoulder around 1180 cm’. The error is due solely to the propagation since a particle spectrum calculated from every fourth value of the autocorrelation function calculated with the 0.02 Ps time step is identical to the reference spectrum. Consequently, the time step should always be chosen as large as possible, while still ensuring numerical stability and keeping calculated frequencies well below the Nyquist frequency. The number of simulation steps can then be adjusted to obtain the desired spectral resolution. 20 Figure 1.1 Calculated absorption spectrum at constant resolution, using different time steps, illustrating the onset of numerical instability on the example of a small (CHF3)2 cube (2x2x2 unit cells). Accurate spectrum (solid), At= 0.02 ps, N 1000; spectrum to demonstrate the effect of a slight numerical error (dashed), At = 0.08 ps, N = 250. The line indicates the mean energy E/hc = 1141.7cm’. Cl) C D Li CD C CD I— 0 U) .0 CD 1100 1120 1140 1160 v (cm1) 1180 1200 1220 1.2.3 Particle model 1.2.3.1 Shape The aim of this work is to understand how intrinsic particle properties affect their absorption spectra and to establish propensity rules for their occurrence. This accordingly has an impact on the choices made in modeling the particles in terms of shape, phase and structure. Thus, the choices of interest in shape are simple, namely limiting cases such as globular (spheres, cubes, cuboctahedra), elongated or slab-like shapes. An equally important factor in choosing 21 which particle shapes to model is the experimental relevance. The first two shape types are the most interesting as several studies [16,38,40,57] have revealed that in the initial stages of particle formation the shape is globular, while elongated particles emerge over time, although the exact mechanism of formation is still unclear. Slabs were used to model thin films (several molecular layers thick) in ref. [8,35]. Amorphous particles discussed in the next section are rather unlikely to form corners or edges and are best approximated by spherical shapes. 1.2.3.2 Phase Aside from the relevant crystalline phases, partially or completely amorphous particles are considered. The latter phase tends to dominate when the experimental conditions are such that freezing happens too fast for relaxation to take place. Model amorphous configurations are always generated starting from a fully crystalline structure cut from the crystalline bulk. In a first step, all molecules within the particle region designated as amorphous are randomly shifted and rotated. The resulting structure will contain physically implausible, that is to say very high energy configurations, such as two atoms being very close, since the randomization process does not include any energetic or structural bias. These unfavorable configurations are removed by simulated annealing using classical molecular dynamics (MD). As energy is transferred into the system and then gradually removed repeatedly, the artificial heating and cooling of the particle allows for the relaxation of strained configurations. The simulated annealing process must not be mistaken for a description of real annealing processes; it is only an algorithm to obtain physically plausible structures. Rather than to describe the detailed temporal evolution, in the context of the present work MD simply serves as an effective means to sample the physically relevant configuration space. Its 22 application to a large variety of complex systems is well established [61-63] which makes classical MD the method of choice to treat the large molecular aggregates investigated in chapters 2-5. Classical MD consists of solving Newton’s equations for the motion of atoms under the influence of the forces acting between them. The core elements of a MD simulation are the potential function used to evaluate the forces and the time integration scheme, i.e. the propagator. The size of the simulated particles limits the choice of potential functions to effective pair potentials. The forces are derived and implemented in analytical form, to maximize the computational efficiency. The evaluation of the forces is the most expensive part in MD so that among the many variants of propagators, we chose the Verlet algorithm [61,62]. It works with a minimal number of potential evaluations and shows a particularly good conservation of energy when Lennard-Jones potential functions are involved [61], which are among the standard choices to describe exchange and dispersion effects in effective pair potentials, e.g. [6 1,64-66]. To characterize the ‘amorphousness’ of a particle we define a deviation of its structure from the initial crystalline arrangement. It is quantified as the standard deviation of all atoms from their positions in the crystal: = inf{1xi _x2} (1.29) where N is the total number of atoms in the particle, x is the position vector of atom I in the amorphous particle and x its analogue in the initial crystalline structure. The infimum function indicates that the standard deviation is minimized with respect to an overall relative rotation of the amorphous particle relative to its crystalline structure. The minimization is performed as a non-linear least square fit using a Levenberg-Marquardt algorithm [67]. For linear molecules we 23 found it more informative to separate the information on rotation and translation in terms of the standard deviations of the molecular orientations and of the molecular centers of mass: o =inf_a’J CM =inf[/1(Axi)2J, (1.30) where N is the number of molecules, Ax. is the center of mass shift of molecule i and a, is the angle between the molecule’s orientation before and after amorphization. Thus a particle’s internal structure can be verified or analyzed in detail by dividing it into volume elements, e.g. spherical shells, and applying Eq. (1.29) or (1.30). An important property that must be considered in the structural analysis is the molecular symmetry. For example, the random rotation of a molecule may lead to a symmetrically equivalent orientation, i.e. physically indistinguishable from the previous one. These symmetry effects are treated individually for each system studied to avoid artificial increases in the standard deviation. 1.2.3.3 Architecture The question of particle architecture arises when dealing with multiple phases or molecular species within a particle. In this thesis, only combinations of the crystalline and amorphous phase are considered. A particularly plausible hypothesis for the architecture of partially amorphous structures is based on the particle formation process: if the cooling process is fast enough, the initial condensation nuclei will first be amorphous. Then, as the particle grows molecules will condense upon the surface of the nucleus. As a result, part of the condensation energy will be transferred to the core. This annealing process allows the particle to crystallize 24 while the continuously cooled surface remains frozen in an amorphous state. Overall then this particle adopts a spherical, crystalline core-amorphous shell architecture. When investigating partially amorphous particles in the subsequent chapters, this model will play a central role. When dealing with multiple molecular species, both core-shell architectures and statistical mixtures must be considered. Core-shell particles are generated using the following algorithm: particles composed each of a single substance are superimposed, placed with the center of mass in the origin of the coordinate system, e.g. an SF6 sphere and a smaller CO2 sphere. Once an (arbitrary) order of these particles is defined, e.g. the CO2 sphere is the first, the SF6 sphere second, all molecules of the second particle overlapping with the first are removed. This process can be repeated for any number of substances to generate an onion-like structure and is not restricted to spherical shapes. The criterion of molecular overlap is met if the distance between two atoms from different molecules is smaller than the sum of their van der Waals radii. When performing statistical substitutions, there is no general information on how the host lattice will rearrange itself locally to accommodate guest molecules, leaving several options. Each of the following approaches has its limitations however, restricting the systems accessible for study. One might attempt to correct the particle structure by using experimental data on mixed bulk crystal phases but these are rarely available. Even if this data were at hand, the transfer of bulk data to small particles is non-trivial. Alternatively, quantum chemical calculations can be performed on small model clusters, but the transferability is again not straightforward, this time because the model systems are too small. Annealing the trial structure of a particle is an option only with reliable many-body potential functions. In cases where these approaches cannot be used, the only option left to extract general trends for the effects of statistical mixing on the vibrational spectra of particles is to vary all pertinent structural parameters (e.g. mixing ratio, lattice parameters) in a systematic fashion. 25 1.2.4 Correlating spectral and structural properties — state and excitation densities Locating where spectroscopic features originate in a particle is an important step towards a better understanding of intrinsic particle properties. Such a correlation would allow one to determine whether a contribution observed in a vibrational absorption spectrum can be attributed to a particle’s surface or core, to amorphous or crystalline regions or whether features characteristic of a given shape are localized in key regions. State and excitation densities are introduced as tools to analyze the correlation between the spectroscopic and structural properties of the particles. 1.2.4.1 State density Projecting the eigenstates I) of the Hamiltonian on the local excitations m) (in the product basis of molecular oscillators with mode m excited on molecule i) for a given frequency V and spherical coordinates (R, Q) (radius and solid angle respectively) one obtains amplitudes v(V,R,Q)= (IIo(i)o(R—Ri>(cQi)im) (1.31) I,i,m where V1 is the eigenvalue of I) in wavenumbers and (R1,c21) are the center of mass coordinates of molecule i. Thus p(V, R,Q) = +‘(V, R,c2 = — —R1)8(( — QjCimI (1.32) I,i,m 26 represents the local density of states at frequency P and position (R, Q). Cimi are the eigenvector coefficients of eigenstate I). The analyzed particles are usually divided into volume elements 5Vk and the local densities discussed in the subsequent chapters refer to the integrals over öVk. With a few exceptions, a division into spherical shells is used throughout the following chapters: p(,R)=f’P(v,R,c2dQ. (1.33) In a discrete representation, p(17, R) takes on the simple form: p(V,R)= CjmjH P— < zi7, R1 —R <AR. (1.34) To be able to compare results between different volume elements, we normalize the state density to obtain the average density per molecule or per volume. The two choices for normalization used are practically equivalent: the volumes 617k or the number of molecules in the volume. For spherical shells (volume 4,rR2dR), 1R-RHAR. (1.35) For graphical representations, the density is smoothed by substituting the &distribution in Eq. (1.32) by suitable Gaussian or Lorenzian distributions. 27 1.2.4.2 Excitation density The excitation density can be regarded as an intensity weighted state density and represents the contributions of the individual volume elements öVk to the total spectrum. Starting from the transition moments of the eigenstates of the Hamiltonian (M1 = (0 ,u I)), inserting the overall dipole moment operator (p = p1) and expanding the eigenstates in basis functions ( I) = c’1,,,1 sm)) yields M1 = Cimi(0I/JiIZm) (1.36) and a local transition moment can be defined m1(R,Q)= Cimi(0lI1iö(R — Ri)8(2_Qilm) . (1.37) Inserting m1 (R, c) in Eq. (1.20) defines the excitation density as u(E,R,)=m1(R,Q)M *f(E_E1) (1.38) which can be regarded as the contribution to the spectrum arising from the volume element at (R, c) interfering with the entire particle. As with the state density, if the particles are divided into spherical shells, (E, R, Q) is once again integrated over the angular component Q. Expressing the energy in wavenumbers, the definitions used in chapter 2 are obtained: cr(i7,R)= (mj(R)fmj(R)Rf2dR1)*f(c_ij) (1.39) m1(R)= Lmj(R,c2)dc. (1.40) 28 In discrete form, using fiim = (OIpjm), o(V, R) becomes u(i,R)= /4jmCjmjfljnCjnj R —R<AR. (1.41) I,i,j,m,n Again, to compare results for different volume elements it is necessary to divide by their respective volumes, i.e. in the case of spherical shells, consistent with Eq. (1.35), each term in the sum in Eq. (1.41) must be divided by R. The expressions of the state and excitation densities presented so far require the eigenvectors of the Hamiltonian. When the infrared spectra of the particle are calculated from the dipole correlation function, the density of states is no longer directly accessible and the excitation density must be derived in a different but mathematically equivalent way. To this end, we write the time evolution operator matrix representation in the molecular oscillator product basis i,,, )} (all molecular oscillators are in the ground state except for mode m excited on molecule i): Uimjn = Qrn e”j) = exp_-tcJ,I. (1.42) Inserting Ujmjn into the correlation function Eq. (1.23) gives &(t) = g(t) (OuI m)Uim,jnKJn 0), (1.43) i,m,j,n where g(t) is once again the Fourier transform of the line shape f(E), p is the overall dipole moment operator and o) is the ground state. The contribution of vibrational mode m of molecule i interfering with all other modes can then be written as 29 &im(t)= (1.44) Applying the inverse Fourier transform one obtains: °im (E) = $eEtajm (t)dt = (o C1,,I fg(t)e_1_E1 )t/hdtCi (i, p 0) =Cim,i(0Ipjm)(Ip0)5(E_Ei)*f(E_Ei) (1.45) Summation over all molecules i and all modes m yields the spectrum as defined in Eq. (1.20). This demonstrates that the total correlation function ö(t) can be decomposed into partial correlation functions &jm (t) whose Fourier transforms will give the local contributions of the molecules to the overall spectrum. Thus for any partitioning of a particle into volume elements SVk, the excitation density can be obtained by Fourier transform of the local partial correlation functions k (t), with Uk(t)_ öim(t)• (1.46) i€ô This does not involve any additional numerical work. Strictly speaking the excitation density is not a true density, as the interferences between vibrational modes can also be destructive. With increasing number of molecules in the volume elements c5V, however, the negative contributions tend to average out. 30 1.2.5 Remarks on the experiments I was not involved in the experiments conducted by our research group to obtain the data analyzed in chapters 2-5. Accordingly, only a brief description of the experimental setup used to generate molecular nanoparticles and record their spectra is provided in this section. Detailed descriptions are provided in refs. [68,69]. The particles discussed in this thesis were generated in a collisional cooling cell. It consists of an outer vacuum mantel thermally isolating the inner cell which contains the optics and is filled with a cold bath gas, usually He or N2. A dilute sample gas, with the concentration on the order of ppm, is rapidly injected into the cold bath gas. This rapid cooling leads to supersaturation and thus particle formation. The particles formed are analyzed using a fast-scan infrared spectrometer. This allows us to follow their temporal behavior. In contrast to particle formation in supersonic expansions, collisional cooling offers the advantage of observing the generated particles at thermal equilibrium over periods of hours. Multiple pass White Optics with an adjustable optical path length of 3-16 m are used to detect low concentrations of small particles. A list of substances and vibrational modes investigated in the subsequent chapters is given in Table 1. 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Hexter, Molecular Vibrations in Crystals (McGraw-Hill, New York, 1977) 36 2 Size effects in the infrared spectra of NH3 ice nanoparticles studied by a combined molecular dynamics and vibrational exciton approach1 2.1 Introduction Molecular ice particles in the form of aerosols play an important role in atmospheric processes of planets. They influence the energy balance of planet atmospheres through radiative energy transfer, phase transitions, and chemical and photochemical processes. Nanosized ice particles with dimensions below 10 nm play a unique role in this context since they do not only occur as stable constituents of particle clouds but are also essential intermediates in the formation and degradation of all larger particles. Despite their ubiquitous occurrence, however, their properties are largely unexplored as both their experimental and their theoretical investigation pose significant challenges. In the Earth’s atmosphere, the most important representative of this species are water ice aerosols [1]. In other planet atmospheres, icy aerosol particles consisting of small molecules play a similar important role, for example methane ice in the atmosphere of Neptune and Uranus [2,3]. Ammonia aerosol particles, the subject of the present investigation, were found as constituents of Jupiter’s and Saturn’s atmospheres [4,5]. The spectroscopic characterization of these ice particles is not only a prerequisite for remote sensing. The analysis of infrared spectroscopic features also gives us information about structural and dynamical properties of these particles. The experimental study of ammonia aerosols has been pioneered by R. E. Miller with spectroscopic investigations of particles in the ‘A version of this chapter has been published. G. Firanescu, D. Luckhaus, R. Signorell, I Chem. Phys., 125, 144501 (2006) 37 submicrometer size range [5 a]. Our previous investigations [6,7] have shown that for strong vibrational bands, the infrared spectra of molecular ice particles with sizes between 10 nm and 100 nm, are dominated by resonant transition dipole coupling. For particles with sizes below 10 nm additional spectral features appear, which are no longer based on this type of intermolecular interactions alone [6-11]. In this size range structural changes and surface effects are suspected to play a crucial role in the infrared spectra, as one expects a gradual change in the particle structure with decreasing particle size. Since for particles smaller than 10 nm the amount of surface molecules can exceed 80%, the influence of the surface should be considerable. These aspects are investigated here for the example of nanosized ammonia particles (< 10 nm) using classical molecular dynamics in combination with a novel extended vibrational exciton approach. The experimental infrared spectra in this size range show characteristic size-dependent features. For the umbrella vibration discussed here, we observe the appearance of a shoulder on the high-frequency side of the main peak which becomes more prominent the smaller the particles are. The main peak had also been observed in the spectra of large spherical ammonia particles (> 10 nm) [7]. Our hypothesis is that the “blue” shoulder results from structural differences between the particle’s core and its surface shell, the former presumably being crystalline and the latter more or less amorphous. In order to investigate this hypothesis we have developed an atomistic model which allows us to predict infrared extinction spectra for NH3 ice particles with different sizes and structures. Following related studies [8,12-15] on small molecular aggregates, we generate structures with varying degrees of crystallinity followed by simulated annealing (classical molecular dynamics). Here this procedure is implemented for particles containing up to 16000 atoms. The spectra are simulated using the vibrational exciton model [16], which is extended here to describe the effect of structural variations within the particle. A prerequisite for this approach is a suitable inter- and intramolecular potential model. 38 We have modified existing models to allow for a consistent description of the strong correlation between transition frequencies and intensities. The present study focuses on the v2 (umbrella symmetrical bending) mode of NH3. Of all the vibrations, it is the most strongly affected by the formation of the hydrogen-bonded network in the solid state, where the inversion of the molecule is now completely quenched. 2.2 Experiment Ammonia particles were generated in a collisional cooling cell by injecting dilute NH3/He gas samples into a precooled He bath gas as described previously in ref. [7]. NH3 (99.98%, Messer) was used without further treatment. Commercially available ultra-pure He (99.999%, Messer) had to be dried further by having it flow through consecutive i-N2 cooling traps to minimize its trace water content, which would otherwise lead to impurities in the NH3 ice particles. Sample gas mixtures consisted of 100-4000 ppm NH3 in He. During each experiment the bath gas was kept at constant temperature and pressure. Our experiments covered a temperature range from 20 to 80K and a bath gas pressure range from 50 to 1000 mbar. The collisional cooling cell was equipped with White optics and coupled to a Bruker IFS66v/S FTIR spectrometer to record infrared extinction spectra. The spectrometer was operated in the rapid scan mode. Individual scans took 360 ms at a resolution of 2 cm’. Data acquisition was started with a delay of about 1 s after the sample injection pulse. Background scans were taken immediately before each injection pulse. From the sample and background interferograms extinction spectra were calculated. Typically 50 such spectra were averaged to produce the final extinction spectra. Figure 2.1 depicts typical ice particle spectra recorded as a function of the bath gas temperature in the region of the umbrella (symmetric bending) mode v2. The spectra are further analyzed in section 2.4. 39 Figure 2.1 Experimental infrared spectra in the region of the umbrella vibration of ammonia ice particles. Particles were generated in a collisional cooling cell at the He bath gas temperatures indicated using a sample gas concentration of 400 ppm NH3 in He. The spectra are normalized to equal maximum extinction and offset relative to each other. Dotted line: Particle spectrum at 20 K with sample gas concentrations adjusted (1600 ppm) to generate particles of comparable size to 54K/400 ppm. 0.03 = 0.02 0.01 0.00 2.3 Theory and simulation details An atomistic model for the vibrational dynamics and structural changes underlying the observed behavior of NH3 particles requires three main ingredients: a potential function describing both intra- and intermolecular forces; a model for the internal structure of the particles formed in the collisional cooling process; and a dynamical approach to translate the structure and forces into infrared extinction spectra. The size (10 atoms) and complexity of the systems 960 1040 1120 1200 1280 I cm1 40 under consideration obviously necessitate some compromise between accuracy and computational efficiency as outlined in the following. 2.3.1 Model potential As pointed out, the potential model must be computationally simple to allow for the explicit treatment of 1 O- 1 ü degrees of freedom. At the same time it has to fulfill certain conditions in order to afford a qualitatively correct yet unbiased description of the changes from NH3 as a free molecule to its particulate state. Thus the potential must include a correct description of the free NH3 molecule, viz, its structure, inversion potential, and fundamental wavenumbers and transition moments. Furthermore the intermolecular part should provide the correct optimal arrangement of the NW’ ‘N hydrogen bond, in particular its bond length and strength. It should be noted that the (major) electrostatic contribution to the intermolecular potential is directly related to the transition moments so that intensities cannot be modeled independently from the potential. On the contrary, intensity patterns provide a sensitive and, most importantly, independent check for the quality of the intermolecular part of the potential. Most recently very accurate intramolecular model potentials have been published for NH3 in its electronic ground state [17,18], but they are computationally far too expensive to be used for describing ensembles of thousands of molecules. A less complicated form of the intramolecular potential, still including anharmonicities, was used in ref. [19] to simulate spectra of small clusters. In order to treat the much more complex particulate NH3, however, one has to resort to still simpler potential forms that allow the explicit treatment of tens of thousands of atoms, e.g. in the molecular dynamics runs of the simulated annealing procedure discussed later on. Since the present study will be limited to fundamental excitations a useful potential does not 41 have to include particularly accurate ariharmonicities. We therefore follow the common approach of including (diagonal) anharmonicities in an effective harmonic valence force field. This approach has been used to simulate both liquid and crystalline bulk spectra of NH3 [20-23]. As the valence force field used in these studies did not describe the inversion potential of NH3 correctly it had to be reparametrized in order to reproduce the frequency shift relative to the free molecule. Consequently these potentials do not show the correct behavior in the limit of large intermolecular separation (gas phase). We avoid this inconsistency by introducing a totally symmetric term in the bond angle displacements = !- (d,2 + dr? + dr )+ !- (da + da + da1 )+ l (da1 + da23 + da31 )2 (2.1) where dr and da represent the displacements of the bond lengths NH1 and angles H,NH (i,j = 1, 2, 3) from their respective gas phase equilibrium values, re=l.0124A and ae=106.68° [24]. The force constants Icy, kb, and i were adjusted to reproduce the gas phase fundamental transition wavenumbers [25]. Following Diraison and Martyna [20], we combined our intramolecular force field with the intermolecular potential function of Impey and Klein [26]. It consists of Lennard-Jones pair- potentials between the N-atoms of different molecules augmented by electrostatic interactions. The Lennard-Jones potential between molecules n and m is defined in the usual way: V =48 12 . (2.2) IrN (m) — rN(n)I,) jrN (m) — rN (n)I,) rN (m) denotes the position of the N-atom of the m-th molecule and 6 and ci are the Lennard Jones parameters. The electrostatic interactions are modeled by the Coulomb forces between a 42 small number of point charges distributed over the molecules so as to reproduce their static dipole and quadrupole moments. Originally designed for rigid molecules, the intermolecular potential function as used in refs. [19,20] suffers from a major defect: neither does the charge distribution correctly describe the dipole moment of the molecule as it moves towards the planar structure (where it vanishes), nor does it yield the correct transition moments for the fundamental transitions of NH3. We have corrected this behavior by redefining the positions of the point charges as follows: Three equal positive charges q are located on the NH bonds (vectors r) at a constant distance d+ from the N-atom: r r +d (23)÷N A negative charge of -3q is placed at r- at a varying distance from the N-atom: r =r +d bR (2.4) - N where R=Jj (2.5) The length parameter b is chosen so that cL becomes the distance of the negative charge from the N-atom at equilibrium. b= Re (2.6) Re1A 43 The parameters q, d+, and d were adjusted so as to reproduce the equilibrium values of the electric dipole and quadrupole moments and the experimental transition moment of the V2 (umbrella) fundamental of O.24D determined by Koops et al. [27]. The electrostatic potential is thus given by __ 1 — 3 — 3 + 9 (2.7) 4 r,. (m) — r÷(n) r (m) — r_ (n) r_ (m) — r (n) r_ (m) — r_(n) There is a total of ten parameters in our model, for which Table 2.1 lists all the numerical values used in the present study. It should be noted that these parameters were not fitted to the particle spectra. Rather they are all determined by independent experimental, or high level theoretical information as discussed above. Table 2.2 compares several monomer, dimer, and bulk properties calculated with the above model potential with corresponding experimental or theoretical (high level ab initio) reference values [21,24,25,27-37]. Dipole () and quadrupole (c9) moments are reproduced by construction as are bending frequencies (V2, V4) and the symmetric stretching fundamental (v3). The AlE splitting of the NH stretching vibration was not fitted, still the experimental value of V3 - V1 = 107cm is reproduced within a few cm’1. Note that the value for k accounts for the stretch- bend Fermi resonance coupling in an effective way: if this effect were to be described explicitly, the potential would have to be extended correspondingly. The inversion barrier height provides another independent test of our potential model. With 1593 cm1 its value lies about 10% below the best available theoretical estimate of 1769 cm’ by Leonard et a!. [38]. Since this quantity 44 Table 2.1 Potential parameters for the intramolecular (see Eq. (2.1)) and the intermolecular (see Eqs. (2.2) to (2.7)) contributions. Intramolecuiar Intermolecular parameters parameters k=40.0 eV/A2 d.=O.15118 A kb=3.945 eV/rad2 d--0.7368 A l=-.2515 eV/rad2 q==0.82334 e a 1 06.68 [24] [26] A re = 1.0124 [24] A = 0.0121 [26] eV was not fitted, the agreement must be considered very satisfactory given the simplicity of the potential model. The adjusted potential of Diraison and Martyna would have given 2259 cm’. The largest relative error occurs for the vibrational transition moments of the stretching fundamentals. They might be improved by allowing d+ (Eq. (2.3)) to vary as a function of the NH bond length. Since the present study focuses on the umbrella fundamental (v2), we have not pursued this further. The transition moment for V2 is reproduced by construction as it was included in the adjustment of the potential parameters. Our model slightly underestimates the strength of the NH N Hydrogen bond by 17% compared with the highest level theory available for the ammonia dimer [301. This is somewhat surprising as the bond length is even slightly shorter (1%) and closer to the collinear arrangement (10°). The optimum relative orientation, however, of the monomers is perfectly reproduced [30]. The absolute blue shift of their umbrella vibrations relative to the free monomer is underestimated by about 15 cm1. More important for our purposes is the splitting between the hydrogen bond donor and acceptor. Here the agreement is perfect (25 vs. 23 cm’). 45 Table 2.2 Comparison of monomer, dimer, and crystalline bulk properties derived from the model potential with experimental and theoretical reference values: dissociation energy De, hydrogen bond length RN..H, nitrogen distance RN..N, V2 fundamental wavenumber frequencies, and zero wave vector phonon wavenumbers ct)ph. The experimental monomer transition wavenumbers and dipole moments are averaged over the two tunneling components. NH3 monomer Exp./Theor. Model(Gas) p/D 1.5 [28] 1.5 &/DA 2.12[291 2.12 v / cm [i(0—*vi) / D] 3337 [0.026] [25] 3336 [0.009 1] V2 / cm [L(0—v2) / D] 950 [0.24] [25] 951 [0.23] V3 / cm1 [i(0—*v3)/ D] 3444 [0.018] [25] 3437 [0.014] V4 / cm [i(0—*v4) / DJ 1627 [0.084] [25] 1627 [0.14] Nil3 dimer Exp./Theor. Model(Gas) De/kJ/mor’ 12.23 [30] 10.14 RN..H / A 2.31 [30] 2.292 RN..N/A [31] 3.297 ZNNH/° 20.7 [30] 57 V2a,b / cm4 979, 1004 [32] 966, 989 Crystalline Solid Exp. Model (crystal) Model (gas) re/A 1.012 [24] ae/ ° 106.7 [24] E/kcalmol’ 8.7 [21,35] 7.1 7.1 o)ph(A)/cm 313 [36] 315 305 ‘107 [36j a)ph(A)/cm “ . 112 108 uncertain COPh (E) / cm” 298[36] 278 264 a)ph(E)/cm 107[36] 103 100 wph(F)/cm 532[36] 513 489 coph(F)/cm 426, 358[361 336 318 Coph (F) / cm4 259[36] 256 248 o)ph(F)/cm 183[36] 180 180 0ph (F) / cm’ 140[36] 136 137 46 Combining the experimental crystal geometry of Hewat and Riekel [34] with the lattice constants given by Olovsson and Templeton [33], we obtain a slight change of the molecular geometry (relative to the gas phase) upon crystallization. The NH bond contracts while the bond angle widens by 1-2% although there is some uncertainty in these values. This small but apparently systematic effect is not reproduced by our simple model potential. However, since the intramolecular part of the potential does not depend on the equilibrium values re and ae, and the intermolecular part does only weakly so, we do not expect any significant effect on the simulated spectra. These will only depend on the difference between the re and ae values used in the potential function and the actual equilibrium geometry of the system, so that these parameters may even be adapted to the bulk values. We refer to this model as “crystal-potential” with re = 0.99A and ae = 108° (as opposed to “gas-potential” using the original gas phase values of re and ae quoted in Table 2.1). Thus the experimental crystal geometry with a crystal-potential yields the same simulated spectrum as a crystal of molecules in gas phase geometry with the gas-phase potential. Similarly using any combination of model potential and crystal geometry leads to virtually identical simulated spectra once the geometry has been optimized. The lattice energy calculated with our potential for the experimental crystal geometry is underestimated by 1.6kcal/mol [2 1,35], which can be traced back to the error in the dimer dissociation energy. The zero wave vector frequencies of the phonon spectrum are in reasonable agreement with experimental data [36] considering that they were not included in the potential refinement. In our calculations we used the formalism of ref. [3 8a] with periodic boundary conditions, Ewald summation for the electrostatic interactions [38b], and direct summation of the Lennard-Jones interaction. The unit cell was defined by the experimental crystal geometry [34] and lattice constant [33]. The choice of molecular reference geometry (“crystal” vs. “gas”) does not affect the lattice energy, nor does it change the phonon spectrum significantly. 47 2.3.2 Structural models As will be seen in section 2.4, the variations observed in the extinction spectra of ammonia particles are caused by the disruption of the crystalline H-bond network, the extent of which depends on the conditions of particle formation. Considering the energies involved (a few kcal/mol) the geometry of individual molecules will remain largely unaffected, which leaves the orientational and translational degrees of freedom as variables of the model. Given the more or less isotropic conditions of nucleation and particle growth, we assume a core-shell architecture of spherical particles either with crystalline core and amorphous shell or vice versa. Without any further information, we construct the particles starting from a perfectly crystalline arrangement and by applying random rotations and translations to the molecules located in the presumed amorphous regions (core or shell). This procedure yields implausibly abrupt core-shell boundaries. It can also produce unrealistic highly strained configurations (excessively small or large nearest neighbour distances). We removed these “artefacts” by annealing the raw structures using the following molecular dynamics (MD) scheme [39]. The kinetic energy was distributed in the system, corresponding to an initial temperature of 50 K or 100 K. Then it was gradually reduced in steps of 0.01% or 0.1% of the current value every 0.1 fs, for a simulation time of 5 ps. For the MD calculations we used the standard “leap-frog” Verlet algorithm [39]. The procedure was repeated until a local minimum of the potential energy was found and the calculated extinction spectra converged. The number of runs necessary to achieve convergence varied depending on the initial particle structure and size. Particles with less then 150 molecules usually required three or more runs regardless of structure. When starting with a completely crystalline particle composed of several hundred molecules, a single run sufficed. For particles consisting of up to about a thousand molecules extinction spectra were averaged over an ensemble of annealed 48 particles to account for statistical variations. For much larger particles the statistical variations were negligible. It turned out that the only effect of translational randomization of the raw structures was a broadening of the calculated extinction spectra which was almost completely reversed by the annealing procedure. To ascertain that this effect did not simply result from a bias of our “amorphization” protocol we checked it against another approach: molecules were randomly added to an initially small sphere, with the condition that no two atoms come closer than 3 A to each other. Once the number of failures exceeded a given threshold, the sphere radius would be expanded, thus creating the particle by shells. Although the annealing took much longer for these particles it finally converged to the same spectra. Thus without any loss of generality we limited our model of amorphous ammonia ice to orientationally randomized structures. We quantify the degree of crystallinity — or rather the deviation from it — by the infimum of the standard deviation of atomic Cartesian coordinates x from their values xC in the crystalline particle: 1NA = inf{_xi - } (2.8) N is the number of atoms in a particle. In practice we determine the infimum in Eq. (2.8) by transforming to the centre-of-mass frame and minimizing the square-root expression with respect to the three Euler angles using a Levenberg-Marquardt algorithm [40). The maximum value of 0A for completely amorphous particles lies between 1.3 and 1.4 A. 49 2.3.3 The extended vibrational exciton model The contents of this section have already been presented in detail in section 1.2, except for a few system specific aspects. The vibrational exciton model has proven an extremely powerful tool for modeling infrared extinction spectra of molecular nanoparticles containing thousands of molecules. It applies to those cases where the spectra are dominated by resonant transition dipole-coupling, involving transition frequencies and dipole matrix elements as the only molecular parameters [6,7]. The vibrational exciton Hamiltonian is given by Eq. (2.9). 1 N 3p .r..).r..—p. .r...r.)H=H — p. ‘ U (2.9)° 4n0 I is the vibrational Hamiltonian of the isolated molecule, p is the point dipole operator located at the site of molecule 1, and is the vector pointing from molecule ito moleculej. In its previous implementation all molecules in the particle are treated as identical with a common transition dipole and frequency (i.e. I eigenvalue), which can be determined completely independently, e.g. from standard gas phase absorption spectra. The particle spectra are then completely determined by the distance and relative orientation of the molecules. The vibrational exciton model is applicable to crystalline solids, where all molecules are equivalent by symmetry as is the case for the cubic crystalline phase of ammonia. The highly anisotropic hydrogen bond interaction between the ammonia molecules, however, is on the order of vibrational excitation energies. Thus any perturbation of the perfect crystal structure can produce significant shifts of vibrational frequencies of individual molecules. The Extended Vibrational Exciton Model (EVE) introduced here accounts for this effect through explicit intra- and intermolecular potential and dipole functions, while retaining the 50 computational efficiency and conceptual elegance of the original approach. The potential and dipole functions enter the model through transition frequencies and dipole matrix elements for each individual molecule, which we calculate within the local double harmonic approximation. A normal mode analysis is performed for each molecule in the field of all other molecules, yielding local normal coordinates, harmonic frequencies, and dipole derivatives. These are then used as parameters in a standard exciton calculation. The EVE model therefore accounts for all terms beyond resonant dipole coupling within the harmonic approximation, which should be sufficient for the fundamental excitations studied here. (Note that the potential functions use locally quadratic forms Eq. (2.1) that effectively account for diagonal anharmonicities.) To validate our approach, we compare it with a full normal mode analysis (NMA) for annealed spherical crystalline particles. Figure 2.2 shows the EVE [panel a)] and NMA [panel b)] spectra for a series of spherical crystalline particles consisting of 25 to 1015 molecules. For the smallest particles — rather molecular clusters — the overall band structure is described correctly by the EVE model while small differences remain in the details of the intensity distribution. With increasing particle size the agreement quickly improves until the EVE and NMA spectra become virtually indistinguishable for particles containing more than a thousand molecules. This is a consequence of the long-range character of the resonant dipole coupling, which increasingly dominates over other interactions as the particle radius increases. The only difference remaining for larger particles is a small blue-shift of the main peak by about 10 cm’ relative to the NMA spectrum. This difference amounts to the peak width (FWHM) or about 10% of the overall blue shift with respect to the free molecule absorption. Given the simplicity of the model (and the absence of any adjustable parameters) this must be considered very satisfactory. It is particularly important for the purposes of the present study that the comparison between EVE and NMA spectra comes out 51 Figure 2.2 Comparison between extinction spectra calculated for spherical particles with radii as indicated, using a) the Extended Vibrational Exciton (EVE) model and b) a complete normal mode analysis (NIvIA). The particles were generated as spheres cut from the perfectly crystalline bulk and then annealed. a) b) (NH3)95 _____ Cl 92n . I (NH3)140 C 0 C Lii the same for all architectures of interest (crystalline, amorphous, crystalline core/amorphous shell). Except for the same blue-shift of 10 cm’, the EVE spectra are virtually indistinguishable from the NMA spectra. We also note that the band shape is almost entirely determined by the frequency variation as an EVE calculation assuming transition dipoles of constant length is almost indistinguishable from the full EVE calculation. The difference in computational effort of EVE and NMA calculations is enormous. For N molecules with Na atoms per molecule the NMA calculations consist of setting up and diagonalizing a force constant matrix of dimension 3Na•N. The EVE calculations by contrast only involve force constant matrices of dimension 3Na and the diagonalization of a final matrix of dimension N. For Na = 4 and N 1015 this translates into a speedup of almost three orders of 52 950 1000 1050 1100 1150 950 1000 1050 1100 1150 Icm1 Icm’ magnitude! Moreover NMA calculations would become difficult to apply to particles larger than that. With the EVE model the computational effort is practically independent of the molecules’ size NA. Vibrational exciton calculations with 10000 molecules [6,7] only take a few hours on a modem PC. For our present calculations, the construction of particle architectures by simulated annealing becomes the limiting factor. In addition to the calculation of extinction spectra, the analysis of the EVE wavefunctions allows us to correlate spectral features with structural properties of the particles (see section 2.4). For this purpose we introduce two auxiliary quantities — the radial density of states p(P, R) and the radial excitation density cr(i7, R). The radial density of states is defined by d2 (2.10) where F is the projection of the spectrum of the EVE Hamiltonian (Eq. (2.9)) onto the local excitations —v1)8(R —R1)8(Q— 2i) (2.11) I I) are the eigenfunctions of the vibrational Hamiltonian (Eq. (2.9)), associated with a transition frequency i relative to the ground state. R are the radial and 2 the angular coordinates with respect to the particle’s centre of mass. Indexed coordinates refer to the translations of individual molecules. In the EVE model the molecules’ positions are frozen reducing the angular integral to a discrete sum. Subdividing the frequency and radial axes into discrete bins leads to the discrete analog of Eq. (2.10) with bin sizes ziv and zIR, respectively: 53 v1 —vI < Av, R1 —RI <AR (2.12) where c1,j is the contribution of the excitation of the i-th local oscillator i) to I). Thus Eq. (2.12) represents the density of locally excited molecular states as a function of the distance from the particle centre of mass and of the frequency. As not all excited states I I) are accessible by optically allowed transitions, we introduce the excitation density o(i7, R) to analyze the origin of individual features in the extinction spectrum (for real valued wavefunctions). o(i’ R) = (mi(R) jmi (R)Rr2dRt)* f(i — i), (2.13) wherefis the line shape the excitation density is convoluted with and m1(R)=m1(R,Q)dQ, (2.14) m1 is the transition dipole density of state at frequency i1 derived from the overall dipole operator i mj(R,Q)=(IIi)(OIp6(R — Rjo(2 —c2i). (2.15) The discrete analog of Eq. (2.13) is given by Eq. (2.16). u(V,R)= vi —VI <Av, R —RI <AR (2.16) 54 R) represents the contribution from all the molecules located in a given radial shell to the absorption strength at frequency i. Note that the sum over j extends over the whole particle accounting for the interferences of a given shell with all others. Summing cr(i7, R) over all radial shells yields the total absorption spectrum. 2.4 Results 2.4.1 Experimental spectra Figure 2.1 shows infrared extinction spectra recorded after particle formation was complete at 21, 37, 54, and 80 K using a sample gas concentration of 400 ppm NH3 in He. The spectra are normalized such that the extinction maxima are all equal. The 80 K spectrum consists of a relatively narrow main peak centered at 1065 cm’ and a weak shoulder around 1100 cm’. The main peak coincides with spectra of large spherical particles (>20 nm) obtained in the limit of large concentrations [7]. With decreasing bath gas temperature, the high frequency shoulder grows relative to the main peak in Figure 2.1, which broadens. Virtually identical series of spectra can be obtained at higher bath gas temperature (I) if the sample gas concentration is lowered correspondingly, and vice versa, as illustrated for one example in Figure 2.1: the spectrum recorded at 54 K with a sample gas concentration of 400 ppm NH3 in He is almost identical with the spectrum recorded at 20 K with a sample gas concentration of 1600 ppm N}13 in He. This experimental observation will serve to show that the series of spectra in Figure 2.1 correspond to particles with increasing size from top to bottom. As all experimental spectra in Figure 2.1 apparently consist of a narrow and a broad contribution in varying ratios, we have tried to decompose the spectra into two components. We represent them as linear combinations of the broadest (21 K! 400 ppm) and the narrowest (80 K / 55 400 ppm) experimental spectrum observed so as to maximize the broad contribution to the former and the narrow contribution to the latter without producing unphysical band shapes. We find that all different experimentally observed spectra can be reconstructed within the experimental signal to noise ratio from the same two T-independent components shown in Figure 2.3a. One component (upper trace) is narrow (FWHM 19 cm1) and slightly asymmetric, the other one (lower trace) is broad (FWHM = 72 cm’) and symmetric. The latter is blue-shifted relative to the former by about 19 cm’. The relative contributions b (in %) of the broad component as a function of temperature are listed in Table 2.3. Figure 2.3 Spectral decomposition. a) Components derived from experimental spectra. b) Components derived independently for two particle sizes from spectra calculated for particles with different amorphous shell volume fractions. Solid line: (NH3)1015 , R = 2.0 nm. Dashed line: (NH3)1512, R 2.4 nm. The dotted line represents the decomposition of a (NH3)4075 particle spectrum into its core (34%) and shell (66%) contributions using Eq. (2.16). See also Table 2.4. a) b) 1’I•I•I• 1000 1050 1100 1150 950 1000 1050 1100 1150 (cm1) (cm1) 56 Table 2.3 Spectral decomposition (broad component b in %) of the experimental particle spectra (Figure 2.1). r is the estimated amorphous shell thickness assuming core/shell particles with a crystalline core and an amorphous shell. T is the temperature, [NH3] the sample gas concentration, and R is the estimated particle radius. T (K) [NH3] (ppm) R (nm) broad component b (%) rs (nm) 21 400 2 92 1.1 37 400 4 75 1.5 54 400 6 60 1.6 80 400 10 46 1.9 20 1600 6 60 1.6 By construction the 8% contribution of the narrow component to the broadest observed spectrum (21 K / 400 ppm) represents the largest value compatible with a smooth broad residue (i.e. without kinks or negative extinction). Consequently the broad contribution b quoted is a lower bound for the 21 K / 400 ppm spectrum and an upper bound for the 80 K / 400 ppm spectrum. The last column of Table 2.3 also lists the estimated shell-thickness r for a core/shell architecture assuming that the narrow component arises from the core and the broad component from the shell. This assumption will be discussed further below and in the next subsection. In principle, one might envisage several explanations for the observed temperature dependence of the spectra shown in Figure 2.1: (i) two different types of particles with T independent characteristics are formed in varying ratios with the “broad” type more abundant at lower T; (ii) only one type of particle is formed at a given temperature, but its characteristics, such as phase and shape, vary continuously with T; (iii) the observed spectral changes are attributed to the differences in the size of particles formed under different conditions. Cases (i) through (iii) are of course only idealized limiting cases for the dependence of the particle size-, 57 shape-, and phase-distribution on the experimental conditions. The decomposition into T independent components would apparently favour explanation (i). This, however, is hard to reconcile with the observation of very similar behaviour at rather different experimental conditions as shown for the example of the 54 K I 400 ppm and the 20 K / 1600 ppm spectrum. If explanation (ii) were correct, it would be equally hard to rationalize why the phase or the shape of the particles should show the same T-dependence under different experimental conditions. Nor do we observe the low frequency shoulder characteristic of crystalline NH3 ice particles with elongated shape [7]. Rather we have chosen experimental conditions that have previously been demonstrated to favor spherical particle shapes [7]. For amorphous particles, no shape effect would be expected at all as demonstrated in Figure 2.4 by simulations for different particle shapes. Figure 2.4 Extinction spectra calculated for amorphous (N}13)32 particles of different shape as indicated. The spectra were calculated after annealing the particles. I • I ‘ I • I • 950 1000 1050 1100 1150 V (cm1) 3x4x12 unit cells 58 This only leaves the variation of the particles’ size (iii) to explain the experimental observations. There are currently no direct experimental ways to determine reliable radii for aerosol ice particles in the range below 10 to 20 nm. Devlin et al. used CF4 coatings to determine the size of supported particles [8,40aj. This method, however, cannot be used for our free aerosol particles. Even in the case of supported particles, its accuracy may be limited by agglomeration and coagulation. In the case ofN20 ice particles we were able to derive particle radii in that range indirectly from the observed spectra, exploiting features that could be assigned to the surface layer and the particle’s core, respectively [10]. Assuming that under the same experimental conditions nucleation and particle growth take a similar course for ammonia and N20, we arrive at the estimates for the particle’s radius R given in Table 2.3. Although there is some uncertainty associated with the underlying assumption, it would be very surprising if these estimates were off by more than a factor of two. It should also be remembered that the values given correspond to averages over a size distribution whose exact shape is not known. Following common theories of nucleation one would then expect the following behavior: at a given sample gas concentration the number of condensation nuclei formed after injection into the cold bath gas increases with the extent of supersaturation, i.e. with decreasing temperature. Consequently, the particle growth will stop at smaller particle sizes. Analogously at a given temperature T the nuclei can grow to larger particles at increased sample gas concentrations. Thus the same particle sizes can be obtained at increased T if the sample gas concentration is lowered appropriately at the same time, which is exactly the situation exemplified in Figure 2.1 for the 54 K / 400 ppm and the 20 K / 1600 ppm spectra. But how does explanation (iii) tally with the decomposition of observed spectra into temperature independent components shown in Figure 2.3 a? The particle size can affect spectra through the inherent size-dependence of extinction spectra as well as indirectly through structural 59 changes in the surface. Molecules close to the surface experience an environment different from those in the core (closer to the bulk). This can have a direct effect on the excitonic structure. In addition, it can also lead to structural relaxation in the surface layer. Smaller particles have a higher percentage of molecules close to the surface than larger particles do. As will be seen in the next section, the simulations of particle spectra using the EVE Hamiltonian support this explanation and reconcile the assumption of a single particle type with the observation of two temperature independent spectral components. 2.4.2 Simulated spectra The comparison between experimental and theoretical spectra focuses on the smallest particles consisting of about 1000 molecules, which corresponds to a particle with a radius in the range of 2 m (21 K spectrum in Figure 2.1). In part this reflects the computational limitations due to the enormous size of the numerical problem. More importantly it enables us to understand what happens for the smallest particles, where the effect on the spectra is largest. With the assumption of a core/shell architecture of the particles, discussed above, we have to consider four different classes. Figure 2.5 compares the corresponding spectra simulated for a) a completely crystalline particle, b) an amorphous core surrounded by a crystalline shell, c) a crystalline core surrounded by an amorphous shell, and d) a completely amorphous particle. The left hand side of Figure 2.5 depicts (Nil3)1015 particle spectra before, and the right hand side after annealing. These spectra illustrate the general trend for different particle architectures and document the influence of annealing. As can be seen the general effect of annealing is important. It shifts the broad component to higher frequencies, while the narrow peak arising from the crystalline core remains practically unchanged except for a slight broadening. 60 At first glance, none of the simulations produce a qualitatively satisfactory agreement with the spectra in Figure 2.1. This is not a question of uncertainties in the particles size as we found no qualitative differences in spectra simulated for particles with smaller or larger radii within 20%. (Note that this covers a factor of almost four in volume, i.e. 500-1800 molecules). The experimental spectrum shows contributions from a broad and a sharp component. The latter is completely absent in the spectra simulated for amorphous particles, whether surrounded by a crystalline shell (b) or not (d). Moreover both yield overall band shapes which are significantly too broad compared to the experimental spectra. After annealing the crystalline particle [trace a)] Figure 2.5 The v2 band of ammonia ice particles with different architectures: a) crystalline, b) amorphous core with crystalline shell, c) crystalline core with amorphous shell, d) amorphous particle. The spectra were calculated with the EVE model for spherical (NH3)1015 particles before (lhs) and after (rhs) annealing. The shell volume fraction before annealing was 50% corresponding to a thickness of 0.4 nm. 800 900 1000 1100 900 1000 1100 t(cm) (cm1) 61 yields a spectrum consisting of a narrow peak symmetrically on top of a broader background. Particles consisting of a crystalline core and an amorphous shell [trace c)] give rise to qualitatively similar spectra. After annealing the most significant difference is a more prominent broad contribution as compared to the crystalline particle. Thus the only significant qualitative discrepancy between experimental spectra and simulations for a crystalline core with an amorphous shell is the small blue-shift of the broad background relative to the narrow main peak, which is not reproduced in the simulations. Closer inspection shows a very small blue-shift (3 cm’) in the simulated spectra [trace b)] after annealing. It does not reproduce the observed shift of 19 cm’ (see Figure 2.3 a) quantitatively, however. Further insight into this behavior is gained from an analysis of underlying exciton wavefunctions. Figure 2.6 shows in traces a 1) to a4) infrared spectra before [a 1) and a3)] and after [a2) and a4)] annealing. al) and a2) are for a crystalline particle and a3) and a4) are for a crystalline core-amorphous shell particle (50% shell volume fraction). The radial corresponding excitation densities (Eq. (2.16)) are depicted in traces bi) to b4) and the corresponding radial densities of state (Eq. (2.12)) are shown in traces ci) to c4). For the crystalline particles the radial densities of states [traces ci) and c2)J are characterized by two narrow main exciton modes split by about 20 cm’, of which only one is optically active so that it also appears in the excitation densities in traces bi) and b2). The optically active mode corresponds to the so-called Froehlich mode of the classical limit characterized by a probability of excitation that is constant as a function of the radius [411. Both modes are narrow in the frequency domain, but completely delocalized radially. The corresponding densities remain largely unaffected by annealing. The broad low frequency tail of the crystalline particle spectrum originates in the surface layer. Upon annealing it shifts by almost 62 Figure 2.6 The influence of particle architecture and annealing on calculated excitation densities for (NH3)1015 particles (R 2.0 nm). Rows: a) Extinction spectra. b) Radial excitation densities R2cr(i,R) (Eq. (2.16)). c) Radial densities of state R2p(i7,R) (Eq. (2.12)). Columns: (1,2) Crystalline particle. (3,4) Crystalline core / amorphous shell particle (50% shell volume fraction). Columns (1,3) refer to particles before annealing. Columns (2,4) refer to those after annealing. The dashed line indicates the core/shell boundary. The integral of row b) over R yields row a). 0< 0< a:: 50 cm’ towards higher wavenumbers. The underlying densities broaden slightly towards the particles interior but remain essentially concentrated near the surface. Introducing an amorphous shell structure into the crystalline particle {a3), a4)] leads to analogous behavior under annealing: The amorphous shell gives rise to a broad spectral feature initially centered around the gas phase transition wavenumber (about 950 cm1), which shifts upon annealing by about 100 cm1 to higher 63 950 1050 950 1050 950 1050 950 1050 (cm1) (cm1) (cm1) (cm1) wavenumbers, while its width is retained [panels b3) and b4)]. The sharp peak near 1050 cm1 is clearly due to the crystalline core. Except for a slight broadening, it remains unaffected by the annealing procedure. The bimodal structure of the density of states observed for the crystalline particle [ci), c2)] remains intact in the crystalline core [c3) and c4)], with only the higher frequency mode showing up as optically allowed in the excitation density [b3) and b4)]. The amorphous shell by contrast does show neither the bimodal state density nor any pronounced selectivity for optical excitation: the excitation density (b4) almost matches the density of states (c4). The important result of Figure 2.6 is that the broad component in the spectra of the core/shell particles clearly arises from the amorphous shell whereas the sharp component is caused by the crystalline core. Figure 2.7 illustrates the deviation from crystallinity as a function of radius as defined by Eq. (2.8) before and after annealing in trace a) for a crystalline particle and in trace b) for a core/shell particle with a crystalline core and a 50 vol% amorphous shell. The broad excitation density arising from the outermost layer of the relaxed crystalline particle [trace a)] correlates clearly with the pronounced decrease in crystallinity near the particle’s surface. This correlation is borne out even more clearly in the case of the amorphous shell structure [trace b)]. The initial sharp step in crystallinity (dashed line) smoothes out somewhat upon annealing (solid line), which explains the slight broadening of the core’s crystal-peak. The amorphous character of the shell, however, remains unchanged, which is reflected in the approximately constant spectral width of the broad contribution to the spectrum. Its large blue shift upon annealing (Figure 2.6) must therefore be a consequence of the local reorientation of NH3 molecules and strengthening of H-bonds without introducing any long-range order. In other words, the frustration of nearest neighbour H-bonds is removed upon annealing without significantly alleviating the long-range frustration of the crystalline network. Thus the blue shift 64 Figure 2.7 Deviation from crystallinity o (Eq. (2.8)) for a (NH3)1015 particle before (dashed lines) and after (solid lines) annealing. The maximum value for cr lies between 1.3 and 1.4 A for completely amorphous particles. 1.2- a) crystal 0.8 - 0- 1.2 b) core/shell (50%) 0.8 - b< ‘I 0.4- o.o—--.-- . __., R(A) of v2 in the crystal compared with the gas phase is clearly a short range effect caused by the nearest-neighbour hydrogen bonds. Note that the single hydrogen-bond in the dimer accounts for about half of the shift (see Table 2.2). From the above discussion, one would clearly be inclined to associate the narrow and broad components of the experimental spectra with a crystalline core and an amorphous shell, respectively. Both give rise to extinction signals blue-shifted relative to the gas phase, as observed experimentally. Only the relatively small difference between the shift of the broad and narrow component is not reproduced quantitatively by our model (Figure 2.1 and Figure 2.6). If this picture is justified, the simulated spectra should decompose into two independent components similar to the ones observed experimentally (Figure 2.3a). Moreover, those two I I • I • I 5 10 15 20 65 components should be largely independent of the particle size and of the relative core/shell contribution if we are to deduce an at least semi-quantitative picture of the particles’ architecture, such as the shell thickness. To address this point, we have simulated crystalline core/amorphous shell particles of different size and varying amorphous volume fraction. Figure 2.8 depicts the spectra simulated for particles consisting of 1812 molecules (about 2.4 nm in radius) with systematically increasing thickness of the amorphous shells after annealing. All these simulated Figure 2.8 Calculated extinction spectra for an ensemble of (NH3)1812 particles (after annealing) with a crystalline core and an amorphous shell of varying thickness. The volume fraction of the amorphous phase is defined before annealing. The typical change in the amorphous to crystalline ratio after annealing is shown in Figure 2.7b 950 1150 66% 50% —. —. 33% 16% 1000 1050 1100 (cm) 66 Table 2.4 Spectral decomposition (broad component b in %) of the simulated core/shell particle spectra after annealing. N is the number of molecules in a particle and R is the particle radius. The amorphous shell thickness r is derived from the broad component b. The amorphous volume fractionfs and the core/shell boundary at ro (Figure 2.7 dashed line) are defined by construction before annealing. N= 1015 (R = 2.0 nm) N= 1812(R = 2.4nm) N= 4075 (R = 3.2 nm) fs (%) r0 (nm) b (%) rs (nm) r0 (nm) b (%) rs (nm) r0 (nm) b (%) 1) rs (nm) 1) 83 0.9 96 1.3 1.1 95 1.5 1.4 66 0.6 82 0.9 0.7 81 1.0 1.0 66(73) 1.0(1.1) 50 0.4 63 0.6 0.5 61 0.6 0.7 33 0.2 35 0.3 0.3 35 0.3 0.4 16 0.1 9 0.1 0.1 4 <0.1 0.2 spectra can indeed be decomposed into one narrow and one broad component in a way completely analogous to that discussed for the experimental spectra. The results are collected in Table 2.4 (N=1812). To test whether or not this decomposition is size-dependent, we have performed the decomposition into a narrow and a broad component independently for another particle size, corresponding to about half the volume (N 1015). The resulting spectral components are shown in Figure 2.3b where the solid line refers to (NH3)1015 particles and the dashed line to (NH3)1812 particles. The two pairs are almost indistinguishable. Linear combinations of either of these pairs also reproduce the spectra calculated for even larger particles. Up to the maximum of about 4000 molecules (particle radius 3.2 nm), those reconstructions agree with the original spectra to within statistical variations of the amorphous ensemble. In other words, the same spectral decomposition works independently of both the 67 thickness of the amorphous shell and the particle size (see Table 2.4), the only parameters by which particles can differ within our model. These results clearly confirm the qualitative correlation of the spectral decomposition into a narrow and a broad component and the spatial division of the particles into a crystalline core and an amorphous shell. With the underlying wavefunctions, the exciton model allows us to go even one step further and make this correlation quantitative. From the excitation densities (Eq. (2.16)) such as the ones depicted in Figure 2.6 [panels bl)-b4)], we can accurately separate the core contributions from the shell contributions to the total extinction spectrum by integrating o-(V, R) over the respective ranges of the radius R. For this purpose, we use the same core/shell boundary used to construct the amorphous shell before annealing (dashed line in Figure 2.6). For a (NH3)4075 particle spectrum with a 66% amorphous shell volume fraction, the dotted lines in Figure 2.3b show the core spectrum in the upper and the shell spectrum in the lower panel. The agreement of the core and shell spectra with the narrow and broad spectral components, respectively, is evident. 2.5 Summary Comparing the analysis of the experimental and simulated spectra we come to the conclusion that the observed spectra follow the behavior expected for particles consisting of an amorphous shell grown on a crystalline core. The widths of the broad and the narrow spectral components derived from our simulations (FWHM=14 cm’, 69 cm’) are very similar to the experimental ones (FWHM19 cm’, 72 cm1 ±1 cm’), except for the blue-shift of the broad component (3 vs. 19 cm’) relative to the narrow one. We have demonstrated that the narrow spectral component arises from the crystalline core while the broad component arises from an orientationally disordered amorphous shell. In particular, we showed that these components are independent of particle size 68 and shell thickness. This enables us to simulate realistic extinction spectra not only for the particle sizes accessible to direct computations, but over the whole size range considered here. Figure 2.9 compares the result with the set of experimental spectra (left hand side). For the simulations (right hand side), we used the spectral decomposition of our calculated spectra, with the broad component blue-shifted by 19 cm1. The relative contributions of the components were adjusted for best agreement with experiment. The fact that the numerical values are almost identical to the experimental ones confirms our interpretation. On the basis of the combined MD/EVE model, we have thus been able to analyze particle extinction spectra in terms of the particles’ architecture. We come to the conclusion that particles are formed with a crystalline core surrounded by an amorphous shell whose thickness is approximately constant at 1-2 nm, independent of the particle size. This result conforms with a plausible picture of isotropic particle growth: as NH3 molecules condense on the shell of a nucleus, the heat of condensation will be partly absorbed by the core before it eventually dissipates into the bath. As a consequence, the core is continuously annealed during the growth process, while the outermost shell is immediately frozen by the bath gas. An approximately constant thickness of this amorphous growth layer of a few unit cells seems perfectly consistent with this picture. Thus the core-shell architecture of the particles is largely controlled by the kinetics of the particle growth. There is of course a small thermodynamic contribution reflected in the structural changes upon annealing of purely crystalline particles. A less ordered surface layer is clearly more stable than the perfect crystal. But it is clear from Figure 2.5a and Figure 2.7a that the deviation from crystallinity is too small and the surface layer too thin to account for the observed spectral broadening. We note that the annealed crystal (Figure 2.7a) is by far the most stable structure and lies about 8 eV (or about 1 kJ/mol NH3) below the annealed core-shell 69 particle ensemble shown in Figure 2.7b. The convergence of the annealing procedure is estimated to be better than 0.5 eV. Figure 2.9 Comparison between a) experimental (see also Figure 2.1) and b) simulated extinction spectra. The particle radii and the contributions of the broad spectral component are indicated for each observed spectrum. The simulations were obtained by linear combination of the spectral components derived from calculated extinction spectra, with the ratios adjusted for best agreement as indicated. The broad contribution was blue-shifted by 19 cm’ as explained in the text. a) b) The model for NH3 ice particles we have presented here provides a realistic qualitative representation of the structural variations at a molecular level that determine the infrared extinction spectra for particle radii of a few nm. We expect that further refinement of the NIf hydrogen bond description would remove most of the remaining small quantitative discrepancies regarding the absolute shift of the extinction maxima relative to the gas phase values and the 4 nm 75% 72% I • I • 1000 1050 1100 1150 1000 1050 1100 1150 Icm1 Icm1 70 difference in the shift between crystalline and amorphous contributions. We would like to stress, however, that these are very small discrepancies on the order of only 1% of the vibrational excitation frequency. Given the simplicity of the model even the quantitative agreement must be considered very satisfactory. 71 2.6 References [1] J. H. Seinfeld and S. N. Pandis, Atmospheric Chemistry and Physics, (Wiley & Sons, New York, 1998) [2] S. G. Gibbard, I. de Pater, H. G. Roe, S. Martin, B. A. Macintosh, and C. E. Max, Icarus, 166, 359 (2003). [3] G. F. Lindal, J. R. Lyons, D. N. Sweetnam, V. R. Eshleman, D. P. Hinson, and G. L. Tyler, I Geophys. Res., 92, 14987 (1987) [4] 5. K. 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Huffman, Absorption and Scattering of Light by Small Particles, (Wiley-Interscience, New York, 1998) 74 3 Phase, shape and architecture of SF6 and SF6ICO2 aerosol particles: infrared spectra and modeling of vibrational excitons2 3.1 Introduction Infrared spectroscopy is one of the central methods to characterize molecular aerosols [1- 9]. In principle, vibrational spectra contain a wealth of information about intrinsic properties of aerosol particles. This includes information about the shape, the size, the surface structure, and the architecture of the particles, as well as their composition and their phase behavior. It is, however, far from standard to extract this information from experimental aerosol spectra. Modeling plays the key role here. Models based on classical scattering theory (e.g. Mie theory) have been in use for many years [10-12]. They rely, however, on empirical input data, such as refractive index data, which have to be extracted from additional experimental data. As a consequence, these classical models are not very accurate. Furthermore, they are restricted to the treatment of larger and basically single component particles. The reason is that they cannot account for variations that happen at a molecular scale in the particles. This is for example the case for the surface region in very small particles where the structure can change on a molecular scale or for multi-component particles where the composition can likewise vary on a molecular scale. The usefulness of these classical models in interpreting experimental infrared spectra is thus very limited. The only solution here are microscopic models that take into account the essential intramolecular and intermolecular interactions - a seemingly impossible challenge for the thousands to billions of molecules contained even in comparatively small aerosol particles with radii below 100 nm. It is clearly impractical to take all intra- and intermolecular interactions 2 A version of this chapter has been published. G. Firanescu, D. Luckhaus, R. Signorell, J Chem. Phys., 128, 184301 (2008) 75 into account when modeling such a complex system, but it turns out that this is not necessary in order to extract valuable information about particle properties from experimental spectra. Research of the last couple of years has shown that intrinsic particle properties, such as shape and architecture, exhibit a particularly strong influence on vibrational bands with large molecular transition dipoles ([8] and references therein). These bands are veritable sensors for intrinsic particle properties, while all other, weaker vibrational bands are much less sensitive to particle properties and are thus of minor interest in this context. Our research group has demonstrated that the quantum mechanical vibrational exciton model is extremely successful in modeling such bands for aerosol particles [8,13,141. Similar approaches had previously been used to describe bulk systems, where periodic boundary conditions apply, and for treating small molecular aggregates [1,15-19]. But the huge number of vibrational degrees of freedom of an aerosol particle and the fact that periodic boundary conditions cannot be applied had long been major obstacles preventing the use of exciton models for modeling molecular aerosol particles. In the past few years, we have investigated several molecular aerosol systems combining infrared spectroscopy with the vibrational exciton model [8,14,20-23]. This has enabled us to uncover characteristic infrared spectroscopic features that arise from the particle shape, the particle architecture, the composition of the particles, and their internal structure. Most importantly, our quantum mechanical model provides truly microscopic explanations for the phenomena observed in our experiments. This has allowed us to formulate propensity rules for the occurrence of strong particle effects in infrared spectra [8,24]. In a previous contribution, we extended our original exciton model [13] to account for structural variations within the particles [14]. The present contribution deals with two further extensions. We present a new computational scheme to increase the number of degrees of freedom that can be treated simultaneously, which is necessary to converge spectra of particles larger than a few nanometers. Furthermore, we account 76 for the effect of the molecular polarizability on the aerosol particle spectra. SF6 aerosol is an ideal model system in this context. Its triply degenerate vibrational bands increase the total number of degrees of freedom to be treated by a factor of three (note that the particle boundary breaks the high symmetry both of the molecules themselves and of the lattice). Furthermore, SF6 has a high polarizability, which is expected to lead to noticeable contributions on top of the leading transition dipole interaction. Our goal is to find out more about the phase and the shape of SF6 aerosols. We also investigate two-component particles of SF6 and CO2. This includes particles that are statistically mixed at a molecular level, as well as particles with a core-shell architecture. 3.2 Theory and numerical approach The contents of this section have already been presented in detail in section 1.2, except for a few system specific aspects. We present two extensions to our vibrational exciton approach to modeling JR spectra of molecular nanoparticles [8,13,14,20-231. Firstly, the effect of molecular polarizability is included in the vibrational exciton Hamiltonian, since these effects have been shown to be significant in the spectra of small SF6 clusters [25-27]. Secondly, we implement a time-dependent approach to the calculation of absorption spectra via the electric dipole autocorrelation function. This becomes necessary to model particle spectra with more than a few thousand oscillators ( = number of molecules x vibrational modes per molecule), as the direct diagonalization becomes prohibitively expensive. These developments enable us to calculate JR spectra for pure and composite SF6 particles with different shapes, architectures, and compositions for particles with up to 30000 oscillators. For SF6 particles, this corresponds to particle diameters of about 12 nm, roughly the minimum value required to “converge” the band widths and band structure of particle spectra with respect to particle size. As we have shown in previous studies [13,28], these characteristic features in the JR spectra do not change once the 77 particles dimensions have reached 5-10 nm. Beyond that they remain constant for particle dimensions up to 100 nm, so that including 1 SF6 molecules (=3 xlü oscillators) in our model is sufficient to analyze the experimental spectra observed in the present study. Within the original exciton model, interactions between molecules were limited to the resonant dipole-dipole terms HDD. The Extended Exciton Model [14] explicitly treated intra- and intermolecular interactions in terms of appropriate force fields to account for structural changes and local molecular variations of transition frequencies and dipole moments, but the subsequent calculation of the spectrum still relied on the resonant dipole-dipole interaction: HDDppJ (3.1) The sum extends over pairs of molecules, p. is the dipole moment vector of molecule I and , is a scaled projection matrix: (i — / — t! 3,, +_U4 3\ uu y where R is the distance between molecules i and j, and e the unit vector pointing from the center of mass of molecule ito that of moleculej. While the (resonant) dipole-dipole interaction generally dominates vibrational spectra of molecular nanoparticles, previous investigations [25- 27] have demonstrated the influence of the molecules’ polarizability on the spectra of small clusters of SF6. To our knowledge there are no systematic studies on corresponding effects on the spectra of molecular aerosol particles. We have therefore included the corresponding dipole induced dipole terms ‘DJD in an extension of our previous exciton model. 78 ‘DID = — ikPkJ (3.3) where a, is the polarizability tensor of molecule k. This yields the overall Hamiltonian — + p7A,p1 (3.4) with = — -L + Ikak2jk ft is the Hamiltonian for uncoupled molecular oscillators. ft is diagonalized in a direct product of molecular oscillator eigenfiinctions im) with the restriction to resonant interactions. Within the double harmonic approximation, the Hamiltonian is completely defined by wavenumbers and transition moments = (0 im). Shifting the zero energy to the uncoupled ground state energy (0 I1 0) further simplifies the Hamiltonian matrix elements: (km l in) = 6klSmnhCi + 2pAklplfl (3.5) km) represents the product function with level m excited on molecule k and all other oscillators in the ground state. 10) represents the overall ground state. For particles in the nanometer range comprising > 1 oscillators, the full diagonalization of the Hamiltonian is no longer viable. (Note that here we need the full spectrum of the Hamiltonian, (Eq. (3.4)), not just a small fraction of it.) Instead we take a time dependent approach to calculate absorption spectra directly from the dipole autocorrelation function. Let {E1 } be the set of eigenvalues of ft with corresponding 79 eigenvectors I I) and transition moments M1 = (Op1 I) (overall dipole function p = Then the absorbance spectrum is proportional to (E)=JM1I2f(E—E), (3.6) where f(E) is an appropriate line shape. cr(E) is related to the time-dependent dipole autocorrelation function C(t) through the Fourier transformation CQ)— fu(EtdE/h —g(t) (km (37) k,m,l,n with g(t) = ff(E)edE I h. To calculate C(t), we employ a second order time-propagation scheme (noting that c(— t)= C*(t)). = e’) —i) + (3.8) The overall accuracy is limited by the time propagation scheme requiring time steps significantly below the Nyquist limit (A < 20 fs) for the typical spectral band width of 100 cm1. The propagation starts with a real cu0) = p km). The apodization g(t) is derived from a Gaussian line shapef(E) and determines the maximum propagation time ± t. A full line width of 6 cm1 requires t,, 20 ps to converge the spectrum u(V). The influence of the dipole-induced dipole terms on infrared spectra is depicted in Figure 3.1 for the example of an elongated SF6 particle with an axis ratio of 1:1:6 and a cubic 80 crystal structure [29]. The thin line shows the calculated spectrum with dipole coupling (HDD) only. The thick line includes both terms HDD and HDID. Figure 3.1 Vibrational exciton calculations for an SF6 particle with an axis ratio of 1:1:6 and a cubic crystal structure [29]. Thin line: Only dipole coupling (‘DD) is taken in to account (see Eq. (3.1)). Thick line: Dipole coupling (DD) and dipole-induced dipole coupling (‘DJD) are taken into account (see Eq. (3.3)). i7 is the wavenumber in cm1. C I C 0 900 920 940 960 980 1000 1020 I cm1 The picture illustrates that dipole-induced dipole coupling leads to band shifts and changes in the band strengths on the order of 10%. The agreement between experimental and calculated spectra of SF6 particles is markedly improved upon inclusion of HDJD (Eq. (3.3)) so that we have included these terms for all simulations in this chapter. All the necessary parameters for our calculations are listed in Table 3.1. 11srat1o1:1:6 81 Table 3.1 Spectroscopic parameters used in the vibrational exciton calculations of SF6 and CO2 particles: transition wavenumbers V, transition dipoles p, and polarizabilities a. e is the vacuum permittivity. V (cm) p (D) a (A3) 4,rs0 SF6 V3 9481 0.3882 5.6 CO2 V3 2355 0.32 2.191,4.357 1 ref. [37] 2ref [38] 3ref. [39] as cited in ref. [271 [40]. V slightly blue-shifted from the value given. a, a11 , ref [41]. For the interpretation of observed spectral features in terms of the particle structure, the local spectral density introduced in [14] has proved extremely valuable. In essence it partitions the overall spectrum into contributions o (E) from individual volume elements 8J’. (E)=o1(E), (3.9) with = f(E — EI)Pkmb1M, (3.10) köV,m I where b1 = (km 1). With Eq. (3.7) we have 1(E)= fCj(t)e_tdt (3.11) with 82 C1(t)= g(t) (km (3.12) kESVm,l,n Interference between individual oscillators can make o negative but will cancel once cW comprises a sufficient number of oscillators. To account for differences in the size of individual volume elements, we define the normalized excitation density as (3.13) N where N is the total number of molecules in the volume V. Up to interference effects, o(E) represents the average contribution of individual molecules to the overall spectrum. It should be noted that the direct diagonalization approach used previously yields identical results. We have checked this for particles containing up to a thousand molecules. 3.3 Pure SF6 aerosol particles Figure 3.2 shows the infrared spectra of SF6 aerosol particles directly after the particle formation (to = 0 s) in the bottom spectra, 33 seconds later (ti) in the middle spectra, and 750 seconds later (t2) in the top spectra. The region of the threefold degenerate v3 mode is depicted in panel a) and the region of the threefold degenerate v4 mode is shown in panel b). The aerosol was prepared and spectroscopically investigated in our collisional cooling cells in the same way as previously described in refs. [8,22] for other aerosol systems. The radius of the particles investigated here lies around 50 nm (1O molecules per particle) and the particle number density is about 106 particles/cm3(see refs. [2 1,30] for more details). 83 Figure 3.2 Infrared spectra of SF6 aerosol particles. The temperature during particle formation in the cooling cell was 78 K. a) Region of the threefold degenerate v3 band. b) Region of the threefold degenerate v4 band. The bottom traces were recorded directly after particle formation at time to = 0 s. The middle and top traces show the spectral evolution of the same ensemble of particles after t1 = 33 s and t2 = 750 s, respectively. 0.8 C 0.4 0 0.0 0.8 0.6 0.2 0.0 600 610 620 630 I cm’ The goal is to clarif’ the following points with the help of predictions from our exciton model. i) Bulk SF6 undergoes a phase transition from cubic [29] above 96 K to monoclinic [31] below this temperature. For the particles, however, formed at temperatures between 78 K and 110 K, it is not clear in which crystal phase they are formed initially and whether they subsequently undergo a phase transition. Infrared spectra of SF6 aerosols in the region of the v3 band have previously been recorded at a temperature of 90 K by Gough and Wang [3], who assumed a cubic 84 880 920 960 1000 structure for the particles. This assumption, however, was not based on any comparison or modeling of the two phases, but solely on the observation that no additional splitting or broadening was observed. We will try to further elucidate this aspect with the help of our exciton model. ii) The exact shape of the particles remains to be determined. Our previous studies on other aerosol systems generated in cooling cells clearly show that aerosol particles are globular immediately after their formation [20-23]. The simulations allow us to be more specific about the exact shape (cubic-like, spherical) of freshly prepared globular SF6 particles (bottom spectra measured at to = 0 s in Figure 3.2. As elaborated below, the exciton model even allows us to classify the observed spectroscopic features in terms of surface and core excitations. iii) Finally, we want to understand the time evolution of the spectra depicted in Figure 3.2. SF6 aerosols show the same overall time evolution we have previously observed for other aerosol systems with prominent side bands evolving over time [8,20-22]. In those investigations, the behavior could be clearly traced back to the formation of elongated particles. It remains to be seen whether what we are observing here is a more general feature of particle growth. 3.3.1 Phase and shape of freshly prepared particles The first step is to identify the crystal structure of the particles. As mentioned above, bulk SF6 changes from a cubic [29] to a monoclinic structure [311 when cooled below 96 K. We have recorded particle infrared spectra at various temperatures between 25 K and 110 K, but to our surprise we did not observe any significant changes in the spectroscopic features with temperature. In particular, the infrared spectra remained unchanged around the temperature of the phase transition (96 K). There are two possible explanations for this behavior: either no phase change occurs, or the infrared spectra of the two phases are too similar to distinguish clearly 85 between them. At this point only our microscopic model can bring clarification by calculating infrared spectra of particles with either a cubic structure or a monoclinic structure and otherwise identical properties (shape, size, etc.). Figure 3.3 shows exciton calculations for particles with different globular shapes for the two different crystal structures. Figure 3.3 Traces al) and bi) show the experimental infrared spectrum of freshly prepared SF6 particles (to 0 s in Figure 3.2). Traces a2) to a4) show simulated spectra for particles with a cubic crystal structure [29] for three different particle shapes (cube, quasi-octahedron, sphere). Traces b2) to b4) show the same for particles with a monoclinic crystal structure [31]. CUBIC MONOCLINIC ‘II __ ju-ocfahedrf\ 900 950 I cmt I I 1000 900 950 1000 I cm1 86 Traces a 1) and b 1) contain the experimental spectrum of freshly prepared particles (see also Figure 3 .2a, t0 = 0 s). The different calculated spectra for a cubic crystal structure [29] are shown in traces a2) to a4) and those with a monoclinic crystal structure [31] are depicted in traces b2) to b4). Comparing the spectra calculated for the two different phases, it becomes clear that for a given particle shape the infrared spectra for the cubic and the monoclinic crystal are almost indistinguishable, except for a small but systematic difference in the band widths. Monoclinic particles feature broader bands (12-15%), while the narrower spectra of the cubic phase agree a littler better with the experiment [traces al) and hi)]. Given, however, the experimental uncertainties and the fact that we observe an ensemble of particles, we consider that the differences are too small to distinguish unambiguously between the two phases. With the crystal phase of SF6 particles uncertain, it must remain open for the time being whether or not SF6 particles do undergo a phase transition subsequent to their formation. Figure 3.3, however, does make clear that the lack of spectral changes near the bulk transition temperature must not be taken as evidence for the absence of a corresponding phase change in the nanoparticles. In fact, there is circumstantial evidence to the contrary from previous spectroscopic investigations where we have indeed observed the clear signatures of analogous phase transitions in the infrared spectra of other icy aerosol systems such as CR4 and CHF3 [32-34]. In these cases, the spectra of the two phases involved were easy to distinguish. We think that the amazing spectroscopic similarity between the cubic and monoclinic phases of SF6 is due to the octahedral symmetry of the molecules. Within the exciton Hamiltonian, this becomes equivalent to spherical symmetry, so that the spectrum is largely determined by the radial density distribution. Hence it is less sensitive to local structural differences between the cubic and the monoclinic phase. This lack of sensitivity also encompasses local defects in the crystal structure: on one hand translational 87 disorder is unlikely due to the near isotropic intermolecular interactions. Moreover, test calculations show no significant effects on the spectra beyond a slight overall broadening. On the other hand, orientational disorder would not affect the exciton spectra of triply degenerate fundamentals of SF6. All calculations for SF6 particles in the following sections assume a cubic crystal structure as it appears to be in slightly better agreement with the experiment. We note, however, that the results are not significantly affected by this assumption. Although we cannot specify the crystal structure of the particles, Figure 3.3 still allows us to specify the shape of freshly prepared SF6 particles. First of all, only globular particle shapes are consistent with the experimental observation, which confirms expectations from previous studies of icy aerosol systems [20-231. Elongated particles, for example, lead to pronounced side bands on both sides of the main peak (see Figure 3.5 and section 3.3.2) that are not observed for aerosol particles immediately after their formation. The experimental spectrum in Figure 3.3 does, however, show a weak shoulder on the high-frequency side of the main band, which turns out to be characteristic for globular particles. The best agreement between experiment and simulation is obtained for cubes [traces a2) and b2)] or for quasi-octahedra [octahedra resulting by cutting off the corners of a cube perpendicular to its space-diagonal; traces a3 and b3)]. The latter are only compatible with observed spectra if overall less than 20% of the space-diagonal of the cube is cut off (10% at each corner), since the characteristic high-frequency shoulder starts to disappear for larger cut-off values. Spectra in traces a3) and b3) were calculated for cube octahedra with a 10% overall cut-off. The spectra of cubes and quasi-octahedra are too similar to distinguish between these two shapes on the basis of our exciton model. Moreover, the SF6 aerosol consists of a particle ensemble for which the shape is very unlikely to be completely uniform. Therefore, we conclude that our particle ensemble is very likely to consist of a mixture of cubes and quasi-octahedra (with an overall cut-off of less than 10%). Other globular shapes 88 such as spheres [see traces a4) and b4)j can be ruled out as their spectra lack the characteristic high-frequency shoulder. Incidentally this also proves the particles to be in the solid state. We also note the importance of including the polarizability in our calculations (see Eq. (3.3) in section 3.2). Without it, the calculated spectra of globular particles are about 10% broader than the experimental ones. Our microscopic model allows us to find out more about the origin of the features observed in the infrared spectra. Our quantum mechanical model lets us determine which region within the particle (molecules in the surface of the particles, molecules in the core of the particles) contributes to a certain spectral signature. Such an analysis is depicted in Figure 3 .4a for a cubic SF6 particle while Figure 3 .4b shows the same for a sphere for comparison (both with cubic crystal structure, see above). The graphs in Figure 3.4 show the infrared spectra in the upper panels and the normalized excitation density (density per molecule) as a function of the distance from the particl&s center in the lower panel. Shell index 0 corresponds to the center of the particle and shell index 10 indicates the particle surface. Dark regions indicate strong vibrational excitation and light regions indicate weak vibrational excitation. The exact definition of the normalized excitation density is given in section 3.2, Eqs. (3.1 1-13). The excitation density in Figure 3 .4b reveals that the infrared spectrum of a spherical SF6 particle is governed by a narrow band of modes (width of about 16 cm’) that are delocalized over the whole particle (same excitation density at a given transition wavenumber for all shell indices). This behavior changes for a cubic SF6 particle, where the modes now spread over a broader wavenumber region. In the core of the particles (shell index 0 to 7), the excitation density is a smooth function of the wavenumber, contributing a broad relatively weak background to the spectrum in accordance with the small core-volume fraction. Towards the surface (region with shell index from 7 to 10), the excitation density becomes highly structured in the wavenumber domain as a consequence of 89 the cubic shape of the surface. In the outermost shell interference effects originating near the cube’s corners even produce negative values at some transition wavenumbers. Figure 3.4 Upper panels: Simulated infrared spectra. Lower panels: Contour plots and single shell cuts of the normalized excitation densities as defined in Eqs. (3.11-13) in section 3.2. The shell index refers to concentric spherical shells around the center of mass of the particle. Trace a): for a particle with a cubic shape and a cubic crystal structure. Trace b): for a spherical particle with a cubic crystal structure. xa) C C) U) 900 950 Icm1 1000 900 950 1000 Jcm 10 1o 8 _______________ 6 4 _ __ 2 ___ 900 950 ‘1000 1cm’ I • I • 900 950 1000 1cm’ 90 While this is generally characteristic of singularities (corners, edges) in the surface curvature the effect appears exaggerated in the bottom panel of Figure 3 .4a by the use of spherical shells. The outermost shell only contains a relatively small number of molecules (viz, those near the corners) so that its contribution to the overall excitation density is less dominant than Figure 3 .4a might suggest at first glance. Notwithstanding this caveat, the comparison with the overall spectrum provides clear evidence that it is the contribution to the excitation density originating from the surface layers of the particle which determines the characteristic spectroscopic signatures of the cubic particle shape. 3.3.2 Time evolution While the absence of strong sidebands clearly established the overall globular shape of freshly formed SF6 nanoparticles, it is that same feature which indicates the subsequent time evolution of the particles’ shape in the spectra shown in Figure 3.2. A few seconds after the particles have formed in our cooling cells, two pronounced bands begin to show up in the spectra on either side of the main peak and grow with time. This observation mimics the behavior of other icy aerosol systems subsequent to their formation in cooling cells e.g. CO2.NH3 and N20 [8,20-22]. In these cases, the spectral time evolution could be traced back unambiguously to the formation of elongated particle shapes. While at the beginning mainly globular particles were formed, the fraction of elongated particles increased systematically over time, although the mechanism behind this process remains unclear. Particles with elongated shape could be formed by agglomeration and coagulation of globular primary particles, but it is equally possible that small (globular) particles evaporate (Kelvin effect) while recondensation onto larger particles increases the fraction of elongated particles over time. Even though it is not clear which route 91 particle growth actually takes, the striking similarity between the behavior of SF6 and those previous observations makes us confident that we are seeing here the same spectral signature of a generic feature of particle growth. Figure 3.5 shows in trace a) the experimental spectrum recorded at time t2 = 750 s (see Figure 3.2a) together with a simulated spectrum in trace b). Figure 3.5 Trace a): Experimental infrared spectrum of SF6 aerosol particles recoded at t2 750 s after particle formation (Figure 3.2a). Trace b): Calculated infrared spectrum. This spectrum is a linear combination of a spectrum of a cubic particle (46%) and spectra of elongated particles with different axis ratios (18% with an axis ratio of 1:1:3, 18% with an axis ratio of 1:1:6, 18% with an axis ratio of 1:1:12). The two side bands marked by arrows arise from the elongated particles. C .0 Cu Cu U C CU .0 0U, .0 Cu a) Exp. \ 880 920 960 1000 880 920 960 1000 Icm1 92 The calculated spectrum is a combination of a spectrum of a cubic particle (46%) and elongated particles with different axis ratios (18% with an axis ratio of 1:1:3, 18% with an axis ratio of 1:1:6, 18% with an axis ratio of 1:1:12). The general trend observed in the experimental spectrum is reproduced by this simulation with the characteristic side bands arising from elongated particles marked by arrows. These side bands grow with increasing fraction of elongated particles (not shown) so that we conclude that the temporal evolution of the aerosol spectra shown in Figure 3 .2a arises indeed from the fraction of elongated aerosol particles increasing over time after particle formation. The agreement between experiment and simulation is only qualitative, but could be improved by fitting a broader range of particle shapes (aspect ratios) to the spectra. Beyond a merely aesthetic improvement there is little to be gained here from such a procedure, so we refrain from any further fitting attempts. 3.4 Two-component SF6/C02aerosol particles 3.4.1 Core-shell particles To illustrate the influence of different particle architectures on the infrared spectra, we investigated various core-shell particles, either with SF6 in the core and CO2 in the shell or vice versa. For both types, we varied the ratio of shell thickness to core radius. These core-shell particles were generated in our cooling cells using a dual inlet system, which introduces the core substance first. After a delay of some hundred ms to allow core-particles formation, the shell substance is introduced and condenses onto the core-particles. Details of the procedure and the setup are described in refs. [22,35]. 93 Figure 3.6 Experimental infrared spectra of a SF6-C02core-shell particle. Traces a) and b) show two different spectral regions. Figure 3.6 shows one example of such a core-shell particle spectrum with the region of the v3 bands of SF6 and CO2 in panel a). For comparison panel b) shows the spectrum in the region of the v4 bands of SF6 and the v2 band of CO2. C 0, 0 1.2 — 0.8 — 0.4- 0.0 — a) SF : core6 CO : shell v -band 23 v -band 880 2340 I Ii 960 1040 SF6: core v4-band 2400 C02: shell v2band 0.4 C 0 0.0 600 620 640 660 680 700 -1 v I cm These spectra have been recorded for particles with SF6 in the core and CO2 in the shell, with a shell-to-core ratio of the number of molecules of 53:47. The particle core gives rise to essentially single bands similar to those observed for pure particles (see Figure 3.3a1/bl). By contrast, the infrared bands of the shell are split into two. The third component observed for the v2 band of 94 CO2 is a consequence of its twofold degeneracy [36], while the shell architecture is only responsible for the splitting of the doublet centered around 670 cm’. We have found the same qualitative behavior (not shown here) for particles with the roles of SF6 and CO2 interchanged (i.e. the former in the shell and the latter in the core). In that case, the SF6 bands show the characteristic splitting of the shell architecture. The relative intensities of the doublet components changes as a function of the shell thickness relative to the core radius. We do not show the spectra here since the trend is the same as observed previously for‘3C02-’ core-shell particles (see for example Figure 8 of refs. [22]). The spectrum of the core-shell particle shown in trace a) of Figure 3.6 is well reproduced by the simulated spectrum in Figure 3.7. Figure 3.7 Upper panel: Calculated infrared spectrum of a SF6-C02 core-shell particle. The corresponding experimental spectrum is depicted in Figure 3.6a. Lower panel: Excitation densities as defined in Eqs. (3.11-13) in section 3.2. 10 8 x C 2 880 960 1040 I cm1 2340 2400 T I cm1 95 This holds both for the split band of the CO2 shell and for the single band of the SF6 core. Although the latter is similar to that of a pure globular SF6 particle (Figure 3.3a1,bl), its main peak has a less pronounced high-frequency shoulder (Figure 3.6), which was the signature of a cube-like particle shape. From the discussion in section 3.3, it is clear that this indicates a larger deviation from the cube-like shape expected for the SF6 core-particles immediately after their formation. Assuming a quasi-octahedron, more than 20% of the space diagonal must be cut off at the corners. For the simulations in Figure 3.7, we assumed a 36% cut-off. Apparently the formation of the CO2 shell is accompanied by a smoothing of the SF6 core towards a more “spherical” shape. Presumably this is a consequence of the heat released by the condensation of CO2 onto the surface of the SF6 core particle. It heats up the core and thus promotes the reduction of surface tension by smoothing its corners. The lower panel in Figure 3.7 shows the excitation density as a function of the distance from the particle’s center (see section 3.2, Eqs. (3.11-13)). The shell index runs from 0 at the center of the particle to 10 at its surface. Dark regions indicate strong vibrational excitation and light regions indicate weak vibrational excitation. The core-shell structure of the particles is clearly reflected in the excitation density. Vibrational excitation of CO2 molecules extends over the whole shell region, while the vibrational excitation of SF6 molecules is limited to (and delocalized over) the core of the particle. 3.4.2 Statistically mixed particles To form particles for which SF6 and CO2 are homogeneously mixed at a molecular level, a premixed gas sample is introduced into the cell as described in ref. [22]. Figure 3 .8a shows a 96 spectrum that results from the injection of a premixed gas sample consisting of 56% SF6 and 44% Co2. Figure 3.8 a) Experimental infrared spectrum for statistically-mixed particles. b) Exciton calculation for statistically-mixed particles. For the simulations, we assumed a quasi-octahedral shape of the particle with a 36% overall cut-off. SF6: C02: 0.8 - a) v3-band v3-band 880 960 1040 2340 2400 I /f I 880 960 1040 2340 2400 -4 v 1 cm The major difference to the infrared bands of core-shell particles in Figure 3.6a or of pure particles in Figure 3.3a1/bl is the band width. The bands of the statistically mixed particles in Figure 3.8a are much broader by about a factor of four. This broadening of the absorption bands is reproduced by the exciton calculations in trace b). In contrast to the calculated spectra, the 97 experimental spectra still show some structure of the bands. For example, similar as for the core- shell particles the CO2 band has a shoulder at higher wavenumbers (see Figure 3 .6a). We believe that the additional structure in the experimental spectra arises because of the difficulties to form perfect statistically-mixed particles in the cooling cell. SF6 and CO2 have slightly different sublimation points and different intermolecular interactions which might lead to the formation of small fractions of core-shell-like particles, in addition to statistically mixed particles. 3.5 Summary Vibrational exciton coupling has been exploited to analyze infrared spectra of pure SF6 and mixed SF6/C02 aerosol particles. To model the triply degenerate vibrational bands of these complex systems, we have extended our previous exciton model to include dipole-induced dipole terms, which account for the high polarizability of SF6 and C02, and by implementing a time- dependent approach to the simulation of spectra. Linear absorption spectra are obtained directly from the Fourier transform of the dipole autocorrelation function, which circumvents the need of full diagonalization of huge matrices. A very concise formulation of matrix elements in terms of structure factors and transition dipoles (Eq. (3.5)) lends itself to a future implementation of direct algorithms which avoid the storage of the full Hamiltonian, so that 1 O or more oscillators could be treated simultaneously. We have determined the shape of pure SF6 particles by comparison of experimental spectra with predictions from our quantum mechanical model. Solid particles with a cube-like shape are formed directly after particle formation. With increasing time, the shape changes to elongated particles. The analysis of the vibrational eigenfunctions in terms of local excitation densities reveals how the shape determines the dominant spectroscopic features. Our microscopic model also explains why the infrared spectra of SF6 nanoparticles do not show any evidence for 98 an analogue of the cubic-to-monoclinic phase transition of bulk SF6 near 96 K (see also discussion in ref. [3]): The octahedral symmetry of the molecules, which translates into spherical symmetry within the exciton model, makes infrared particle spectra for the two phases almost indistinguishable. Mixed particles of SF6 and CO2 allow us to study the influence of various particle architectures on infrared spectra. Core-shell particles show a characteristic split band for the shell. Statistically mixed particles lead to broad infrared bands that are only slightly structured. This chapter shows that the infrared spectroscopic signature of aerosol particles can be understood on a truly molecular level despite the many degrees of freedom of these complex systems. 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[1-121 and references therein). The lively interest in this area encompasses aspects of both fundamental and applied science. The former ranges from confinement effects on ensemble properties and non-equilibrium phenomena, e.g. super-cooling of droplets, to the microscopic mechanism of phase transitions, e.g. surface vs. volume freezing [1,6,8,13-181. The discussion has found a broad echo in the atmospheric sciences since the phase behavior of aerosols determines weather processes by affecting the energy balance of the atmosphere. On Earth, water and its ices are the obvious focus of interest, but recent high-profile space missions have found other simple molecules, such as ammonia, methane, or ethane, play a similar role in the atmospheres of certain planets and moons of our solar system (see ref. [19] and references therein). Most notably, the ongoing Cassini—Huygens mission has found methane and ethane cloud and rain formation on Saturn’s moon Titan that mimic weather processes on Earth, albeit at much lower temperatures (see ref. [20] and references therein). On a more applied note, the phase behavior of aerosols is crucial for technical processes involving aerosol formation, such as the Rapid Expansion of Supercritical Solutions (RESS). [21,221 This micronization technique exploits the formation of mixed solvent—solute aerosols to generate extremely fine powders of non-volatile solute substances. For example, if the A version of this chapter has been published. 0. Sigrirbjomsson, G. Firanescu, R. Signorell, Phys. Chem. Chem. Phys., 11, 187 (2009) 103 solute is a drug, this offers a promising way to improve the bioavailability of pharmaceuticals [22). 4.2 Experiment and calculations 4.2.1 Experiment We provide here only a brief description of the experimental setup. Additional details can be found in ref. [23] and [24]. Fluoroform aerosol particles were formed by bath gas cooling in our collisional cooling cell at a temperature of 78 K. Prior to particle generation, the pre cooled cell was filled with He bath gas, which after some minutes thermally equilibrated with the cell (78 K). The particles were then formed by rapid injection of a CHF3/ e gas mixture (300— i05 ppm CHF3 in He) into the cold He bath gas. The sample gas was injected through a stainless steel tube (7 mm diameter). The pulse duration (< 1 s) was controlled by a Burkert solenoid magnetic valve. This rapid cooling led to supersaturation of the fluoroform gas and thus to condensation of particles. This technique has considerable advantages over the more common laboratory generation of weakly bound molecular aggregates by supersonic expansions using nozzles of varying design (slit nozzles, Lava! nozzles). Collisional cooling affords much longer observation times (minutes to hours compared with ms at best in supersonic expansions) and, unlike supersonic expansions, it allows measurements under thermal equilibrium over a wide range of temperatures (4-298 K) and for a wide range of particles sizes (less than 1 nm to hundreds of micrometers). In the present investigation, the mean particle sizes varied between 20 and 700 nm, which allowed us to perform size-dependent experiments. We did not, however, observe any unusual size-dependent effects. Thermal equilibrium between particles and the 104 surrounding gas phase and a well-defined temperature are crucial for the study of phase transitions. At a bath gas temperature of 78 K, CHF3 aggregates were formed close to the freezing point of bulk fluoroform (T=l 18K). As shown in section 4.3.1, at the very beginning CHF3 gas condensed to supercooled liquid aerosol droplets. The ensuing crystallization dynamics of these droplets and the shape change of the resulting solid particles were recorded in situ with a rapid- scan Fourier transform infrared spectrometer (Bruker IFS 66 v/S) as a function of time. The time resolution was 30 ms at a spectral resolution of 2 cm1. As explained in section 4.3, the crystallization of the super-cooled liquid droplets and the shape change of solid particles happen simultaneously for pure fluoroform particles. To separate the time-scale of the two processes, we mixed traces of water (0.1% to 6% of the fluoroform content) into the gaseous fluoroform samples in order to speed up crystallization. When these gas mixtures were introduced into the cold cell, water condensed first to tiny solid ice particles which served as condensation nuclei for the formation of supercooled liquid CHF3 droplets. Once the liquid droplets were formed, the tiny solid water cores acted as nuclei for the heterogeneous crystallization of fluoroform droplets, which proceeds much faster than the crystallization via homogeneous nucleation prevalent in pure CHF3 particles. This allowed us to separate the crystallization temporally from the subsequent slower shape change of the crystalline particles. 4.2.2 Calculations The contents of this section have already been presented in detail in section 1.2, except for a few system specific aspects. The crystallization of supercooled liquid droplets to solid 105 particles is accompanied by a change in the shape of the particles. The band structure of infrared bands with strong molecular transition dipoles (>0.1-0.2 D) are sensitive to the shape of the particle [23,25-28] and can thus be used as a probe. The vibrational exciton model allows us to calculate and therefore predict infrared band structures of such bands for different particle shapes when transition dipole coupling is the dominant interaction. The shape and how it changes can then be determined from a comparison of the experimental infrared spectra with exciton calculations for different particle shapes. For the present contribution, the overall vibrational Hamiltonian includes resonant dipole-dipole and dipole-induced dipole terms (1st and 2nd term in Au, respectively, see ref. [27-28] for further details): (4.1) with A = —— + 2kakAJk is the Hamiltonian for uncoupled molecular oscillators. The sum in Eq. (4.1) extends over pairs of molecules, p1 is the dipole moment vector of molecule i and is a scaled projection matrix: (1— o..) = (3ee; — i) (4.2) where R is the distance between molecules i and], and the unit vector pointing from the center of mass of molecule i to that of molecule]. a is the polarizability tensor of molecule k. Within the double harmonic approximation, the Hamiltonian is completely defined by transition wavenumbers im transition moments Pin, = (oIpJ im), and polarizability tensors a,. im) is the 106 mth eigenfunction of the jth molecular oscillator. Shifting the zero energy to the uncoupled ground state energy (0 i 0) further simplifies the Hamiltonian matrix elements: (kmI:Iln)=ökl8mnhCI7Ic,,l +2pAklplfl. (4.3) km) represents the product function with level m excited on molecule k and all other oscillators in the ground state. 0) represents the overall ground state. The input parameters used for the calculations in the region of the stretching vibrations v5 / v2 of fluoroform [30,31] are listed in Table 4.1. Table 4.1 Spectroscopic parameters used in the vibrational exciton calculations of the CHF3 particles in the region of the v5 / v2 stretching vibrations: transition wavenumbers IY, transition dipoles p, and polarizabilities a. is the vacuum permittivity. V (cm) p (D) a (A3)4rs0 V5 1143.6 a 0.29 a 2.89, 2.61 b V2 1136.3a 0.18a a) ref. [30]. V correspond to the mean values measured in the different liquids. b) a = ar,, ref. [31]. The transition wavenumbers and the transition dipoles in Table 4.1 correspond to the mean values of the different solutions measured in ref. [30]. We used solution values rather than gas-phase values [32] because they provide a better representation of the influence of a dielectric surrounding on a molecule. Calculations with gas phase values (not shown here) lead to similar 107 results with only slightly broader bands shifted to higher wavenumbers than in calculations performed with the solution values. The CHF3 aerosol particles consist of tens of thousands of molecules and therefore tens of thousands of oscillators so that the full diagonalization of the Hamiltonian is no longer viable. Instead we take a time-dependent approach to calculate absorption spectra directly from the dipole autocorrelation function. {E1 } is the set of eigenvalues of I with corresponding eigenvectors I) and transition moments M1 = (0 1u1 I) (overall dipole function p = Then, the absorbance spectrum is proportional to cr(E)=jM12f(E—E (4.4) where f(E) is an appropriate line shape. o(E) is related to the time-dependent dipole autocorrelation function C(t) through the Fourier transformation (h = h / 2r): C(t)= fu(E1EtdE/h =gQ) (km (4.5) k,mj,n with g(t) = ff(E)etdE / h To calculate c(t) we employ a second order time-propagation scheme. çti) = + (4.6) For the interpretation of observed spectral features in terms of the particles’ structure (see section 4.3.2), the local spectral density used in ref. [28] and [29] has proved extremely valuable. It partitions the overall spectrum into contributions o (E) from individual volume elements 6J’. 108 u(E)=cr,(E) (4.7) with a1(E) = f(E — Ei)p(km 11)M = $Cje_1Etcit (4.8) kEöVm I The C1 (t) are obtained from an equivalent partitioning of C(t). C1(t)= g(t) (km (49) k€öVm ,l,n Division by the number of molecules N1 contained in the volume element 8V yields the normalized excitation density = u1(E) (4.10) N1 which represents the average contribution of individual molecules to the overall spectrum. 4.3 Results and discussion 4.3.1 Spectral features of crystallization and shape change Infrared spectroscopy is a particularly powerful method for investigating the crystallization behavior and shape changes of fluoroform aerosol particles. All mid-infrared bands (the CF3 deformation mode v3, the CH bending vibration v4, and the CF3 stretching modes v5 /v2) show band structures that characteristically depend on the internal structure (phase). Under the present conditions, the phase is either a completely disordered state or the crystalline monoclinic phase also found in the bulk. [33] However, only the v5 / v2 band system 109 around 1140 cm’ is also sensitive to the shape of the particle and can thus provide information about shape changes. The band structures of the v3 vibration around 696 cm’ and the v4 vibration around 1378 cm’ are not influenced by the shape of the aerosol particles. The reason why the three band types behave so differently lies in their molecular transition dipoles. The strong transition dipole of the v5 band (see Table 4.1 and ref. [30]) leads to strong exciton coupling between all molecules within a certain aerosol particle. Since the v2 band lies close to the v5 band, it is also involved in this coupling. As explained in refs. [27-29], this dipole coupling lifts the degeneracy of the uncoupled molecular eigenstates and leads to vibrational eigenfunctions that are delocalized over the whole aerosol particle. It is this delocalization which explains the shape sensitivity. The v3 and v4 vibrations, by contrast, have only small transition dipoles. [30,32] Exciton coupling and shape effects are thus negligible for the band structures of these two bands. The different behaviors of these mid-infrared bands allow us to distinguish between the crystallization process of aerosol particles and the shape changes of the crystalline particles once formed. Figures 4.1-4.3 show both processes as a function of time in the region of the v5 / v2 band, the v4 band, and the v3 band respectively. Panel a) in each case illustrates the spectral evolution during the crystallization process followed in panel b) by the spectral evolution caused by a subsequent change in the shape of the crystalline particles. All spectral changes happen continuously with increasing time after particle formation (t = 0 s). For clarity, we only show snapshots of the particle spectra recorded at twenty different times, although in principle we can observe the whole crystallization process in great detail with a time resolution of ms. For the particular example shown in Figures 4.1-4.3, the two processes — crystallization and shape change — are clearly separated in time. The shape change starts (t = 71 s) after crystallization is 110 complete (t = 38 s). As explained in section 4.2.1 and further discussed below, this separation of time scales was achieved by the presence of traces of water-ice nuclei, which does not, however, affect the general conclusions. We start with a discussion of the crystallization process (Figures 4.la-4.3a). The spectra measured directly after particle formation at time t = 0 s in Figure 4. la-4.3a exhibit broad unstructured bands reflecting the high degree of disorder in the particles. These particles are either amorphous solid spheres or supercooled liquid droplets. Several considerations clearly point to the formation of supercooled liquid droplets. The first one is the fact that the temperature in the cooling cell is not too far from the melting point of fluoroform (118 K) and that fluoroform is a liquid over the broad temperature range of 73 K. Secondly, throughout particle formation the actual CI{F3 partial pressure in our cell lies 1—3 orders of magnitude above the extrapolated vapor pressure of liquid fluoroform at 78 K [34], which in turn probably lies much higher than the •vapor pressure over the solid. The final argument comes from the comparison of the infrared spectrum calculated for an amorphous solid particle (not shown here) with the experimental spectrum in Figure 4.1 a for t = 0 s. The calculated spectrum of the v5 / v2 band of the amorphous solid particle still shows some structure which does not agree with the completely structureless experimental spectrum. The arguments taken together clearly favor the formation of supercooled liquids. During the crystallization process (t = 0-38 s), the absorption bands split into several sub bands. After crystallization is complete (t = 38 s), the v3 and v4 bands show the same splitting pattern as found in the crystalline bulk [35]. From these splitting patterns, we conclude that the 111 Figure 4.1 Time-dependent infrared spectrum of CHF3 aerosol in the region of the v5 / v2 bands. (a) Temporal evolution during the crystallization of the particles in the presence of trace amounts of water ice nuclei. (b) Temporal evolution during the change of the particles’ shape from cube- like to elongated particles. t is the time after particle formation (t = 0 s). a t .0 (U 0 4-, C 4-, x a) a) Crystallization 1080 1120 1160 1200 (cm1) 1080 1120 1160 1200 (cm1) b) Shape Change 112 C 0 a C x Figure 4.2 The same as in Figure 4.1, but in the region of the v4 band. a) Crystallization b) Shape Change 1360 1380 1400 1360 7 (cm4) 1380 (cm’) 1400 113 Figure 4.3 The same as in Figure 4.1, but in the region of the v3 band. a) Crystallization b) Shape Change t = I 09s t—20 t188s t = 346s 35frj\A 680 690 700 680 690 700 t (cm1) (cm1) 114 particles have the same crystal structure as the crystalline bulk [33] (monoclinic space group P21/c with four molecules per unit cell). The behavior of the v5 / v2 band system during the crystallization differs from what has just been described for the v3 and v4 bands. Its band structure is determined by strong transition dipole coupling between the molecules that make up the particle. Hence this band system is not only sensitive to tiny details in the internal structure of the particles, but also to the shape of the particles. As discussed in the next section, the analysis of this band allows us to determine the particle shape right after crystallization is complete. Between t = 38 s (bottom trace of panel a) in each figure) and t = 65 s no further changes can be observed in the mid-infrared spectrum. After that time period, however, the v5 / v2 band system (Figure 4. ib) exhibits a second systematic change, whereas the v4 band (Figure 4.2b) and v3 band (Figure 4.3b) do not change any further. For the v5 / v2 band, two side bands at about 1108 cm1 and between 1145-1170 cm’ grow with increasing time (see arrows in Figure 4.4) relative to the central band region between 1125—1140 cm’. The fact that only the v5 / v2 band is sensitive to this second change clearly indicates that this change must influence the exciton coupling scheme within an aerosol particle. In principle this could happen by a change in the internal structure (phase change), a change in the particles’ size, or a change in the particles’ shape. The first two possibilities, however, can be ruled out. The stability of the v3 and v4 bands clearly speaks against a second phase change. The size of the particles cannot be the reason because the band structure is not sensitive to the particular size in the size range considered here. This is due to the resonant dipole coupling, which scales with the 3’ power of the distance, reaching convergence as a function of size (see detailed explanations in ref. [27] and [36]). The only possibility left is a change in the particles’ shape. The next section shows by comparison with exciton calculations that between t = 71 s and t = 960 s the particle ensemble transforms 115 from one with mainly globular particles to an ensemble with an increasing contribution from elongated particles. The time scale for this shape change depends on the experimental conditions. We find that the shape change starts earlier and happens faster at low bath gas pressures (200 mbar) compared with high bath gas pressures (800 mbar). The same trend is also found for higher sample gas concentrations (50 000 ppm) compared with low sample gas concentrations (600 ppm). For the particular example shown in Figures 4.1-4.3, the crystallization process is temporally separated from the subsequent shape change of the crystalline particles. To achieve this separation of the two different processes, we had to speed up the crystallization process of fluoroform droplets and in addition, to delay the onset of the shape change as much as possible. This was achieved by altering the experimental conditions. The onset of the shape change was delayed by choosing a high bath gas pressure in the cooling cell (800 mbar). The crystallization process was accelerated by introducing a minor amount (up to 5.8% of the fluoroform content) of very small water ice nuclei, onto which fluoroform could condense. The heterogeneous crystallization that becomes possible through the presence of the water ice nuclei is much faster than the predominant homogeneous nucleation in the case of pure fluoroform droplets. Since the ice nucleus is tiny compared with the fluoroform particle as a whole, they do not alter any of the spectroscopic features of pure fluoroform particles. In both cases, the features of crystallization and shape changes look exactly the same. The only difference is that for pure CHF3 particles crystallization and shape change overlap in time because crystallization happens much more slowly, as illustrated in Figure 4.4. This spectrum was recorded 1165 s after particle formation. Traces a) and b) show the v4 band and the v5 / v2 band, respectively. The splitting pattern of the v4 band is not yet completely resolved indicating that some particles of the ensemble are still liquid in line with the broad features in the v5 / v2 band system. At the same time the two side 116 = 0.1 0 0 0 0.0 1080 1120 1160 (cm) bands in that system (see arrows) indicate that some crystalline particles of the ensemble are already growing into elongated particles. At a comparable time (t = 960 s, Figure 4.1) particles with ice nuclei are already completely crystalline and a large fraction of these crystalline particles already have an elongated shape. We note, however, that by deconvolution of the infrared spectra we can still distinguish between the two processes even in the case of pure fluoroform aerosol. Figure 4.4 Experimental infrared spectrum of pure fluoroform particles without ice nuclei. The spectrum was recorded 1165 s after particle formation. a) Region of the v4 band. b) Region of the v5 / v2 band. The long tails towards lower wavenumbers are due to elastic scattering of the light by the particles, which have in this case sizes in the upper nanometer range. a) 680 690 700 b) 0 0 0) 0 1.0 0.5 0.0 1200 117 It is in principle possible to derive the kinetics of the crystallization process as well as the kinetics of the shape change of crystalline particles from time-dependent infrared spectra. We have already reported crystallization rate constants of pure fluoroform aerosol droplets in ref. [17]. The emphasis of that contribution was to clarifj whether it is possible to distinguish between surface nucleation and volume nucleation. We clearly demonstrated that this is not the case, both because of experimental uncertainties and because of approximations used in the derivation of the crystallization kinetics. For pure CHF3 droplets, we derived homogeneous crystallization rates ofJ = 1O81Ob0 cm3s’ if volume nucleation is assumed to dominate and J3 = 1 1 cm2s’ if surface nucleation is assumed to dominate. The presence of water ice nuclei leads to much faster heterogeneous crystallization rates that can be larger by orders of magnitude. Their exact value depends on the number and the size of the ice nuclei, which was varied for different experiments. For the change in the shape of the crystalline particles observed in Figure 4. lb, we have not derived any rate constants. As discussed in the next section, we can only speculate about possible processes responsible for these shape changes. Without better knowledge of the processes involved and their mechanisms, it is impossible to derive meaningful rate constants for the shape change. 4.3.2 Analysis of the particle shape by exciton calculations Since the v5 I v2 band system is dominated by exciton coupling, it is sensitive to the shape of the aerosol particles. The comparison of exciton calculations for different particle shapes with the experimental infrared spectra thus allows us to find out more about the evolution of the particle shape as a function of time. There are two types of shape changes in Figure 4.1-4.3. The 118 first is associated with the crystallization of the supercooled liquid droplets [see traces a)]. The second shape change [traces b)] has already been described in the previous section. It happens after crystallization is complete and thus corresponds to a shape change of crystalline fluoroform particles. We begin our discussion with the first type. At time t = 0 s, liquid supercooled droplets are formed with a spherical shape. Crystallization of these droplets is likely to alter this shape. Since individual particles of the ensemble crystallize instantaneously, they will presumably retain a shape with very similar axis ratios directly after crystallization. Figure 4.5 compares two exciton calculations (see section 4.2.2) for a crystalline spherical particle [trace a)] and a crystalline particle with a cube-like shape [trace b)] with the experimental spectrum directly after crystallization is complete [trace c), identical to the bottom trace of Figure 4.1 a]. For the simulation in trace b), we used a monoclinic parallelepiped with equal side lengths to represent what we call a cube-like particle. Note that other cube-like shapes, such as perfect cubes or quasi- octahedra, show almost identical spectral features and are therefore not treated separately. The experimental spectrum consists of a broad band (70 cm’) with a maximum of the extinction around 1130 cm’ and pronounced secondary maxima. The exciton structure for a spherical particle [trace a)] does not reproduce this band structure, which is a clear hint that after crystallization is complete the particles are no longer perfectly spherical. The experimental band structure is very well reproduced by exciton calculations of cube-like particles [trace b)]. The main features are clearly reflected in the calculated spectrum so that we conclude that during the crystallization process the shape changes from spherical droplets to cube-like crystalline particles. It is worth noting that we cannot expect perfect agreement between calculation and experiment in the finer details of the complex v5 / v2 band system, primarily because our calculations represent true predictions and not fits to the experimental spectra (none of the input parameters were fitted!). Their structure is determined by exciton coupling of the weak non- 119 degenerate v2 transition and the strong doubly degenerate v5 transition involving each of the thousands of molecules in an aerosol particle. Figure 4.5 Infrared spectra in the region of the v5 / v2 band. a) Vibrational exciton calculation for a crystalline spherical particle. b) Vibrational exciton calculation for a crystalline cube-like particle. c) Experimental infrared spectrum (see t = 38 s in Figure 4.1). 0 .0 C 0 C 4..’ x 1080 1120 1160 (cm1) 1200 1240 The result is correspondingly sensitive to the exact values of the input parameters. In the exciton model described in section 4.2.2, all molecules in a unit cell are assigned the same transition dipole moment and the same transition frequency (see Table 4.1). For the monoclinic crystal structure this is an approximation which we have to make because the values of these parameters for individual molecules in the unit cell are not known (typical differences in the transition a) 0 120 wavenumbers are expected in the region of several cm’). These are the reasons why details in the spectra cannot be reproduced perfectly by the present calculations. A more elaborate model, such as the Extended Exciton Model [29], might possibly afford better agreement as it treats intra- and intermolecular interactions in terms of appropriate force fields to account for structural changes and local molecular variations of transition frequencies and dipole moments. The development of a sufficiently accurate force field for crystalline CHF3 would be a major undertaking. Although interesting in its own right, we do not expect that the results would affect our current conclusions. Further analysis with our microscopic model allows us to get more insight into the molecular origin of the observed band structures in Figure 4.5. From our quantum mechanical model, we can determine which region within an aerosol particle (molecules in the surface, or molecules in the core) contributes to a certain spectral feature. Such an analysis is depicted in Figure 4.6 for a spherical crystalline particle on the left and a cube-like crystalline particle on the right. The upper panels show again the infrared spectra. The lower panels show the normalized excitation density (see Eq. (4.10)) as a function of the distance from the particle’s center. Shell index 0 corresponds to the center of the particle and shell index 10 denotes the surface. Dark regions indicate strong vibrational excitation and light regions indicate weak vibrational excitations. The infrared spectrum of the crystalline sphere (left) is governed by three modes at 1130, 1150, and 1162 cm1 that are completely delocalized over the particle. The band at 1130 cm1 mainly arises from the v2 transition while the other two bands derive their intensity from the doubly-degenerate v5 transition. Note, however, that this gives only a general trend because both transitions are coupled since they are near resonant. More interesting is the excitation density of the cube-like particle on the right hand side in Figure 4.6, because it allows us to 121 Figure 4.6 Left panels: crystalline spherical particles. Right panels: crystalline cube-like particles. Upper panels: calculated infrared spectra in the region of the v5 I v2 band. Lower panels: normalized excitation densities as defined in Eq. (4.10). kri. x a) C ci) C’, understand the features of the experimental spectrum for t = 38 s (Figure 4.1 a and Figure 4.5 c). In the core of the particle (shell index 0 to about 6), the excitation density arises from several modes completely delocalized over the core. In this region, the density is a smooth function of the wavenumber contributing to a weak background in the infrared spectrum. The major structure in the infrared spectrum arises from the highly structured excitation density in the particle’s surface (shell index 7 to 10), which is a consequence of the cube-like shape, mainly its corners, which represent singularities in the surface curvature. 1110 1140 1170 1200 1110 12001140 1170 L 1140 - v (cm ) 1110 1140 1170 (cm) 1200 122 The characteristic of the second type of shape change is the growth of two side bands on both sides of the main peak with increasing time (see Figure 4. ib). As we have reported for several other aerosol systems in previous publications, [23-25,28,37] the evolution of side bands is indicative of the formation of elongated aerosol particles from initially mainly “globular” particles. The formation of elongated particles also shows the same qualitative dependence on the experimental conditions found in our previous studies: (i) elongated particles are formed quicker at lower bath gas pressures than at higher bath gas pressures; (ii) elongated particles are more easily formed at higher sample gas concentrations than at lower concentrations. As reported previously [23,24,28], the mechanism behind the growth process remains unclear. Both agglomeration and coagulation of cube-like particles as well as evaporation of small particles and recondensation onto larger particles are possible explanations. These similarities between the behavior of the fluoroform aerosols with previous observations are strong hints that we observe the same shape change here, i.e. the formation of elongated particles from initially mainly cube- like particles. Figure 4.7 confirms this interpretation. It shows as a thick line in trace b) the experimental spectrum recorded at t = 960 s (see Figure 4.1), which exhibits pronounced side bands. For comparison, the thin dashed line shows again the spectrum of the cube-like particle recorded at t = 38 s. The pronounced side bands that appear at t = 960 s reflect the fact that the fraction of elongated particles in the ensemble has increased over time. This trend is reproduced by the simulated spectra in trace a). The thick line represents a linear combination of cube-like and elongated particles with axis ratios of 1:1:9, 1:9:1, and 9:1:1 (a:b:c principal crystal axis ratio, see ref. [33]). Particles with all three different aspect ratios are equally included in the simulations because we do not know whether growth happens preferentially along one particular 123 Li I 0 U Figure 4.7 Infrared spectra of crystalline particles in the region of the v5 / v2 band. a) Vibrational exciton calculations. Thick line: a 1:1 mixture of cube-like and elongated particles. Thin dashed line: pure cube-like particles (see Figure 4.5). b) Experimental spectra. Thick line: a mixture of cube-like and elongated particles (estimated ratio 1:1) after 960 s of particle growth. Thin dotted line: cube-like particles immediately after crystallization is complete (see Figure 4.5). elongated + cube-like elongated + cube-like 1080 1120 1160 1200 (cm1) 1240 axis. The ratio of cube-like to elongated particles (the three different aspect ratios contribute equally) is 1:1. The thin dashed line shows again the exciton calculation for the pure cube-like particle from Figure 4.5b. The general trends observed in the experimental spectra are clearly reproduced by these simulations. The characteristic side bands arise from elongated particles and 124 are thus more pronounced for higher fractions of elongated particles in the ensemble. The simulations also reveal that the side band between 1140—1170 cm’ is unique to the elongated particles with an axis ratio of 1:9:1, which stresses the importance of growth along the b axis. The other two types of elongated particles do not show this feature. Since they also do not exhibit any other characteristic features, we cannot conclude from our simulations whether growth along the a and c axis is important in reality. From the above results, we conclude that cube-like crystalline particles formed directly after crystallization grow to elongated particles with increasing time. The simulations show that growth along the b axis is important for fluoroform particles. However, they do not provide information about the importance of growth along the a and c axis. We would like to mention at this point that the qualitative agreement between experiment and simulation could easily be improved by fitting a broader range of particle shapes (aspect ratios) and by adjusting the parameters in Table 4.1. With little more physical insight to be gained, however, we abstain from doing so. 4.4 Summary The understanding of aerosol spectra is still in its infancy. The major reason is the complexity of these systems with their many degrees of freedom. To go beyond a mere speculative interpretation of spectral features, it is essential to analyze the experimental data with appropriate model calculations. The present study demonstrates how infrared spectra of molecularly-structured aerosol particles can be understood from molecular properties using the quantum mechanical vibrational exciton model. We have shown for the example of fluoroform aerosol that the combination of time-dependent infrared spectroscopy and quantitative modelling allows us to obtain a detailed understanding of dynamic processes such as crystallization and shape changes of aerosol particles. 125 In addition to more fundamental spectroscopic aspects, the data found here for fluoroform aerosol, i.e. crystallization rate constants as well as infrared spectroscopic features of supercooled liquid droplets and crystalline particles of different shapes, are important for a better understanding of the Rapid Expansion of Supercritical Solutions. Because of its low critical data, fluoroform is an attractive supercritical solvent, e.g. for the micronization of drugs by RESS. During the expansion process, the solvent fluoroform can co-condense with the drug to form an aerosol. For this reason, the phase behavior of fluoroform aerosol particles can crucially influence drug particle formation. 126 4.5 References [1]. 1. 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Of small pure CO aggregates apparently only the dimer received broader attention, which focused on its highly non-rigid nature [10,11 and refs. therein]. Beyond this smallest complex, very little work has been done in analyzing larger molecular aggregates of CO. The few exceptions are, to the best of our knowledge, a theoretical investigation of the trimer by Ford [12], the study of Kandler et. al. [13] of clusters of up to 320 molecules by mass spectrometry, and the experimental work of Bauerecker and coworkers [6,14] on aerosol particles in the 100 nm size range. Similar (unpublished) data for aerosol particles were recorded in our research group. With the present contribution, we intend to bridge the current gap between small CO clusters (about 100 molecules) and large aerosol particles (104108 molecules) by predicting the infrared spectra of these CO aggregates using a molecular model. We also compare our theoretical results with the experimental data available for large aerosol particles, which allows for a far more detailed analysis of the observed spectral features than previously possible. Furthermore, our model allows us to predict refractive index data for various CO ices, i.e. “A version of this chapter has been published. G. Firanescu and R. Signorell, I Phys. Chem. B, 113, 6366 (2009) 130 crystalline, amorphous, and partially amorphous ices. Our calculation of infrared spectra is based on the vibrational exciton model proposed in the pioneering work of Fox and Hexter [15] for the treatment of bulk infrared spectra. Resonant transition dipole coupling between the molecular dipoles in the bulk gives rise to collective vibrational modes, referred to as vibrational excitons. Cardini et al. [16] and Ewing and coworkers [171 were the first to use this model for small molecular aggregates. From these investigations, however, the suitability of the model remained uncertain. In the past few years, our lab has undertaken systematic studies, in which we compared experimental data with exciton calculations for clusters and aerosol with various properties [18 and refs. therein]. For vibrational bands with a strong molecular transition dipole moment (> 0.1 D) resonant transition dipole coupling turned out to be an excellent sensor for probing the shape, size, phase, and architecture of molecular aggregates. When this type of interaction dominates, it greatly simplifies the computational problem and makes the prediction of infrared spectra from first principles possible even for particles with tens of thousands of degrees of freedom [19,20]. Combined with a molecular dynamics approach to determinate of the internal structure of the particles, our model allows us to understand infrared spectra of CO aggregates and identify the characteristics specific to their makeup. 5.2 Computational approach 5.2.1 Exciton model The contents of this section have already been presented in detail in section 1.2, except for a few system specific aspects. The vibrational exciton model used in this work to predict the infrared spectra of CO aerosol particles is a quantum mechanical model that mainly describes the resonant transition dipole coupling between all molecules within a molecular aggregate 131 [16,21,22]. The current implementation — which we call Extended Vibrational Exciton (EVE) model - contains several extensions to our original implementation [22,23], which have also been discussed in detail in refs. [19,20]. The EVE model is briefly reviewed in this section. The first extension to pure dipole coupling refers to the inclusion of polarization effects which were shown to play a significant role in small SF6 clusters [24-26] (about 10% of the total contribution to the frequency splitting in the SF6 dimer), followed by similar findings for SF6 particles [20] and CHF3 particles [27]. The molecular polarizability of CO is roughly half that of CHF3 or SF6 so that we expect similar if somewhat less pronounced effects for CO particles. This is confirmed in Figure 5.1 by the noticeable redistribution of intensities in the spectra of small particles (here a crystalline sphere with a radius of 2 nm). A second general effect of induced dipoles is a slight bathochromic shift of the vibrational band as was observed for SF6 aerosol particles. Figure 5.1 EVE spectra of a (CO)748 crystalline sphere (r = 2.0 nm): with (solid) and without (dashed) polarization effects. The spectra are convoluted with a 0.2 cm’ FWHM Gaussian line shape. 0 C 0 C 0 0 0 .0 0 21 II II II II II II II I’ I’ (cm1) 132 With the inclusion of polarization effects the vibrational Hamiltonian takes the form: E=f20+p7Ap1 (5.1) with 4 = _-L[2q + (5.2) and (1— o..) = (3ee; —i) (5.3) 21•80 g where Et is the Hamiltonian for uncoupled molecular oscillators, 2, are scaled projection matrices, ak is the polarizability tensor of molecule k, p1 is the dipole moment of molecule i, e is the unit vector pointing from the center of mass of molecule i to that of molecule j and R is the corresponding distance. In the product basis of molecular oscillator eigenfunctions im) the Hamilton matrix elements take the form: Qrn ViJn) = 8mnh17im + 2pAp1 (5.4) where im and Pun are the wavenumber and transition dipole moment, respectively, for transitions from the ground state to the mt level of the jth molecule. m) represents the product function with level m excited on molecule i and all other oscillators in the ground state. Here we 133 have set the zero energy at the uncoupled ground state (OIHI 0). The basis is limited to near- resonant single molecule excitations. The second extension in the EVE model accounts for local structural variations within an aerosol particle when it is no longer justified to assume the same values of and ujm for all molecules of a given type (as in the original vibrational exciton model). Instead we assign individual transition wavenumbers i7m and transition dipoles fljrn to each molecule based on the explicit potential and dipole functions described in the next section. For the treatment of particles with more than 10000 degrees of freedom a direct diagonalization of the Hamiltonian (Eq. (5.1)) becomes impractical and particle spectra u(i7) are computed by Fourier transformation of the dipole autocorrelation function: C(t)= g(t) (km (5.5) k,m,l,n where g(t) is the Fourier transform of a suitable window function (line shape). The correlation function C(t) is calculated using a 2m1 order propagation scheme: = + &) (5.6) In addition to the calculation of the infrared extinction spectra the analysis of the vibrational wave functions from the EVE model allows us to correlate spectral features with structural properties of an aerosol particle. We can, for example, decide whether a certain spectral feature has its origin in the particle surface or in the core (see section 5.3.1). To this end we partition the spectrum (I7) into contributions o (i7) associated with individual volume elements 134 6J’ in the aerosol particle [19]. Technically the cr1(E) are obtained from an equivalent partitioning of C(t) [19]: (V)=cr1(V)= jgQ) (km (5.7) 1 k€8V m,1,n To account for the different sizes of the volume elements we use the “normalized excitation density” (5.8) where N1 is the number of molecules in 8J’. With a local density of states defined as [19]: p1 (i7)= c12 with 1E /hc—i7 iW (5.9) k€SV, m,I cr1 can be interpreted as the local density of states weighted by the transition probability. Here c1 is the contribution of km) to the eigenvector I I) with eigenvalue E1 of the Hamiltonian (Eq. (5.1)) and A V is the spectral resolution. Note that the above definition of the local excitation density (Eq. (5.7)) is equivalent to that of ref. [19]. 5.2.2 Potential model The EVE model uses as input individual transition dipoles p1,, and transition frequencies for each molecule in the particle, which depend on the intramolecular potential and the potential field generated by all other molecules in the particle. Suitable intra- and intermolecular potential models must be computationally efficient to treat particles with tens of thousands of 135 degrees of freedom. This limits the choice for the intermolecular potential to two-body potentials. The most advanced dimer potential has been developed by Vissers et al. [28]. For our purposes, however, this potential is too complicated. Although the literature provides several simple intermolecular potential models for CO [29-34], a few adjustments were necessary for the application to our CO nanoparticles. The starting point for intermolecular interactions is the model of Nutt and Mewly (NM) [29]. Dispersion and exchange are modeled using a Lennard Jones potential: V = 4e[ - (5.10) The C-O interaction parameters are derived as usual: = (o + 00)/2 and = . (5.11) The electrostatics are described by a point charge array composed of 3 partial charges placed at the atomic sites (qc and qo) and in the center of mass (qcM) with qCM=—(qCqO)• (5.12) In extension to previous models [30] the partial charges are allowed to vary with the bond length: q=a0+1r+23. (5.13) Fitted to the high level ab initio dipole and quadrupole functions of Maroulis [35,36] the variable charges provide a very good description of the electrostatic interactions. There seem to be 136 typographic errors in Table 1 of reference [29] so that we have re-determined the expansion coefficients (ao-a3) given in Table 5.1. Table 5.1 Adapted Nutt & Mewly (ANM) potential parameters: harmonic frequency V, Lennard Jones parameters, and coefficients for the variable partial charges placed on the C and 0 atoms. To assess the quality of the potential parameters, we compared its description of the dimer and the crystalline solid with results taken from the literature (Table 5.2 and Table 5.3). As a reference for the dimer we refer to the high level ab initio potential of Vissers et al. [28] (see also ref. [ii]). The potential has two minima at planar slipped anti-parallel configurations, Si (carbon atoms closest) and S2 (oxygen atoms closest) (see Table 5.2 and Figure 5.2a). The latter represents only a very shallow local minimum at the top of the disrotatory path that connects equivalent global minima Si. The disrotatory S2—>S 1 conversion is thus almost barrierless while the conrotatory conversion involves a barrier of about 60 cm1 with respect to S2 (from Fig. 2 of [ii]). The NM model qualitatively reproduces these general features, but with the roles of Si and S2 interchanged (Figure 5.2b1, Table 5.2). As a consequence, the dissociation energy for Si is about 40% too small, which also contributes to the error in the crystal lattice energy, which is calculated with the NM potential 15% below the experimental value (Table 5.3). We decided to refine the potential model with respect to the Si and S2 dissociation energies and the lattice energy. Since the description of the electrostatics is already validated independently, we only adjusted the Lennard-Jones parameters to improve the potential. We obtained satisfactory results with a single set of Lennard-Jones parameters for both C and 0. 137 Figure 5.2 a: Dimer structures corresponding to local minima on the ab initio potential surface of Vissers et al. [28], see text. b: Potential energy surface for planar CO dimer as a function of molecular orientation. 0A and 8B are the angles between the molecular axes of the monomers and the axis that connects their respective centers of mass. 8B 0 correspond to trans (like atoms on opposite sides) and B 0 to cis configurations (like atoms on the same side). At each point the energy is minimized with respect to the intermolecular distance R. b 1: Original Nutt & Mewly surface, [29]. b2: Adapted Nutt & Mewly surface, this work. al) a2) o C 0 C \A \ /8A / \R \4 / R//98 C 0 C 0 bi) b2) 180 120 60 0 — 0 -60 -120 -180 I 800 60 120 180 0 60 120 GAo GAo 138 Table 5.2 Dimer geometry and dissociation energy (De) for the slipped anti-parallel configurations (Si and S2, see Figure 5.2a). R is the vector connecting the molecular centers of mass, 6A and &B are the angles between R and the respective molecular axes. 8B 0 corresponds to trans (like atoms on opposite sides) and 0B 0 to cis configurations (like atoms on the same side). The C-O bond length is 1.128323A. IRI (A) 8A (°) De (cm1) SI high level ab initlo [28] 4.31 136.1 43.9 -148.4 high level abinitio[11] 4.33 134.2 45.8 -139.0 ANM model 4.05 123.3 56.7 -131.5 NM model [29] 4.26 121.8 58.2 -84.7 S2 high level abinitio[28] 3.66 63.6 116.4 -121.8 high level abinitio[11] 3.67 61.0 119.0 -123.0 ANM model 4.00 52.8 123.6 -105.7 NM model [29] 3.88 51.5 128.9 -141.5 The optimal values given in Table 5.1 are very close to those derived from thermodynamic data by Stoll [37], which provides further validation for our approach. The resulting “Adapted NM” (ANM) potential model improves the dissociation energy of the Si configuration and restores the correct energetic ordering (Figure 5.2b2, Table 5.2). The disrotatory 52—>S1 barrier, which was already tiny in the ab initio potential, has now completely disappeared, so that S2 is now a true saddle-point. More importantly, disrotatory motion is correctly described as only weakly hindered (max. energy difference of ca. 25 cm1 compared with ca. 20 cm1 ab initio). To determine the quality of the model in describing the crystalline solid, we compare experimental results for the a-phase of CO with the original NM potential and our ANM potential in Table 5.3. 139 Table 5.3 Crystal minimum energy configuration (lattice constant, molecular center of mass shift from the fcc symmetry positions; fractional coordinates: (0,0,0), (0,Y2,V) (0,Y2,V) (0,Y2,Y)) [46], corresponding lattice energy contributions [33,55] and phonon frequencies as assigned from experiment [38-40] and reassigned from theoretical work [33,34]. Symmetry Experiment Theoretical NM model ANM model Geomet,y lattice constant (A) 5.646 5.7200 5.627 shift towards 0 (A) 0.1607k; 0.0004k 0.1128 0.0559 xC 0.0495; 0.0659 -0.0537 -0.0604 xO -0.0659; 0.0495 0.0602 0.0554 Energies (kcal/mol) Electrostatic -0.509 -0.566 exchange & dispersion -1.625 -1.919 Total -2.480 -2.134 -2.485 Phonons(cm1) F 905b 905d,e 123 128 F 85a,86c 8586d,e 99 100 F 49a1505c2b 58d 49/50.5/52e 71 88 F 38 49/50.5/52 d e 52 60 E 645d,e 86 89 E 44d38e 55 66 A 645b 58e 60 65 + the crystal is inverted relative to the other publications, which is taken into account ++ not assigned in the experiment a) [39] b) [40], c) [38] d)[33]e)4 In both cases the lattice energy was minimized as a function of lattice constant and center of mass shift of the CO molecules from the fcc symmetry positions (fractional coordinates: (0,0,0), (0,Y2,V) (0,Y2,’4), (0,V2,V)). The change in CO bond length is negligible. By design, the ANM model yields very good agreement with the experimental lattice constant and lattice energy. While phonons are not explicitly treated in the EVE model, their frequencies represent another 140 check of how realistic the intermolecular potential is. While there is some uncertainty in the assignment of experimental data [38-40] compared with more recent phonon calculations [33,34], the frequency range is clear and well described both within the NM and ANM model. The calculated results lie approximately 20 cm’ higher which is in part consistent with the uncertainties in our potential model. Further the anharmonic red shift is not included since the phonon frequencies were calculated in the harmonic approximation. For the EVE model, the intermolecular potential has to be combined with an explicit intramolecular potential function. Instead of the anharmonic intramolecular potential function for CO available in the literature [41], we employ an effective harmonic stretching potential parameterized to reproduce the experimental gas phase stretching wavenumber. This approach neglects small anharmonic frequency shifts resulting from the change of bond lengths of individual CO molecules in different environments. Our model potentials (see above) yield fully relaxed structures with CO bond lengths shortened by a few 1 0A regardless of phase (amorphous, crystalline). This would translate into wavenumber shifts on the order of 10 cm’. Our choice of an effective harmonic CO stretching potential is justified by the observation that those anharmonic shifts are essentially the same for all molecules within the particulate phase so that there is no net contribution to the spectral band shapes. Note that local transition wavenumbers Vh,,, still depend on the relative position and orientation of individual molecules. They are derived from a normal mode analysis for each molecule in the field of all others. The same analysis yields local transition dipole moments ,uh,, (within the double harmonic approximation) as the corresponding 1St derivative of the overall dipole moment function, i. e. the transition dipole moments are obtained directly from the electrostatic contribution to the potential function. In our model, overall band shifts remain uncertain to within anharmonic couplings between intra- and intermolecular vibrations. From the above, they are expected to be on the 141 order of several cm’ and largely independent of the molecular environment. We account for the effect a posteriori by shifting all calculated exciton spectra by 5 cm1 to lower wavenumbers. This constant offset was determined by comparing of the calculated spectra with the experimental aerosol spectra of our own research group (see section 5.3.2), as well as experimental aerosol spectra of ref. [6]. 5.2.3 Particle model and molecular dynamics simulations The particle properties to be studied in the infrared spectra are the size, the shape and the internal structure (phase, architecture). In the particle ensembles typically formed in experiments (collisional cooling, supersonic expansions), these observables are usually distributed over a certain range, which depend on the particle generation method and on the experimental conditions. The purpose of the present contribution is to predict general trends. We discuss the influence of the size and also the size distribution on CO aerosol particle spectra. For the particle shapes, we consider some specific cases in our simulations: equal aspect ratio geometries such as spheres or cubes, which are likely candidates during initial particle formation, and elongated shapes, which are often formed as particles grow over time [20,23,27,42,43]. In terms of internal structure, we consider particles ranging from fully crystalline to fully amorphous. The intermediate cases are assumed to be core-shell particles with a crystalline core and an amorphous shell. In the very first stage of particle formation, condensation will initially produce more or less amorphous aggregates if cooling is fast enough. During subsequent growth, as additional molecules condense upon this aggregate, some of the energy released is transferred to the already existing aggregate. Under the right conditions, this might lead to annealing of the existing core and eventually to the formation of partially amorphous particles composed of a 142 crystalline core surrounded by an amorphous shell. We have observed the formation of such core- shell particles in the case of nanosized NH3 aerosol particles [19] (chapter 2). Bulk CO crystallizes in two phases: the high temperature hexagonal fl-phase (61.5-68.1 K), where molecules rotate freely [44] and the low temperature cubic P213 a-phase (below 61.5 K), where rotation is hindered [45-47]. Since no signs for the occurrence of the fl-phase has been found in the available experimental data for CO particles and since it is relevant only in a very narrow temperature range, we will focus entirely on the a-phase. The crystalline particles used in the simulations are cut directly from the bulk structure which is constructed with the ANM model parameters (Table 5.1). Based on experimental findings [47], head-tail flips of the CO molecules within the crystal are to be expected. In the EVE model, CO flips around the molecular centre of mass produce only minute changes in the spectra which arise from variations in the local force fields. The spectra of particles with random head-tail flips are virtually indistinguishable from the fully ordered a-phase. Therefore, we limit our studies to the latter case. The fully amorphous particles as well as the amorphous shells of the core-shell particles are constructed by randomizing the orientations of the molecules and their center of mass positions followed by simulated annealing to relax configurations with unphysically high energy. The range of the centre-of-mass shifts is limited to 10% of the nearest neighbor distance (0.23 A) to avoid extremely high energy configurations that would lead to evaporation in the annealing procedure. The simulated annealing does not attempt to reproduce the actual annealing process mentioned previously. It is merely a tool to create physically plausible amorphous structures and is carried out by classical molecular dynamics using a Verlet propagator and the ANM potential, as described in detail in chapter 1.2. For all annealing runs, the following parameters were being used unless otherwise specified: an initial kinetic energy of l.SkBTkIfl is microcanonically 143 distributed in the system, where Tk1 = 50 K. The system evolves with a time step of 0.1 fs and the kinetic energy is reduced by 0.1% of its current value at each time step. Except for slight relaxation, crystalline particles are energetically the most stable configurations so that the annealing ultimately leads to complete crystallization, which must be avoided in order to generate crystalline core-amorphous shell particles. The relaxation of unphysically high energy configurations is so much faster than the eventual recrystallization that the two can be temporally separated. We define a structure as sufficiently relaxed if its spectrum has converged on the relaxation time scale (on the order of 102 fs) while still displaying a sharp core-shell boundary. This point was found to be reached universally when the average binding energy per molecule in the amorphous region reached 60% of the value in a fully crystalline shell. As the particle size decreases (below 1 molecules), ensemble averages become necessary to account for statistical variations in both their spectral and structural properties. To analyze the internal structure of our particles, we measure their crystallinity, or rather the deviation from it, as the standard deviation of the molecular centers of mass and of the molecular orientations from the perfect crystal (inf over macroscopic rotations) as defined in chapter 1.2: CM =inf[-(&i)2 =inf[1a (5.14) where Ax is the shift of the center of mass of molecule I relative to the perfect crystal, a1 is the angle of CO relative to its orientation in the perfect crystal, and N is the number of molecules. Based on the aforementioned randomization algorithm values of above 0.23 A for CM and 60° for cia indicate essentially amorphous structures. For example, Figure 5.3 shows the analysis of 144 the internal structure for an ensemble of 10 spheres with a radius of 4 nm and a 40 vol% amorphous shell annealed for 50 fs. The particle was divided into 1 nrn thick shells. The dips close to the center of the particle (r = 0 A) arise from empty shells. The crystalline-to-amorphous boundary is a clearly distinguished step function at 33 A. Some reorientations and displacements can also be observed in the crystalline core. They arise from the initial kinetic energy distributed in the system and are independent of the core and shell dimensions. Figure 5.3 Standard deviation of the molecular centers of mass (upper trace) and of the molecular orientation (lower trace) from the crystalline structure (see Eq. (5.14)) for an ensemble of 10 spheres with a radius of 4 nm and a 40 vol% amorphous shell after 50 fs of simulated annealing. 0.3 - 0.2- C) b 0.1- 0.0 60 0) . 40 20 0 0 5 10 15 20 25 30 35 40 0 5 10 15 20 (A) 25 30 35 40 145 5.3 Results and discussion 5.3.1 Size, shape and surface effects in nanosized particles (<10 nm) In the first part of our study we focus on small particles, i.e. up to about 10 nm in radius, for which vibrational bands with strong exciton coupling are dominated by size, shape, and surface effects [18,48,49]. These result from the high surface to volume ratio so that the properties of the particle surface increasingly dominate the spectra with decreasing size. 5.3.1.1 Crystalline particles We begin our discussion with completely crystalline particles, which have characteristic size and shape dependent spectra in this size range. The number of molecules per aerosol particle spans three orders of magnitude, ranging from a few 100 (2 nm radius) to about 30000 molecules (7 nm radius). At the upper limit the particles reach the size region for which the spectra are size-converged. The effect of size on particle spectra is shown for a series of spheres with radii of 2 nm (750 molecules), 4 nm (6000 molecules) and 6 nm (20000 molecules) from top to bottom in Figure 5.4a. Each particle size has a unique spectrum. The structure in the spectra of the small particles is gradually lost as they become larger. Similar trends with increasing size are also observed for other particle shapes such as cubes (depicted in Figure 5.5a for the same particle volumes as in Figure 5.4a) or elongated particles (not shown here). The comparison of Figure 5.4a with Figure 5.5a also demonstrates that for a given size (volume) different shapes give rise to unique spectra. 146 Figure 5.4 EVE spectra of crystalline spheres as a function of size. Absorbances are scaled to the same value (normalized to the maximum). a) Individual particles with increasing radii of r = 2, 4 and 6 nm from top to bottom. b) Lognormal distributions of particles with increasing mean radii of = 2, 4 and 6 nm from top to bottom. o- = 1.3 for all distributions. The spectra are convoluted with a 0.2 cm’ FWHM Gaussian line shape. a)J rj\_ r 4 nm r=6nm II 2136 2138 2140 2142 2136 2138 2140 2142 V (cm’) (crn1) 147 Figure 5.5 Same as Figure 5.4 for crystalline cubic particles, a) Single size particles with the same volumes as in Figure 5 .4a. b) Lognormal size distributions with the same average volumes as in Figure 5.4b. 2136 2138 2140 2142 2136 2138 2140 2142 (crTi’) (cm1) Experiments, however, rarely generate particles of a single size. Most experimental methods, such as collisional cooling or supercritical expansion, produce particles with a size distribution. As a consequence the spectral characteristics of individual sizes blend together, resulting in broader, less structured bands. To investigate the effect of size distributions on CO particle spectra, we calculate infrared spectra of particle ensembles described by a log-normal size distribution with mean radius F and standard deviation a [501. Figure 5.4b shows the spectra 148 resulting for spherical particle ensembles with c=l .3 and values of 2, 4 and 6 nm to match the sharply defmed particle sizes of Figure 5.4a. As expected, the information specific to individual particles is lost, resulting in a single structureless band (full width at half maximum (FWHM) = 0.8-0.4 cm’ from top to bottom). Except for being slightly broader, the spectra are very similar to the size-converged spectrum of a large, individual sphere (r 6 nm, in Figure 5.4a). Thus the size distribution of a particle ensemble leads to an almost complete loss of information about the mean particle size. We have repeated this analysis for cubic (Figure 5.5b) as well as elongated particles (not shown here) using the same size distributions in which the spheres were replaced by corresponding particles of equal volume. The results are the same: the spectra for the size distributions (Figure 5.5b) always resemble that of the largest individual particle (bottom of Figure 5.5a), clearly identifying their shape. Consequently, even if the size information is lost in the ensemble, different particle shapes can still be distinguished. Thus a single symmetric band is specific to spheres (Figure 5.4b), while the cubic particles (Figure 5.5b) are characterized by a broad band at 2139.5 cm’ and a broad shoulder around 2141 cm’. Ensembles of elongated particles feature a sharp band at 2138.0 cm’ with a broad shoulder at 2140.6 cm1 (not shown here). In a previous study of the infrared spectra of nano-sized NH3 aerosol particles [19], we were able to distinguish infrared active and inactive components of the local density of states p (i7) (Eq. (5.9)) of the umbrella vibration (-1060 cm’) in crystalline spherical particles. Figure 5.6 shows again the comparison of p1 (ii) (Figure 5.6c1) of a crystalline NH3 sphere (r = 2nm) 149 Figure 5.6 Comparison of crystalline NH3 (1) and CO (2) spheres in (r 2 nm). a) Calculated EVE JR spectra without polarization effects. b) Corresponding excitation density (Eq. (5.7)). c) Corresponding local density of states (Eq. (5.9)). The CO data were convoluted with a 0.2 cm1 FWHM Gaussian line shape to resolve the details of the very narrow spectral region, whereas a 6 cm’ FWHM was used for the broad NH3 band. NH3 Co ThL 2145950 1000 1050 2135 2140 -1 — -4 v(cm ) v(cm 150 with the corresponding excitation density o (i7) (Eq. (5.7)) (Figure 5.6b1). p1 (i7) shows two distinct maxima (arrows around 1030 cm’ and 1048 cm’), which are delocalized over the particle’s core. But only the higher frequency one corresponds to a maximum in the excitation density in Figure 5 .6b 1 and thus determines the infrared spectrum (Figure 5 .6a 1). The crystal structures of a-CO and NH3 belong to the same space group (P213) and both the NH3 umbrella and the CO stretch vibration are dominated by the coupling of molecular transition dipoles in the same orientations. Hence we expect a similar behaviour for a CO as for NH3, although less pronounced because of the smaller transition dipole (0.11 D for CO compared to 0.23 D for NH3). The analogous results for a crystalline 2 nm CO sphere are depicted in Figure 5.6 a2, b2, c2. (Note that the CO data were convoluted with a 0.2 cm’ FWHM Gaussian line shape to resolve the details of the very narrow spectral region, while for the broad NH3 band 6 cm1 FWHM was used). The comparison of the state density (Figure 5.6c2) and excitation density (Figure 5.6b2) of the CO sphere in Figure 5.6 shows a partial overlap indicating the same effect to occur for crystalline CO spheres as well. However, the effect is less obvious in the case of CO spheres because of the weaker coupling between the CO molecules, which leads to a much narrower band structure. We have ascertained that artificially increasing the transition dipole strength of CO from 0.11 D to 0.23 D to equal that of NH3 reproduces the behavior of the ammonia particles. 5.3.1.2 Amorphous particles and partially amorphous core-shell particles Completely amorphous particles are best approximated by spherical shapes, as they lack edges and corners. Moreover the spectra of amorphous particles with other shapes show only minor deviations from those of amorphous spheres so that we only consider the latter. In contrast 151 to their crystalline counterparts, individual amorphous particles display the same broad featureless band regardless of size and therefore also regardless of size distributions. Thus, aside from clearly identifying the phase no further information on the particles can be extracted if completely amorphous particles are formed. Between the limiting cases of purely crystalline and completely amorphous particles lies the wide range of partially amorphous structures such as core-shell architectures. Based on the considerations outlined in section 5.2.3, we restrict our discussion to particles with crystalline cores and amorphous shells, for which we assume a spherical shape as the physically most plausible choice. This leaves only two variables: the particle size and the core-shell ratio. To determine their individual influence, we calculate the spectra for a series of spheres with radii of 2 nm, 4 nm and 6 nm and we vary the amorphous percentage of the shell for each size between 20-80%. All particles were annealed for 50 fs, yielding binding energies per molecule in the amorphous shell of around 60-62% of the value in a crystalline environment. Results for the 2 nm and 4 nm spheres were averaged over an ensemble of 40 and 10 elements, respectively. No averaging was required for the larger spheres. As an example, Figure 5.7 depicts the evolution of the spectra for a 4 nm sphere with increasing thickness of the amorphous shell. Compared with a crystalline 4 nm sphere (Figure 5 .4a), the distinctive feature of all particles with amorphous shells is an asymmetric broad spectrum. With increasing amorphous percentage, the bands broaden and a pronounced shoulder emerges at the low-frequency side of the main peak. The crystallinity (Eq. (5.14)) shows a sharp separation between the crystalline core and the amorphous shell as demonstrated for the 4 nm sphere with a 40 vol% amorphous shell in Figure 5.3, where the shell extends from 3.4 to 4 nm. This allows for a clear differentiation between core and shell region. The corresponding 152 excitation density (Eq. (5.8)) presented in Figure 5.8 confirms that the origin of the asymmetry and the long tails on both sides of the main band lies in the amorphous shell. Similar features are Figure 5.7 EVE spectra for ensembles of spheres with radii of 4 nm. All spectra are normalized to max. absorbance. Each ensemble is composed of 10 spheres with a crystalline core-amorphous shell architecture with a shell volume of 20% (thick line), 40% (dashed), 60% (thin line) and 80% (dashed-dotted). The spectra are convoluted with a 0.2 cm’ FWFTM Gaussian line shape. C,) CU ci) C) C CU -Q 0 Ci) CU also observed for the 2 nm and the 6 nm particle as a function of the volume fraction of the amorphous shell. Varying the second parameter, the particle size, for constant amorphous volume fraction, the bands become slightly narrower (by 0.3-0.8 cm’) as the size increases because the contribution of the spherical crystalline core narrows down (see Figure 5.4a). These differences, however, disappear if size distributions are included. 0 0’ / I I’ / / / 0 ,0 \ \ 2136 2138 2140 2142 2144 (cm’) 153 Figure 5.8 EVE spectrum and excitation density (Eq. (5.8)) for an ensemble of 10 spheres with radii of 4 nm and crystalline core-amorphous shell architecture. The asymmetry in the spectrum and the long tails on both sides of the main peak are clearly associated with the 40 vol% amorphous shell extending from 3.4 to 4 nm. The spectra are convoluted with a 0.2 cm1 FWHM Gaussian line shape. Based on our findings, crystalline core-amorphous shell particles can be identified (as distinct from fully amorphous or fully crystalline particles) both by the asymmetry and by the width of the infrared bands for amorphous shell volume fractions between about 20% and 80%. However, particle size and shell thickness cannot always be extracted independently because different combinations can lead to similar band structures. This is especially true for the interpretation of experimental data when distributions in both properties must be taken into account. 2136 2138 2140 2142 2144 (cm1) 154 5.3.2 Phase and shape effects in large particles (>10 nm) For large particles, i.e. in the region between 10-100 nm, the band shapes in the infrared spectra are converged as a function of size (see refs. [18,22,42]), leaving only structure and shape as influencing factors. For large CO aerosols, experimental infrared spectra from collisional cooling cells have been measured by us (see Figure 5.9c) and by Bauerecker et al. [6,14,5 1]. This provides us with the opportunity to compare our theoretical results with experimental data and to analyze the experimental data based on the EVE model. There are three different types of experimental spectra reported: i) spectra recorded directly after particle formation, which show a single symmetric peak with a FWHM of 1.8-2.5 cm1 as depicted in Fig. 4 of ref. [6]. These spectra were thought to arise from spherical crystalline particles, but as our analysis will show this interpretation is not correct. ii) Spectra with a broad (FWHIVI -5 cm’) band (see Fig. 4 in ref. [14] and Fig. 3 in ref. [51]). It was suggested that these spectra arise from amorphous CO particles, but since no molecular model was available this could not be confirmed. iii) A series of spectra recorded as a function of time after CO aerosol particle formation as depicted in Fig. 2 of ref. [6]. With increasing time, two side bands at higher and lower frequency evolve. The corresponding simulations are shown here in Figure 5.10 and the origin of this spectral change is discussed below in section 5.3.2.2. As spectra in this size region (‘— 10-100 nm) are converged with respect to size, it is sufficient to reach the convergence threshold for the particles in our simulations, i.e. 20000 — 30000 molecules. Given the experimental resolution of the spectra, we consistently convoluted all calculated spectra in this section with a Gaussian function with a full width at half maximum of 0.5 cni1. 155 5.3.2.1 Phase of the particles We start our discussion with spectra of type i) (see Fig. 4 of ref. [6]). In ref. [6], it was argued that these spectra are for spherical crystalline particles because of the symmetric band shape. The band width and the fact that the band lies between the transversal and longitudinal optical modes of the bulk state was taken as a strong hint that these particles are crystalline. The following analysis with our EVE model demonstrates that both interpretations are not correct. In a first step, we assume that the particles are crystalline and have a spherical shape. This leads to the calculated spectrum in Figure 5.9a. With a FWHM of 0.7 cm’, it is much narrower than the 1.8-2.5 cm’ FWHM of the experimental spectra, already a first hint that the particles are not crystalline. The second argument has to do with the crystal structure and the particle shape. Crystalline a-CO has a cubic crystal structure [46], which makes it very unlikely that spherical particles are formed. Much more plausible particle shapes are cubes or cube-like shapes (i. e. cuboctahedra). Both lead to infrared bands with a characteristic shoulder at the high-frequency side of the main peak as depicted in Figure 5.9b. As demonstrated in previous studies of SF6 particles [20] and CHF3 particles [27], cube-like shapes can thereby clearly be distinguished from spherical shapes, experimentally as well as in the calculations. Since the experimental spectra of the CO particles do not show the high frequency shoulder and since these spectra are broader by more than a factor of two than the calculated spectra for crystalline particles, we conclude that the CO particles under discussion are not crystalline. 156 Figure 5.9 Infrared spectra for different large CO aerosol particles: a) Calculated spectrum of a crystalline spherical particle. b) Calculated spectrum of a crystalline cubic particle. c) Experimental spectrum measured in a cooling cell at a temperature of 12 K (this work). d) Calculated spectrum of a 100% amorphous spherical particle. The calculated spectra are convoluted with a 0.5 cm’ FWHM Gaussian line shape. 0 4-’ D I Cu -Q 1. 0 0 0 CU 0 Cu Cu C) C Cu 0 0 Cu 246 If not completely crystalline, the particles might be amorphous. The comparison with the corresponding calculated spectrum in Figure 5.9d rules out the possibility of completely amorphous particles. The calculated spectrum of a large amorphous sphere (amorphous particles are unlikely to support corners and edges) reveals a much broader band (FWFIM 5.lcm’) than observed in the experiments. In addition, the experimental widths are not constant. By changing the experimental conditions, the width can be varied. We have added as an example an a) c) b) 2142 21’34 21’38 d) 2142 2146 2134 (cm1) ; (cm1) 2138 2142 157 experimental spectrum from our own group (Figure 5.9c) with a width around 3 cm1,which lies slightly above the values reported by Bauerecker and coworkers. All this evidence taken together is a strong hint that in all those cases the particles are partially crystalline with varying amorphous contributions, which determine the observed bandwidths. There are several possibilities of how these partially crystalline particles could look like. Since in cooling cell experiments particles ensembles are measured, it is possible that the partially amorphous spectra arise from an ensemble with a certain fraction of crystalline and another fraction of amorphous particles which would determine the amorphous contribution to the spectra. Another possibility is the formation of amorphous shell-crystalline core particles. Here the shell volume fraction determines the amorphous contribution. Alternatively, crystalline inclusions could form in an amorphous matrix or vice versa. Lastly, partially crystalline structures could form on a molecular level, e. g. some but not perfect order as in the crystal on a molecular level. We can simulate this last case by controlling the length of the simulated annealing runs. With simulations for the different cases we have tried to distinguish between them. However, for all cases we fmd essentially a single band and band widths ranging from 1.6 to 4 cm1 depending on the amorphous contribution, similar to what is observed in the various experiments. Since the general trends are reproduced for all possibilities it is impossible to say more about the nature of the partially amorphous particles. To give but one quantitative example, the band width of 1.8 cm’ observed in Fig. 4 of ref. [6] is reproduced by a large core-shell particle with a 40 vol% amorphous shell. The very broad spectra in Fig. 4 for‘3C’60(Case ii)) are indeed very likely to arise from almost completely amorphous particles. This is confirmed by the simulation of a completely amorphous particle in Figure 5 .9d. The calculated as well as the experimental spectrum have a FWFIM around 5 cm’. Note that the band widths are the same for different isotopomers. 158 5.3.2.2 Shape effects Case iii) deals with the evolution of the infrared spectra observed as a function of time (see Fig. 2 of ref. [6]), i. e. the formation of two prominent side bands on the high and the low frequency side of the main peak with increasing time after particle formation. In several previous contributions for various aerosol particles [20,23,27,42,43,48,521, we have shown that analogous spectral changes were due to a change in the particle shape with increasing time. The analysis with our quantum mechanical model revealed in all these cases that the particle shape changed from an initial shape with equal axis ratios (cubes, spheres) to particles with an elongated shape. This makes it very likely that the same shape effects are also the correct explanation in the case of CO particles. Phase effects, such as a change from the a to the /3 crystalline phase, can be excluded because the measurements were performed well below the corresponding transition temperature of 61.5 K. We know from our previous studies on shape effects that initially most particles of the ensemble are likely to form shapes of equal axis ratio. We thus assume the same behavior also for CO aerosols. The influence of a spherical and a cubic shape on crystalline CO particle spectra was already presented in section 5.3.1.1 where size convergence was reached for the largest particles (see lower traces of Figure 5.4a, & Figure 5.5a). The spectra of these shapes with equal axis ratio are dominated by a single band. Based on our previous investigations, we also assume that with increasing time elongated CO particles are formed and that their fraction steadily increases with time at the expense of the spherical/cubic particles. Elongated CO particles have spectra with three peaks. The central peak lies close to the band of the spherical/cubic particle and the other two appear at higher and lower frequency of this middle peak (not shown here). In addition, we find that for higher axis ratios the peaks on the lower-frequency side are further 159 away from the central peak. Figure 5.10 shows a series of calculated spectra based on this scenario, i. e. with a systematically increasing fraction of elongated particles (solid thick line to dashed line) which produces ever more pronounced side bands. These spectra reproduce the experimentally observed trend very well. Figure 5.10 Calculated EVE spectra for mixtures of crystalline particles with different shapes. Spheres, cubes, and cuboids with an axis ratio of 1:1:3 and 1:1:9 are included. The ratio of the different shapes are: solid thick line: 21:34:22:23%, dash-dotted line: 14:34:24:26%, solid thin line: 7:32:28:33% and dashed line: 7:21:31:41%. The spectra are convoluted with a 0.5 cm1 FWHM Gaussian line shape. C’) 4-. ci) C-) C C C,) 2135 2143 II,’.-’ 2137 2139 2141 (cm1) 160 We thus conclude that it is indeed the evolution of the particles’ shape which determines the spectral change in the experiment. This statement is based on true predictions from our EVE model, which do not involve any fitting to the experimental data. This is in marked contrast to classical scattering calculations (such as Mie theory or Discrete Dipole Approximation DDA [53,54]) using experimental refractive index data. Our calculations are not only independent of experimental results, but also provide a molecular explanation of the origin of shape effects in infrared spectra, viz, the transition dipole coupling of all molecules in the aerosol particles, which lifts the degeneracy of the uncoupled vibrational states of individual molecules and leads to vibrational eigenfunctions that are delocalized over the whole particle thus probing its shape. Compared with the experimental data (Fig. 2 of ref. [6]), our calculated spectra are slightly more structured. There are several reasons for this: the simulations include only two types of elongated particles (axis ratios 1:1:3 and 1:1:9), whereas in the experiment there will probably be a distribution of different axis ratios. The inclusion of more axis ratios would smooth the structure in the low frequency peak around 2138 cm1. Furthermore, the analysis in section 5.3.2.1 tells us that the particles formed in the experiment are not completely crystalline. Shape effects are less pronounced for partially as opposed to fully crystalline particles. Without information on the exact internal structure of the particles, we have performed the simulations for fully crystalline ones. To some extent we account for deviations from perfect crystallinity by including not only cubes but also spheres in our simulations since partially crystalline particles are unlikely to form perfect cubes. The inclusion of partially crystalline particles would also smooth the band structures as well as slightly broaden the bands. The agreement with experimental spectra could doubtless be perfected by including partially crystalline structures and additional intermediate axis ratios, but we refrain from doing so, as no further physical insight would be gained from such fitting procedures. 161 5.3.3 Refractive index data Finally, the EVE model allows us to calculate refractive index data for CO ice with a well defined phase/internal structure. There are several refractive index data sets available in the literature for CO ice [2-6]. The derivation of these data is based on fits to experimental infrared spectra, for which the phase/internal structure is not known. It is therefore not clear whether these data are for crystalline, amorphous, or partially crystalline CO ices. With the EVE model, we can now provide optical data for completely amorphous and for completely crystalline CO ices as well, and in principle for any intermediate case. For this purpose, we calculate a spectrum for a large spherical particle with the desired phase/internal structure with the EVE model and then use Mie theory and a Kramers Kronig inversion [53] to extract the refractive index data. The resulting refractive index data for four different phases/internal structures are presented in Figure 5.11. Panel a) shows the real part n and the imaginary part k of the refractive index of completely amorphous CO ice. Panel d) shows the same for completely crystalline CO ice. Panels b) and c) are derived from amorphous shell-crystalline core particles with 80 vol% and 40 vol% amorphous shells, respectively. With decreasing amorphous contribution, the maximum values of k and n increase and the minimum value of n decreases. In addition, the “bands” in Figure 5.11 become sharper. Classical scattering theory [53] shows that particle spectra have strong infrared absorption bands and show pronounced shape effects for all particle shapes in regions where n has values close to zero while k varies strongly. Our EVE model reveals that the classical criterion is equivalent to the case when transition dipole coupling is strong and the molecules in the aerosol particle are well ordered. It is therefore not astonishing that for CO, the refractive index data for crystalline particles in Figure 5.11 d fulfill the “classical” criterion best. 162 Figure 5.11 Refractive index data derived from the calculated EVE spectra of large spherical CO particles, a) Completely amorphous CO ice. b) CO ice with an 80% amorphous contribution. The spectrum is derived from a particle with a crystalline core/amorphous shell architecture with an 80 vol% amorphous shell. c) CO ice with 40% amorphous contribution. The spectrum is derived from a particle with a crystalline core/amorphous shell architecture with a 40 vol% amorphous shell. d) Completely crystalline CO ice. a) - b) - C) - d) 2150 2130 2150 2130 2150 2130 2150 2130 (cm’) (cm’) (cm4) (cm’) Some experimental optical constants from the literature [2-5] are depicted in Figure 5.12 These data were derived from thin film spectra at 10-16 K. The comparison of our calculated optical data with the literature data [2-6] reveals that none of these data sets apply to fully 163 crystalline particles. The literature values are for amorphous or partially amorphous CO ices with different degrees of amorphous contribution. Figure 5.12 Refractive index data from the literature derived from experimental spectra of thin CO films at temperatures of 10-16K: a) ref. [2], b) ref. [3], c) ref. [4], d) ref. [5]. 4 3 2- 1- 0- 4- 3- 1- 0 5.4 Summary We have combined our extended vibrational exciton model with a molecular dynamics approach to predict how the shape, size and internal structure (surface, phase, architecture) of CO aerosol particles manifest themselves in infrared spectra. We distinguish two different size regions: i) particles with diameters between 1 and 10 nm, for which the spectra are sensitive to 164 a) b) ci) I I I I A*A*A, 2150 2130 2150 2130 2150 2130 2150 2130 (cm1) (cm1) (cm1) (cmj size, shape, surface, and phase. Experimental data are not available for this size range so that our calculations provide important predictions for such experiments. ii) Particles between 10 and 100 nm, whose spectra no longer depend on the size (in a non trivial way) or the surface (surface contribution too small), but are dominated by the shape and phase of the particles. In the 1-10 nm region, we find that individual crystalline particles exhibit characteristic size and shape dependent spectra, while spectra of particle ensembles with a certain size- distribution retain only information on the shape but not on the mean size. Amorphous particles in contrast, display a broad asymmetric band regardless of size only allowing for the determination of the phase. For partially amorphous particles the extent of the amorphous contribution can be estimated from an analysis of the band width of the spectra. In the 10-100 nm size range, we find a similar influence on the spectra for the phase and shape of the particles, while there is no distinct size dependence in this range. The comparison with experimental data shows that the particles generated in collisional cooling cells always have varying amorphous contributions depending on the experimental conditions. The observed change in the band structure as a function of time is identified as a shape effect due to a transformation from globular particles into elongated ones. The success of the EVE model in analyzing the infrared spectra of pure CO particles would warrant an analogous investigation for astrophysically relevant ice mixtures which contain CO. 165 5.5 References [1] G.E. Ewing and G. C. Pimentel,i Chem. Phys., 35, 925 (1961) [2] D. M. Hudgins, S. A. Sandford, and L. J. Allamandola, A. G. G. M. Tielens, Astrophys. I Suppi. Ser., 86, 713 (1993) [3] G. A. Baratta and M. E. Palumbo, I Opt. Soc. Am. A, 15, 3076 (1998) [4] M. E. Palumbo, G. A. Baratta, and M. P. Collings, Phys. Chem. Chem. Phys., 8, 279 (2006) [5] P. Ehrenfreund, A. C. A. Boogert, P. A. Gerakines, A. G. G. M. Tielens, and E. F. van Dishoeck, Astron. Astrophys., 328, 649 (1997) [6] E. Dartois and S. Bauerecker, I Chem. Phys., 128, 154715 (2008) [7] S. A. Sandford, L. J. Allamandola, A. G. G. M. Tielens, and G. J. 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Travis, Light Scattering by Nonspherical Particles (Academic Press, San Diego, 2000) [55] K. K. Kelley, USBur. Mines, 383, 34 (1934) 168 6 Conclusions 6.1 Summary and discussion The influence of intrinsic particle properties (size, shape, architecture, phase) on mid- infrared spectra of small molecular aggregates in the size range of 1-100 nm was systematically investigated within this work for several model systems. The vibrational infrared spectra of these particles contain a wealth of information on their intrinsic properties. However, due to the high complexity of these spectra, the use of theoretical models is crucial in unraveling this information from the spectra. The only quantum mechanical molecular model that can currently be used to model infrared spectra of small molecular aggregates from the sub-nanometer region up to sizes comparable to the wavelength of the probing infrared light (‘100 nm / molecules) is the vibrational exciton model [1-6, chapters 2-5]. Of particular importance is that the calculated spectra are not fits, but predictions based on intra- and intermolecular interactions. Resonant transition dipole coupling turns out to be the dominant interaction that leads to a strong influence of intrinsic particle properties on infrared spectroscopic features. In other words, only strong infrared bands (molecular transition dipoles> 0.1-0.2 D) are sensitive to these properties — weak bands do not contain information on intrinsic particle properties. Chapters 2-5 demonstrate how the vibrational extended exciton model can be used to identif’ and differentiate between the influences of different particle properties on infrared spectra. 6.1.1 Differentiation between shape and phase effects Large particles, defined here as 10-100 nm in size (-10-10 molecules), make the most suitable starting point for the investigation of intrinsic particle properties. Dominated only by 169 shape and phase effects, these two can be differentiated and studied combining experimental information with predictions made by the exciton model. In the case of CHF3 (see chapter 4), an easy differentiation between phase and shape effects was possible by comparing the behavior of strong and weak infrared bands. In this case, phase effects affect the structure of both band types while only strong bands are sensitive to the particle’s shape because of exciton coupling. A completely different situation is found for SF6 particles (chapter 3). SF6 particles undergo a phase transition from cubic to monoclinic around 96 K, yet no change in the structures of the infrared bands was observed in experiments [7, chapter 3]. Exciton calculations reveal that both the cubic and monoclinic phase lead to almost identical spectra regardless of the phase. As an important result we thus find that in spite of the absence of changes in the experimental spectra as a function of temperature there can still be a phase transition of the particles around 96K. The microscopic origin of this behavior is also explained as follows: SF6 is spherically symmetric with respect to exciton coupling and therefore the molecular orientation plays no role in the coupling scheme. In addition, different positions of the molecules in the crystal lead only to a slight broadening of the spectra (section 3.3.1). Since the two phases differ only by the molecular orientation and position of the molecules they cannot be distinguished. Different particle shapes, by contrast, can be distinguished for SF6 particles, similar to CHF3 particles. 6.1.2 General shape effects From the systems studied in this work and previous studies [1-5,8], system independent trends can be established for shape effects in infrared spectra of particles composed of a single substance. While small differences in shape are difficult to distinguish, major changes have distinct spectral signatures. For example spheres, cubes or elongated particles can be distinguished clearly. Small changes in the particle shape however, lead to very similar spectra. 170 The spectrum of a cube for example, will look very similar to that of a cuboctahedron obtained by cutting off its corners at 10% of the space diagonals. Similarly, spectra of different elongated particles can only be distinguished if the axis ratio differs markedly, for example 1:1:3 vs. 1:1:6. Characteristic band shapes uncovered so far are: a single symmetrical peak for spheres for substances with a cubic crystal structure. This can change for crystals with lower symmetry, such as CHF3,for which the monoclinic structure leads to two peaks for both the doubly degenerate v5 and the non-degenerate v2 modes. Equal axis ratio particles with edges and corners, such as cubes or cuboctahedra, display a main peak accompanied by a shoulder at higher frequency (chapters 3-5). Elongated particles show shoulders at higher and lower frequencies on both sides of a main peak. 6.1.3 Time evolution of the particle shape The infrared spectra of these large aggregates show a characteristic, more or less system independent time evolution. The exciton model has revealed that this can be attributed to a temporal change of the particle shape. Based on the information summarized above, the general trend is as follows: all particles start out with a ‘globular’ shape, i.e. a shape with equal axis ratio, possibly with edges and corners. The same characteristic main peak with a high frequency shoulder is observed for all systems. Then, a gradual transformation into elongated particles takes place over time (sections 3.3 and 4.3.1), which is characterized by pronounced band shoulders both at higher and lower frequency. The mechanism behind the shape change remains, however, unclear. Evaporation of small particles in the ensemble and recondensation onto large particles, agglomeration or coagulation of particles are all plausible explanations. 171 6.1.4 General phase effects: crystalline and amorphous particles In contrast to crystalline particles, amorphous particles are unlikely to have corners or edges. They form more or less spherical particles. For this reason, amorphous particles do not show shape effects in infrared spectra. Spectra of amorphous particles can easily be distinguished from crystalline particles due to the very broad shapes. With increasing amorphous percentage, partially amorphous particles show broader spectra and a gradual loss of information on intrinsic particle properties. 6.1.5 Size and surface effects The spectral analysis increases in complexity for small particles (< 10 nm / -i04 molecules), for which in addition to shape and phase, size and surface effects play an important role. With the knowledge gained from large particles on the influence of shape and phase on vibrational spectra, size and surface effects can be identified and studied. The effect of the dipole coupling, which is inversely proportional to the cube of the distance between coupling molecules, is not yet converged as a function of size for small particles. Thus, for each combination of shape and size a unique spectrum is observed, as discussed for the example of CO in section 5.3.1.1. Nowadays, infrared spectra of small particles can only be recorded for particle ensembles because of the sensitivity. This however, means that particle sizes are distributed over a certain range. Since spectra of small particles strongly depend on the size, size distributions have to be taken into account. In contrast to size, the distribution over different shapes is often narrower in the experiment, so that shape distributions do not have to be taken into account. The exciton calculations show that for particle ensembles with size distributions the size specific spectral 172 features overlap resulting in broader, less detailed bands and often only information on the shape but not the size is retained in the spectra. Surface effects appear more pronounced with decreasing particle size because the surface contributions become more and more important. Surface effects were studied for the example of NH3 in chapter 2, where it was found that those particles are composed of a crystalline core surrounded by an amorphous shell of (size independent) constant thickness. With decreasing particle size the amorphous shell begins to dominate the spectrum in the region of the umbrella vibration. A similar architecture was also found for CO particles (section 5.3.1.2). Such a core-shell architecture seems plausible for particles formed quickly at sufficiently low temperatures. The crystalline core is formed by annealing, using part of the energy released as the particle grows through condensation. The surface stays amorphous because of the lack of energy to rearrange itself. It remains to be established whether the formation of such core-shell particles is a general phenomenon for particles formed by collisional cooling. 6.1.6 General comments on the computational approaches used In highly symmetric crystalline environments molecules are equivalent and such systems are well described using the same transition frequency and transition dipole for all molecules as input for the exciton model. When molecules are no longer equivalent, each experiencing a different local environment, e.g. in the amorphous phase or at the surface of particles, local transition frequencies and transition dipoles have been derived using an explicit potential and dipole function. When an explicit potential function is available, one could envision an alternative approach to the vibrational exciton model for the computation of particle infrared spectra. The infrared spectra could be obtained by calculating the dipole correlation function 173 using classical molecular dynamics (MD). The major drawbacks of this method are the computational effort involved, as well as the accuracy of absolute intensities and frequencies. Whereas in the exciton model only the vibrational modes of interest are considered, the classical method treats all 3N molecular degrees of freedom and requires certain corrections for quantum phenomena [9]. We have attempted to calculate the spectra for particles composed of NH3 by MD. However, even taking full advantage of various enhancements to MD such as multiple time step integration [10-12] or freezing vibrational or lattice modes [13], the largest tractable number of molecules per particle was still an order of magnitude below the systems investigated in this work. The high complexity of the systems studied limits the choice in potentials to effective pair potentials. Limited to a small number of parameters, e.g. a total of 10 for NH3 (section 2.3.1) or 13 for CO (section 5.2.2), a potential’s performance may vary depending on the task and must be tested carefully before application for spectroscopic purposes. For example, a potential parameterized to give a good description of the thermodynamic properties of a substance may give a poor description of vibrational frequencies. Even if the potentials are simple, it is vital that a good and consistent description of structural and vibrational properties be derived from independent molecular properties alone and not from a fit to the quantities that are modeled. As an example for the failure of the latter approach, we mention the ad hoc adjustment to the NH bond length in the intramolecular potential of NH3 in reference [14]. This slight change of the bond length corrects the bending frequencies in the liquid and works in that specific context, but it does not provide a consistent description of the vibrational behavior of NH3 from the gas phase to the particulate state. In the present thesis by contrast the potential was fitted to independent molecular properties so that the particle spectra calculated with the exciton model are still true 174 predictions. In the same spirit, the dipole function is directly derived from the potential function in the present thesis. This allows for another independent check using the calculated intensities. 6.2 Outlook The investigation of intrinsic particle properties in this thesis and in the literature [1- 6,15-21] is mainly focused on pure substances. Many molecular aggregates of atmospheric and interstellar relevance, however, are composed of several molecular species. One such example are interstellar clouds, where newly formed gas phase molecules are expected to freeze onto icy grains immediately upon contact, due to the low temperature in these environments (T < 30 K) [22]. The mantles formed around the grains in this process depend in composition on the hydrogen abundance in the cloud. In environments rich in H or H2, the molecules formed are hydrides such as H20, NH3 or CH4. At low hydrogen abundances, reactive species such as 0 and N will form 02 and N2 molecules. The complexity of such core-mantle particles is further increased by photochemical processes in the mantle leading to the formation of new molecular species. Therefore, one of the most important future goals will be to study such multi-component particles with vibrational exciton and MD approaches. The few exciton calculations performed so far for binary mixtures (chapter 3, refs. [1,4-6]) have already shown that for statistically mixed (homogeneously mixed on a molecular level) particles the information on intrinsic properties is lost due to the increasing distance between (near) resonant oscillators. This contrasts with two component core-shell particles for which characteristic spectral features have been identified (chapter 3, refs. [1,4,5,8]). However, this still leaves a large variety of multi-component particles for study. Layered or shell (onion like) architectures with more than two molecule types, inclusions of one type of molecules in the matrix of another type of molecules, or statistical 175 mixtures where the different molecular species happen to have (near) resonant vibrations are very interesting candidates. A major challenge in studying the vibrational dynamics of such multi-component particles will be the generation of physically plausible structures. For that purpose suitable potential functions will be required, which are computationally efficient, but still describe interactions between all types of molecules properly. For such complex systems a full characterization may also exceed the accessibility by the current implementation of the exciton model. The main limitation in calculating the particles’ infrared spectra is currently the memory storage requirement for the Hamiltonian matrix. One possible solution to this problem would be to alter the numerical implementation such that the Hamiltonian is calculated on the fly. As a direct extension of the current work and given that CO is one of the most abundant molecular species in interstellar space, mixtures of CO with polar (H20, NH3) and apolar (N2, 02, CO2) types of ice would make an excellent starting point, as these have already been studied experimentally [23-25]. 176 6.3 References [1] R. Signorell and M. K. Kunzmann, Chem. Phys. Lett., 371, 260 (2003) [2] M. Jetzki, A. Bonnamy, and R. Signorell, J. Chem. Phys., 120, 11775 (2004) [3] A. Bonnamy, R. Georges, E. Hugo, and R. Signorell, Phys. Chem. Chem. Phys., 7, 963 (2005) [4] R. Signorell, M. Jetzki, M. Kunzmann, and R. Ueberschaer, I Phys. Chem. A, 110, 2890 (2006) [5] R. Signorell and M. Jetzki, Faraday Discuss., 137, 51(2008) [6] 0. Sigurbjornsson, G. Firanescu, R. Signorell, Annu. Rev. Chem. Phys., 60, 127 (2009) [7] T. E. Gough and T. Wang, I Chem. 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