Glass transition in charged colloidal suspensions

Volltext

(1)Dissertation zur Erlangung des akademischen Grades des. Doktors der Naturwissenschaften (Dr. rer. nat.) vorgelegt an der U NIVERSITÄT KONSTANZ. Mathematisch-Naturwissenschaftliche Sektion Fachbereich Physik von. Herbert Kaiser. Glass transition in charged colloidal suspensions. Konstanz, Oktober 2016. Tag der mündlichen Prüfung: 12.10.2016 1. Referent: Prof. Dr. Georg Maret. 2. Referent: Prof. Dr. Matthias Fuchs. Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-0-372356.

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(3) Contents Introduction. 1. 1. Theoretical Background 1.1 Charged colloids . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Dynamic and static correlation functions . . . . . . . . . . . . . . . 1.2.1 Particle density . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Van Hove function and radial distribution function . . . . . 1.2.3 Intermediate scattering function and structure factor . . . . 1.2.4 Direct theoretical calculation of 𝑆(𝑞) and 𝑔(𝑟): MPB-RMSA 1.2.5 Mean squared displacement . . . . . . . . . . . . . . . . . 1.2.6 Connection to light scattering experiments . . . . . . . . . 1.3 Mode coupling theory . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Mori-Zwanzig formalism . . . . . . . . . . . . . . . . . . . 1.3.2 Mode-coupling equations . . . . . . . . . . . . . . . . . . . Physical picture . . . . . . . . . . . . . . . . . . . . . . 1.3.3 MCT features and predictions for the glass transition . . . . Power law divergence of the 𝛼-relaxation time . . . . . . 𝛼-relaxation: Time temperature superposition principle . 𝛽-relaxation: Master function and scaling laws . . . . . Power law for the 𝛽-relaxation time . . . . . . . . . . . Approximation of the structure factor by a single peak . 1.4 Definitions for binary mixtures and multicomponent systems . . . .. 2. Results from mode coupling theory 2.1 Critical parameters and features at MCT transition lines . . . . . . . . . . . . . . . . 2.1.1 Computation of the transition lines . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Ideal MCT Glass transition lines . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Critical non-ergodicity parameters and critical structure . . . . . . . . . . . 2.1.4 Critical localization lengths and critical exponent parameters . . . . . . . . . 2.2 Dynamics near the MCT glass transition . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Computation of density correlators and mean squared displacments . . . . . 2.2.2 Density correlators and mean squared displacements near the glass transition 2.2.3 Time temperature superposition principle and factorization law . . . . . . . 2.2.4 MCT power laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Realistic colloidal systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Unreachable transition line . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Reentrant transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Phase diagrams for varying the salt concentration 𝑐11 . . . . . . . . . . . . . 2.3.4 Power laws for varying the salt concentration 𝑐11 . . . . . . . . . . . . . . . 2.4 Binary systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Monte Carlo simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Glass transition in binary mixtures . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. 7 7 9 9 9 11 13 15 16 17 17 19 21 22 22 22 23 24 24 24. . . . . . . . . . . . . . . . . . .. 27 28 28 28 32 35 37 37 37 38 40 42 42 44 45 45 46 47 49 i.

(4) Contents 2.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55. 3. Particle detection 3.1 Detection of particles according to Crocker and Grier . . . . . . . . . . 3.1.1 Background and noise reduction . . . . . . . . . . . . . . . . . 3.1.2 Sub-pixel precision . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Summary and critique . . . . . . . . . . . . . . . . . . . . . . 3.2 Detection of particles with SIFT . . . . . . . . . . . . . . . . . . . . . 3.2.1 Scale space and difference of Gaussians . . . . . . . . . . . . . 3.2.2 Detection and localization of local minima . . . . . . . . . . . 3.2.3 Rejection of bad candidates . . . . . . . . . . . . . . . . . . . Depth of the DoG minimum . . . . . . . . . . . . . . . . . Curvature in DoG space . . . . . . . . . . . . . . . . . . . Relative intensity criteria, noise removal . . . . . . . . . . . Overlap removal . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Sizing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Different resolution in 𝑧 direction, inflation . . . . . . . . . . . 3.2.6 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Tests of the SIFT algorithm . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Tests on simulated images . . . . . . . . . . . . . . . . . . . . Radius determination of particles on a grid . . . . . . . . . Radius determination of particles on random positions . . . Positioning accuracy in dense particle distributions . . . . . 3.3.2 Tests on a rigid sample . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Comparison to Crocker and Grier’s method . . . . . . . . . . . 3.4 Finite exposure time problem . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Positioning error for moving particles: Brownian motion . . . . Larger error in 3D . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Recomputing 𝑥 and 𝑦 coordinates from the 2D slices . . . . . . 3.4.3 Application to a binary system with glassy dynamics . . . . . . 3.4.4 Is the 2D exposure time short enough for MSD measurements? . 3.4.5 How to do a correction for the 𝑧 direction? . . . . . . . . . . . 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4. Particle tracking 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Ambiguities using Crocker and Grier’s method in dense systems 4.2.1 Crocker and Grier’s algorithm . . . . . . . . . . . . . . 4.2.2 Violation of time-reversal symmetry . . . . . . . . . . . 4.2.3 Tracking the small particles of a binary mixture . . . . . 4.3 Tracking according to Karrenbauer and Wöll . . . . . . . . . . . 4.3.1 Linear optimization problem . . . . . . . . . . . . . . . 4.3.2 Cost function . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Joining . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Implementation . . . . . . . . . . . . . . . . . . . . . . 4.3.5 Tracking the small particles of a binary mixture . . . . . 4.4 Iterative tracking . . . . . . . . . . . . . . . . . . . . . . . . .. ii. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 57 58 58 60 61 62 62 65 66 66 66 67 67 68 68 69 70 70 71 72 72 74 77 80 80 81 82 82 85 87 88. . . . . . . . . . . . .. 91 92 93 93 93 95 96 96 99 100 100 101 101.

(5) Contents 4.4.1 4.4.2. 4.5 5. Multipass tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Track joining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Virtual track endings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Search for connectable non-overlapping tracks . . . . . . . . . . . . . . . Search for connectable overlapping tracks . . . . . . . . . . . . . . . . . . Search for conflicts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cost function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modifications to the optimization step: Conflicting links and “eating” links Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Iterative tracking procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . First step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . High flexibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Iterative tracking of small particles in a binary mixture . . . . . . . . . . . . . Comparison of iterative tracking to single pass tracking . . . . . . . . . . . . . . . . .. Confocal microscopy: Experimental details 5.1 Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 PMMA particles . . . . . . . . . . . . . . . . . . . . . . . Size determination . . . . . . . . . . . . . . . . . . . . Incorporation of rhodamine . . . . . . . . . . . . . . . Regrafting the stabilizer . . . . . . . . . . . . . . . . . Storage, washing and drying of particles . . . . . . . . . Origin of particle charges . . . . . . . . . . . . . . . . . Micro-electrophoresis, estimation of the charge number . 5.1.2 Solvents . . . . . . . . . . . . . . . . . . . . . . . . . . . . Purification of CHB . . . . . . . . . . . . . . . . . . . Conductivity of the solvents . . . . . . . . . . . . . . . Sample preparation and density matching . . . . . . . . Density difference of small and big particles . . . . . . . 5.1.3 Sample cells . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Confocal Microscope . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 System parts . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Recording images . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Image quality . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Experimental issues and challenges . . . . . . . . . . . . . . . . . . 5.3.1 Aggregation, unstable samples . . . . . . . . . . . . . . . . 5.3.2 Mixing particles of different particle types . . . . . . . . . . 5.3.3 Non-controllability of particle interactions . . . . . . . . . . 5.3.4 Dynamic instability: Drift . . . . . . . . . . . . . . . . . . Computation of collective particle drift . . . . . . . . . Periodic drift in long-time-measurements . . . . . . . . Abandoning measurements due to inhomogeneous drift . 5.4 Searching for the glass . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Crystallization in charged monodisperse samples . . . . . . Criteria for crystallinity . . . . . . . . . . . . . . . . . 5.4.2 Monodisperse glasses? . . . . . . . . . . . . . . . . . . . . 5.4.3 Binary mixtures . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. 103 104 104 104 104 105 106 106 107 108 108 108 109 110. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 115 116 116 117 117 118 118 118 118 119 120 120 121 121 122 123 124 125 127 127 127 128 129 130 131 131 132 132 132 133 135 137 iii.

(6) Contents. 5.5. Relation between charge number and size of the colloids . . . . . . . . . . . 140 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141. 6. Confocal microscopy: Results and discussion 6.1 Monodisperse systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 General comment on comparisons to MCT . . . . . . . . . . . . . . . . . . 6.1.2 Separation from the glass transition, mean squared displacement, localization 6.1.3 Structure of glassy systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.4 Comparison to MCT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Localization length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mean squared displacement - MCT error estimation . . . . . . . . . . . Density correlators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.5 Glassy dynamics, dynamical heterogeneity . . . . . . . . . . . . . . . . . . Individual localization lengths and particle trajectories . . . . . . . . . . Mobile and less mobile regions . . . . . . . . . . . . . . . . . . . . . . Mobility and local density . . . . . . . . . . . . . . . . . . . . . . . . . Mobility and local structure . . . . . . . . . . . . . . . . . . . . . . . . Correlation between mobility and crystallinity? . . . . . . . . . . . . . . 6.2 Binary systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Separation from the glass transition, mean squared displacement . . . . . . . 6.2.2 Structure of the systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Comparison to MCT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Localization length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mean squared displacement . . . . . . . . . . . . . . . . . . . . . . . . Density correlators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Glassy dynamics, dynamical heterogeneity . . . . . . . . . . . . . . . . . . Mobile and less mobile regions . . . . . . . . . . . . . . . . . . . . . . Time evolution of mobile regions . . . . . . . . . . . . . . . . . . . . . Comparison to a hard sphere binary system . . . . . . . . . . . . . . . . Van-Hove functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mobility and local density . . . . . . . . . . . . . . . . . . . . . . . . . Correlation between mobility and local structure . . . . . . . . . . . . . Other systems, other correlations? . . . . . . . . . . . . . . . . . . . . . 6.2.5 Aging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 143 143 143 144 147 148 149 151 152 153 153 154 155 156 158 160 160 161 163 163 165 165 167 167 167 169 169 171 172 174 177 181. 7. DWS: Theory and experimental methods 7.1 Theoretical background . . . . . . . . . . . . . . . . . . . . 7.1.1 General form of the autocorrelation function . . . . Summary of approximations . . . . . . . . . . . 7.1.2 Diffusion equation and path length distribution 𝑃 (𝑠) 7.1.3 Transmission geometry . . . . . . . . . . . . . . . . 7.1.4 Backscattering geometry . . . . . . . . . . . . . . . 7.1.5 Mixtures with different particle sizes . . . . . . . . . 7.2 Dynamic light scattering by non-ergodic media . . . . . . . 7.2.1 Siegert relation . . . . . . . . . . . . . . . . . . . . 7.2.2 What happens in the non-ergodic case? . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. 183 184 184 187 188 188 189 191 191 192 192. iv. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . ..

(7) Contents. 7.3. 8. 7.2.3 Replacement for the Siegert relation . . . . . . . . . . . . . 7.2.4 Properties of the time averaged correlation function 𝑔2 (𝑡) . . 7.2.5 Caveats in measurements of 𝑔2 (𝑡) . . . . . . . . . . . . . . Experiments and methods . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Light scattering setup . . . . . . . . . . . . . . . . . . . . . Incoming beam . . . . . . . . . . . . . . . . . . . . . . Sample cell . . . . . . . . . . . . . . . . . . . . . . . . Single speckle measurements . . . . . . . . . . . . . . Multi speckle measurements . . . . . . . . . . . . . . . 7.3.2 PS particles . . . . . . . . . . . . . . . . . . . . . . . . . . Characterization . . . . . . . . . . . . . . . . . . . . . Sample preparation – Deionization . . . . . . . . . . . . Determination of 𝛾 . . . . . . . . . . . . . . . . . . . . 7.3.3 Brute force measurements on non-ergodic systems . . . . . 7.3.4 Multi speckle spectroscopy . . . . . . . . . . . . . . . . . . Correction for non-uniform illumination and noise . . . Determination of the coherence factor 𝛽 . . . . . . . . . Measuring at different time scales simultaneously . . . . Correction for instabilities exploiting the memory effect 7.3.5 Time resolved correlation measurements (TRC) . . . . . . . 7.3.6 Speckle visibility spectroscopy . . . . . . . . . . . . . . . . Speckle visibility and speckle contrast . . . . . . . . . . Contribution of noise to the speckle visibility . . . . . . Varying the exposure time . . . . . . . . . . . . . . . . Long-time speckle visibility measurements . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. DWS: Results and discussion 8.1 Finding the glass in binary mixtures . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Attempts with monodisperse systems . . . . . . . . . . . . . . . . . . . . 8.1.2 Glassy binary mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.3 Glasses at 3 % volume fraction . . . . . . . . . . . . . . . . . . . . . . . . 8.1.4 Glassy binary mixtures at 5 % volume fraction . . . . . . . . . . . . . . . . 8.2 Comparison to results from mode coupling theory . . . . . . . . . . . . . . . . . . 8.2.1 Fitting 𝑔1 (𝑡) with theoretical curves . . . . . . . . . . . . . . . . . . . . . 8.2.2 Localization lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 𝛽-relaxation times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Aging of a charged colloidal glass . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Aging time-time superposition and power law for the 𝛼-relaxation time . . 8.3.2 Measuring the growth of the correlation – Obtaining laws in a different way 8.3.3 Visibility spectroscopy – Influence of aging on the 𝛽-relaxation . . . . . . 8.4 Melting a charged colloidal glass across the transition . . . . . . . . . . . . . . . . 8.4.1 Aging and melting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Melting – Time temperature superposition . . . . . . . . . . . . . . . . . . 8.4.3 Power law for the 𝛼-relaxation time? . . . . . . . . . . . . . . . . . . . . . 8.4.4 Decrease of the correlation – Linear increase of the salt concentration . . . 8.4.5 TRC validation of 𝜏𝛼 power law and master function . . . . . . . . . . . . 8.4.6 MCT square-root law for the plateau heights? . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. 193 194 194 196 196 196 197 197 198 198 199 200 200 201 202 203 205 205 206 209 210 210 211 211 212. . . . . . . . . . . . . . . . . . . . .. 215 216 216 216 217 217 218 218 220 222 223 224 226 227 229 230 231 232 234 236 237 v.

(8) Contents. 8.5. 8.4.7 Short time dynamics from visibility spectroscopy . . . . . . . . . . . . . . . . . 238 8.4.8 Attempt to control the salt concentration in the sample cell . . . . . . . . . . . . 240 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242. Conclusions and outlook. 243. Zusammenfassung. 247. A Formulas for the particle detection algorithm 251 A.1 Differentation with Savitzky-Golay filter coefficients . . . . . . . . . . . . . . . . . . . 251 A.2 Difference of Gaussians for a perfect sphere . . . . . . . . . . . . . . . . . . . . . . . . 251 B Brownian motion in a harmonic potential B.1 Theoretic model . . . . . . . . . . . . . . . . . . . . . . . B.2 Measured position versus real position and its distribution . B.3 Measured MSD . . . . . . . . . . . . . . . . . . . . . . . B.4 Free particles . . . . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 255 255 256 256 257. C Computation of important physical quantities from particle positions C.1 Pair distribution function 𝑔(𝑟) . . . . . . . . . . . . . . . . . . . . . C.2 Structure factor 𝑆(𝑞) . . . . . . . . . . . . . . . . . . . . . . . . . C.3 Mean squared displacement . . . . . . . . . . . . . . . . . . . . . . C.4 Coherent intermediate scattering function 𝐹 (𝑞, 𝑡) . . . . . . . . . . C.5 Self intermediate scattering function 𝐹𝑠 (𝑞, 𝑡) . . . . . . . . . . . . . C.6 Self-part of the van Hove function 𝐺𝑠 (𝑟, 𝑡) . . . . . . . . . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. 259 259 261 264 264 265 265. D Useful knowledge for multi-speckle DWS D.1 Finite exposure time . . . . . . . . . . . D.1.1 Zero lag time – speckle visibility D.1.2 Arbitrary lag time . . . . . . . . D.2 Discrete Fourier transform . . . . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 267 267 267 268 269. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. Bibliography. 271. Danksagung. 281. vi.

(9) „Es ist nicht das Wissen, sondern das Lernen, nicht das Besitzen, sondern das Erwerben, nicht das Dasein, sondern das Hinkommen, was den größten Genuss gewährt.“ Johann Carl Friedrich Gauß (1777 - 1855).

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(11) Introduction Glass plays an important role in our daily live. This becomes most obvious by its use for windows and for the storage of liquids, but also for optics like in eye-glasses, cameras, microscopes or telescopes. Without glass we would still live in houses that lack real windows and we would have no idea about things that are too small to be seen by the naked eye. Historians believe that the first man-made glass dates back to about 3500 BC [1, 2]. At that time it was artificially produced in Egypt and Eastern Mesopotamia and used for glazes on pots and vases. Since then, and especially in the last hundred years, production processes have developed rapidly and today glasses are available for a huge variety of applications. Within the scientific definition of glass as a “non-crystalline amorphous solid” many materials are included. Not only the well known silicate glass, which is used for tableware and windows. Most plastics, like PET (polyethylene terephthalate) used for food and drink packaging or PC (polycarbonate) used for housings of electronic devices, are organic polymer glasses. This in mind, one can truly say that glasses are almost everywhere in our live. However, despite their importance the physics of glasses is far from being understood. Only some years ago New York Times author K. Chang summarized the situation in his title: “The Nature of Glass Remains Anything but Clear” [3]. From a physical point of view glasses are solids with a liquid-like, disordered structure.1 They form by cooling or densification of a liquid when crystallization is avoided. In principle, any liquid can be supercooled into an amorphous solid state as long as the cooling is fast enough.2 This is achieved when the rearrangement processes driving the system to an equilibrium crystalline state become so slow that the system remains trapped in the amorphous state. For common silicate glasses, cooling rates of 0.1 K∕s are sufficient, but it was only two years ago that a pure metal could be vitrified from a liquid, which required an extraordinary high rate of 1014 K∕s [5].3 Looking at the structure given by the particle positions it is almost impossible to tell whether a system is a liquid or a glass. It is a topic of fundamental research to decide what “almost” means in this context. In a glass there is no long-range spatial order like in a crystalline solid. The most intuitive way to identify a glass is its ability to sustain shear stress. As C. A. Angell [6] stated: “Glass [. . . ] is a liquid that has lost its ability to flow”. Furthermore, unlike in a common phase transition, there is no divergent thermodynamic quantity. For instance, in the typical transition from crystal to liquid there is a certain temperature 𝑇melt , where the system can absorb an amount of energy without temperature increase. This energy called “latent heat” is used to actually melt the crystal. The specific heat capacity, which is the energy required to increase the temperature by a certain amount, therefore diverges at 𝑇melt . But such a divergence is not seen in the vicinity of the glass transition. A common way to define the glass transition is from the rapid increase of the viscosity 𝜂 within a small temperature range. For molecular or atomic glasses a value of 𝜂g = 1012 Pa⋅s is commonly used to define the glass transition temperature 𝑇g . Technically, at such a high viscosity one cannot speak of a liquid any more. A similar idea is to determine dynamical relaxation times (e.g. from the measurement of density fluctuations) and to define systems that exceed a certain threshold to be glassy. Such arbitrary thresholds in characterizing the glass transition are somehow unsatisfying, but lacking a clear divergence one cannot give an exact definition. And even worse, 𝑇g also depends on This actually applies to “structural glasses”. Other types are spin glasses or orientational glasses, where the disorder is not in the structure but in the particle interactions or in the orientation of the molecules. 2 Strictly speaking, this statement is only valid for three and more dimensions. Already in 2D systems the locally favoured structure is the same as that of the crystal, which means that even the highest cooling rates lead to the formation of polycrystals. [4] 3 To realize such large rates the sample was rather tiny (∼ 5 nm), enhancing the cooling by a high surface to volume ratio. 1. 1.

(12) Introduction the history of the system, namely on the cooling rate when going from liquid to glass. Another possibility is to examine the response to an applied shear stress. Elasticity corresponds to the glass, yielding to the liquid phase. But also with this definition the transition point depends on the heating/cooling rate and on the waiting time for the response. There are many theories that try to explain the phenomena at the glass transition. An early approach was the “free volume theory” (1959, [7]) that links the mean free volume accessible to a single diffusing particle to the relaxation times of the system. Adam-Gibbs theory (1965, [8]) connects the temperature dependence of the relaxation with the size of cooperatively rearranging regions. Popular more recent approaches are random first order transition theory (RFOT) [9, 10] and mode coupling theory (MCT) [11, 12]. RFOT is strongly related to the dynamical heterogeneities found in supercooled and glassy systems.4 The system is described as having a mosaic structure, being decomposed into patches each corresponding to a minimum of the free energy. Two transition temperatures are predicted, the first one corresponds to the appearance of well-defined metastable free energy minima, at the second one the system is thermodynamically trapped in a such a minimum (the size of the patches diverges). While RFOT is able to qualitatively describe many of the phenomena at the glass transition, mode coupling theory (MCT) has a very strong (also quantitative) predictive power. In the framework of MCT it is possible to compute the dynamical behaviour, described by quantities such as the time dependent density correlation functions, purely from the static structure of the supercooled or glassy system. The glass transition emerges from tiny, almost unmeasurable changes of the structure. Most important result is the prediction of a dynamical bifurcation. The long time behaviour changes from ergodic to non-ergodic, i.e. from a state where all possible configurations of the system are dynamically accessible to a frozen state where this is not possible any more. The physical picture is the so-called cage effect. In the liquid state all particles can move freely. With increasing particle density more and more particles get trapped in the cage built by their nearest neighbours. Only small movements within this cage are allowed, they correspond to the 𝛽-relaxation. As long as the system is only supercooled, in the long run the cages may break by collective particle movements, which corresponds to the 𝛼-relaxation at longer time scales. In the ideal MCT glass, even these collective movements become impossible. The relaxation time of the system diverges and it is able to transmit applied stresses across the whole system despite its disordered structure. In order to test theories of the glass transition it is important to get as much information as possible about the individual particle movements being the microscopic origins of glassy phenomena. This is very hard to achieve in atomic or molecular glasses because atoms and molecules are too small and their motions are too fast. A great step forward has been made with the introduction of colloidal suspensions as model systems. In the most general sense, colloidal systems are particles or droplets suspended in a continuous medium. The size of these particles corresponds to the mesoscopic scale of 10 nm up to 10 µm. Large colloids are thus observable in microscopes, smaller ones can still be addressed with dynamic laser light scattering techniques. Examples are milk, mud, blood and even more complex bio-materials, but also artificial materials like dispersion paint, toothpaste and many cosmetics. Therefore, the study of colloidal systems also yields valuable contributions for applications in medicine, biology and industry. Commonly used colloidal suspensions in the investigation of phenomena like crystal nucleation and the glass transition are spherical plastic particles. They are dispersed in a solvent that has the same (or at least a very similar) density as the particles, which means they can float freely without feeling gravitation. Being subject to Brownian motion the particles diffuse in the solvent so that the system explores the 4. 2. Dynamical heterogeneities emerge as regions of the system where particles are more mobile compared to other regions..

(13) Introduction available configuration space as the time passes. Compared to atomic systems, “solid” colloidal systems (crystals, glasses) are much softer. This can be explained by an estimation of the shear modulus as binding energy per volume, with the volume being the cubed length scale [13]. Typical binding energies in atomic crystals are in the range of 1 eV. In a colloidal crystal they are not much smaller being around 𝑘B 𝑇 ≈ 1∕40 eV. But the length scale in colloidal systems defined as the mean interparticle distance is given by ∼ 1 µm, while it is a factor of 104 smaller in atomic systems (∼ 1 Å). Therefore, the shear modules in crystalline or glassy colloidal suspensions is about 12 order of magnitudes smaller. This does not only explain the softness of the system but also the possibility to observe density fluctuations at experimentally accessible time scales. Also phase transitions can be investigated at moderate temperatures. Pioneering works on colloidal glasses were done by Pusey and van Megen using monodisperse particles (all having the same radius) with (almost) purely hard core interactions [14]. The only parameter in such a system is the volume fraction Φ occupied by the particles. A transition into the glassy phase was observed upon quenching the system to Φ ≳ 56 %. Later it turned out that a small polydispersity (relative standard deviation of the particle size distribution) of around 7 % is required to prevent crystallization [15]. Dynamic light scattering (DLS) experiments investigating the dynamic density correlation functions revealed a good qualitative and quantitative agreement with MCT [16, 17, 18]. However, one main difference compared to purely theoretical works remains. MCT using theoretically computed structure factors as input underestimates the value for the location of the glass transition yielding Φ ≈ 52 % [11, 19]. Hard core interactions do not occur in atomic or molecular glasses. Charged colloidal suspensions possessing electrical charges at the surface resemble them more closely. In total, the systems are electrically neutral. Like the delocalized electrons in atomic metals the counterions to the colloidal charges are allowed to move more or less freely in the liquid solvent. The resulting particle interactions in charged colloidal glasses are therefore similar to those in metallic glasses. They can be described by a screened Coulomb potential [20, 21]. Besides the density or volume fraction of the colloids and the number of charges on each particle the screening length becomes the third important system parameter. It is possible to decrease the screening length by increasing the concentration of ions with the addition of salt to the solvent. In principle, this allows for an in-situ melting of charged colloidal glasses. At sufficiently high ion concentrations the interactions again become pure hard sphere repulsions. First rheological experiments on charged colloidal glasses were already reported in 1982 by Lindsay and Chaikin [22]. The structure of a glassy phase was first measured in 1989 by Sirota et al. [23] using X-ray diffraction. Light scattering experiments by Meller and Stavans [24] confirmed the non-ergodic dynamic behaviour in the glassy phase of binary mixtures of charged colloids. Furthermore, they varied the mixing ratio of the two particle species and reported that systems predominantly consisting of one species only exhibit a crystal phase. Glassy samples are rather obtained for an equal particle density of both species. Later works also performed comparisons of experimental results from dynamic light scattering measurements to mode coupling theory [25, 26]. A qualitative agreement was confirmed, however, the results were restricted to a small number of different system parameters, all being in a moderately charged regime where the glass transition is observed around Φ = 20 %. Furthermore, there are also some purely theoretical works considering the glass transition in charged colloidal suspensions. Lai et al. [27] and Yazdi et al. [28] computed the ideal MCT transition lines providing predictions for charge numbers and screening lengths expected in monodisperse glassy systems. However, numerical simulations confirmed that relatively high charge polydispersities ( 20 %) are required to avoid crystallization [29]. Simulations could confirm and further investigate the glassy phase in systems with screened coulomb potentials [30, 31, 32]. A more recent work also showed qualitative agreement with MCT and RFOT predictions in a simulation of a binary charged system [33]. 3.

(14) Introduction Within this thesis two qualitatively different charged colloidal systems were established, both allowing the study of glassy behaviour. Moderately charged PMMA (polymethyl methacrylate) particles in a mixture of CHB (cyclohexyl bromide) and decaline solvents provide samples where the particle motions can be observed directly in 3D in a confocal microscope. A complementary system of highly charged PS (polysterene) particles in water provides glassy systems at very low volume fractions (Φ ≲ 5 %). The dynamics is accessible in diffusing wave spectroscopy (DWS) measurements, which proves to be a powerful technique able to address a range of time scales covering up to 10 orders of magnitude. Particle trajectories measured in the PMMA system enable the determination of structural and dynamic properties and allow for real quantitative comparisons to mode coupling theory. Further investigations address the connection of dynamical heterogeneities to the structure of the system. Additionally, the phenomena of ageing is studied for a system deep in the glassy phase. In order to compare results from the DWS measurements on the PS system to MCT, Monte Carlo simulation were performed to obtain the static structure factors of those binary systems. The latter were used as input to MCT for the computation of correlation functions describing the dynamics of the system that is experimentally determined with DWS. Unprecedented measurements of a sample during the melting from glass to liquid make it possible to do even more detailed comparisons to MCT. Studies of an aging sample cover the growth of the structural relaxation by four orders of magnitude. Chapter 1 introduces the theoretical model for charged colloidal suspensions and presents definitions and discussions on the structural and dynamical quantities that were measured and computed within this thesis. The derivation of the mode coupling theory equations is summarized together with the most important predictions. Chapter 2 presents purely theoretical MCT results for monodisperse and binary charged systems. The structure factor input to the MCT equations stems from calculations in the modified Poisson-Boltzmann rescaled mean spherical approximation (MPB-RMSA) [34] or from Monte Carlo simulations (in the binary case). Ideal glass transition lines as well as static and dynamical properties of systems in the vicinity of the glass transition are discussed with a focus on comparability to experiments. A special interest is also the influence of the hard core repulsion in weakly charged systems. Chapter 3 deals with the detection of particle positions in 3D and 2D confocal microscopy images. A new algorithm is presented that allows for a reliable determination of the particle sizes. The implementation is tested on simulated and real experimental data discussing the advantages compared to the traditionally used algorithm by Crocker and Grier [35]. The influence of the finite exposure time in capturing microscopy images is elaborated. In Chapter 4 a new iterative method is established to connect the detected positions to complete particle trajectories. In a first step only positions with a low spatial and temporal distance are connected resulting in a large number of short but reliable trajectories. Using a newly invented track joining algorithm the short tracks are further mended to the full trajectories, employing the knowledge of the already existing tracks. This is a highly flexible technique, that can be adapted to produce reliable results in situations where particles are not continuously detectable and the dynamics is heterogeneous. Chapter 5 presents the colloidal system and the experimental setup used for the confocal microscopy experiments. Experimental challenges are discussed as well as the methods used to characterize the samples. The search for samples with glassy behaviour is summarized.. 4.

(15) Introduction In Chapter 6 the results of confocal microscopy measurements on monodisperse and binary glassy systems are discussed in full detail. Comparisons to MCT are given and the appearance of dynamical heterogeneities in supercooled and glassy systems is confirmed. Connections between structure and mobility of the particles are investigated and results of a study of an aging glassy sample are elaborated. Chapter 7 gives an introduction to the theory behind diffusing wave spectroscopy (DWS) and presents the experimental setup as well as the methods to evaluate the data. The focus lies on the measurement of non-ergodic systems and the techniques to obtain ensemble-averaged data of those. In Chapter 8 the results of DWS measurements on highly charged binary colloidal glasses are presented. Comparisons of the short and long time dynamic behaviour to MCT predictions are carried out. Unprecedented studies of an ageing and a melting charged colloidal glass yield new interesting results.. 5.

(16) 6.

(17) Chapter 1. Theoretical Background This chapter starts with an overview of the physical model of charged colloidal suspensions. Some important static and dynamic correlation functions, often used for the description of liquid and glassy systems, are presented and discussed in more detail. Then follows an introduction to mode coupling theory (MCT) including its most important predictions for systems near the glass transition. Important details for the discussion of mixtures consisting of two and more particle species are given in the end.. Contents 1.1 1.2. 1.3. 1.4. 1.1. Charged colloids . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamic and static correlation functions . . . . . . . . . . . . . 1.2.1 Particle density . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Van Hove function and radial distribution function . . . . . 1.2.3 Intermediate scattering function and structure factor . . . . 1.2.4 Direct theoretical calculation of 𝑆(𝑞) and 𝑔(𝑟): MPB-RMSA 1.2.5 Mean squared displacement . . . . . . . . . . . . . . . . . 1.2.6 Connection to light scattering experiments . . . . . . . . . Mode coupling theory . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Mori-Zwanzig formalism . . . . . . . . . . . . . . . . . . . 1.3.2 Mode-coupling equations . . . . . . . . . . . . . . . . . . . 1.3.3 MCT features and predictions for the glass transition . . . . Definitions for binary mixtures and multicomponent systems . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. 7 9 9 9 11 13 15 16 17 17 19 22 24. Charged colloids. It is well known that most colloidal particles are electrically charged, either by adsoprtion of ions from the solvent or by dissociation of ionizable groups on their surface [36]. An important example for charged colloids are polysterene particles (PS). They posses carboxylic (-COOH) and sulfate (-SO4 H) groups, of which many will readily give away an H+ ion when getting in contact with water, leading to a net negative charge of the surface. The hydrogen ions act as counterions, together with ions from the autodissociation of water (H2 O ⇄ H+ +OH− ) and any additional salt ions. They screen the colloid charges, so that beyond the so-called screening length (𝜅 −1 ) the particles do not experience repulsions from each other any more (see Figure 1.1). This picture is known by the name “electrical double layer” consisting of the charges on the surface and the screening ions around it. Besides the electric repulsion there is also the short-range Van der Waals attraction stemming from the interplay of dipoles in the dielectric media of particles and solvent [37]. In strongly repulsive systems the particles are never close enough to feel the Van der Waals forces, they can usually be neglected. In less repulsive systems one usually adds a steric stabilizer in order to prevent aggregation, but even there the short range of the Van der Waals forces makes them negligible as long as the steric stabilizer works well enough and the difference in the dielectric constants of solvent and particles is not particularly large. 7.

(18) Chapter 1: Theoretical Background. +. +. - - +- - 1 + + - + + - + - + + - + + - 2 - + + - + + -. 2.0. +. - - + - 3 - -1+ - +. +. -. Double-layer repulsion. 1.5 pot ent ial energy [ kBT]. +. -. + +. -. +. 1.0 second minimum. 0.5. 0.0 first minimum. 0.5. +. 0.0. van der Waals attraction. 1.5 0.5 1.0 2.0 surface dist ance of t he spheres [ µm ]. 2.5. Figure 1.1: Left panel: Model of the electrical double layer for negatively charged colloids of diameter 𝜎 with a screening length 𝜅 −1 in the solvent. Ions on the colloid surfaces and their counterions are blue, salt ions are red. Colloids 1 and 2 repell each other, while colloid 3 is too far off to feel the others. Right panel: Curves calculated using the DLVO potential as given in [37]. The full DLVO potential as a function of the surface distance is shown as the red line. It decomposes into a Van der Waals term and a repelling term from the partially screened Coulomb interaction. The Van der Waals attraction is strongly exaggerated for purpose of illustration. For colloids with sizes in the micrometer range the second minimum is usually smaller than 0.01 𝑘B 𝑇 .. With these arguments one does not need the full DLVO (Derjaguin Landau Verwey Overbeek [20, 21]) pair potential to describe charged colloidal systems. Instead one can simply stick to its repulsive part or Debye-Hückel part, which is often called hard sphere Yukawa (HSY) potential: ⎧∞ ⎪ 𝛽 𝑢(𝑥) = ⎨ 𝑒−𝑘𝑥 ⎪𝛾 𝑥 ⎩. for 𝑥 = 𝑟∕𝜎 < 1. (1.1). for 𝑥 ≥ 1. For this dimensionless form one introduces the dimensionless distance 𝑥, rescaling the real distance 𝑟 with a division by the hard-core diameter 𝜎. Consequently, 𝑘 is the dimensionless screening parameter 𝑘 = 𝜅𝜎. The potential is multiplied by 𝛽 = 1∕(𝑘B 𝑇 ), where 𝑇 is the temperature and 𝑘B Boltzmann’s constant. The coupling constant 𝛾 𝑙 𝛾= B 𝜎. (. 𝑒𝑘∕2 1 + 𝑘∕2. )2. 𝑍ef2 f ,. with. 𝑙B =. 𝛽 𝑒2 4𝜋𝜖0 𝜖. (1.2). is strongly related to the effective charge 𝑍ef f and the Bjerrum length 𝑙B , the distance where the interaction potential between two elementary charges 𝑒 in the medium with dielectric constant 𝜖 equals the thermal energy scale 𝑘B 𝑇 . The screening parameter 𝜅 is an important result of the Debye-Hückel theory, where it is derived from a linearized Poisson-Boltzmann equation. One often refers to 𝜅 −1 as the Debye length of the system. It is given by the ion number densities 𝑛𝑖 and their charge numbers 𝑧𝑖 : 𝑘2 = (𝜅𝜎)2 = 4𝜋𝑙B 𝜎 2. ∑ 𝑖. 𝑛𝑖 𝑧2𝑖 =. ) 𝑙B ∕𝜎 ( 24 Φ |𝑍ef f | + 8𝜋𝑁A 𝑐11 𝜎 3 1−Φ. (1.3). Here we assume monovalent counterions, whose density is defined via the volume fraction Φ of the colloids as 𝑛𝑐 = |𝑍ef f | 6Φ∕(𝜋𝜎 3 ) and we also assume monovalent additional salt ions with a density 8.

(19) 1.2 Dynamic and static correlation functions 𝑛𝑠 = 2𝑐11 𝑁A , where 𝑐11 is the molar concentration of the added electrolyte (e.g. 10 µmol/l K+ Cl− ); 𝑁A is the Avagadro constant. As defined here 𝑘−1 gives the screening length in units of the hard sphere diameter 𝜎, the actual screening length is 𝜅 −1 = 𝜎∕𝑘. The factor (1−Φ) in the computation of 𝑘 accounts for the fact that microions (counterions and salt ions) cannot penetrate macroions (colloids), so that only a fraction (1 − Φ) of the total volume is accessible to them, see [38] for a discussion on that topic. The three main parameters of charged colloidal systems are henceforth the volume fraction, the effective charge and the concentration of added salt (Φ, 𝑍ef f , 𝑐11 ). While Φ can be determined by weighing the ingredients of a sample (e.g. before and after drying) and 𝑐11 e.g. by conductivity measurements the charge is not easy to determine. The number of surface groups on the colloids is not sufficient to know. One cannot tell how many of the groups actually have dissociated and further one does not know how many other ions are adsorbed on the particle surfaces. Both depends on the pH value in the solvent and also on the number density of particles and ions. That’s why the charge in the HSY potential is always an effective charge 𝑍ef f . A good way to measure it is indirectly via the static structure factor, which is either obtained in light scattering experiments or with the particles’ positions determined from confocal microscopy images. The relation between potential and structure will be discussed later in section 1.2.4. For a discussion on the relation between bare surface charges and effective charges see [39].. 1.2. Dynamic and static correlation functions. A very common and convenient way to describe the structure and dynamics in liquids and glasses is by correlation functions in space and time. In this section some definitions are given together with the discussion of some of the features of commonly used correlation functions.. 1.2.1. Particle density. For a given system of 𝑁 particles being at positions 𝑟𝑘 (𝑡) at time 𝑡, one can define the particle density 𝑛(⃗𝑟, 𝑡) using Dirac delta functions: 𝑛(⃗𝑟, 𝑡) =. 𝑁 ∑. 𝛿(⃗𝑟 − 𝑟⃗𝑘 ). 𝑘=1. FT. ←←←←←→ ←. 𝑁 1 ∑ −𝑖𝑞⋅⃗ 𝑟⃗𝑘 (𝑡) 𝑛(𝑞, ⃗ 𝑡) = √ 𝑒 𝑁 𝑘=1. (1.4). √ To the right we define the Fourier transformed version of 𝑛 with an additional division by 𝑁, which is useful for other definitions, as we will see later. The average particle density 𝑛 is given by the integral 𝑛=. 1.2.2. 1 𝑁 𝑛(⃗𝑟, 𝑡) d⃗𝑟 = 𝑉 ∫ 𝑉. (1.5). Van Hove function and radial distribution function. The space and time-dependent version of the density correlation function is the van Hove function. It is actually the probability density to find a particle at time 𝑡 at a distance 𝑟⃗ if there was a particle (not 9.

(20) 1.0. 1.6. t =∞. Gd (r,t =const.)/n. Gs (r,t)/Gs (r =0,t), t =const.. Chapter 1: Theoretical Background. 0.8 0.6 0.4 0.2 0.0. 0. self 1. 2. 3. r. 4. 5. 6. t =10−3 t =0.1 t =0.5 t =10 t =100. 1.4 1.2 1.0 0.8 0.6. distinct 0. 1. 2. 3. r. 4. 5. 6. Figure 1.2: Normalized self and distinct parts of the van Hove function 𝐺𝑠 (𝑟, 𝑡) (left panel) and 𝐺𝑑 (𝑟, 𝑡) (right panel) in a liquid for several constant delay times 𝑡. Normalization as indicated in the 𝑦-axis labels. At 𝑡 = 0.5 a particle has on average diffused its own diameter 𝜎 = 1. Curves are results from mode coupling theory (MCT) calculations on a monodisperse liquid-like charged colloidal suspension.. necessarily the same) at the origin (⃗𝑟 = 0) at time 𝑡 = 0. The definition is as follows: ⟩ ⟨ 𝑁 𝑁 ) 1 ∑∑ ( 𝛿 𝑟⃗ − [⃗𝑟𝑗 (𝑡) − 𝑟⃗𝑘 (0)] 𝐺(⃗𝑟, 𝑡) = 𝑁 𝑗=1 𝑘=1 ⟨ ⟩ 𝑁 𝑁 ) ( ′ ) ′ 1 ∑∑ ( ′ = 𝛿 𝑟⃗ + 𝑟⃗ − 𝑟⃗𝑗 (𝑡) 𝛿 𝑟⃗ − 𝑟⃗𝑘 (0) d⃗𝑟 𝑁 ∫ 𝑗=1 𝑘=1 ⟨ ⟩ ⟩ 1⟨ 1 ′ ′ ′ ⃗ 0) 𝑛(⃗𝑟 + 𝑟⃗, 𝑡) 𝑛(⃗𝑟 , 0) d⃗𝑟 = 𝑛(⃗𝑟, 𝑡) 𝑛(0, = 𝑁 ∫ 𝑁. (1.6). The angular brackets ⟨… ⟩ denote the canonical average which actually involves a full integration over all states in phase space weighted by their probability. In practice, considering measurements in an experiment or a numerical simulation, one would use the time average ⟨𝐴⟩𝑡 if the system is ergodic and/or an average over as many configurations as possible ⟨𝐴⟩𝐸 if the system is non-ergodic: 𝑇. ⟨𝐴⟩𝑡 = lim. 𝑇 →∞ ∫0. 𝐴(𝑡) d𝑡. and/or. 𝑃 1∑ ⟨𝐴⟩𝐸 = lim 𝐴𝑝 𝑝→∞ 𝑃 𝑝=1. (1.7). The van Hove function is often separated into two terms the self and distinct parts 𝐺 = 𝐺𝑠 + 𝐺𝑑 : ⟨ ⟩ ⟨ ⟩ 𝑁 𝑁 𝑁 ) ) 1 ∑ ( 1 ∑∑ ( 𝐺𝑠 (⃗𝑟, 𝑡) = 𝛿 𝑟⃗ − [⃗𝑟𝑗 (𝑡) − 𝑟⃗𝑗 (0)] , 𝐺𝑑 (⃗𝑟, 𝑡) = 𝛿 𝑟⃗ − [⃗𝑟𝑗 (𝑡) − 𝑟⃗𝑘 (0)] (1.8) 𝑁 𝑗=1 𝑁 𝑗=1 𝑘=1 𝑘≠𝑗. Now 𝐺𝑠 (⃗𝑟, 𝑡) d⃗𝑟 gives the probability to find particle 𝑗 in a volume d⃗𝑟 around position 𝑟⃗ given that the same particle 𝑗 was at the origin at time 𝑡 = 0, while 𝐺𝑑 (⃗𝑟, 𝑡) d⃗𝑟 gives the probability to find a different particle 𝑘 at that place. Of particular interest is the delay time 𝑡 = 0: 𝐺𝑠 (⃗𝑟, 0) = 𝛿(⃗𝑟). and. 𝐺𝑑 (⃗𝑟, 0) = 𝑛 𝑔(⃗𝑟). isotropy. =. 𝑛 𝑔(𝑟). (1.9). While the self part simply becomes a 𝛿-function (all particles stay at their position), the distinct part at 10.

(21) 1.2 Dynamic and static correlation functions 𝑡 = 0 gives us the definition of the pair distribution function 𝑔(⃗𝑟). In the isotropic case for liquids and glasses it becomes the radial distribution function 𝑔(𝑟), depending only on the modulus 𝑟 = |⃗𝑟| and not on the direction of 𝑟⃗. Simplifying the statement above, we can say that 𝑛𝑔(𝑟) is the frequency per volume at which a pair of particles with the distance 𝑟 is observed. That’s why also 𝑔(𝑟) is often simply called pair distribution function. The shape of 𝑔(𝑟) gives a very good picture of the distances between particles that occur in the system, e.g. the first peak stems from the distance to the nearest neighbours, the second one to the next-nearest neighbours and so on. For large distances 𝑟 the correlation vanishes so that 𝐺𝑑 (𝑟 → ∞, 0) = 𝑛 and 𝑔(𝑟 → ∞) = 1. Note that the height of the structure peaks in 𝑔(𝑟) does not depend directly on the average particle density 𝑛, but it is obvious that a density increase indirectly leads to a growing of those peaks because the lack of space in denser systems will lead to more ordered particle arrangements. In an ergodic system, where the particles can move freely, the peak in the self part will broaden to a bell shaped curve for times 𝑡 > 0 until it finally becomes a flat line 𝐺𝑠 (⃗𝑟, 𝑡 → ∞) = 1∕𝑉 1 . For all times the area below the non-normalized curve stays constant, because ∫ 𝐺𝑠 (⃗𝑟, 𝑡) d⃗𝑟 = 1 for all 𝑡. The structure peaks in the distinct part 𝐺𝑑 will get washed out more and more with increasing 𝑡 until we have a flat distribution 𝐺𝑑 (⃗𝑟, 𝑡 → ∞) = 𝑛. In the non-ergodic case these limits are never reached and both 𝐺𝑠 and 𝐺𝑑 are stuck somewhere in between.. 1.2.3. Intermediate scattering function and structure factor. The straightforward definition of the intermediate scattering function 𝐹 (𝑞, ⃗ 𝑡) is via the spatial Fourier transform of the van Hove function 𝐹 (𝑞, ⃗ 𝑡) = FT𝑞⃗[𝐺] =. ∫. ⟨ ⟩ ⃗𝑟 𝐺(⃗𝑟, 𝑡)𝑒−𝑖𝑞⋅⃗ d⃗𝑟 = 𝑛(𝑞, ⃗ 𝑡) 𝑛∗ (𝑞, ⃗ 0) ,. (1.10). here we use Equation 1.6 where we can see in the last line that 𝐺(⃗𝑟, 𝑡) is the convolution of the particle density 𝑛(⃗𝑟, 𝑡) with 𝑛(⃗𝑟, 0). Application of the convolution theorem yields the product of 𝑛(𝑞, ⃗ 𝑡) with the complex conjugate of 𝑛(𝑞, ⃗ 0).2 This makes clear why 𝐹 (𝑞, ⃗ 𝑡) is actually the density correlation function. As we will see later, the name intermediate scattering function is related to the signal measured in dynamic light scattering experiments. The separation 𝐹 = 𝐹𝑠 + 𝐹𝑑 into self and distinct parts can be done as well for the intermediate scattering function to obtain the two parts: ⟨ ⟩ 𝑁 1 ∑ −𝑖𝑞⋅[⃗ 𝐹𝑠 (𝑞, ⃗ 𝑡) = FT𝑞⃗[𝐺𝑠 ] = 𝑒 ⃗ 𝑟𝑗 (𝑡)−⃗𝑟𝑗 (0)] 𝑁 𝑗=1 ⟨ ⟩ 𝑁 𝑁 1 ∑ ∑ −𝑖𝑞⋅[⃗ ⃗ 𝑟𝑗 (𝑡)−⃗𝑟𝑘 (0)] 𝐹𝑑 (𝑞, ⃗ 𝑡) = FT𝑞⃗[𝐺𝑑 ] = 𝑒 (1.11) 𝑁 𝑗=1 𝑘=1 𝑘≠𝑗. The self part 𝐹𝑠 is often denoted as the incoherent scattering function or the tagged particle correlator. In light scattering it is the signal one obtains when only a dilute set of tagged particles, uncorrelated with each other, contribute to the signal. Therefore it is a good quantity to describe the average motion of single particles, analogous to the self part of the van Hove function. 1 2. here 𝑉 is the volume of the system the factor 𝑁1 disappears due to our definition of 𝑛(𝑞, ⃗ 𝑡). 11.

(22) Chapter 1: Theoretical Background 1.5. 0.5 0.0. t =0 t =0.1 t =0.5 t =10. 0.5 1.0 0. 5. 10. 15. q. 20. 25. glass critical curve. 0.8. F(q,t)/F(q,t =0). 1.0. Fs/d(q,t =const.). 1.0. self part distinct part. 30. 0.6. liquid. 0.4. strongly supercooled. supercooled. 0.2 0.0. 4. 2. 0. 2. log t. 4. 6. 8. 10. Figure 1.3: Left panel: Intermediate scattering function in an isotropic system for constant lag time 𝑡 split into self and distinct parts. Right panel: Normalized intermediate scattering function for constant scattering vector 𝑞 (𝑡-dependence of distinct and self parts is qualitatively the same). The curves in the left panel correspond to the green curve (liquid) in the right panel and to the van Hove functions in Figure 1.2. Black arrows indicate the location of 𝑡 and 𝑞 values corresponding to the curves in the left and right panel. The average time for a particle to diffuse its own diameter 𝜎 = 1 is 𝑡 = 0.5. Further curves in the right panel show what happens when crossing the idealized MCT glass transition point. All curves are results from MCT calculations on a monodisperse charged colloidal suspension.. Again, the delay time 𝑡 = 0 is of particular interest, it defines the static structure factor: ⟩ ⟨ 𝑁 𝑁 1 ∑ ∑ −𝑖𝑞⋅[⃗ ⃗𝑟 𝑒 ⃗ 𝑟𝑗 −⃗𝑟𝑘 ] = 𝐹𝑠 (𝑞, ⃗ 0) + 𝐹𝑑 (𝑞, ⃗ 0) = 1 + 𝑛 𝑔(⃗𝑟) 𝑒−𝑖𝑞⋅⃗ d⃗𝑟 𝑆(𝑞) ⃗ = 𝐹 (𝑞, ⃗ 0) = ∫ 𝑁. (1.12). 𝑗=1 𝑘=1. Here we used the self and distinct parts of the van Hove functions at 𝑡 = 0 to get the last equality. The 𝛿-function from 𝐺𝑠 yields the base line 1 and the distinct part provides the structure peaks which, in contrast to the radial distribution function 𝑔(𝑟), now scale with the average density 𝑛. In the dilute case 𝑛 → 0 all the peaks vanish and we are left with 𝑆(𝑞) = 1. Similarly one can argue that 𝐹 (𝑞, ⃗ 𝑡) = 𝐹𝑠 (𝑞, ⃗ 𝑡) in the dilute limit. The evolution of 𝐹𝑠 and 𝐹𝑑 in time for an ergodic system can be derived from the Fourier transformed van Hove functions discussed above. Being 1 for all 𝑞⃗ at 𝑡 = 0, for 𝑡 > 0 the self part 𝐹𝑠 develops into a bell shaped curve that goes to 0 for 𝑞 → ∞. The value at 𝑞⃗ = 0 stays constant, it is 𝐹 (𝑞⃗ = 0, 𝑡) = 1 for all 𝑡. The bell shape evolves into a peak at later times until it converges to a 𝛿-function for 𝑡 → ∞. The structure peaks that we have in 𝐹𝑑 for 𝑡 = 0 shrink with increasing time 𝑡 until 𝐹𝑑 (𝑞⃗ ≠ 0, 𝑡 → ∞) = 0. Only at 𝑞⃗ = 0 we still have a 𝛿-peak which is there for all times due to the base line of 𝐺𝑑 (⃗𝑟, 𝑡) with the value 𝑛. In the isotropic case the intermediate scattering function 𝐹 and the self part 𝐹𝑠 only depend on the modulus 𝑞 = |𝑞|. ⃗ Often the intermediate scattering function is divided by the static structure factor to give 𝜙(𝑞, 𝑡) = 𝐹 (𝑞, 𝑡)∕𝐹 (𝑞, 0) = 𝐹 (𝑞, 𝑡)∕𝑆(𝑞) and 𝜙𝑠 (𝑞, 𝑡) = 𝐹𝑠 (𝑞, 𝑡). (1.13). Now both correlators are 1 for 𝑡 → ∞ which is common sense for correlation functions. 𝜙 is often named (coherent) density correlator and 𝜙𝑠 incoherent or tagged particle correlator.. 12.

(23) 1.2 Dynamic and static correlation functions Another simplification in the isotropic case is the reduction of the 3D Fourier transform to a 1D sine transform in the relation between the structure factor 𝑆(𝑞) and the radial distribution function 𝑔(𝑟): ∞. ∞. 4𝜋𝑛 𝑆(𝑞) = 1 + [𝑔(𝑟) − 1] 𝑟 sin(𝑞𝑟) d𝑟 and 𝑞 ∫. 1 𝑔(𝑟) = 1 + 2 [𝑆(𝑞) − 1] 𝑞 sin 𝑞𝑟 d𝑞 2𝜋 𝑛𝑟 ∫. 0. (1.14). 0. The subtraction of 1 from both 𝑔(𝑟) and 𝑆(𝑞) is necessary to remove the baseline that otherwise would produce a 𝛿-function causing unnecessary numerical errors in the computation.. 1.2.4. Direct theoretical calculation of 𝑆(𝑞) and 𝑔(𝑟): MPB-RMSA. The static correlations between particle positions are obviously a result of the particle interactions. Therefore there should be a direct connection between the effective pair potential 𝑢(𝑟) between two particles and the pair correlation functions 𝑔(𝑟) and 𝑆(𝑞). To get this connection, one splits the total correlation function ℎ(𝑟) = 𝑔(𝑟) − 1 into two parts by according to the Ornstein-Zernike equation: (1.15). ℎ(𝑟) = 𝑐(𝑟) + 𝑛 𝑐(𝑟′ ) ℎ(|⃗𝑟 − 𝑟⃗′ |) d⃗𝑟′ ∫. Here the first part is the direct correlation function 𝑐(𝑟) and the second (indirect) part is 𝑐 ∗ ℎ the convolution of 𝑐(𝑟) with ℎ(𝑟). Equation 1.15 is actually a recursive definition for ℎ(𝑟): Following the recursion one time by inserting ℎ(𝑟) into the integral, one receives ℎ = 𝑐 + 𝑐 ∗ 𝑐 + 𝑐 ∗ 𝑐 ∗ ℎ. Doing it a second time one gets terms with 𝑐 ∗𝑐 ∗𝑐 and so on. ℎ(𝑟) is expanded into an infinite series of terms of higher and higher order in 𝑐(𝑟). While the total correlation ℎ(𝑟) decays very slowly with increasing 𝑟, the direct correlation 𝑐(𝑟) decays much faster, already after several mean interparticle distances. To be able to solve the Ornstein-Zernike equation for ℎ(𝑟) one needs another equation as a closure relation. Many of these relations are derived from the link between the total correlation function to the potential of mean force 𝑤(𝑟), the average work needed to bring two particles from infinite separation to a distance 𝑟: (1.16). ℎ(𝑟) = 𝑔(𝑟) − 1 = exp [−𝛽𝑤(𝑟)] − 1. For long-range potentials an approximation is to take the first order term of exp[−𝛽𝑢(𝑟)] − 1 as the direct correlation function and assume that all the rest is more or less in the indirect part 𝑐(𝑟) ≈ −𝛽𝑢(𝑟),. 𝑟>𝜎. and. 𝑐(𝑟) = −1,. 𝑟 < 𝜎,. (1.17). this closure has the name mean spherical approximation (MSA) and 𝜎 is the diameter of an impenetrable spherical shaped particle. Other important closures are hypernetted-chain (HNC), Percus-Yevick (PY) and Rogers-Young (RY), but they will not be discussed here, since they were not used here. Equations 1.15 and 1.17 together with the HSY potential 1.1 can be solved analytically for 𝑔(𝑟) and 𝑆(𝑞) (see for example in [40]) but good accuracy is limited to higher particle densities 𝑛 and weak Yukawa repulsion [34]. At lower densities and stronger repulsion the closure yields unphysical negative values for 𝑔(𝑟) which is easy to verify: As we go to the limit 𝑛 → 0 we get 𝑔(𝑟 > 𝜎) = 1 − 𝛽𝑢(𝑟) (neglecting higher orders in 𝑐(𝑟)), which is negative for strong repulsions. To circumvent this problem Hansen and Hayter [41] introduced a rescaling procedure (→rescaled MSA or RMSA), where the impenetrable diameter 𝜎 is inflated to 𝜎 ′ = 𝜎∕𝑠, with the rescaling parameter 0 < 𝑠 ≤ 1 until the so-called Gillan condition 𝑔(𝑟 = 𝜎+′ ) = 0 is fulfilled, so that negative values of 𝑔(𝑟) disappear. An inflation of the radius also rescales the coupling parameter 𝛾 ′ = 𝛾𝑠, the screening 𝑘′ = 𝑘∕𝑠 and the packing fraction Φ′ = Φ∕𝑠3 . The rationale behind this idea is that the repulsion 13.

(24) Chapter 1: Theoretical Background. Figure 1.4: Radial distribution function 𝑔(𝑟) and static structure factor 𝑆(𝑞) of charged colloidal systems with no additional salt ions 𝑐11 = 𝑛𝑠 = 0 at volume fractions Φ as indicated. Open symbols are MC simulation data, colored lines are results from different closures of the Ornstein-Zernike equation, insets are magnifications of the principal peak of the most dense system. Other parameters of the potential are 𝑍ef f = 100, 𝑙𝐵 = 5.62 nm, 𝜎 = 200 nm. These plots are taken from Heinen et al. [34].. between the particles is so strong that the hard sphere part of the potential is negligible anyway so that an increase of the diameter does not really change the interactions but still has an influence on the static correlation functions. A problem of RMSA is that it tends to underestimate the principal peaks of 𝑔(𝑟) and 𝑆(𝑞) when compared to results from simulations, one has to increase the effective charge 𝑍ef f or the coupling parameter 𝛾 artificially to get good quantitative agreement [34]. As a further improvement Snook and Hayter [42] argued that in the deriviation of the DLVO pair potential the point-like microions (counterions and salt ions) actually move freely in the whole volume, so that they are actually considered as being able to penetrate the large macro-ions (charged colloids). This leads to an effective decrease of the charge by a factor 1 − Φ that can be compensated with a rescaling to 𝑍ef∗ f = 𝑍ef f ∕(1 − Φ).. (1.18). This correction has the name penetrating background RMSA (PB-RMSA), taking into account the penetration abilities of the microions. Heinen et al. [34] found that even PB-RMSA still underestimates principal peaks and they argue that the inclusion of a correction for the filling fraction Φ of the colloids in the computation of the screening parameter 𝑘 (see Equation 1.3) together with the PB increase of the effective charge (also influencing 𝑘) would be a double correction of 𝑘. To this end they propose to modify 𝑘 again as √ 𝑘mod = 𝑘 1 − Φ (1.19) withdrawing the correction in Equation 1.3 [43, 34] and essentially increasing the screening length, which is proportional to 1∕𝑘. The coupling parameter 𝛾 is also affected as it depends on 𝑘. This modified PBRMSA (MPB-RMSA) scheme performs better compared to the other MSA schemes and is also in quite good agreement with Monte-Carlo simulations. Only compared to the Rogers-Young scheme (RY), which combines both HNC and PY closures and directly imposes thermodynamic consistency, the agreement with simulations is slightly worse in some situations: MPB-RMSA produces a kink at the rescaled diameter 𝜎 ′ which is smoothed out with the RY scheme and MPB-RMSA slightly underestimates the principal peak of 𝑔(𝑟) at high particle densities. On the other hand, also at high densities, the principal peak of 𝑆(𝑞) according to MPB-RMSA is sometimes in better agreement with simulations than that computed 14.

(25) 1.2 Dynamic and static correlation functions within the RY scheme [34] (see Fig. 1.4). Since the height of the principal peak is particularly important for MCT calculations, MPB-RMSA is thus a good choice for the purpose of this work. Another big advantage is the fact that it is an analytic solution, so the computation is a lot faster than doing a scan for numerical solutions of the RY equations. For this work the author is very grateful having received the C-code for MPB-RMSA computations directly from M. Heinen. For more detailed descriptions and performance comparisons the reader is referred to [34].. 1.2.5. Mean squared displacement. A very common quantity that is often determined in microscopy experiments or in molecular dynamics simulations, where particle trajectories can be recorded directly, is the mean squared displacement (MSD): ⟨. 𝑁 ⟨ ]2 ⟩ 1 ∑ [ 𝑟 (𝑡) = 𝑟⃗𝑗 (𝑡) − 𝑟⃗𝑗 (0) 𝑁 𝑗=1 2. ⟩. (1.20). It is a measure for the amount of space a particle has explored on average after travelling for a time 𝑡. For very short times (before any collisions) the position of each particle is determined by the initial velocity, which means 𝑟⃗𝑗 (𝑡) = 𝑣⃗𝑗 𝑡. This results in a ballistic MSD ⟨𝑟2 (𝑡)⟩ ∝ 𝑡2 . Concerning colloids suspended in a liquid solvent, for times 𝑡 > 1 µs that are much longer than the average time between two collisions with solvent particles, we are in the diffusive regime with ⟨ 2 ⟩ 𝑟 (𝑡) = 2𝑑 ⋅ 𝐷 ⋅ 𝑡. with 𝐷 =. (1.21). 𝑘B 𝑇 , 6𝜋𝜂𝑅. Here 𝑑 is the dimensionality of the vectors in Equation 1.20 and 𝐷 is the diffusion coefficient, given by the Stokes-Einstein-Sutherland equation for spheres of radius 𝑅 in a solvent with dynamic viscosity 𝜂 at temperature 𝑇 . The MSD can also be seen as the squared average displacement of a particle after the time 𝑡 has passed. This gives a direct link to the self part of the van Hove function 𝐺𝑠 (𝑟, 𝑡), which is the probability distribution for a particle for a displacement of length 𝑟 after the lag time 𝑡. At least in the limit of many 1. liquid. supercooled. strongly supercooled. 1. critical curve. glass. . log r2 (t). ®. 0. 2. 3 4. 4. 2. 0. 2. log t. 4. 6. 8. 10. Figure 1.5: Mean squared displacement (MSD) corresponding to the correlation functions in Figure 1.3 for a system that crosses the glass transition point. Analogue to the correlation function, the MSD develops an intermediate plateau and a two step decay coming closer to the transition. In the ideal MCT glass particles are trapped in the cages of their next nearest neighbours so that the MSD cannot rise more than the plateau value 2𝑑𝑟2𝑠 . This defines a localization length 𝑟𝑠 , which is the size of the cage.. 15.

(26) Chapter 1: Theoretical Background particles 𝑁 → ∞ and for ergodic systems, we can apply the Gaussian approximation [44] and assume this distribution to be Gaussian: ( ) ( )−𝑑∕2 2𝜋 2 𝑑 𝑟2 𝐺𝑠 (⃗𝑟, 𝑡) = ⟨𝑟 (𝑡)⟩ exp − 2 (1.22) 𝑑 2⟨𝑟 (𝑡)⟩ Here again, 𝑑 is the number of dimensions. In this case the self part of the intermediate scattering function (tagged particle correlator) can be written as ( ) ( ) 1 𝐹𝑠 (𝑞, 𝑡) = exp −𝑞 2 2𝑑 ⟨𝑟2 (𝑡)⟩ = exp −𝑞 2 𝐷𝑡 , (1.23) where the last equality is valid in the diffusive regime. For short times 𝑡 even glassy colloidal systems are in this diffusive regime, which means that we can do the expansion (1.24). 𝐹𝑠 (𝑞, 𝑡) = 1 − 𝑞 2 𝐷𝑡 + (𝑡2 ). One can also apply it to the complete intermediate scattering function (density correlator) for short times: 𝐹 (𝑞, 𝑡) = 𝑆(𝑞) − 𝑞 2 𝐷𝑡 + (𝑡2 ) and. 𝜙(𝑞, 𝑡) = 1 − 𝑞 2. 𝐷 + (𝑡2 ) 𝑆(𝑞). (1.25). Here one assumes the distinct part 𝐹𝑑 , describing correlations between distinct particles, to be more or less constant for small 𝑡. However, in denser systems the effective diffusion coefficient decreases. An additional friction from collisions or repulsive interactions and also hydrodynamic interactions with the neighbour particles start to play a role.. 1.2.6. Connection to light scattering experiments. The intermediate scattering function is strongly related to the intensity measured in light scattering experiments. In a common setup one measures the intensity of light scattered by a sample to an angle 𝜃 relative to the incident beam from a coherent light source. For the average intensity ⟨𝐼(𝑞)⟩ one finds the proportionality: ⟨𝐼(𝑞)⟩ = ⟨𝐸(𝑞, 𝑡) 𝐸 ∗ (𝑞, 𝑡)⟩ ∝ 𝐹 (𝑞) ⋅ 𝑆(𝑞). with 𝑞 =. 4𝜋𝑛ref sin(𝜃∕2) 𝜆. (1.26). The modulus of the scattering vector 𝑞 is computed from the refractive index in the surrounding medium 𝑛ref and the light wavelength 𝜆. Here 𝐹 (𝑞) is the form factor that describes the scattering by single particles, it can be measured at low particle densities, where we have 𝑆(𝑞) = 1 (as argued above).With the knowledge of 𝐹 (𝑞) and the proportionality in 1.26 it is possible to measure the static structure factor. As indicated in Equ. 1.26 the intensity is the squared absolute value of the electric field 𝐸(𝑞, 𝑡). Neglecting the influence of the form factor 𝐹 (𝑞), one can understand the relation of 𝐼(𝑞) to the structure factor 𝑆(𝑞) = 𝐹 (𝑞, 0) and the intermediate scattering function 𝐹 (𝑞, 𝑡) = ⟨𝑛(𝑞, 𝑡) 𝑛∗ (𝑞, 0)⟩ by interpreting the summands 𝑒−𝑖𝑞⋅⃗ 𝑟⃗𝑘 (𝑡) in the definition of 𝑛(𝑞, ⃗ 𝑡) (see Equ. 1.4) as interfering plain waves of the electric ⃗ 𝑟𝑘 (𝑡) describes the scattered field from particle 𝑘 at the position 𝑟 field. The wave 𝐸𝑖 ∝ 𝑒−𝑖𝑞⋅⃗ ⃗𝑘 . All the interfering waves together yield the sum in the definition of 𝑛(𝑞, 𝑡). The field autocorrelation function 𝑔1 (𝑡) is hence directly related to the intermediate scattering function or, more precisely, to the density correlator Φ(𝑞, 𝑡): ⟨𝐸(𝑞, 𝑡)𝐸 ∗ (𝑞, 0)⟩ 𝐹 (𝑞, 𝑡) 𝑔1 (𝑡) ≡ = = Φ(𝑞, 𝑡) (1.27) ∗ ⟨𝐸(𝑞)𝐸 (𝑞)⟩ 𝐹 (𝑞, 0) 16.