Overview ‧Dynamics ‧Dilute & Concentrated Suspensions ‧Polydispersity Effects

1. Light Scattering Theory - Dynamics Overview ‧Dynamics ‧Dilute & Concentrated Suspensions ‧Polydispersity Effects 2007-01-16 *This presentation file includes topics appearing in Chapter 3 of A. D. Pirie’s Ph.D. thesis *A. D. Pirie’s Ph.D. thesis can be downloaded free from the internet http://www.era.lib.ed.ac.uk/handle/1842/279

2. Motivations • In the early days of DLS, the effects of interparticle interactions were removed by extrapolation of data to zero concentration • There has been growing interest in interactions since concentrated particle suspensions provide a challenging many-body problem to which modern statistical mechanical and hydrodynamical theories can be applied • The many body problem is also of considerable practical importance because many processes in industry (paints, foods, detergents, etc) and biology involve concentrated suspensions

3. Dynamics • As the particles undergo Brownian motion the speckle pattern fluctuates, as the conditions for constructive and destructive interference in the far field change FIG. Experimental setup To monitor the speckle fluctuations a detector is placed in the far field with a sensitive area adjusted to be approximately the size of one speckle If the system is ergodic, the Time averaged correlation function is equivalent to the Ensemble averaged correlation function

4. The Intensity Correlation Function: Y Scattering volume • There are ca. 1,428 particles in the VS • Average particle-particle separation d: 9,000 nm (structural relaxation time) 然而，上方例子告訴我們，當粒子夠大時，在Low Q即 能探測粒子擴散一個粒徑長度的可能 The Siegert Relation in the modified form: Conventionally the detector aperture is adjusted so that β~1, i.e., only one coherence area (speckle) is measured

5. Dilute Suspensions • From measurements in the dilute limit one can obtain the free diffusion coefficientD0 and consequently the particle radius a • Without structural correlations g(1)(Q,τ) simply reduces to: 解法至少有三法，以下為其一

6. Concentrated Suspensions • For systems at high concentration (φ>0.05) or of strong particle interactiong(1)(Q,τ) no longer decays as a single exponential • The origin of such complex relaxations is due to: • Direct interparticle potential - The time scale τIis approximately the duration of a particle collision ~ 10-4 s • Indirect hydrodynamic force -The hydrodynamic time τHis ~ 10-7 s (the time taken for the fluid to reach a steady state) More info see P. N. Pusey and R. J. A. Tough in Ch4 of: Dynamic Light Scattering, R. Pecora (editor), Plenum, New York (1985) The Effective Short Time Diffusion CoefficientDeff(Q)is: ‧The short time limit ofg(1)(Q,τ) is determined by dynamics taking place on a time scale much less than τI ‧In the absence of hydrodynamics Deff(Q) is highly sensitive to the S(Q) in concentrated suspensions Deff(Q) can be obtained experimentally: Deff(Q) =D0 in the case of a dilute suspension

7. For Hydrodynamically Interacting Particles: Hydrodynamic factor (can be determined experimentally) In the high-Q regime (S(Q)~1) the length scales probed are approximately that of a single particles and Deff(Q) reduces to the short time self diffusion coeff. High-Q measurements probe the motion of single particles perturbed by the presence of all the other particles In the high-Q limit g(1)(Q,τ) can be written as:

8. In the low-Q limit (Q0) the behavior of the system on large length scales can be measured Deff(0) can be regarded as the diffusion coeff. describing the decay of density fluctuations of macroscopic extent

9. Polydispersity Effects • The effect of polydispersity nullifies the decoupling approximation of the following eq. • For hard sphere suspensions the effect of polydispersity on S(Q) is striking (see Fig.) ‧The main peak in S(Q) moves to smaller Q and pair correlations decrease sharply as the σ is increased ‧The polydispersity “washes out” the information available from S(Q) as the scattered intensities from the particles of different sizes interference destructively CONSTANT volume fraction The particle size distribution is assumed to be a Schulz distribution

10. Appendix: Data Analysis • . • . • . (1) Photocurrent Autocorrelation: Number of terms in the sum E.G.: (NS=1,000) When j=1 When j=2 When j=3