Phase separation and dynamics of concentrated colloids : Diffusing wave spectroscopy and brownian dynamics simulations

Abstract

The purpose of this chapter is to summarize the principal results and conclusions reported in this thesis and also to point out the scope of further work. The focus of this thesis is on static and dynamic aspects of the liquid-to-crystalline transitions and the liquid-to-metastable glassy state in the aqueous suspensions of charged polystyrene spheres (polyballs) and on the flocculation of large particles induced by the presence of small particles in a nearly hard-sphere system. The investigations are both experimental and theoretical in nature. The experimental techniques include the recently developed technique of Diffusing Wave Spectroscopy (DWS), where the light is multiply scattered, and the direct observations of the particle behavior by optical microscopy. The theoretical studies include both the analytic calculations and the Brownian Dynamics (BD) computer simulations. The results from the analytic calculations provide insight and understanding of our experimental results on the polarization dependence of the intensity autocorrelation functions of the multiply scattered light from crystallizing colloids as well as on the flocculation of nearly hard-sphere polyballs. BD simulations reveal important information on the static and dynamic properties near the liquid-to-crystal as well as liquid-to-glass transition in these “soft-sphere” systems. In Chapter I, we have introduced our system of interest, the polyballs, and the techniques involved in studying different physical properties in the liquid, crystalline and glassy states of the system. A brief review of the pertinent literature is also presented in order to focus the motivation of the present research work. The design, fabrication and operation of our light scattering and the photon correlation spectroscopy set-up and also the preparation and characterization of our samples are detailed in Chapter 2. In the following, we summarize the principal results and conclusions of this thesis. In Chapter 3, we have provided the first unambiguous experimental evidence using optical microscopy of purely entropic phase separation of chemically identical species in which at least one component remains fluid. An initially homogeneous binary aqueous suspension of nearly hard-sphere polystyrene particles with different diameters spontaneously phase separates into aggregates of predominantly large diameter particles and fluid phase rich in smaller particles. The energetics alone cannot favor the phase separation of the hard spheres. The effect is a consequence of entropic “depletion force” which originates as follows: if two large particles (radius rB) get within 2(rA + rB) of each other, a smaller particle (radius rA < rB) cannot enter the region in between, causing a net imbalance in the osmotic pressure of the small particles on the larger ones driving them together. Simple geometrical calculation shows that this effective attractive interaction between the larger polyballs mediated by the small particles has a depth of Udep = 1.5 kBT ?A rB / rA and a width of 2rA, where ?A is the smaller particle volume fraction. Thus, for the diameter ratios of 5, flocculation is expected (Udep ~ kBT) only if ?A ~ 15%. This value of ?A is easily reached in the sediment which leads to the flocculation of the large particles. To confirm this explanation, we have checked that no flocculation is seen (a) in single-component suspensions and (b) in density-matched (and hence non-sedimenting) two-component suspensions in a D?O + H?O mixture with parameters as above. We have also provided simple density functional arguments to show that once the ratio rB / rA is big enough, the resulting growth in the osmotic compressibility of the large particles will cause them to condense. This is confirmed from the increased magnitude of the large-particle structure factor Seff by the presence of the small particles. Further experimental and theoretical studies of the phase diagram over a range of radius ratios and volume fractions are necessary to see whether a gas-liquid phase boundary can be found for some range of radius ratios. In Chapter 4, we present our Diffusing Wave Spectroscopy results of a striking and altogether unexpected polarization dependence of the multiply scattered signal from crystallizing colloidal suspensions and provided a theoretical model which rationalizes our observations. The main results and conclusions are as follows: 1. While freezing of the colloidal crystallites we find that Cvv(t) = - 1 (polarized) decays to a nonzero constant, as expected for a frozen phase, but Chh(t) = — 1 (depolarized) decays to zero at essentially the same rate as in the liquid which too eventually saturates to a non-zero value. 2. This extraordinary intermediate state of non-decaying Cvv(t) and decaying Chh(t) is also observed while melting a well-formed crystal by adding HCl, followed by an ordinary liquid-like decay of the correlations. 3. We have presented a new phenomenological model for this behavior based on decoupled translational and orientational fluctuations in a weakly depolarizing medium which is also of interest as a new treatment of waves in random media. Our study highlights the greater sensitivity of depolarized diffusing-wave spectroscopy as a probe of the dynamics of the medium. 4. The origin of this unexpected behavior lies in the fact that the intermediate stage is an imperfect crystal of some kind, containing optically anisotropic entities such as interfaces between crystallites, vacancies etc. The orientational correlations can decay to zero at large times even though the translational correlations may be frozen. The former contribute mainly to the depolarized scattering causing the decay of correlations in HH channel when the polarized VV component shows a saturation. These defects freeze out at the later stages of crystallization thereby. Decphasing Cvv and Chh in essentially the same way. 5. It has been suggested that in solid (crystalline or glassy) phase Cvv(t) ? particle in time t. We have fitted our data in the crystal phase by taking for (?r²(t)) a model of overdamped, harmonically bound particles. The resulting parameter ? is found to be much lower than that in the liquid phase. Keeping in mind that ? is inversely proportional to the transport mean free path l*, l* has been computed using Mie scattering theory for the form factor F(q). The interparticle interactions have been increased via the structure factor S(q) obtained using the Percus–Yevick (PY) approximation. The l* increases continuously with the interaction, consistent with the observed decrease of ? in the interacting colloids. We feel that small-angle X-ray or neutron scattering study of these colloidal suspensions in the various stages of crystallization could reveal much better insight of the underlying origin of such an anomalous behavior and allow us to refine the phenomenological model of the medium that we have presented here. It is also worthwhile to check if such an intermediate stage could be reproduced in the crystallization studies via computer simulations. In Chapter 5, we have reported our DWS study on a 1:1 binary mixture of two different diameter polyball suspensions. Our prime results and conclusions are as follows: 1. We have observed that for a total volume fraction ? = 5%, the mixture freezes into a crystalline phase when the coion concentration is reduced. For total ? = 10%, the resulting state is glassy. 2. The intensity autocorrelation functions of the supercooled liquid while freezing towards a glassy state show two different relaxation behaviors which are similar to the ? and ? relaxations seen in conventional organic glasses. This behavior has been seen in computer simulations and has been discussed in the mode-coupling theory of the ideal liquid-to-glass transition. These relaxations are absent in the data towards freezing into the crystalline phases. In Chapter 6, we have reported the results of our Brownian Dynamics (BD) simulations of a binary system of colloids with different radii and charges. The aim has been to study the structure and dynamics of liquid phase, crystalline phase and, in particular, the glassy state. The chapter is divided into three parts for the sake of clarity in presenting the results. PART A focuses on the structural and transport properties of the system. Our principal results are as follows: 1. The mixture freezes into a crystalline phase for ? = 0.2 by reducing the coion concentration while the final state for ? = 0.3 is a glass. This result is in qualitative agreement with our experiments with binary mixtures, presented in Chapter 5. 2. The structural quantities studied are the average and the partial pair distribution functions g(r) and their spatial Fourier transforms, the static structure factors S(q), the time-averaged g(t) and the Wendt–Abraham parameter Rg = gmin/gmax. All these parameters show that by lowering the effective temperature (achieved by reducing the coion concentration) at ? = 0.2, liquid freezes into a body-centered cubic (BCC) crystal with imperfect sublattice ordering whereas at ? = 0.3 a glassy state is reached. The structural quantities change suddenly at the crystal transition, expected of a sharp first-order phase transition. On the contrary, the glass transition is slow and kinetic in nature as suggested by very slow and gradual changes of these static parameters. 3. The temporal evolution of the average and the partial mean square displacements (?r²(t)) and the corresponding time-dependent diffusion constants D(t) = (?r²(t)) / 6t of the supercooled liquid show a short-time (higher in magnitude) and an asymptotic (lower in magnitude) diffusion constant. These two are distinctly separated by an intermediate “subdiffusive” regime which spans in time as the system solidifies. Near the glass transition, the mean square displacement is predominated with “staircase” behavior. These observations support the “cage” concept: the short-time behavior may be due to the initial-time vibrations of the particles in some sort of cages formed by their nearest neighbors before it diffuses through more than one interparticle separation. The staircase behavior indicates repeated arrest of a particle in cages and intermediate hop-jumps from one cage to the other. In PART B, we have presented our study of the self-part Gsa(r,t) and the distinct part Gsp(r,t) of the van Hove density correlation functions, the self-intermediate scattering functions F*(q,t) [the spatial Fourier transform of G*(r,t)], the non-Gaussian parameter ??, the nonergodicity parameters f(q) = F*(q,t ? ?) [?, ? — 1 or 2] and the individual particle trajectories. Our main results and conclusions are as follows: 1. It is known that Gsa(r,t) in the hydrodynamic limit for liquid or harmonic limit for frozen systems can be well represented by a Gaussian form which scales with D?t, where D? is the asymptotic diffusion constant for particles of species a. We have shown that this scaling starts holding good even at the initial times far before the hydrodynamic limit (t ? ?) is reached for the temperatures where the asymptotic value is reached. For temperatures lower than the above, the “subdiffusive” regime spans over the full length of simulation so that the asymptotic D?t is not reached and hence this scaling cannot be checked for. 2. As the supercooled liquid is cooled towards the glass transition, a second peak (and at times a third peak) in Gsa(r,t) at the first and the second nearest-neighbor positions, respectively, evolve in time, at the expense of the area under the first peak. This clearly shows that the particles execute activated jump motions rendering the system extremely non-Gaussian as determined in terms of the non-Gaussian parameters ??. The strong cooperative behavior of these hop motions near the glass transition (? = 0.3) makes ?? shoot up much more than that near the crystal transition (? = 0.2), where the hop motions are rare. 3. The most significant result of our simulation is that at a temperature (T* = 0.0313) very near to the glass transition (T* = 0.0312), Gsa(r,t) shows an increase of the first peak height at some later time (at the cost of the second peak). This clearly indicates that some particles must be hopping back to their original positions from their respective hopped positions. This must take place extremely cooperatively to show up in the statistically averaged quantity G*(r,t). The individual trajectories and displacements of the particles are followed to find that about 2% of particles show cooperative hopping back and forth and the motion of these particles are interconnected. The initial state and the configuration after the cooperative hop can be viewed as two states of the “two-level” system known to occur in the glassy states. 4. Our simulation results confirm the following predictions of the Mode Coupling Theory (MCT) for an ideal glass transition: (a) The temporal evolution of the self-intermediate scattering functions F*(q,t) show two different relaxation processes, which are called the ? and ? relaxations in the conventional atomic systems. The gradual slowing down of the collective dynamics as probed by Gdp(r,t) shows two time scales usually associated with the aforesaid ? and ? relaxations present in the system near the glass transition. (b) In the ? relaxation regime, plotting F*(q,t) at various temperatures as functions of a properly scaled variable t/?(T*), where ?(T*) is a temperature-dependent scaling time, all the data fall on a “master curve” (time-temperature superposition principle), which is better fitted by a KWW stretched-exponential law than a simple power law. This has been checked for different values of q. (c) In the ? relaxation regime, the difference of Gsp(r,t) from its plateau value obeys a factorization property. (d) The nonergodicity parameter f(q) = F*(q,t ? ?) when plotted as a function of T* follows a scaling behavior near the glass instability. In PART C, we study the translational order as well as the bond-orientational order of the binary colloidal mixture while reducing its effective temperature from a liquid to a crystal (? = 0.2 and varying) and also to a glass (? = 0.3 and ? varying). 1. We quantify the translational order in terms of the two-point C?(t) and four-point C?(r,t) density autocorrelation functions. The nearest-neighbor orientational order is quantified in terms of the quadratic rotational invariant Q? and the normalized bond-orientational autocorrelation functions g?(t). 2. The value of C?(r,t) being nearly equal to [C?(t)]² for all temperatures when a liquid is quenched to a metastable glass shows that there exists no correlation length in the system which is growing to diverge near the glass transition. On the other hand, the difference between these two quantities increases as the crystal transition is approached and eventually goes to zero in the crystal state of the system indicating that there is a finite growth of correlation length as a liquid is cooled to a crystal. 3. The Z-dependence of Q? in the crystalline state (? = 0.2) does not match with that of a perfect BCC (with simple cubic sublattice ordering). This suggests that the crystalline state, whose g(r) shows BCC order, has distortions present in its structure. The changes of the descending order of magnitude from Q?, Q?, Q??, Q?, Q? in the liquid to Q?, Q?, Q?, Q??, Q? in the glass and Q?, Q?, Q??, Q?, Q? in the crystal indicate that icosahedral (Z = 6) order predominates at the cost of simple cubic order (Z = 4) near the glass transition as well as the crystal transition. 4. The normalized C?(t) grows faster than g?(t) as a liquid is cooled either towards a crystalline or glassy state. These correlation functions for a supercooled liquid freezing towards a glass fit well to the Kohlrausch–Williams–Watts (KWW) stretched-exponential form exp[?(t/?)^?] supporting the existing notion that stretching or multi-relaxation processes could be a universal feature of the dynamics near glass transitions. 5. The average translational and bond-orientational relaxation times [?(t)] are extracted from the fitted stretched-exponential functions as a function of T*. The temperature dependence of ? is Arrhenius [?(T*) = A exp(B/T*)] for liquid-to-crystal transition while it can be approximated reasonably well by the Vogel–Tammann–Fulcher (VTF) law [?(T*) = exp(CT?*/(T* ? T?*))] for liquid-to-glass transition. The value of the parameter C points out that the colloidal suspensions are “fragile” glass formers like the organic and molecular liquids. The Gaussian Random Energy Model (GREM) fails to explain the temperature dependence of ?.