Phase Separation in Colloidal
Suspensions by Collective Growth of Domains
Colloidal suspensions are fluid materials
which contain myriads of micrometer-size solid particles in a solvent. These materials find ubiquitous application in the chemical and food industries (various kinds of paints, ink, cosmetic creams, toothpaste, coatings for various purposes); even bullet-proof vests are produced using such colloidal dispersions. Often one is interested in suspensions that can separate in several phases and are able to coexist with each other. For instance, if the solvent also contains polymers, i.e. long flexible macromolecules, which have a random coil-like structure with a radius comparable to the colloid particles, an attractive force between colloidal particles arises from entropic effects, the so-called “depletion attraction”. By a suitable choice of polymer concentration and molecular weight these interactions and hence the state of the suspension (homogenously mixed or separated into colloid-rich and polymer-rich regions) can be manipulated.
Of course, for processing such dispersions one uses containers, and then the particles also interact with the
confining walls, which often leads to the formation of colloid-rich “wetting layers” at the walls. Such effects have a particularly pronounced influence in applications of microfluidic devices, where the dispersion is confined in tube-like or slit-like pores, with distances between the walls exceeding the colloid diameter at most by 2 or 3 orders of magnitude. The interplay of wetting layer formation and lateral domain formation (in the directions parallel to the walls of the slit pore) is a challenging problem. Experimental studies of such problems are on-going (e.g. ) and clearly show that the kinetics of phase separation in such systems is a multi-stage process. However, in order to be able to experimentally control the resulting domain morphologies, a better theoretical understanding is urgently needed: one would like to know how different effective potentials between the colloids and a wall (which one could modify by suitable wall coating) affect the domain growth, and one also would like to understand the role of hydrodynamic (i.e., flow-mediated) interactions, whose strength could be controlled by changing the solvent viscosity.
However, for such soft matter systems it is impossible to formulate a theoretical first-principles approach, and hence the method of choice to address such problems is large-scale computer simulation. Colloidal suspensions are a true multi-scale problem. Only by a substantial investment of supercomputer resources at the massively parallel supercomputer HERMIT progress could be obtained. Remember that the fluid molecules in the solvent are of nm
(10-9m) size. The polymer coils can be approximated as soft spheres and the colloids as (almost) hard spheres, both of size (10-9m). However, the domain structures that one wishes to study are in the range of 10 to 100m!
Thus, even on petascale computers a treatment taking full account of molecular details of the solvent molecules is impossible, but it is also not necessary: the physical principle that can be exploited to simplify the problem is the separation of time scales for the dynamics of solvent molecules versus colloids. The latter move many orders of magnitude slower than the solvent molecules. On the typical time scale of colloid motion, all correlations in the motions of solvent particles are fully relaxed, and hence it is possible to replace the solvent by an effective fluid, particles undergoing random collisions, but otherwise behaving like a dense ideal gas. This is achieved by the so-called “multiparticle collision dynamics” method (MPC) . With the appropriate choice of the density and hence the
viscosity of this effective fluid, the hydro- dynamic interactions in the motion of colloid particles and polymers are properly included.
When we take a colloid diameter as our unit of length, a typical linear
dimension of the simulated system is 256 in both directions parallel to the confining walls, which are 10 colloid diameters apart. The walls repel both particles with a smooth potential, decaying with the 12th power of distance, and a range of 50% of the respective particle diameters (polymers were assumed to have a diameter of 0.8 relative to colloids; see  for further
details of the model). Such a system may contain 236859 colloid particles, about a million polymers, and 52 million effective solvent particles. Using the domain decomposition scheme and message passing (MPI), 4096 CPU cores could be used in parallel to study such systems, with almost no loss of efficiency in comparison to simulations without solvent particles .
Figure 1: Phase diagrams of the model colloid-polymer mixture in the bulk (open symbols) and confined by two planar repulsive walls a distance D=10 colloid diameters apart (full symbols). The locations of the full squares and diamonds indicate the packing fractions , of the colloids and polymers in the coexisting polymer-rich (squares) and colloid-rich (diamonds) phases, respectively. These coexisting phases are connected by “tie lines” (broken straight lines: a state on such a line is a mixture of the two coexisting phases with volume fraction 1-x, x proportional to the length along the line from the state point to the coexisting phases on the left and right, respectively (“lever rule”)). Arrows show three “quenching experiments” (cf. text). From .
Unlike in experiments , one does not need to worry about small perturbations due to gravitational forces acting on the colloids, and one can obtain the equilibrium phase behavior of the studied system, both with and without confining walls, by separate simulations, using completely different methods, namely Monte Carlo simulations in the grand-canonical ensemble. Of course, the ideal gas-like solvent particles do not affect the static phase diagram, and can in this context simply be omitted. Figure 1 shows the resulting phase diagrams, using the “packing fractions” , of polymers and colloids as variables (the packing fraction gives the percentage of the total volume of the system taken by the particles multiplying the spherical volume of one particle by their number density). One sees that both in the bulk (a bulk system is simulated by choosing a cubic box with periodic boundary conditions in all three directions) and in the confined geometry the system is miscible if either or is sufficiently small (or both). The symbols represent the phase boundary, separating the miscible system (below) from the demixed region (above). One sees already that confinement enhances miscibility on the polymer-rich side, where is small (), but not on the colloid-rich side of the phase diagram (there, the full and open symbols in Fig. 1 almost coincide). This happens because for small the few colloids in the system get almost completely attracted to the walls, forming there enrichment layers with enhanced colloid density, and hence there is no need for unmixing in the remainder of the system any more: it is lateral phase separation parallel to the wall that matters for the confined system, in the direction normal to the walls some inhomogeneity always occurs. The so-called “tie lines” (shown as dashed straight lines in Fig. 1) indicate the
respective two phases that coexist with each other in each case where lateral phase separation has occurred.
This complete knowledge about the phases that coexist in equilibrium, in relation to the precise knowledge on interactions, is a feature by which our simulations clearly reach beyond the corresponding experiment .
The arrows in Fig. 1 now indicate three “quenching experiment”-simulations, i.e. reduction of the system volume at constant ratio /. The initially homo-
geneously mixed system becomes thermodynamically unstable, concentration fluctuations grow spontaneously, and various pattern of growing domains form (Fig. 2). These patterns indeed have great similarity to what one finds in corresponding experiments, but we have the bonus that we can isolate the effect of hydrodynamic interactions
clearly and thus elucidate their role in domain growth. This is done by an appropriate choice of hydrodynamic boundary conditions for our solvent fluid particles at the walls, all the way from perfect slip to perfect stick
(Fig. 2). For perfect slip, hydrodynamic interactions are not affected in the
directions parallel to the walls, but for
perfect stick, they are partially screened
(the screening increases when the
distance between the walls decreases). One sees that hydrodynamic interactions clearly speed up the domain growth. For stick boundary condition the pattern at t=6000 Molecular Dynamics time units resembles the pattern at t=2400 if slip boundary conditions are used. (Using the Stokes-Einstein formula for diffusion of colloids in liquids, one estimates that the time unit of the simulation corresponds to at
least 1 sec of real time, illustrating once more that a truly molecular simulation, for which the time unit is on the picosecond scale, 10-12 s, is completely hopeless.) As an extreme case, one can turn off hydrodynamic interactions completely, and finds then that the growth proceeds considerably slower. Thus, all the factors controlling the domain growth can be “isolated” in our simulations, and hence their role eluci-
dated in detail. Thus, we arrive at a more firm interpretation of the details of these complex processes than is possible in the corresponding experiments, where one always has to consider all factors influencing the domain growth together.
Figure 2: a) Snapshots of the (coarse-grained) domain pattern for the quench #3 in Fig. 1, showing only the (x,y) coordinates of polymers as black dots, for the system of size 256x256x10 at times t=12, 240, 2400 and 6000 Molecular Dynamics time units, as indicated. The coordinates of colloid particles and of the solvent fluid particles are not shown. The data refer to different treatments of hydrodynamic interactions: slip (left) and stick (middle) boundary conditions, and no hydrodynamic interactions (“NOHI”, right).
b) Side views of the same system with slip boundary conditions, at times t=120, 324, 984 and 1680, from top to bottom. From .
Of course, it is very satisfactory that there is a large similarity between the snapshot pictures recorded in experiment  and in our simulation (Fig. 2). But we can go further and enhance the resolution, so that we not only see coarse-grained domain patterns, but visualize individual particles (Fig. 3).
In this way, it will become possible to study in more detail (i.e., on the scale of individual colloid particles) what precisely happens e.g. when two domains
coalesce and the domain shape changes.
Studying such phenomena in more
detail is still a challenge for future work.
One aspect that has received considerable attention in the theoretical literature
for decades  is the question whether in the late stages of domain growth a
scaling regime emerges (i.e., the domain pattern at different times is “self-similar”),
where the characteristic length scale grows as a power law of time: this shows
up as straight line on a double-logarithmic plot. Fig. 4 shows that we indeed found some evidence for such a behavior, but the growth exponent does depend on the hydrodynamic boundary condition: for the slip case, the exponent is 2/3 as theoretically predicted , while for the other cases the standard Lifshitz-Slyozov-Wagner growth law  with exponent 1/3 is recovered.
Figure 3: Snapshot of a section of the colloid-polymer mixture during the demixing process, visualizing the three different length scales: the point-like solvent particles (blue), the colloids and polymers (shown in yellow and grey, respectively), as well as the length scales of the polymer-rich and colloid-rich domains. The snapshot corresponds to an ultrathin film of D=1.5, deeply in the two-phase region. For a clearer view, a layer containing many solvent particles on the top is removed.
In conclusions, our study has shed light on the role of hydrodynamic boundary conditions on domain growth in systems
that undergo phase separation. We
expect that this study will help to better
interpret existing experiments, stimulate new experiments, and ultimately help to better control confinement effects
on the processing of colloidal suspensions in various applications.
We thank the Deutsche Forschungs-
gemeinschaft (DFG) for support (grant No TR6/projects A4 and A5) and the Jülich Supercomputer Center (JSC/NIC),
where this project was begun at the JUROPA supercomputer. We are particularly grateful to the Höchtsleistungs-
rechenzentrum Stuttgart (HLRS) for a generous grant of computer time at the Hermit supercomputer.
Figure 4: Log-log plot of the reduced domain size versus time, for a deep quench of an ultrathin film (D=1.5). From .
 Jamie, E.A., Dullens, R.P., Aarts, D.G.
J. Phys. Condens. Matter 24 (2012) 284120.
 Gompper, G., Ihle, T., Knoll, D.M., Winkler R.G.
Adv. Polymer Sci. 121 (2009) 1.
 Winkler, A., Virnau, P., Binder, K., Winkler, R.G., Gompper, G.
EPL 100 (2012) 16003.
 Winkler, A.
Dissertation, Johannes Gutenberg-Universität Mainz (2012).
 Binder, K., Fratzl, P.
„Spinodal decomposition“, in Phase Transformations in Materials, edited by Kostorz, G. (Wiley-VCH, Weinheim, 2001) p.409.
• Alexander Winkler (1)
• Peter Virnau (1)
• Kurt Binder (1)
• Roland G. Winkler (2)
• Gerhard Gompper (2)
(1) Institut für Physik,
(2) Institute for
tion & Institute for