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Phase Transitions in Confined Colloidal Suspensions

Geometrical confinement has long been known to influence even qualitative aspects of materials' phase behavior [1]. Freezing point suppression in fluid-filled pores [2] and crystallization of confined fluids under shear [3] are examples with ramifications for practical applications such as catalysis and lubrication. Often, the microscopic mechanisms by which geometrical confinement affects phase behavior are not accessible to direct observation. Our understanding of phase transformations, furthermore, is not yet complete even for unconfined systems. Charge-stabilized colloidal suspensions provide one instance of a system whose structural phases are strongly influenced by confinement and yet whose structure and dynamics can be analyzed with `atomic' resolution using optical microscopy.

The charge-stabilized colloid in this study consists of 0.3250.005 m diameter polystyrene spheres suspended in water. Each sphere has a large number of ionizable groups chemically bonded to its surface which dissociate when the spheres are dispersed in water. The resulting surface charge engenders mutual repulsions among the spheres which are moderated by the counterions in solution. This electrostatic interaction stabilizes the suspension against flocculation driven by van der Waals attractions. The most widely accepted theory describing the pair-wise interaction between colloidal microspheres was formulated almost 50 years ago by Derjaguin, Landau, Verwey, and Overbeek (DLVO) [4]. In addition to the electrostatic interaction, the full form of the DLVO potential includes contributions from both van der Waals and hydrodynamic forces. These, however, play a relatively minor role in the behavior of highly charged colloid in suspensions of low ionic strength such as the ones we discuss below. In the limit that the Poisson-Boltzmann equation for the distribution of counterions can be linearized and under the assumption that the charge on a sphere is fixed at Z, the dominant electrostatic contribution to the DLVO potential has the form [4]


Here r is the center-to-center separation between spheres of radius a, is the Debye-Hückel screening length which measures the range of electrostatic interactions in an electrolyte of z-valent counterions at concentration n, and is the dielectric constant of the water. Eqn. (1) resembles a Yukawa repulsion, with the term in brackets accounting for exclusion of counterions from the spheres' interiors.

The pairwise interaction between colloidal microspheres has been measured directly only very recently. Measurements on colloidal polystyrene spheres [5,6] and ferrofluid emulsion droplets [7] find excellent agreement with eqn. (1) at least in the strongly interacting limit. The single-sphere charges extracted from these and related neutron-scattering experiments on micellar suspensions [8] are orders of magnitude smaller than their titratable charges [9]. This large discrepancy is consistent with a recently proposed charge renormalization theory [10] which relates a sphere's effective charge to its size as


provided its bare charge is sufficiently large. The Bjerrum length, , measures the equilibrium separation between counterions, and C is a constant both predicted [10] and measured [8,5,6] to be around 10. For an electrolyte with z = 1 at C, = 0.715 nm.

The ensemble of colloidal microspheres in a charge-stabilized suspension undergoes phase transitions among fluid and ordered phases as a function of the spheres' volume fraction, , and the screening length, [11,12]. The fluid phase appears in low density suspensions with short screening lengths while crystals form in more strongly interacting suspensions. Of the two ordered phases, crystals of face-centered cubic (fcc) symmetry appear in dense suspensions while body-centered cubic (bcc) crystals occur in the strongly-interacting dilute limit. The conditions for bcc crystallization approximate those of the one-component plasma model [13] and thus provide a practical realization of the classical Wigner crystal [14]. There also is experimental evidence that charge-stabilized suspensions can fall into a glassy state at particularly high volume fractions [12]. All features of this phase diagram except for the glass have been reproduced at least qualitatively in molecular dynamics simulations of Yukawa particles [15].

This relatively simple phase diagram becomes extremely rich when charge-stabilized suspensions are sandwiched between clean glass surfaces [16,17]. The ensemble of spheres in a single layer acts as an ideal two-dimensional system whose ordered phase undergoes a two-stage second-order Kosterlitz-Thouless melting transition [18]. As the separation between glass surfaces is increased and out-of-plane motion becomes possible, the single-layer triangular crystal is replaced by a two-layer crystal with square symmetry in the plane [17]. The transition between and structures involves strongly correlated out-of-plane fluctuations [17] which have been analyzed by Chou and Nelson [19]. As the separation between confining walls is further increased, the suspension undergoes a cascade of phase transitions in the following sequence [17]:

This sequence proceeds to roughly 10 layers before the bulk structure is recovered. The structural fluctuations at the transition points have been studied in detail only for the transition [20].

In the sections which follow, we study the series of local structural transformations which occur when a crystal is shear-melted to a supercooled confined fluid and then is allowed to freeze. Rather than achieving its equilibrium structure in one step, the ensemble of spheres first freezes to a buckled single layer triangular crystal and then passes through a martensitic [21] phase transition. The triangular crystal's presence in the reaction path is an example of the more general phenomenon of one symmetry's intervention in the evolution of a structure with another symmetry. The formation of quasicrystals during splat cooling of metal alloys provides a dramatic example [22], while metallic glass phases prevent supercooled metallic fluids from crystallizing at all [23].

Next: Confinement and Shear Up: Martensitic Transition in a Previous: Martensitic Transition in a

David G. Grier
Wed Feb 15 13:32:44 CST 1995