Particle Trapping and Banding in Rapid Colloidal Solidification (original) (raw)

We derive an expression for the nonequilibrium segregation coefficient of colloidal particles near a moving solid-liquid interface. The resulting kinetic phase diagram has applications for the rapid solidification of clay soils, gels, and related colloidal systems. We use it to explain the formation of bandlike defects in rapidly solidified alumina suspensions. The rapid solidification of the solvent phase of a colloidal or nanosuspension is receiving increasing interest in the materials science community owing to the novel micro-aligned structures that can be produced and the relatively benign and inexpensive nature of the process [1-3]. The freezing of particulate systems is also of scientific interest in natural phenomena such as ice growth at the base of glaciers [4] and for applications in biotechnology including cellular cryopreservation [5] and tissue engineering [6]. In order to understand and control these diverse processes, a predictive and general theory is required for the number of particles trapped by an advancing solidification front as a function of front velocity and particle radius. Thermodynamic modelling has focused on the slow velocity limit, in which the ice-suspension interface is locally in thermodynamic equilibrium and all particles are rejected by the ice phase [7,8]. A stability analysis of the interface determines the transition from a stable planar interface to an unstable nonplanar morphology [8,9]. It is known from experiments and theory, however, that at faster solification velocities the solid (typi-cally ice) interface engulfs the particles, trapping them in a nonequilibrium manner into the ice phase [10-13]. Recent experiments on the rapid solification of aqueous suspensions of alumina particles have been undertaken at speeds near to, or greater than, the measured critical en-gulfment velocity V c , casting doubt on the validity of the assumption of local equilibrium at the freezing interface [3]. In this high velocity regime interesting phenomena occur, such as banding, whereby the interface velocity fluctuates in time leaving behind significant defects in material properties [3]. Similar banding structures are observed in the basal layer of glaciers [4] and in food cryopreservation systems [14]. In order to model such far from equilibrium behavior we use the Boltzmann velocity distribution of a particle to derive an expression for the nonequilibrium segregation coefficient k v at an ice-colloidal suspension interface as a function of the freezing interface velocity V and particle radius R. This equation reflects the fact that particles are rejected by the ice at small V and engulfed at high V, as well as experimental observations that particles are trapped over a range of V and not just at a single critical velocity V c [12,15,16]. We use this expression to construct a kinetic version of the thermody-namic phase diagram. The phase diagram consists of the solidus line (particle volume fraction in the solid as a function of temperature) and the liquidus line (particle volume fraction in the liquid as a function of temperature). The constructed kinetic phase diagram captures the expectation that at low V the interface temperature is determined by the equilibrium phase diagram, while at sufficiently large V the solidus and liquidus coincide and the ice phase engulfs all particles (no segregation occurs). By considering the interface temperature as a function of V in a freezing alumina suspension the kinetic phase diagram provides an explanation for the interface velocity fluctuations and the formation of bands. Figure 1 shows a schematic diagram of a particle moving at speed V x along the x direction, near a solid-liquid interface advancing at speed V. We define the segregation coefficient by the equation k v ¼ s = l , where s and l are the particle volume fractions in the solid and liquid phases, respectively, at the interface. To derive an expression for the segregation coefficient, we assume that the particles are moving in the x direction with the Maxwell-Boltzmann velocity distribution, with the probability that a particle has the x direction velocity of V x given by p x : δt V δt V x x solid liquid FIG. 1 (color). Schematic diagram showing the velocity V x of a colloidal particle near a solid-liquid interface moving at V. PRL