Number 11 / 27 Particle trapping and banding in rapid solidification by

Solidification of suspensions of small particles, from nanometer to colloidal (submicrometer) sizes, produces biomimetic materials with novel microstructure and expanding applications in microfluidics, nanotechnology and tissue engineering. To facilitate understanding and control of the solidification process, a thermodynamically consistent theory is here developed. We use the Boltzmann particle velocity distribution to determine the probability a particle is engulfed by an advancing solid-liquid interface and obtain the resulting kinetic phase diagram. We demonstrate use of the theory by predicting the formation of bands in rapidly solidified alumina suspensions, in quantitative agreement with experiment. 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 microaligned structures that can be produced and the relatively benign and inexpensive nature of the process [25, 8, 7]. The freezing of particulate systems is also of scientific interest in natural phenomena such as ice growth at the base of glaciers [3] and for applications in biotechnology including cellular cryopreservation [10] and tissue engineering [23]. 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 1 the slow velocity limit, in which the ice-suspension interface is locally in thermodynamic equilibrium and all particles are rejected by the ice phase [17, 18]. A stability analysis of the interface determines the transition from a stable planar interface to an unstable nonplanar morphology [18, 19]. It is known from experiments and theory, however, that at faster solification velocities the solid (typically ice) interface engulfs the particles, trapping them in a non-equilibrium manner into the ice phase [21, 4, 14, 20]. 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 engulfment velocity Vc, casting doubt on the validity of the assumption of local equilibrium at the freezing interface [7]. 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 [7]. Similar banding structures are observed in the basal layer of glaciers [3] and in food cryopreservation systems [15]. In order to model such far from equilibrium behaviour we use the Boltzmann velocity distribution of a particle to derive an expression for the non-equilibrium segregation coefficient kv at an ice–colloidal suspension interface as a function of 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 Vc [16, 11, 14]. We use this expression to construct a kinetic version of the thermodynamic 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