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Experimental Verification of Morphological Instability In PHYSICAL REVIEW LETTERS week ending PRL 100, 238301 (2008) 13 JUNE 2008 Experimental Verification of Morphological Instability in Freezing Aqueous Colloidal Suspensions S. S. L. Peppin1 and J. S. Wettlaufer1,2 1Department of Geology and Geophysics, Yale University, New Haven, Connecticut 06520, USA 2Department of Physics, Yale University, New Haven, Connecticut 06520, USA M. G. Worster Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, Cambridge University, Cambridge CB3 0WA, United Kingdom (Received 12 February 2008; published 9 June 2008) We describe an experimental test of a new theory of the unidirectional freezing of aqueous colloidal suspensions. At low freezing speeds a planar ice lens completely rejects the particles, forming a steady- state compacted boundary layer in the liquid region. At higher speeds the planar interface becomes thermodynamically unstable and breaks down geometrically to trap bulk regions of colloid within. The theoretical stability threshold is determined experimentally, thereby demonstrating that colloidal suspen- sions can be treated analogously to atomic or molecular alloys. DOI: 10.1103/PhysRevLett.100.238301 PACS numbers: 82.70.Dd, 64.75.Xc Structure formation during crystal growth is an impor- into account the strong concentration dependence of the tant example of self-organization in systems driven away governing parameters in colloidal systems (diffusivity, from equilibrium [1]. Owing to their intrinsic complexity osmotic pressure, etc.) [5,6]. Theory predicts that even in and technological importance, pure materials and molecu- the absence of ionic solutes the freezing interface can lar solutions have been the focus of intense theoretical and become morphologically unstable owing to the effects of experimental studies over the past several decades [2]. the colloidal particles alone. In the present work we test Recently attention has been paid to colloidal systems [3– this theory on colloidal bentonite. We first discuss the 6], which also display a fascinating variety of patterns upon overall geometry of the experiment, the equilibrium phase freezing (Fig. 1). Besides presenting new challenges to our diagram, and dynamic properties of the colloid. Measure- understanding of colloidal physics, the self-organization of ments of the compressibility (osmotic pressure) are used freezing colloids plays an important role in many natural to predict the freezing point depression as a function of and technological processes. The phenomenon underlies particle volume fraction. Finally, measurements of the frost heave and patterned ground [7,8], influences the permeability are employed to predict the concentration- success of cryopreservation [9], and provides a mechanism dependent diffusivity. These properties enable us to con- for the remediation of contaminated clay [10,11]. At low struct a quantitative model of the solidification process, initial particle concentrations it is possible to remove the and to determine conditions under which constitutional segregated ice by freeze drying to yield microaligned supercooling occurs. porous materials with uses in bioengineering and micro- We consider a system in which a layer of colloid at fluidics [4,12]. Despite the large number of applications, initial volume fraction 0 is placed to height L0 in a many questions remain about the fundamental mechanisms water-filled glass cell between two fixed temperature underlying the solidification of colloidal systems. In liquid solutions the phenomenon of constitutional supercooling, in which the temperature of the melt directly (a) (b) (c) in front of the freezing interface is below its equilibrium freezing temperature, governs the onset of morphological instability [2,13]. Although it is known experimentally that colloidal particles can profoundly affect the nature of the instability [14], it has not been clear how best to account for colloidal effects, especially if the particle concentration is high. Most theoretical and experimental studies have 1.0 mm 0.5 mm 1.5 mm therefore focused on the interaction between an isolated colloidal particle and the solid–liquid interface [15,16]. FIG. 1 (color online). Segregated ice (darker regions) formed Recently, it has been recognized that in concentrated during the unidirectional solidification of a colloidal suspension systems the colloidal suspension itself can become con- (bentonite). Depending on the particle concentration, tempera- stitutionally supercooled [5,17]. A theoretical framework ture gradient, and freezing rate, the ice (a) rejects the particles, describing this phenomenon has been developed by taking (b) forms aligned dendrites, or (c) forms polygons. 0031-9007=08=100(23)=238301(4) 238301-1 © 2008 The American Physical Society PHYSICAL REVIEW LETTERS week ending PRL 100, 238301 (2008) 13 JUNE 2008 blocks (Fig. 2). A temperature gradient GT is maintained hydrodynamic radius R 0:5 m [21] and kB is by holding the blocks at temperatures TH and TCwith TH > Boltzmann’s constant. The empirical coefficients bk ac- Tf 0 >TC, where Tf 0 is the freezing temperature of count for long range electrostatic, structural, and van der the bulk suspension. The cell is moved through the blocks Waals interactions between the colloidal particles, while at a fixed speed V. We consider speeds V Vc, where Vc the 1 ÿ =p factor allows for excluded volume effects is the critical engulfment speed of an individual colloidal near the shrinkage limit. The solid curve in Fig. 3(a) shows particle. (For submicron hydrophilic particles Vc > the fit to the data, and the resulting prediction for Tf is 10 msÿ1 [15,18].) All of the particles will therefore be compared with experimental measurements in Fig. 3(b). pushed ahead by the growing ice and, if the interface The agreement is excellent for temperatures near or above remains stable, a continuous layer of ice will grow at steady ÿ5 C, which are relevant to our analysis. The dashed line state, pushing ahead a consolidated colloid boundary layer. in Fig. 3(b) is a reminder that pore ice may exist below In order to obtain the concentration-dependent freezing ÿ8 C [19], in which case alternative methods can be used temperature, we consider a system in which a portion of ice to predict the liquid fraction as a function of temperature is in equilibrium with a quantity of unfrozen colloid [19]. [22,23]. As the temperature is lowered, the ice phase grows and the Figure 4 shows measurements of the permeability of water content of the colloid decreases. The freezing tem- bentonite. The solid curve is a fit of the data in the form 8 2:6 ÿ1 2 perature can be obtained by equating the chemical poten- k k01 3:1 10 where k0 2R =9. tial of water in the unfrozen colloid with the chemical As shown previously [5], in colloidal suspensions one is potential of pure ice [5,20]. For temperatures near to the free to choose either Darcy’s law or Fick’s law to describe freezing temperature of pure water Tm, this leads to the the colloid mass flux. The two equations are related by a relation generalized form of the Stokes-Einstein relation k @ Tf Tm 1 ÿ ; (1) D ; (3) ‘Lf @ T;P where ‘ and Lf are the density and latent heat of fusion, where D is the particle diffusivity, is the dynamic respectively, of water. In order to use Eq. (1), knowledge of viscosity of the fluid, T is the absolute temperature, and the osmotic pressure is required. Figure 3(a) shows P is the mixture pressure. Given measurements of the measurements of obtained from several sources. To permeability and osmotic pressure, Eq. (3) determines fit the data over the full range in we use a virial-type the diffusivity as a function of volume fraction. expression in the form A typical experiment with freezing velocity V ÿ1 k 0:1 ms is shown in Fig. 5. In Fig. 5(a) we observe vp 1 bk the consolidation of the boundary layer above the ice- ; (2) kBTm 1 ÿ =p colloid interface as the system approaches steady state. 4 3 where vp 3 R is the volume of a bentonite particle of 8 0 10 (a) (b) T 6 H 10 −5 Π Tf o z water (Pa) ( C) 104 −10 suspension L0 2 ice 10 −15 0 0.2 0.4 0.6 00.2 0.4 0.6 φ φ TC FIG. 3. (a) Measurements of the osmotic pressure of bentonite V as a function of particle volume fraction (4 [20]; * [26]; ᮀ [27]; [28]). The solid curve is a fit of the data to Eq. (2) using b3 9 10 10 FIG. 2. Schematic of the unidirectional solidification stage. 8 10 , b4 ÿ2 10 , b5 1:3 10 , and p 0:64. The height, width, and depth of the cell between the blocks (b) Prediction from Eq. (1) (solid curve) and measurements are 6 cm, 12 cm, and 0.5 cm, respectively. A detailed description (symbols) of the freezing point depression of bentonite (4 of the experimental apparatus is given in Ref. [25]. [29]; + [19]; [30]; * [31]). 238301-2 PHYSICAL REVIEW LETTERS week ending PRL 100, 238301 (2008) 13 JUNE 2008 −15 10 where z^ z=0L0, Pe V0L0=D0 is the Peclet number, D^ D=D0 is the dimensionless diffusivity, and D0 kBTm=6R is the Stokes-Einstein diffusivity of an iso- lated colloidal particle. A boundary condition for (4)atz^ k 2 0 can be obtained by using global mass conservation in the (m ) form Z −20 i Pe D^ d; (5) 10 0 where i is the volume fraction at the ice-colloid interface. Once the concentration profile is determined from (4) 0 0.2 0.4 0.6 and (5), the freezing temperature profile can be obtained φ from (1). For a linear temperature gradient the steady-state temperature profile in the colloid is given by FIG. 4. Measurements of the permeability of bentonite as a T^ z^T^ G z;^ (6) function of particle volume fraction ( , Mesri and Olson [32]; +, f i T Kirby and Smiles [33]).
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