XXIV ICTAM, 21-26 August 2016, Montreal, Canada
FREEZING SOLUTIONS AND COLLOIDAL SUSPENSIONS
1
M. Grae Worster ∗1 , J.S. Wettlaufer2, 3, 4 , and S.S.L. Peppin2
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, CMS Wilberforce Road,
Cambridge CB3 0WA, UK
2
Yale University, New Haven, CT 06520, USA
3
Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
4
Nordita, Royal Institute of Technology and Stockholm University, SE-10691 Stockholm, Sweden
Summary We present a theoretical model of solidification of a colloidal suspension when a partially frozen region (mushy layer) forms.
The relative motion between solvent and colloidal particles within the mushy layer can be described as diffusion of particles, as regelation
of the particles or as flow of the interstitial solvent through the porous matrix of particles driven by a cryosuction pressure. We illustrate
these ideas using similarity solutions of the governing equations.
INTRODUCTION
When mixtures are frozen, their constituents tend to become segregated. This is equally true whether the mixture is a
molecular solution or a suspension of solid particles. The physical character of the segregated, partially frozen material can
differ markedly between solutions and suspensions (figure 1), though similar structures to those produced from solutions
(figure 1a) are sometimes seen when the suspension is dilute and colloidal (figure 1b).
Any segregation requires relative motion (transport) between the constituents. In the case of solutions, such relative
motion is typically described by Fickian diffusion, with fluxes proportional to concentration gradients. In the case of dense
suspensions, particularly those near to being close-packed, such relative motion is typically described by Darcy’s law of fluid
motion through porous media, with fluxes proportional to pressure gradients. For example, in the case illustrated by figure 1c,
water must flow from the upper layer of original, colloidal suspension, through the intermediate porous medium formed by
the compacted suspension, whether unfrozen or partially frozen, to form the ice lenses below.
Peppin, Elliott & Worster [1] showed that Fickian diffusion and Darcy flow are equivalent and are end members of a unified
thermodynamic model of relative motion. They used their formulation [2] to calculate freezing rates and segregation at planar
interfaces of ice growing from suspensions of different particle sizes, representing the two extremes of dilute solutions and
concentrated suspensions. Both are prone to constitutional supercooling in a layer of rejected solute (colloid), which can lead
to morphological instability of the interface [3, 4], as originally quantified for solutions by Mullins & Sekerka [5].
Here we extend this unified thermodynamic approach to the solidification of mixtures by analysing the formation of mushy
layers from colloidal suspensions, simply using a phase diagram appropriate for colloidal suspensions and following the
approach established for solutions [6]. The unified approach allows connections to be made between studies of solidification
of binary alloys and studies of freezing in soils through the intermediate cases of colloidal suspensions, particularly the
equivalence of solute diffusion to regelative fluxes in partially frozen systems.
(a)
(b)
(c)
Figure 1: Segregation during solidification upwards of solutions and suspensions. (a) Dendritic microstructure of ammonium chloride solidifying
from aqueous solution (photo MA Hallworth). (b) Dendritic microstructure of ice solidifying from a dilute aqueous suspension of bentonite (5%
by mass) (photo SSL Peppin). (c) From bottom to top, there is a layer of ice lenses (dark coloured) separating lenses of frozen colloid (light
coloured); a continuous layer of frozen colloid (also light coloured); a thicker layer of unfrozen, compacted colloid (dark grey); all forming from
an aqueous suspension of alumina (27% by volume of approximately 3µm particles) (uppermost, light grey layer)[7].
∗ Corresponding
author. Email: [email protected]
MATHEMATICAL FORMULATION
Central to the mathematical formulation is an expression for the osmotic pressure
kB Tm
z(φ),
Π(φ, R) = φ
vp (R)
(1)
where φ is the volume fraction of solute (colloid), kB is Boltzmann’s constant, Tm is the absolute freezing temperature of
the solvent, vp = 34 πR3 is the (effective) particle volume for each element of the solute of (effective) radius R and z(φ)
is a dimensionless ‘compressibility factor’ that accounts for particle–particle interactions. From the osmotic pressure, the
equilibrium freezing temperature (liquidus) is calculated from the Clausius-Clapeyron equation and the diffusion coefficient
is determined from the generalised Stokes-Einstein
relation
as
Π(φ, R)
k ∂Π(φ, R)
Tf (φ, R) = Tm 1 −
and
D(φ, R) = φ
(2)
ρL
µ
∂φ
respectively, where k(φ, R) is the permeability of the suspension, µ is the dynamic viscosity of the solvent, ρ is density and
L is the latent heat of fusion. The solute flux, both in unfrozen and partially frozen (mushy) colloid, can be written in three
equivalent forms as
k ∂p
k ρL ∂Tf
∂φ
=φ
=φ
,
(3)
−D(φ)
∂z
µ ∂z
µ Tm ∂z
where χ is the volume fraction of unfrozen colloid and p = P − Π is the ‘pervadic pressure’ [1], which is the difference
between the bulk pressure P and the osmotic pressure and can be interpreted as the pore pressure in a close-packed colloid or
porous medium. In solutions and dilute colloidal suspensions, it is convenient to work with and think in terms of the diffusive
flux (3a) but as the close-packed limit is approached the diffusivity diverges while the concentration gradient tends to zero and
it is more convenient to work with and think in terms of the Darcy Flux (3b). Within a mushy region T = Tf and the diffusive
flux is equal to the regelative flux (3c) driven by ‘thermodynamic buoyancy’ [8].
RESULTS
We have used these fundamental ideas to determine similarity solutions for solidification from a cooled boundary within
partially frozen colloidal suspensions (e.g. figure 2). As particle size increases, the interfacial concentration increases and the
concentration more quickly reaches the close-packed limit within the mushy layer.
0.25
0.402
0.41
0.401
0.4
0.4
0.2
0.39
(a)
0.15
0.28
0.3
0.32
0.34
?
0.36
0.38
0.38
(b)
0.399
0.3
0.35
0.4
0.45
0.5
?
0.55
0.398
(c)
0.3
0.35
0.4
0.45
0.5
0.55
0.6
?
Figure 2: The particle volume fractions (solute concentration) as functions of vertical position for particle sizes (a) 0.5nm, (b) 2nm and (c)
10nm in the vicinity of the mush–liquid interface (dashed line): mush below (red); unfrozen colloid above (blue). Note the different axis scales.
CONCLUSIONS
Regelation within a partially frozen colloidal suspension or soil results from a balance between thermodynamic buoyancy,
driving particles towards warmer temperatures, and viscous drag forces associated with flow through the partially frozen
porous medium. It is fundamentally equivalent to diffusion of solute or of particles within a molecular solution or dilute
suspension of small particles. We have developed a unified model that illustrates this equivalence and provides a means of
computing the characteristics of freezing colloids of arbitrary particle size and concentration.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
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