Morphological Stability of a Cylinder

Abstract

The stability of the shape of a solid cylinder crystallizing in a supercooled liquid is treated. The effects of solute diffusion, slightly anisotropic surface tension and interface kinetics are included. The resulting stability equations are applied to the specific case of ice cylinders.

1. Introduction

The stability of the shape of a solid growing by diffusion or heat flow was first studied by Mullins and Sekerka [1]¹, who determined the stability criteria for a sphere. The cylindrical geometry has been treated by Coriell and Parker [2] and Kotler and Tiller [3]. These studies assumed that the interface properties were isotropic. Cahn [4] has taken account of slightly anisotropic surface free energy for the sphere. Recently, we have made experimental studies [5, 6] of the morphological stability of ice cylinders. In connection with this experimental program, it appears desirable to work out the stability of a cylindrical shape taking account of slightly anisotropic surface tension and interface kinetics. In this paper we treat the case of a cylinder crystallizing from a binary melt; it is assumed that the surface tension and kinetic coefficient are slightly anisotropic. For the case of isotropic interface properties, the problem reduces to the case previously treated by Kotler and Tiller [3].

Although our calculation is general, we are particularly interested in the case of an ice cylinder growing from pure water and from water with impurities added. We present some specific calculations for these cases.

2. Formulation and Calculation

We wish to solve Laplace’s equation [1, 2] for the temperature T and concentration C of impurity. The subscripts S and L denote solid and liquid, respectively, while a subscript I denotes that the quantity is to be evaluated at the interface. We assume a slightly perturbed cylinder of shape [2]

$$r=R+\text{δ}{e}^{ik\Phi }{e}^{i2\pi z/\lambda },$$ (1)

where r, Φ, and z are the usual cylindrical coordinates, R is the radius of the unperturbed cylinder, δ is the amplitude of the perturbation (δ/R ⪡ 1), and k and λ determine the shape of the perturbation. Neglecting diffusion in the solid, the equations to be solved are

$${\nabla }^2{T}_L={\nabla }^2{T}_S={\nabla }^2{C}_L=0$$ (2)

with the auxiliary equations

$$T_L\left(R_b\right)=T_b$$ (3a)

$$T_S\left(0\right)-\text{finite}$$ (3b)

$$T_{LI}=T_{SI}$$ (3c)

$$C_L\left(R_c\right)=C_b$$ (3d)

$$v=\frac{k_{\text{S}}}{L_v}{\left(\frac{\partial T_{\text{s}}}{\partial r}\right)}_I-\frac{k_{\text{L}}}{L_v}{\left(\frac{\partial T_{\text{L}}}{\partial r}\right)}_I$$ (3e)

$$v=\left\{\sum _k{\mu }_k{e}^{ik\Phi }\right\}\left\{T_e-T_{LI}\right\}$$ (3f)

$$v=\frac{-D_{\text{L}}}{C_{\text{LI}}\left(1-j\right)}{\left(\frac{\partial C_{\text{L}}}{\partial r}\right)}_I$$ (3g)

$$v=\dot{R}+\dot{\delta }{e}^{ik\Phi }{e}^{i2\pi /\lambda }$$ (3h)

$$T_e=T_{\text{M}}-T_{\text{M}}K\left\{\sum _k{\Gamma }_k{e}^{ik\Phi }\right\}-m{C}_{LI}$$ (3i)

and

$$K=\frac{1}{R}+\frac{\text{δ}}{R^2}\left(k^2+k_z^2-1\right)e^{ik\Phi }{e}^{i2\pi z/\lambda }.$$ (3j)

In the above equations R_b is the bath radius, T_b (a constant) is the temperature at R_b, R_c is the radius at which the concentration of impurity is C_b (a constant), v is the velocity of the interface, k_S and k_L are the thermal conductivities of the solid and liquid, respectively, L_v is the latent heat per unit volume of the solid, the $\left(\sum _k{\mu }_k{e}^{ik\Phi }\right)$ is the slightly anisotropic linear kinetic coefficient,² T_e is the equilibrium temperature of the interface, D_L is the solute diffusion coefficient, j is the partition coefficient (the ratio of the equilibrium concentration of solute on the solid side of the interface to that on the liquid side of the interface), $\dot{R}$ and $\dot{\delta }$ are the rate of growth of the unperturbed cylinder and the perturbation, respectively, T_M is the melting point of a flat interface, K is the curvature of the perturbed cylinder, the $\left(\sum _k{\Gamma }_k{e}^{ik\Phi }\right)$ is the slightly anisotropic capillary constant, m is the freezing point lowering constant, and k_z = 2πR/λ. The solution of Laplace’s equation in cylindrical coordinates is of the form C₁ + C₂ ln r + e^ikΦe^i2πz/λ [C₃I_k (k_zr/R) + [C₄K_k (k_zr/R)] where the C’s are constants and I_k and K_k are modified Bessel functions. A solution of this form is written for T_L, T_S, and C_L and the constants are determined by the boundary conditions. Since we are treating small perturbations and slightly anisotropic surface tension and interface kinetics, we neglect all terms that are greater than first order in δ, μ_k, and Γ_k (k ≠ 0), i.e., we omit terms containing δ², δμ_k, δΓ_k, Γ_kμ_k, etc., (k ≠ 0). Although the calculation is tedious, it is straightforward and we will not reproduce the details here. We define A_b = ln (R_b/R), A_c= ln (R_c/R),

$$\Delta T=T_M-\left(T_M{\Gamma }_0/R\right)-T_b,\Delta \theta =\Delta T-m{C}_b$$

$${\beta }_A=\left(k_L/R{L}_v{\mu }_0\right)+A_b,\xi =\left[\left(1-j\right)k_L{A}_C\right]/\left[D{L}_v\right]$$

$$A_1=\left[K_k\left(k_z{R}_b/R\right)/I_k\left(k_z{R}_b/R\right)\right],$$

$$D_1=\left[K_k\left(k_z{R}_c/R\right)/I_k\left(k_z{R}_c/R\right)\right],$$

$$J_A=-k_z\left\{\left[K_k^{\prime }\left(k_z\right)-A_1{I}_k^{\prime }\left(k_z\right)\right]/\left[K_k\left(k_z\right)-A_1{I}_k\left(k_z\right)\right]\right\},$$

$$J_D=-k_z\left\{\left[K_k^{\prime }\left(k_z\right)-D_1{I}_k^{\prime }\left(k_z\right)\right]/\left[K_k\left(k_z\right)-D_1{I}_k\left(k_z\right)\right]\right\},$$

$$H_I=k_z{I}_k^{\prime }\left(k_z\right)/I_k\left(k_z\right),$$

where the prime indicates the derivative with respect to k_z. The results of the calculation are

$$\dot{R}=\left(k_L/R{L}_v{A}_b\right)\left(DT\right),$$ (4a)

where (DT) is given by

$$\left(DT\right)=A_b\left[\frac{1}{2\xi }+\frac{\Delta T}{2{\beta }_A}\right]-\left|A_b{\left\{{\left[\frac{1}{2\xi }+\frac{\Delta T}{2{\beta }_A}\right]}^2-\frac{\Delta \theta }{{\beta }_A\xi }\right\}}^{1/2}\right|.$$ (4b)

For the ratio $\left(\dot{\delta }/\delta \right)/\left(\dot{R}/R\right)$ we obtain

$$\frac{\dot{\delta }R}{\delta \dot{R}}={\left\{1+\frac{\alpha }{{\mu }_0}+\frac{m\alpha }{\nu }\right\}}^{-1}\left\{J_A-1+\frac{m\alpha }{\nu }\left(J_D-1\right)+\frac{\left[J_A+\left(k_S/k_L\right)H_I\right]A_b}{\left(DT\right)}\left[\frac{\dot{R}R{\mu }_k}{\delta {\mu }_0^2}-\frac{T_M{\Gamma }_k}{\delta }-\frac{T_M{\Gamma }_0\left(k^2+k_z^2-1\right)}{R}\right]\right\},$$ (4c)

where α = (k_L/L_VR) (J_A + k_SH_I/k_L) and

$$\nu =\frac{k_{\text{L}}{A}_C}{L_{V}R{C}_b\xi }\left[1-\frac{\xi \left(DT\right)}{A_b}\right]\left[J_D-\frac{\xi \left(DT\right)}{A_C{A}_b}\right].$$

In the above equations Γ_k and μ_k(k ≠ 0) are zero if k_z ≠ 0, i.e., the anisotropic terms only effect Φ-type perturbations and have no effect on perturbations along the axis of the cylinder. This follows, for example, since there are no terms of the form Γ_{k, kz}e^ikΦe^i2πz/λ in eq (3i) (see footnote 2).

Thus one of the important results of this calculation is that the growth rate of z-perturbations is independent of slightly anisotropic surface tension and kinetic coefficient.

Discussion

The general results of our stability analysis are given by eqs (4a–c). In this section, we apply these equations to the special case of ice cylinders (oriented with the c-axis of ice parallel to the cylinder axis) growing in slightly supercooled water (ΔT ≅ 0.1 °C). We also consider the case in which impurities have been added to the water. We are interested in the case where R lies between 0.04 cm and 0.25 cm; the bath radius R_b in most of the experimental work was 0.875 cm. Under the above conditions we can simplify the equations by neglecting interface kinetics, i.e., we take $1+\frac{\alpha }{{\mu }_0}\cong 1.$ This implies that (k_L/RL_V) (J_A + H_Ik_S/k_L) ⪡ μ₀; typically (J_A + H_Ik_S/k_L) < 100 and the above approximation is valid if μ₀ ⪢ .05 cm deg⁻¹ sec⁻¹. Although the interface kinetic coefficient for ice growing perpendicular to the c-axis is not known, a lower bound can be obtained from bath undercooling measurements. For example, Lindenmeyer et al. [7], observed a growth velocity of 1 cm sec⁻¹ at a bath undercooling of 2 °C. Assuming a linear law, this gives μ₀ > .5 cm sec⁻¹ deg⁻¹ [8]. Since μ₀ may be orders of magnitude greater than this, it appears valid to neglect interface kinetics. It follows that β_A ≅ A_b.

It is almost always (and certainly under the experimental conditions of interest) valid to neglect the A₁ and D₁ terms in the definition of J_A and J_D. Defining $H_K=-k_z{K}_k^{\prime }\left(k_z\right)/(K_k\left(k_z\right),$ we then have J_A ≅ H_K≅ J_D. With this approximation and the neglect of interface kinetics, eqs (4b) and (4c) can be written

$$\left(DT\right)=\frac{1}{2}\left(\left[\left(A_b/\xi \right)+\Delta T\right]-\left|{\left\{{\left[A_b/\xi )+\Delta T\right]}^2-4\Delta \theta A_b/\xi \right\}}^{1/2}\right|\right)$$ (5a)

and

$$\begin{array}{l}\frac{\dot{\delta }R}{\delta \dot{R}}=H_K-1+\frac{\left[H_K+\left(k_{\text{S}}/k_L\right)H_I\right]A_b}{\left(1+m\alpha /\nu \right)DT}\\ \times \left[\frac{\dot{R}R{\mu }_k}{\delta {\mu }_0^2}-\frac{T_M{\Gamma }_k}{\delta }-\frac{T_M{\Gamma }_0\left(k^2+k_z^2-1\right)}{R}\right].\end{array}$$ (5b)

For pure water, there is further simplification, viz, taking m = 0 and ξ → 0,

$$\left(DT\right)=\Delta T$$ (6a)

and

$$\begin{array}{l}\frac{\dot{\delta }R}{\delta \dot{R}}=H_K-1+\frac{\left[H_K+\left(k_{\text{S}}/k_L\right)H_I\right]A_b}{DT}\\ \times \left[\frac{\dot{R}R{\mu }_k}{\delta {\mu }_0^2}-\frac{T_M{\Gamma }_k}{\delta }-\frac{T_M{\Gamma }_0\left(k^2+k_z^2-1\right)}{R}\right].\end{array}$$ (6b)

Recalling that for a k_z ≠ 0 perturbation μ_k = 0 and Γ_k = 0 (k ≠ 0), eq (6b) becomes

$$\frac{\dot{\delta }R}{\delta \dot{R}}=H_K-1-\frac{T_M{\Gamma }_0\left(k^2+k_z^2-1\right)A_b\left[H_K+\left(k_S/k_L\right)H_I\right]}{R\left(DT\right)}.$$ (6c)

This equation has been used in references 5 and 6 to analyze the experimental data.

We now give a brief discussion of the choice of the bath boundary conditions, i.e., the choice of R_b and R_c; we also discuss the use of Laplace’s equation. For an infinite bath, the choice of these parameters has been previously discussed [2]. For an infinite bath R_b = R_λ where R_λ = R/(1.33λ) and λ satisfies

$${\lambda }^2{e}^{{\lambda }^2}Ei\left(-{\lambda }^2\right)+S=0.$$

In the above equation Ei is the exponential integral function and for heat flow S = S_T = C_v(T_I −T_b)/L_v and for impurity diffusion [9] S = S_c = (C_LI − C_b)/C_LI(1 − j). In these equations T_b and C_b are the temperature and concentration at infinity, respectively, and C_v is the specific heat per unit volume of the liquid. In order for Laplace’s equation to be valid, it is necessary that S_T ⪡ 1 and S_c ⪡ 1. Since $S_T\cong \left(\frac{1}{73}\right)\Delta T,$ it is clear that for ΔT < 1, S_T ⪡ 1. Using the solution of the diffusion equation, we may rewrite S_c as S_c = (A_c/A_b)(k_L/DL_v)DT. Taking k_L = 1.33(10⁻³) cal cm⁻¹ deg⁻¹ sec⁻¹, L_v = 73.4 cal cm⁻³, D = 10⁻⁵ cm² sec⁻¹, and (A_c/A_b) ≅ 1, we have S_c ≅ 1.8 (DT). For DT = 0.1, S_c ≅ 0.2, and Laplace’s equation is a reasonable approximation. For larger (DT) and for smaller diffusion constants, there may well be deviations from Laplace’s equation.

In the actual experiments the temperature is maintained at T_b at some radius, say R_a, If R_a < R_λ, it seems reasonable to take R_b = R_a and this has been done [5, 6]. If R_a > R_λ, it is better to take R_b = R_λ. If this were not done, i.e., if instead we let R_b = R_a when R_a > R_λ, then from eq (4a) we would predict that $\dot{R}$ is larger when T_L(∞) = T_b than when T_L(R_a) = T_b. This is obviously wrong and hence for R_a > R_λ we take R_b = R_λ. When R_a < R_λ, there is a certain error in taking R_b = R_a, but this error lies within the experimental error in measuring ΔT and $\dot{R}$. The choice of R_c is even more complicated since the average concentration in the liquid changes as solute is rejected from the ice. Fortunately for many cases the results are not very sensitive to the choice of R_c and for calculational purposes we take R_c = R_a. The proper choice of R_c and R_b can be studied experimentally by comparing the observed growth rate $\dot{R}$ of the unperturbed cylinder with the theoretical $\dot{R}$.

We present some calculations of δ as a function of R for the special case where $\left(\dot{\delta }R/\delta \dot{R}\right)$ is given by eq (6c). Following Cahn [4] we use the relationship $\left(\dot{\delta }R/\delta \dot{R}\right)=\left(R/\delta \right)\left(d\delta /dR\right)$. Denoting the right hand side of eq (6c) by f(R), we have

$$\frac{d\text{ln}\delta }{dR}=\frac{f(R)}{R}.$$ (7a)

Integrating yields

$$\text{ln}\left(\delta /{\delta }_0\right)={\int }_{R_0}^R\frac{f\left(x\right)}{x}dx,$$ (7b)

where δ₀ is the value of δ at R = R₀. The integral has been evaluated numerically for various values of the parameters occurring in f(R). In many cases over small ranges of R, ln δ is to a good approximation linear with R. This is illustrated in table 1 which gives (d ln δ/dR) as a function of R. From table 1, it is seen that (d ln δ/dR) changes very slowly with R. Since in the experimental measurements of δ versus R, R varies by about 0.02 to 0.03 cm and R is greater than 0.120 cm, it is to be expected that plots of the experimental data in the form ln δ versus R will appear linear.

Table 1.

(d ln δ/dR) as a function of R for λ = 0.055 cm; y = 0.018 J/m² and ΔT =0.1 deg.

R(cm)	(d ln δ/dR) (cm−1)
0.100	85.8
.110	84.3
.120	82.9
.130	81.7
.140	80.7
.150	79.7
.160	78.9
.170	78.1
.180	77.4
.190	76.8
.200	76.3

Of interest is the wavelength dependence of the δ versus R curves. This is shown in table 2 for three different wavelengths. We can consider δ/δ₀ as an amplification factor, that is, it gives the relative magnitude of δ at R as compared to δ at R₀. As seen from the table, these amplification factors are very large. Experimentally values of δ of 3(10⁻³) cm have been observed at R ≅ 0.175 cm. Extrapolating to R = 0.05 cm gives a value of δ ≅ 10⁻⁷ cm as the size of the initial perturbation. It is clear from table 2 that the amplification factor is a function of wavelength. For small R, perturbations with λ = 0.045 cm are amplified slightly more than perturbations corresponding to the other wavelengths of the table. For larger R, however, the amplification is greatest for λ = 0.055. Calculations similar to these will be useful in attempting to predict the wavelength of the perturbation which appears on a growing cylinder. However, such a prediction requires some assumption about the initial distribution of the amplitudes of the perturbations of various wavelengths.

Table 2.

Effect of wavelength on δ/δ₀ for λ = 0.018 J/m² and ΔT = 0.1 deg.

R(cm)	(δ/δ0)
	λ = 0.045 cm	λ = 0.055 cm	λ = 0.065 cm
0.050	1.00	1.00	1.00
.075	1.08(101)	1.07(101)	9.88
.100	9.85(101)	9.70(101)	8.16(101)
.125	7.80(102)	7.91(101)	6.07(102)
.150	5.53(103)	5.99(103)	4.22(103)
.175	3.59(104)	4.28(104)	2.80(104)
.200	2.17(105)	2.93(105)	1.80(105)

We wish to discuss the effect of solute on the stability equations. It is interesting that for no anisotropy and zero surface tension, the addition of solute has no effect on $\left(\dot{\delta }/\delta \right)/\left(\dot{R}/R\right)$, i.e., eqs (5b) and (6b) are identical. The addition of solute changes the unperturbed growth rate by changing DT. For a bath undercooling ΔT = 0.1, the value of (DT) for various values of mc_b are given in table 3.

Table 3.

Value of effective undercooling (DT) as a function of added solute for bath undercooling ΔT = 0.1 °C (ξ/A_b = 1.8 deg⁻¹ used in the calculation)

mcb	DT
0.001	0.0988
.01	.0881
.02	.0768
.04	.0556
.06	.0359
.08	.0174

Also of interest is the factor 1 + (mα/ν) appearing in eq (5b). It can be shown that

$$\left(m\alpha /\nu \right)\cong \left(k_L/D{L}_v\right)\left(m{c}_b\right)\left(1+k_S/k_L\right)\cong 9\left(m{c}_b\right).$$

Thus for mc_b < 0.1 deg., 1 + (mα/ν) varies from 1 to 2. Since DT varies more rapidly than this, the main effect of adding solute is to decrease DT and thus to increase the magnitude of the last term in eq (5b). Thus for fixed ΔT adding solute makes $\left(\dot{\delta }/\delta \right)/\left(\dot{R}/R\right)$ smaller, thus stabilizing the cylinder. On the other hand for fixed (DT), which corresponds to not changing the unperturbed velocity $\dot{R}$, the addition of solute increases $\left(\dot{\delta }/\delta \right)/\left(\dot{R}/R\right)$ and consequently makes the cylinder more unstable. The above statements assume that the Γ₀ term is the dominant term inside the bracket of eq (5b) and that the addition of solute does not change the physical properties such as the surface tension. If solute is adsorbed at the interface, one expects a lowering of the surface tension. One method of studying such effects is to add a very small amount of solute, e.g., 10⁻⁴ M NaCl in which case mc_b ≅ 4(10⁻⁴). From the preceding calculations, it is clear that DT ≅ ΔT and 1 + (mα/ν) ≅ 1 so that eqs (6a) and (6b) can be used. Thus any difference between experimental results for distilled water and water containing 10⁻⁴ M of impurities should probably be attributed to a change in the surface tension due to adsorption at the interface.

In summary, we have analyzed the stability of a solid cylinder growing by heat flow into a binary melt and have taken account of any small anisotropy in the interface properties. In particular, we have provided a theoretical framework for the experimental study of the stability of ice cylinders.