PHYSICS

E. A. ARINSHTEIN

ON THE STATISTICAL THEORY OF CRYSTALLIZATION

(Presented by Academician N. N. Bogolyubov on 12 II 1957)

In paper ($^1$), on the basis of N. N. Bogolyubov’s method, a statistical theory of crystallization of one-component systems was constructed. The method developed in that paper admits a generalization to the case of multicomponent systems with short-range forces. In this case the distribution functions are determined by the generating functional

[
\mathscr{L}(u_\alpha)=\int D\Pi\,(1+v_\alpha u_\alpha(i))\,dq_i,
]

containing (n) auxiliary functions (u_\alpha,\ \alpha=1,\ldots,n), where (n) is the number of components ($^2$). The functional equation for this functional takes the form

[
\frac{\delta \mathscr{L}(u_\alpha)}{\delta u_\alpha(1)}
=
(z_\alpha v_\alpha)\,
\mathscr{L}\left(
u_\beta(i)(1+f_{1i}^{\alpha\beta})+\frac{1}{v_\alpha}f_{1i}^{\alpha\beta}
\right).
\tag{1}
]

We obtain the equation for the generating functional of the correlation functions (W=\ln\mathscr{L}) by taking the logarithmic functional derivative of equation (1):

[
\frac{\delta^2 W(u)}
{\delta u_{\alpha_1}(1)\delta u_{\alpha_2}(2)}
=
\frac{\delta W(u)}{\delta u_{\alpha_1}(1)}
\left{
\frac{
\delta W\,u_\beta(i)(1+f_{1i}^{\alpha\beta})+\dfrac{1}{v_{\alpha_1}}f_{1i}^{\alpha\beta}
}
{\delta u_{\alpha_1}(2)}
-
\frac{\delta W(u)}{\delta u_{\alpha_2}(2)}
\right}.
\tag{2}
]

Just as was done in ($^1$), by means of further functional differentiation and successive substitutions one can express all distribution functions in terms of the single-particle functions (F_\alpha(i)) and find for the single-particle functions the equations

[
\ln\frac{F_{\alpha_1}(1)}{z_{\alpha_1}v_{\alpha_1}}
=
\sum_{\Sigma s=1}^{\infty}
\frac{1}{s_1!\cdots s_n!\,v_1^{s_1}\cdots v_n^{s_n}}
\times
]

[
\times
\int
K_{\alpha_1\ldots\alpha_{\Sigma+1}}
(1,\ldots,\Sigma s+1)\,
F_{\alpha_2}(2)\cdots
F_{\alpha_{\Sigma+1}}(\Sigma+1)\,
dq_2\cdots dq_{\Sigma+1}.
\tag{3}
]

The numbers (s_1,\ldots,s_n) indicate how many times the corresponding index occurs among the indices (\alpha_1,\ldots,\alpha_{\Sigma+1}). System (3) is equivalent to the problem of determining the extremum of the functional

[
A(F_\alpha)=
\vartheta
\left{
\sum_\alpha \frac{v}{v_\alpha}
\langle F_\alpha(1)\ln F_\alpha(1)\rangle_{\mathrm{cp}}
-
\right.
]

[
\left.
-
\sum_{\Sigma s=2}^{\infty}
\frac{v}{s_1!\cdots s_n!\,v_1^{s_1}\cdots v_n^{s_n}V}
\int
K_{\alpha_1\ldots\alpha_\Sigma}
(1,\ldots,\Sigma)\,
F_{\alpha_1}(1)\cdots F_{\alpha_\Sigma}(\Sigma)\,
dq_1\cdots dq_\Sigma
\right.
]

[
\left.
-
\sum_\alpha \frac{v}{v_\alpha}
\ln
\frac{(2\pi m_\alpha\vartheta)^{3/2}v_\alpha}{h^3}
\right}.
\tag{4}
]

under the additional conditions (\langle F_\alpha(1)\rangle_{\mathrm{av}}=1). Here (1+(z_\alpha \vartheta_\alpha)) are Lagrange multipliers, (1/v=\sum_\alpha 1/v_\alpha); a function with one index denotes the one-particle distribution function for the corresponding component. The functional (A(F_\alpha)) is, just as in ({}^{(1)}), the specific free energy. Verification of the thermodynamic identities (p=-\partial A/\partial v) and (E=-\vartheta^2\partial(A/\vartheta)/\partial\vartheta) is carried out in the same way as in ({}^{(1)}), with the sole difference that several binary functions (F_{\alpha_1\alpha_2}(i,j)) are introduced.

Expression (4) for the free energy makes it possible to solve the question of crystallization of multicomponent systems. Let us consider the example of a two-component system.

The question of the formation of solid solutions of one component in another reduces to the problem of determining a periodic solution for (F_1) when (F_2\equiv 1) (or conversely), which is identical to the problem considered in ({}^{(1)}). In view of the dependence of the thermodynamic functions on the concentration of the components, the liquidus curve and the solidus curve may differ (see, for example, ({}^{(3)})).

Let us consider the formation of a binary crystal of definite composition and structure, for example a crystal of the (\beta)-brass type: a simple cubic lattice of atoms of one kind, centered by atoms of another kind. We take the distribution functions in the form of a sum of Gaussian distributions ({}^{(1)}):

[
\begin{aligned}
F_1
&=\frac{v}{(2\pi)^{3/2}}
\left{
\frac{1+\xi}{r_1^3(\xi)}
\sum_l \exp\left[-\frac{(\mathbf r-\mathbf r_l)^2}{2r_1^2(\xi)}\right]
+
\frac{1-\xi}{r_1^3(-\xi)}
\sum_l \exp\left[-\frac{(\mathbf r-\mathbf r_l-\mathbf r')^2}{2r_1^2(-\xi)}\right]
\right}
\
&=
\sum_l \exp[i(\mathbf k_l\mathbf r)]
\left{
\left[
\frac{1+\xi}{2}\exp\left(-\frac12 k_l^2 r_1^2(\xi)\right)
+
\frac{1-\xi}{2}\exp\left(-\frac12 k_l^2 r_1^2(-\xi)\right)
\right]\cos\frac{(\mathbf k_l\mathbf r')}{2}
\right.
\
&\qquad\left.
-i\left[
\frac{1+\xi}{2}\exp\left(-\frac12 k_l^2 r_1^2(\xi)\right)
-
\frac{1-\xi}{2}\exp\left(-\frac12 k_l^2 r_1^2(-\xi)\right)
\right]\sin\frac{(\mathbf k_l\mathbf r')}{2}
\right},
\tag{5}
\end{aligned}
]

[
\begin{aligned}
F_2
&=\frac{v}{(2\pi)^{3/2}}
\left{
\frac{1-\xi}{r_2^3(-\xi)}
\sum_l \exp\left[-\frac{(\mathbf r-\mathbf r_l)^2}{2r_2^2(-\xi)}\right]
+
\frac{1+\xi}{r_2^3(\xi)}
\sum_l \exp\left[-\frac{(\mathbf r-\mathbf r_l-\mathbf r')^2}{2r_2^2(\xi)}\right]
\right}
\
&=
\sum_l \exp[i(\mathbf k_l\mathbf r)]
\left{
\left[
\frac{1+\xi}{2}\exp\left(-\frac12 k_l^2 r_2^2(\xi)\right)
+
\frac{1-\xi}{2}\exp\left(-\frac12 k_l^2 r_2^2(-\xi)\right)
\right]\cos\frac{(\mathbf k_l\mathbf r')}{2}
\right.
\
&\qquad\left.
+i\left[
\frac{1+\xi}{2}\exp\left(-\frac12 k_l^2 r_2^2(\xi)\right)
-
\frac{1-\xi}{2}\exp\left(-\frac12 k_l^2 r_2^2(-\xi)\right)
\right]\sin\frac{(\mathbf k_l\mathbf r')}{2}
\right}.
\tag{6}
\end{aligned}
]

Here (\mathbf r') is the displacement vector of the two simple Bravais lattices; (\xi) is the degree of order ({}^{(3)}). Substituting (5) and (6) into (4) and summing over the reciprocal lattice, all products of an odd number of functions (\sin\frac{(\mathbf k_l\mathbf r')}{2}) (for identical and different (k_l)) drop out, and, expanding (r_1) and (r_2) in (\xi), we obtain for the free energy the expression

[
\begin{aligned}
A_\tau
&=A_{\mathrm{l}}
-6\vartheta\Big{
[(\sigma_{11}(k)-1)(1+\alpha)^2x^2
-2\sigma_{12}(k)(1+\alpha)(1+\beta)xy
\
&\qquad
+(\sigma_{22}(k)-1)(1+\beta)^2y^2]\xi^2
+2[(\sigma_{11}(k\sqrt2)-1)(1-\xi^2+(1+\alpha)^2\xi^2)^2x^4
\
&\qquad
+2\sigma_{12}(k\sqrt2)(1-\xi^2+(1+\alpha)^2\xi^2)(1-\xi^2+(1+\beta)^2\xi^2)x^2y^2
\
&\qquad
+(\sigma_{22}(k\sqrt2)-1)(1-\xi^2+(1+\beta)^2\xi^2)^2y^4]
+\ldots
\Big}.
\tag{7}
\end{aligned}
]

The quantities (x, y, \alpha, \beta, \xi), where (\exp\left(-\dfrac{k^2 r_1^2(\xi)}{2}\right)=x(1+\alpha \xi)), (\exp\left(-\dfrac{k^2 r_2^2(\xi)}{2}\right)=y(1+\beta \xi)), are variational parameters in the expansion of the functional (A(F)). The integrals

[
\sigma_{ij}=\frac{4\pi}{v}\int_0^\infty K_{ij}(r)\frac{\sin kr}{kr}r^2\,dr
]

differ from the integrals introduced in ((1)) in that the kernels depend on the component indices.

Even without solving the full system of equations determining the equilibrium of the crystal and the magnitude of the variational parameters, one can find the conditions under which continuous crystallization is possible. In the vicinity of points of continuous crystallization the quantities (x, y) are small; in the vicinity of points of continuous ordering the quantity (\xi) will be small. Crystallization occurs if (A_t \ll A_{\mathrm{l}}).

The degree of ordering (\xi) is equal to zero if the quadratic form ((\sigma_{11}(k)-1)u^2-2\sigma_{12}(k)uv+(\sigma_{22}(k)-1)v^2) is negative definite, for which the conditions

[
\sigma_{11}(k)<1;\qquad (\sigma_{11}(k)-1)(\sigma_{22}(k)-1)>\sigma_{12}^2(k).
\tag{8}
]

must be satisfied.

It follows from (8) that simultaneously (\sigma_{22}(k)<1). If, in addition, the form ((\sigma_{11}(k\sqrt{2})-1)w^2+2\sigma_{12}(k\sqrt{2})wt+(\sigma_{22}(k\sqrt{2})-1)t^2) is also negative in the region of positive values of (w) and (t), then the crystalline structure is absent. For this, in addition to conditions (8), the conditions

[
\sigma_{11}(k\sqrt{2})<1;\qquad
(\sigma_{11}(k\sqrt{2})-1)(\sigma_{22}(k\sqrt{2})-1)>\sigma_{12}^2(k\sqrt{2}),
\tag{9}
]

must be satisfied; or, if the second of conditions (9) is violated, the conditions

[
\sigma_{11}(k\sqrt{2})<1;\qquad
\sigma_{22}(k\sqrt{2})<1;\qquad
\sigma_{12}(k\sqrt{2})<0.
\tag{10}
]

Violation of conditions (8), when conditions (9) or (10) are fulfilled, leads to continuous crystallization into an ordered structure. The points of such a transition are determined by the equation

[
(\sigma_{11}(k)-1)(\sigma_{22}(k)-1)-\sigma_{12}^2(k)=0.
\tag{11}
]

If, conversely, conditions (9) and (10) are violated while conditions (8) are fulfilled, then, provided the following term of expansion (7), not containing (\xi), is positive, continuous crystallization into a disordered structure will occur at points satisfying the equation

[
(\sigma_{11}(k\sqrt{2})-1)(\sigma_{22}(k\sqrt{2})-1)-\sigma_{12}^2(k\sqrt{2})=0.
\tag{12}
]

Equations (11) and (12) define two curves in the (v,T) plane. At their point of intersection the curve (12) terminates, while the curve (11) passes into the curve of transition from the disordered to the ordered state. Expansion (7) contains only even powers of the ordering parameter (\varepsilon), which leads to results known from the thermodynamic theory of second-order phase transitions developed by L. D. Landau and E. M. Lifshitz ((3)).

Tomsk
Civil Engineering Institute

Received
12 II 1957

REFERENCES

1. E. A. Arinshtein, DAN, 112, No. 4 (1957).
2. N. N. Bogolyubov, Problems of Dynamical Theory in Statistical Physics, 1946.
3. L. D. Landau, E. M. Lifshitz, Statistical Physics, 1951.