15972
J. Phys. Chem. 1996, 100, 15972-15985
Informational Models for the Configurational Entropy of Regular Solid Solutions: Flat Lattices Victor L. Vinograd* Theoretical Geochemistry Program, Institute of Earth Sciences, Uppsala UniVersity, S-75236, Uppsala, Sweden
Leonid L. Perchuk Institute of Experimental Mineralogy, Russian Academy of Sciences, 142432, ChernogoloVka, Moscow District, Russia ReceiVed: February 9, 1996; In Final Form: April 23, 1996X
The present study attempts to improve the results of the well-known quasi-chemical (pair) approximation for systems with short-range interactions. Using an analogy between expressions arising in information theory and in statistical thermodynamics, the procedure of derivation of configurational factors for 2D Ising models was found which guarantees nonnegative sign of the configurational entropy at any set of point and pair probabilities. The present approximation predicts values of critical parameters for 2D Ising models of a ferromagnetic with the accuracy of 1-3% in comparison to the exact results for square, honeycomb, and triangular lattices.
Introduction Our interest in statistical modeling of flat lattices has been caused by the need of thermodynamic description of solid solutions in mica and feldspar composition space. In phlogopite-eastonite solid solution, K (Mg3-xAlx)(Al1+xSi3-x)O10(OH)2, Mg,Al and Al,Si mixing occurs in octahedral and tetrahedral sites of mica structure. The arrangement of these sites forms layers which can be closely described by triangular and honeycomb flat lattices. In plagioclase solid solution, (Na1-xCax)(Al1+xSi3-x)O8, Al and Si cations occupy the tetrahedral sites of four-coordinated three-dimensional (3D) framework feldspar lattice. The 2D square lattice, which has the same coordination number, can be considered as a simple analogue of the real lattice and used for a qualitative study of the Al,Si order-disorder effects. Recent 29Si NMR1-6 and OH-stretching IR6 spectroscopy data on these minerals suggested the existence of strong short range order (SRO) effects which mainly result in a significant decrease of the density of Al-O-Al contacts (Al avoidance) among Al,Si and Mg,Al cations. Numerous X-ray diffraction studies of these minerals summarized in the reviews by Bailey7 and Ribbe8 gave evidence for long-range ordering (LRO) of Mg,Al and Al,Si cations at compositions with stoichiometry Mg2Al and Al2Si2. This experimental information allows one to conclude that the appearance of LRO at the specific compositions can be caused by cooperative effects of strong unequal interactions between neighboring like and unlike atoms. Therefore, it seems probable that the apparatus of statistical thermodynamics which has been widely used for the description of cooperative effects in systems with short-range interactions9 (Ising models, regular solutions) can be applied for the thermodynamic description of cationic ordering in these minerals. However, the known exact solutions10,11 for 2D Ising models correspond only to the specific composition (x ) 0.5) of an AxB1-x system (where A and B designate atoms with different spin orientations) and describe only equilibrium case of ordering. The well-known approximate methods such as quasi-chemical12-14 and cluster variation15,16 X
Abstract published in AdVance ACS Abstracts, June 15, 1996.
S0022-3654(96)00416-9 CCC: $12.00
(CVM) are either insufficiently rigorous or mathematically cumbersome. In fact, the application of quasi-chemical and loworder CVM approximations (“pair”, “star”, and “hexagon”) for the description of Al,Si ordering in layer17 and framework silicates18 gives reasonable estimates of the Al,Si configurational entropy only in cases when the composition of the (Al1+xSi3-x) solid solution is far from the stoichiometry Al2Si2. If the composition of the solution approaches this stoichiometry and if the degree of SRO is high, the models predict strongly negative values of entropy. A more rigorous description of atomic distribution in systems with short-range interactions (regular assemblies9) can be done with the help of high-order approximations of the cluster variation method.15,16 These approximations, however, involve an increased number of configurational variables and the necessity of solving systems of complicated nonlinear equations. Despite the remarkable progress in the development of rapid computational algorithms and in generalizations of the cluster-variation method,19-21 there is still a need in more simple statistical-thermodynamic approaches which can be used for a routine modeling of regular solutions. This study aims to develop such an approach for systems of low dimensionality. It can be easily shown that the result of the “pair approximation”, which is known to be correct for 1D Ising model, can be reproduced using the theory of Markov chains.22 In fact, Markov chain process allows one to simulate typical sequence of A and B symbols (text) with a given SRO constraint (like that of avoidance of A-A contacts) and to calculate the value of information23 of the text with respect to one symbol. The calculated value of information appears to be proportional to the configurational entropy of an AxB1-x 1D solution which in quasi-chemical and cluster-variation models is evaluated by combinatorial methods.14,15 Lapides and Lustenberg24-26 have shown that it is possible to define stochastic processes which result in 2D configurations, where A and B symbols are arranged according different topologies and different ordering schemes. We noticed that the entropies of two-dimensional sequences of symbols can be also easily calculated using the informational approach. Moreover, it was found that in cases of square, © 1996 American Chemical Society
Configurational Entropy of Regular Solid Solutions
J. Phys. Chem., Vol. 100, No. 39, 1996 15973 energy and entropy terms allows us to write the expression for the free energy of an isolated system:
Fσ ) Eσ - TSσ
(1)
This expression can be minimized with respect to the order parameters at any given temperature and the equilibrium properties of an isothermal system can be found. Simulation of Atomic Configurations Using Stochastic Algorithms Entropy and Information of an Atomic Configuration. One of the basic concepts of the communication theory23 is the information source which produces a message or a sequence of events (symbols). The discrete information source can be modeled by a certain stochastic process (Markov chain). The output of this process on each step results in a certain event which belongs to a set of possible events {A, B, C, ...}. In the general case the output can depend on previous events. Shannon23 defined the information of the message with respect to one symbol as the mathematical expectation of the logarithm of the inverse probability of a unit event ζi:
H ) cM{logb(1/P(ζi))}
Figure 1. Schemes of construction of different lattices discussed in the study. (A) 1D lattice. (B) Square lattice. (C) Triangular lattice. (D) Honeycomb lattice. Arrows show the direction of lattice growth.
honeycomb, and triangular lattices the parameters of stochastic processes (transition probabilities) can be rigorously defined as functions of point and pair probabilities. That result allowed us to construct entropy expressions which, in contrast to those used in the quasi-chemical model, predict nonnegative values in all space of order and composition parameters. The new approach is constructive and allows one to simulate all the configurations which contribute to the value of entropy. It was found, however, that the 2D configurations resulted from sequential stochastic algorithms have asymmetry on the scale of correlations between next-to-nearest symbols. This paper aims to summarise our previous results27,28 for square and honeycomb lattices and to develop a new pair approximation for the triangular lattice. The results obtained by the informational approach, refereed in the paper as the first (asymmetric) pair approximation, will be then used to develop the second, more accurate, (symmetric) pair approximation, which is nonconstructive, however. In contrast to the usually cited models of random sequential adsorption,29 we define the algorithm of lattice construction in such a way that no defects (voids) can be formed and that at all steps of the process new atoms adsorb in similar local configurations (Figure 1). These rules allow us to construct an ergodic stochastic process which results in configurations characterized by a given set of order and composition parameters {σ} and by a given value of the configurational energy (Eσ). The configurations composed of a sufficiently large number of atoms (N) can be characterized by the same total probability Pσ. The assembly of these configurations corresponds, therefore, to an isolated system at equilibrium. The configurational entropy (Sσ) of the system can be found as a value proportional to the logarithm of the inverse value of Pσ. Combination of
(2)
where c and b are constants which depend on the choice of a unit of information. This expression is similar to the entropy function which commonly arises in statistical thermodynamics. The analogy between informational and thermodynamic variables becomes evident if a certain configuration of A and B atoms in a solid solution (AxB1-x) is considered as a certain text (message) written by A and B symbols according to certain semantic rules. If A and B symbols are placed in the text independently of previous symbols, then P(ζ1) ) PA, P(ζ2) ) PB and the total information of the resulting message consisting of N symbols can be easily found as NH. If natural base of the logarithm is chosen and Boltzmann’s constant is substituted for c, the calculated value of information becomes similar to the wellknown result for the configurational entropy of an ideal solution:
Sid ) kN∑Pi ln(1/Pi) ) k ln i
N!
∏i (PiN)!
(3)
where Pi is the concentration (probability) of i component. The information theory gives the same result for the configurational entropy of an ideal solution avoiding the necessity of calculation of the number of all possible arrangements of atoms in the lattice. In fact, it is enough just to simulate a typical configuration and calculate its information as a sum of logarithms of the inverse probabilities of all unit events. Expression 2 can be used to calculate information (and entropy) in cases of a nonzero short-range order. In these cases, the value of absolute probability (P(ζi)) can be substituted by a certain conditional probability which corresponds to an event of joining of a certain symbol to the text in a certain local arrangement of previously placed symbols. The concept of conditional probability can be easily generalized for cases of two-dimensional sequences of symbols or atoms (Figure 1b,c,d). Thus, the problem of evaluation of entropy reduces to the problem of calculation of a set of conditional probabilities. In the following sections we will show that these probabilities can
15974 J. Phys. Chem., Vol. 100, No. 39, 1996
Vinograd and Perchuk
be rigorously found as functions of conventional order and composition parameters. Markov-Chain Model for an AxB1-x Solution with SRO on 1D Lattice. Consider a transition matrix
A B m 1-m n 1-n
A B
(4)
which defines a first-order Markov chain22,23 with the two states (A and B). Each element of this matrix (Pk/j) corresponds to the probability to find symbol K at the end of the chain under the condition of previously placed symbol J. The process defined by this matrix results in a 1D configuration of A and B symbols (Figure 1A). Except for the special case (m ) 1, n ) 0), the chain is ergodic. It means that all sufficiently long configurations produced by the process have similar statistical properties. The probabilities PA and PB to find atom A or B in a certain place of the chain after an infinite number of steps from the first event take constant values which are independent on the first event. The limiting distribution (PA,PB) represents, in fact, the set of composition parameters of the resulted configuration. A useful theorem allows one to calculate the limiting distribution as a function of the transition probabilities:
Pk ) ∑PjPk/j
(5)
j
Relation 5 also means that if the composition of the resulted configuration is known, then only one of the transition probabilities (m) can have an independent value. The other probability (n) can be easily found as a function of m and PA:
n ) PA(1 - m)/PB
(6)
The parameter m (0 e m e 1), in fact, describes the degree of SRO in 1D configurations. The case of 0 e m < PA corresponds to the tendency of A atoms to surround themselves by B atoms, while the case of PA < m e 1 corresponds to the tendency of A atoms to form clusters. The case of m ) PA corresponds to the absence of SRO. The existence of the limiting distribution implies that the probability of the {K/J} event to occur during the process is exactly equal to PjPk/j. This fact enables one to calculate the mathematical expectation and find the information of the Markov-chain source following Shannon’s expression:23
H ) c∑Pj∑Pk/j logb(1/Pk/j) j
(7)
k
By multiplying this expression by N (number of atoms), choosing natural logarithm base, and substituting Boltzmann’s constant for c, one obtains the expression
S1D ) kN∑Pj∑Pk/j ln(1/Pk/j) j
(8)
k
which in the case of PA e m e 1 coincides with the entropy expression by Kikuchi15 for 1D Ising model of a ferromagnetic. In fact, by using Stirling’s approximation and the notations Pj ) xj, PjPk/j ) yjk, it is easy to show that the expression 8 is similar to the combinatorial expression15
S1D ) k ln
∏j (xjN)!
∏(yjkN)! j,k
XN ) k ln YN
Figure 2. Schemes of LRO in different lattices. Black circles correspond to the sites which are preferentially occupied by A atoms.
It is also interesting to investigate the case of 0 e m
