Chemical Equilibrium: A Convex Optimization Problem

Transcription

1 Chemical Equilibrium: A Convex Optimization Problem Linyi Gao June 4, Introduction The equilibrium composition of a mixture of reacting molecules is essential to many physical and chemical systems, ranging from physiology to chemical plant design. The computation of chemical equilibrium is thought to be challenging, and, as such, only simple ideal cases are introduced in textbooks. Although there exists a substantial body of literature on chemical equilibrium, the convex nature of the problem is seldom emphasized. We show that the computation of chemical equilibrium in closed systems can in general be formulated as a convex optimization problem, leading to an elegant conceptual framework, as well as efficient numerical solutions. This formulation provides a natural generalization of a number of important topics in physical chemistry and thermal physics. 2 Background When a set of molecules are mixed together, two fundamental processes can occur: Mass distribution (i.e., phase transitions) and chemical reactions. Phase transitions. At equilibrium, the mass distribution in a system may not be homogeneous. For instance, a mixture of oil and water will naturally separate into layers. Suppose the system contains n distinct types of species, and let x = (x 1, x 2,..., x n ) T R n + be the composition of the system, where x i is the amount of species i. If the system is separated into k physical phases with composition x (j) R n + in the jth phase, then x = x (1) + x (2) + + x (k), x (j) 0, j = 1, 2,..., k. Chemical reactions. The molecules in a system can not only physically redistribute, but they can also change via chemical reactions. For example, consider the hydrogen combustion reaction 2H 2 + O 2 2H 2 O, or equivalently, 2H 2 + O 2 2H 2 O 0. This reaction converts two molecules of H 2 and one molecule of O 2 into two molecules of water, and vice versa. 1

2 In general, chemical reaction processes in a system consisting of n species X 1,..., X n and m reactions can be described as a 11 X a n1 X n 0 a 1m X a nm X n 0, where a ij is a constant. For each reaction j, j = 1,..., m, if a ij > 0, species X i appears as a reactant if a ij < 0, species X i appears as a product if a ij = 0, species X i does not appear We can collect the coefficients into a matrix A R n m, where A ij = a ij. For instance, for the hydrogen combustion reaction above, we have n = 3, m = 1, X 1 = H 2, X 2 = O 2, X 3 = H 2 O, and A = (2, 1, 2) T. By the conservation of mass,. x = x init + Ay (1) for some y R m, where x init R n + is the initial composition. Physically, y i represents the quantity of reaction i (i.e. the integrated rate). The reactions are assumed to be reversible, so y i can be either positive or negative. In other words, the overall composition of the system is the sum of the initial composition (x init ) and the changes caused by chemical reactions (Ay). No assumptions about the molecular formulas of the species are necessary, provided that the reactions as written are balanced. The constraint (1) already accounts for the elemental conservation of mass. 3 Problem Description According to thermodynamics, a chemical system will spontaneously change in any way possible in order to minimize a particular objective function of the composition [Gib78]. For example, at constant temperature and pressure, the objective function is Gibbs free energy. To compute equilibrium, all possible variations of both mass distributions and chemical reactions must be considered. Problem data. A R n m : description of reactions x init R n +: initial amounts f 1,..., f k : R n R: phase functions, e.g., free energies ( dissatisfaction levels ), usually (but not always) convex 2

3 Outer loop (reaction equilibrium). minimize subject to f(x) x = x init + Ay x 0, (2) where the variables are x R n and y R m. The overall objective f(x) is given by the optimal value of another optimization problem: Inner loops (phase equilibrium). Given a composition x R n +, compute f(x): minimize ˆf1 (x (1) ) + + ˆf k (x (k) ) subject to x (1) + + x (k) = x x (i) 0, i = 1,..., k. (3) If the phase functions f i are convex, then ˆf i f i. However, if f i is not convex, then ˆf i (x (i) ) is the solution to another (non-convex) optimization problem minimize subject to f i (z (i) 1 ) + + f i (z p (i) ) z (i) z p (i) = x (i) z (i) j 0, j = 1,..., p p = 1, 2,.... (4) Physically, z (i) j R n + is the composition of the jth partition (phase) of type i. Conceptually, these loops perform the following task: Of all possible physical partitions of the system given a fixed composition x, find one that minimizes the objective. Problems (3) and (4) have an elegant geometric interpretation: ˆfi is the convex envelope of f i, and f is the convex envelope of min{f 1,..., f k }. Thus, f is always a convex function. This is a generalization of the lever rule of thermal physics. Any optimal point x of (2) is an equilibrium composition of the system. At equilibrium, the composition of each phase of the system is given by solving the inner loop problems (3) and (4) for x = x. (The optimal value f is usually insignificant; the important quantity is the optimal point, i.e., the equilibrium amounts of the species.) If f i, i = 1,..., k, are convex, then (4) does not need to be solved; it will simply return the value of f i (x (i) ). Furthermore, if k = 1 and f 1 is convex (i.e., a single-phase system), then neither (3) nor (4) need to be solved because they will return the value of f 1 (x). In this case, we can set f f 1 and solve the outer loop problem (2) only. In chemistry, these cases are sufficiently common that f 1,..., f k are often assumed to be convex [Smi80], [ZS11]. However, the above formulation is valid with or without this assumption. Remark. Note that we have separated the problem into inner and outer loops in order to emphasize that we are simultaneously considering two distinct physical processes. For computation, (2), (3), and (4) can be combined into a single optimization problem. 3

4 Objective function. The form of the objective function f(x) depends on reaction conditions [PP08]. In chemistry, among the most widely used objectives are those that model ideal systems. Chemistry textbooks almost exclusively use these objectives [Oxt03], though they are rarely explicitly presented. Ideal gas or liquid, constant temperature and pressure: f i (x) = c T x + n j=1 ( xj ) x j log, (5) 1 T x where c is a constant determined by thermodynamics. In particular, c j = µ j(t, p )/RT + log (p/p ) for a gas, and c j = µ j(t, p )/RT for an ideal liquid. Ideal gas, constant temperature and volume: f i (x) = c T x + n x j log x j, (6) i=1 where c j = µ j(t, p )/RT + log (RT/p V ) 1 is constant. Ideal dilute liquid: If the reacting species are extremely dilute, 1 T x is assumed to be constant, and the objective (5) is replaced with (6). In general non-ideal systems in which f i may be non-convex, f(x) is still convex since it is the convex envelope of min{f 1,..., f k }. However evaluating f(x) requires solving the problem (4), which may be NP-hard. Nevertheless, the convex optimization formulation remains valid. In the cases (5) and (6) above, f i (x) is strictly convex. Thus, for a single-phase ideal system (k = 1), f ˆf 1 f 1 is also strictly convex, implying that the equilibrium composition is unique. This simplifies uniqueness arguments presented previously [PP08], [NZS65]. We focused on the objectives of the form (5) and (6), since they are among the most widely encountered. In both these cases, 2 f(x) has structure; in (5), it is diagonal plus rank one, and in (6), it is diagonal. The structure of 2 f(x) was exploited for efficient computation. 4 Solutions Although many chemical equilibrium problems are small (m, n < 10), some problems of recent interest, such as nucleic acid (DNA and RNA) interaction problems, are much larger (n > 1000) and would benefit from efficient computation algorithms. For example, a DNA template strand with s distinct RNA binding sites generates a system with s + 2 s species. 4

5 Figure 1: Computation time, in seconds. Problems were randomly generated instances of ideal gas and liquid equilibria problems of the form (5), and (6), with 15 equality constraints. Revisiting the constraints. The equality constraint (1) can be converted to Ãx = b, (7) where à is any matrix such that N (Ã) = R(A), and b = Ãx init. Computing à can be expensive if A is large, but in many systems, à has a particularly simple form. If n rank(a) = s, where s is equal to the total number of elements (i.e. atoms types) present in the system, then Ãij is simply the number of times element i appears in species j. In practice, this case occurs sufficiently often [Smi80] that some authors assume that the equality constraint is given in the form (7). Since there are only 118 elements in the periodic table, à has at most 118 rows (usually fewer than 20). On the other hand, to compute à in the case that n rank(a) s, we used a full QR decomposition. For the objectives (5) and (6), the inequality constraint x 0 can be removed, since it is implicit in the domain of the objective. Algorithm. Newton s method with backtracking line search was implemented for this problem. A primary challenge was that for many equilibrium problems, x i 0 for some i; in other words, the equilibrium amount of certain species may be close to zero. For instance, for the objective (5), since ( ) ( ) f i (x) = diag,..., 11 T, x 1 1 T x the diagonal entries of the Hessian can become extremely large if x i 0 for some i. If the condition number of the Hessian was sufficiently high, Newton s method failed in MATLAB. To overcome this challenge, we used the matrix inversion lemma, as well as block inversion x m 5

6 of the KKT system, to avoid explicitly computing the Hessian or its inverse [BV04]. The Newton decrement, as well as primal and dual updates, were then successfully computed. A further challenge was evaluating the gradient at each step. For (5), and for (6), f i (x) = c + ( ( x1 ) ( xn )) T log,..., log, 1 T x 1 T x f i (x) = c + (log (x 1 ),..., log (x n )) T + 1. Though any x 0 with x i = 0 for some i is contained in dom f i (by virtue of the convention 0 log 0 = 0), the gradient becomes unbounded as x i 0 +. To overcome this problem, any starting point x init with x i < ε for some i was deemed to be infeasible (we used ε = 10 5 ). All such x i were set equal to ε, and the problem was then solved with either infeasible-start Newton or by choosing a feasible start point by projecting onto the feasible set. Finally, the stopping criterion was based on duality gap; for (5), we have { b T ν if log n i=1 g(ν) = exp( ãt i ν c i ) 0 otherwise, and for (6), g(ν) = b T ν n exp ( ã T i ν c i 1 ). i=1 We tested the algorithm on randomly generated equilibrium problems with 15 equality constraints (of the form Ãx = b, where b = Ãx init). Results are shown in the figure above. The equilibrium compositions of systems containing up to n = species were efficiently computed on a dual-core 2.90 GHz processor; the bottleneck was memory rather than speed. 5 Conclusions In conclusion, we have shown that: The computation of chemical equilibrium (in the general case) can be naturally formulated as a convex optimization problem. Equilibrium compositions of large ideal reacting systems can be efficiently computed if problem structure is exploited. 6

7 References [BV04] [Gib78] S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press, J. W. Gibbs. On the equilibrium of heterogeneous substances. Trans. Conn. Acad. Art. Sci., 3: , , [NZS65] L.S. Shapley N. Z. Shapiro. Mass action laws and the gibbs free energy function. J. Soc. Ind. Appl. Math., 13:353 75, [Oxt03] D. W. Oxtoby. Chemistry: Science of Change. Thomson-Brooks/Cole, [PP08] J. M. Powers and S. Paolucci. Uniqueness of chemical equilibria in ideal mixtures of ideal gases. Am. J. Phys., 76:848 55, [Smi80] R. W. Smith. The computation of chemical equilibria in complex systems. Ind. Eng. Chem Fundam., 19:1 10, [ZS11] F. Zeggeren and S. H. Storey. The Computation of Chemical Equilibria. Cambridge University Press, second edition,