Title: A Deduction of the MulticriticalityConditionsof, Mixturesfrom the Gibbs Tangent Plane Criterion Author: Nélio Henderson Wagner.F. Sacco Raimundo A. Rodrigues Jr PII: S0378-3812(13)00550-5 DOI: http://dx.doi.org/doi:10.1016/j.fluid.2013.09.043 Reference: FLUID 9791 To appear in: Fluid Phase Equilibria Received date: 16-8-2013 Revised date: 20-9-2013 Accepted date: 21-9-2013 Please cite this article as: N. Henderson, Wr.F. Sacco, R.A.R. Jr, A Deduction of the MulticriticalityConditionsof, Mixturesfrom the Gibbs Tangent Plane Criterion, Fluid Phase Equilibria (2013), http://dx.doi.org/10.1016/j.fluid.2013.09.043 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Highlights (for review) We report a systematic deduction of the multicriticality conditions of mixture. The deduction uses the Gibbs Tangent Plane criterion for phase stability analysis. The proof is based on the principle of mathematical induction, being valid for any order. Page 1 of 14
*Revised Manuscript A Deduction of the Multicriticality Conditions of Mixtures from the Gibbs Tangent Plane Criterion Nélio Henderson*, 1, 3, Wagner. F. Sacco 2, 3, Raimundo A. Rodrigues, Jr. 1, 3 1 Instituto Politécnico, Universidade do Estado do Rio de Janeiro, 28625-570, Nova Friburgo-RJ, Brazil 2 Instituto de Engenharia e Geociências, Universidade Federal do Oeste do Pará, 68135-110, Santarém-PA, Brazil 3 Thermodynamics and Optimization Group (TOG), Brazil Abstract. Here, we follow a classification proposed by Griffiths and Widom [1], where the order of a multicritical point in a mixture is equal to the number of phases which simultaneously become identical, considering phases and assuming that of these phases become identical to a given test phase. Thus, employing Rolle s theorem and basic properties of the so-called tangent-plane distance function, we develop a deduction of the multicriticality conditions of mixture from Gibbs tangent plane criterion, which relies on the principle of mathematical induction, being appropriate for any. Keywords: Critical point, Phase equilibria, Stability, Mathematical modeling. 1. Introduction Critical states of thermodynamic mixtures are characterized by points in the configuration space of the mixture, called critical points, where the coexisting phases become identical. The classical forms of the criticality conditions were developed by Gibbs [2], in his fundamental work on the thermodynamics of phase equilibrium. Such conditions are given as two non-linear *Telephone and Fax: +55 (22) 2533-2263, e-mail address: nelio@iprj.uerj.br (N. Henderson) 1 Page 2 of 14
equations, which require the calculation of two determinants associated with higher-order partial derivatives of the molar Gibbs free energy, taken with respect to the compositions. Motivated by the tricritical phenomenon, Batis [3] extended Gibbs original formulation of the criticality conditions to multicritical points, with emphasis given to the tricriticality criteria. A further extension of Gibbs formulation to multicritical points can be found in a work by Mistura [4]. A modern form of the Gibbs Tangent Plane (GTP) criterion was deduced by Baker et al. [5], where the so-called tangent-plane distance function is a fundamental tool for the phase stability analysis. Using an expansion in Taylor series of tangent-plane distance function, Michelsen and Heidemann [6] presented a deduction of the tricriticality conditions from the GTP criterion. These conditions do not require the computation of determinants, and thus are more appropriate to the calculation of tricritical points. However, as can be seen, the deduction proposed by these authors uses the hypothesis that a polynomial of degree six (associated with the above mentioned expansion in Taylor series) admits a particular factorization. In this work, we present a deduction of the multicriticality conditions from the GTP criterion for phase stability analysis, which does not use factorizations of polynomials, being valid for critical points of any order. 2. On the GTP Criterion Let be the compositional space of r-component mixtures, where the mole fraction is obtained from. Given a mixture with composition, the so-called tangent-plane distance function [5] can be written using a modified form proposed by Henderson et al. [7],, (1) 2 Page 3 of 14
where and are, respectively, the temperature and the pressure of the mixture, and represents the chemical potential of component, which also depends on and, i.e.,. Following the GTP criterion for phase stability deduced by Baker et al. [5], a thermodynamic mixture with composition is stable, at temperature and pressure, if and only if, for all. As indicated by Henderson et al. [7], the gradient vector of, denoted by, can be determined by, for all. (2) Thus, from Eqs. (1) and (2), we can note that, (3). (4) Here, we will consider that the tangent-plane distance function is sufficiently continuously differentiable with respect to compositions, and we use the notation to represent the k-form, (5) where is a vector in. 3. Basic Considerations The following definition is based on a classification proposed by Griffiths and Widom [1]. Definition 1: Given an integer, the point, with, is called a multicritical point of order of an r-component mixture, if there exist phases in thermodynamic equilibrium, such that the compositions of each phase become identical to, whenever the 3 Page 4 of 14
temperature and pressure of the multiphase system tend to and, respectively. Particularly, if, we say that is a critical point, and if, then the point is called tricritical, etc. The next result is an obvious corollary of Definition 1. Corollary 1: All multicritical points of order of an r-component mixture are also multicritical points of order of the same mixture, for all. Suppose we have a critical point of an r-component mixture, given by. Since at a critical point two phases become identical, then it is reasonable to assume that there is a nonzero vector, such that two phases with compositions and may coexist in equilibrium at and, so that. As these compositions are points of, a convex subset of the Euclidian space, then there exists a straight line joining the point to, such that the line segment between and is contained in. Thus, at and, we can consider two phases in equilibrium with compositions given, respectively, by and, where is the direction vector of that straight line, and is a nonzero scalar, arbitrarily small. Now, suppose we have a tricritical point of an r-component mixture, given by. Then, it is reasonable to conceive three phases in equilibrium, at and, whose compositions are given by, and, where and are nonzero vectors in, and and are nonzero scalars, arbitrarily small. In this case, it is clear that these three points are not (necessarily) on the same straight line. But we can assume that such points lie on a curve. In fact, as illustrated in Fig. 1, we can consider that the points, lie on a curve, which can be seen as the path of an isobaric and isothermal quasi-static and process, outlined in. In Fig 1, the point lies on the straight line, that passes through, while lies on another straight line, denoted by, which also that passes through. Note that, when and, the secant lines and tend to, the tangent line to the curve at the point, which has direction vector given by. Hence, the direction vectors of and 4 Page 5 of 14
, respectively, and, tend to, when and. Thus, for and arbitrarily small, there exists an unitary vector such that we can consider the approximations and. It is clear that this geometric situation can be extended to include phases in equilibrium on a given quasi-static path, where of these phases become identical to a given test phase, whose composition is. Therefore, we assume the following. Conjecture 1: If, with, is a multicritical point of order of an r- component mixture, then there is an unitary vector,, and nonzero scalar (arbitrarily small), denoted by,, such that it is possible to conceive (at and ) a multiphase equilibrium state with phases, whose compositions of the coexisting phases are given by. Furthermore,, for all. Note that Conjecture 1 emphasizes that any multicritical point is stable. 4. The Deduction Method Theorem 1: If, with, is a multicritical point of order of an r-component mixture, then, for all, (6) where is the unitary vector considered in Conjecture 1. Proof: The demonstration will be done using mathematical induction on multicritical point. Thus, initially, we show that the relations in Eq. (6) are valid for, the order of the. For this, from Conjecture 1, we consider two phases in equilibrium, whose compositions are given by and, where. Since these phases are in thermodynamic equilibrium, then we can write the equilibrium conditions with respect to matter flow,, for all. (7) 5 Page 6 of 14
In order to obtain the condition, we consider the auxiliary function, (8) which is defined for all, such that. From Eqs (1), (7) and (8), we note that. (9) Thus, since is a convex set of, then, from Rolle s theorem, there is between 0 and such that, (10) so that, when, and. From Eqs. (3) and (4), we know that and. Hence, by expanding the function in a Taylor series around, we have. (11) Combining the Eqs. (10) and (11), we get the relationship. (12) Dividing Eq. (12) by, we have. (13) Since, when, then taking the limit on both sides of Eq. (13), when, we obtain. (14) To obtain the criterion, we use the stability condition considered in Conjecture 1. Thus, as,, and, by expanding the function in a Taylor series around, we note that the multicritical point is locally stable if and only if, (15) 6 Page 7 of 14
for all, such that. As the dominant term of this expansion has odd sign, to ensure the local stability of, independently on the sign of, it is necessary that. (16) In order to continue with the induction process, assuming that the result of this theorem is valid for, we show that it is also valid for. For this, we consider the existence of phases in equilibrium, whose compositions are given by,, such that, with, for all. (17) As the multicritical point is of order, then, in agreement with Conjecture 1, we can consider that the multiphase system has an additional phase in equilibrium, whose composition is given by, with. Thus, for this additional phase, we can write the following equilibrium conditions with respect to matter flow,, for all. (18) From Eqs. (1), (8) and (18), we note that. (19) Again, from Rolle s theorem, there is between 0 and such that, (20) so that, when, and. Expanding in Taylor series and using the conditions shown in Eqs. (3), (4), (17) and (20), we can show that Since, from Eq. (21), we have. (21). (22) As, when, taking the limit on both sides of Eq. (22), when, we obtain 7 Page 8 of 14
. (23) In order to obtain the last condition in Eq. (6), from Eqs. (3), (4), (17) and (23), we note that the multicritical point is locally stable if and only if, (24) for all, such that. Finally, as the dominant term of this expansion has odd sign, to ensure the local stability of, independently on the sign of, it is necessary that. (25) This concludes the proof. The following result characterizes the unitary vector. Theorem 2: For any order, the unitary vector,, referred in Conjecture 1 (and consequently in Theorem 1) is an eigenvector of the Hessian matrix of the tangent-plane distance function, associated with the small eigenvalue of this matrix evaluated at, which, in turn, is zero. Proof: Let be the Hessian matrix of evaluated at. As is sufficiently continuously differentiable, with respect to compositions, then the Hessian is a symmetric matrix. Hence, all eigenvalues of are real numbers. We denote these eigenvalues by, (26) where is the small eigenvalue of. Given, since,, and, for all, then by expanding in a Taylor series around, we obtain, (27) for all, such that. Thus, choosing, we can write. (28) 8 Page 9 of 14
Taking the limit on both sides of Eq. (28), when, we note that, for all nonzero vector in. (29) Thus, from the relation in Eq. (29), we can assert that the Hessian is a symmetric positive semidefinite matrix. Hence, we can write We affirm that the unitary vector Theorem 1) is an eigenvector of. (30) referred in Conjecture 1 (and consequently in. In fact, by contradiction, suppose that, As, then, for all (31) However, by Theorem 1,. Then from Eq. (31) we obtain that, for all. Hence, and the Hessian is a symmetric positive definite matrix, i.e.,, for all nonzero vector in. (32) Particularly, for, we obtained, which is an absurd! Therefore, this unitary vector is necessarily an eigenvector of, and consequently is the eigenvalue of the Hessian matrix associated with. 5. Discussion Unlike the classical form originally derived by Gibbs [2], for ordinary critical points, and later extended by Batis [3] and Mistura [4], for critical points of higher order, multicriticality conditions of mixtures obtained from the GTP criterion do not require the computation of determinants, being more suitable for numerical computation, especially when the number of components becomes large. Despite this fact, it can be observed that the article by Michelsen and Heidmann [6] is the only study in the literature that is devoted to obtaining the 9 Page 10 of 14
multicriticality conditions using the GTP criterion, with emphasis on tricriticality conditions. Despite the pioneering work of these authors, we cannot fail to observe that the deduction of the tricriticality conditions presented by Michelsen and Heidmann [6] is based on the existence of a possible factorization of a polynomial of degree six (associated with an expansion in series of Taylor), which proved to be suitable for demonstrating these conditions, but, unfortunately, cannot be theoretically guaranteed. Indeed, in the absence of a theoretical substantiation, these authors simply used the following argument: One could imagine circumstances under which this polynomial can be factored as follows. If such a factorization is possible, see p. 58 of [6]. This communication is not intended to discredit a previous work. In fact, this article should be seen as an effort toward reaching a deduction of the multicriticality conditions (from the GTP criterion) which does not use factorizations of polynomials, providing, this way, a more substantial proof. Thus, in Conjecture 1, it was considered that, near a multicritical point of order, the points representing the compositions of phases are on a given straight line passing through the composition of the test phase. If this conjecture is relaxed, then to proceed with the demonstration described in the proof of Theorem 1, it is required to assume that these compositions are on a curve passing through the composition of the test phase. In this case, the conditions described in Conjectured 1 remain valid, unless errors that depend on the magnitudes of the increments,, which tend to zero, when, for all. As, in the proof of Theorem 1, such limits occur, then one can see that Conjecture 1 is not absolutely necessary to demonstrate this theorem. However, the use of this conjecture simplifies the proof, highlighting the essence of the proposed deduction method. 10 Page 11 of 14
6. Conclusions From a classification proposed by Griffiths and Widom [1], in this work we presented a systematic deduction of the multicriticality conditions of mixtures, obtained via the Gibbs Tangent Plane criterion for phase stability analysis. In this deduction, we used the properties of the tangent-plane distance function and Rolle s theorem to guarantee the existence of roots of this function, located at intervals taken along a particular direction. With these tools, it was possible to develop a demonstration based on the principle of mathematical induction, which is valid for multicritical points of any order. Acknowledgements N.H. and W.F.S. gratefully acknowledge the financial support provided by CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico, Ministry of Science & Technology, Brazil). R.A.R, Jr. was supported by CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior, Brazil). The research by N.H. has been carried out in the framework of project PROCIENCIA-UERJ financed by FAPERJ. References [1] Griffiths, R.B., Windom, B., Multicomponet-fluid tricritical points. Phys. Rev. A, 8 (1973) 2173-2175. [2] Gibbs, J.W., On the equilibrium of heterogeneous substances, part I. transactions of the Connecticut Academy, 3 (1876) 108-248. 11 Page 12 of 14
[3] Bartis, J.T., Thermodynamic equations for tri- or third order critical points. J. Chem. Phys., 59 (1973) 5423-5430. [4] Mistura, L., Critical phases in multicomponent fluid mixtures. J. Phys. A, 9 (1976) 2139-2148. [5] Baker, L.E., Pierce, A.C., Luks, K.D., Gibbs energy analysis of phase equilibria. Soc. Pet. Eng, J., 22 (1982) 731-742. [6] Michelsen, M., Heidemann, R.A., Calculation of tri-critical points. Fluid Phase Equilib., 39 (1988) 53-74. [7] Henderson, N., Freitas, L, Platt, GM., Prediction of critical points: a new methodology using global optimization. AIChE J., 50 (2004) 1300-1314. Figure Captions Fig. 1. A three-phase equilibrium near a tricritical state. 12 Page 13 of 14
Accepted Manuscrip Page 14 of 14 Figure 1