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14УДК 544-971.62
Чарыков Николай Александрович1,2, Чарыкова Марина Валентиновна3, Семенов Константин Николаевич1,3,4, Кескинов Виктор Анатольевич1, Воронков Михаил Евгеньевич1, Шаймарданов Жасулан Кудайбергенович5, Шаймарданова Ботагоз Касымовна5
ЭКСТРЕМУМЫ ПАРАМЕТРОВ СОСТОЯНИЯ В УСЛОВИЯХ ДВУХФАЗНОГО И МНОГОФАЗНОГО РАВНОВЕСИЯ (Обзор)
1Санкт-Петербургский государственный технологический институт (технический университет), Московский пр. 26, Санкт-Петербург, 190013, Россия 2Санкт-Петербургский государственный электротехнический университет «ЛЭТИ», ул. Профессора Попова, 5, Санкт-Петербург, 197376, Санкт-Петербург, Россия 3Санкт-Петербургский государственный университет, Университетская наб., 7/9, Санкт-Петербург, 199034, Россия
4Первый Санкт-Петербургский государственный медицинский университет им. И.П. Павлова, ул. Льва Толстого, 6-8, Санкт-Петербург, 195176, Россия 5Восточно-Казахстанский государственный технический университет им. Д. Серикбаева, ул. Протозанова, 69, г. Усть-Каменогорск, 070004, Республика Казахстан e-mail: keskinov@mail.ru
Рассмотрены общие условия прохождения параметров состояния через экстремумы/ в многокомпонентных многофазных системах. Рассмотрение основано на обобщенных дифференциальных уравнениях Ван-дер-Ваальса в метрике полных и неполных потенциалов Гиббса. Два принципиально разных случая соотношения между параметрами состояния системы, когда температура (при постоянном давлении), давление (при постоянной температуре), химические потенциалы/ компонентов (при постоянной температуре и давлении одновременно), параметры/ фазового состава (также при постоянной температуре и давление одновременно) проходят через экстремумы/. Оба общих правила, касающиеся прохождения параметров состояния через экстремумы/, подтверждаются примерами из диаграмм разных типов фазового равновесия: жидкость-пар, жидкость-твердое тело (диаграммы/ растворимости и плавкости).
Ключевые слова: уравнение Ван-дер-Ваальса, сдвиг фазового равновесия, полный и неполный потенциал Гиббса, векторно-матричная форма, параметры состояния при прохождении через экстремум.
Importance for Applied Science and
Technology
This article discusses the conditions of passing through the extrema of such parameters as: temperature at constant pressure (ТР), pressure at constant temperature (РТ), the chemical potential of the component at constant temperature and pressure (p¡)TpP components concentrations in a certain phase (r) at constant temperature,
Charykov Nikolay A.1,2, Charykova Marina V.3, Semenov Konstantin N.1,3,4, Keskinov Victor A.1, Voronkov Mikhail E.1, Shaimardanov Zhasulan K.5, Shaimardanova Batagoz K.5
EXTREME STATE PARAMETERS IN THE CONDITIONS OF TWO-PHASE AND MULTI-PHASE EQUILIBRIUM (Review)
1St. Petersburg State Institute of Technology (Technical University), Moskovsky pr., 26 Saint-Petersburg, 190013, Russia
2St. Petersburg Electrotechnical University «LETI», ul. Professora Popova 5, 197376, St. Petersburg, Russia 3St Petersburg State University,7/9 Universitetskaya emb., Saint-Petersburg, 199034, Russia 4Pavlov First St.Petrsburg State Medical University,. L. Tolstoy st. 6-8, St Petersburg, 195176, Russia 5D. Serikbayev East Kazakhstan State Technical University, 69, A.K. Protozanov St., Ust-Kamenogorsk, 070004, Kazakhstan. e-mail: keskinov@mail.ru
The common conditions of extreme parameters of state in multicomponent multiphase systems are considered. The consideration is based on the common differential Van der Waals equations in the metric of complete and incomplete Gibbs potentials. Two fundamentally different cases of the ratio between the system state parameters, when temperature (at constant pressure), pressure (at constant temperature), components or compounds chemical potentials (at constant temperature and pressure simultaneously), phase composition parameters (also at constant temperature and pressure simultaneously) pass through the extremums are determined. Both common rules, concerning extreme state parameters are supported by the examples from the different types diagrams of phase equilibrium: liquid-vapor, liquid-solid (diagrams of solubility and fusibility).
Key words: van der Waals equation, phase equilibrium shift, complete and incomplete Gibbs potential, vector-matrix form, specific points, extreme state parameters.
Дата поступления -3 июня 2019 года
pressure or both parameters (X(т) )Т, (X{T) )P, (xw )PtT. The review is conducted in arbitrary conditions - without regard to types of phase equilibrium, number of equilibrium phases, number of components. The presented consideration allows to resolve a number of practical issues, connected with phase equilibrium, namely:
1. To verify phase equilibrium experimental data in the conditions nearby extrema characteristics of state
parameters (for example, azeotropes, Hetero-azeotropes, Van Rijn points, extrema of liquidus temperature of solid solutions, etc);
2. To determine concentration regions of phases compositions, which may be produced in the conditions of "closed phase processes" [1] (when finite masses of all phases are in equilibrium with each other, and no phases are removing from heterogeneous system) at constant parameters of state. The examples of such processes are, for example: single-time distillation of vapor from the solution at fixed Tand P, single-time solid solution crystallization from the melts at fixed T, single-time solid solution crystallization from the solutions at fixed T, P and chemical potential of the solvent (w) - w etc (see, for example [2-6];
3. To determine concentration regions of phases compositions, which may be produced in the conditions of "opened phase processes" [1] (when one or several equilibrium phases masses are infinitely small, and one or several phases are constantly removing from heterogeneous system) at constant change of some parameters of state. The examples of such processes are, for example: multistage distillation of vapor from the solution in distillation columns in the conditions of change TP or PT; multistage solid solution crystallization from the melts in the conditions of change TP (implemented, for example, in processes of materials purification by "zone melting" or in "recrys-tallization columns"), multistage solid solution crystallization from the solutions in the conditions of solvent evaporation or (pw)T change, etc (see, for example [7-11].
If 2 issue determines the available phase compositions at fixed state parameters (in fixed point of phase diagram, which is characterized by nodes simplex [12]), that 3 issue determines available phase compositions at changing state parameters (in whole phase diagram in heterogeneous system).
So, such consideration may be widely used in the planning and development of different technological phase processes in: metallurgy, oil refining, salt industry, wine industry, etc.
Introduction
In XIX-XX centuries huge number of laws and rules, concerning extreme state parameters was established. The vast majority of them have private character. Note the classic investigations of J.W. Gibbs, F.A.H. Schreinemakers, M.S. Vrevskiy, D. Konovalov, H.B. , J.H. Rijn, A.V. , V.T. Jarov, V.K. and coauthors (see, for example [13-23]).
It is well-known, that conditions of phase equilibrium conservation (phase equilibrium shift) may be written in different forms, for example, as the condition of the equality of first differentials of intense parameters of state (temperature, pressure and components chemical potentials) in equilibrium (a,P) phases: dTa)=dT(P); dPa)=dPP); d/j(a)=d/jf^, see, for example bibliography of J. Willard Gibbs, represented in [24].
From these conditions van der Waals got well-known differential equation of phase equilibrium shift for two-phase equilibrium in the binary system [25]. From this equation it is easy to obtain all rules or laws, characterizing two-phase equilibrium in binary and single-component systems: for example: Clausius-Clapeyron equation, Gibbs-Konovalov law, Gibbs-Roseboom rule etc.
A.V. Storonkin spread van der Waals equation to a multicomponent systems [26]. It allowed to obtain easily
thermodynamic laws and rules in ternary, quaternary and more component systems: spread Clausius-Clapeyron equation on the states with the equality composition of equilibrium phases, van Rijn rule, hetero-azeotropes rule, etc.
V.K. and V.A. Sokolov probably the first recorded van der Waals equation in the most convenient vector-matrix form in the metric of Gibbs potential [27, 28].
D.A. Korjinskiy introduce incomplete Gibbs potentials [29], characteristic in the following set of variables: temperature, pressure, moles numbers of some components and chemical potentials of rest components.
Authors [27, 28] also recorded van der Waals equation in vector-matrix form in the metric of incomplete Gibbs potential and prove, that in the conditions of phase diffusional stability matrixes of second concentration derivatives of these potentials are nondegenerate and determined positively (such matrixes of complete Gibbs potential naturally have such properties).
Authors [30] with the help of apparatus, elaborated by authors [27, 28], proved absolute topological isomorphism of different types phase diagrams in the metrics of complete and incomplete Gibbs potentials: diagram of equilibria vapor-liquid at T = const, diagram of equilibria vapor-liquid at P = const, diagrams of fusibility; diagrams of solubility at T,P = const etc. In the cases of fusibility and vapor-liquid phase diagrams one should use complete concentration space of components molar fractions and in the case of solubility diagrams - reduced concentration space (without taking into account solvent content) of dissolved components Yeneke indexes.
The main goal of the article is to get system of differential equations of different type open phase processes, where masses of equilibrium phases are natural variables of state and demonstration of the validity of use of these equations for the calculation of open phase processes in some multicomponent natural or model systems.
Questions, connected with extremal properties of the parameters of state, in the conditions of poly-variant equilibria in heterogeneous systems were investigated and discussed for many times, as in the papers of Gibbs J. Willard himself [13, 24], as in the papers of later physicists and chemists - see, as an example classical monographies [14, 23] or much later [30]. We shall consider following parameters of state in multiphase equilibrium: temperature at constant pressure - TP, pressure at constant temperature - PT, chemical potential of i-th component at constant temperature and pressure (d/Ji)TP, or composition vectors in arbitrary a phase in full concentration space - (X (a))PT, or Yeneke indexes vectors of dissolved components in reduced - excluding solvent, concentration space - (f {a))PT. It was formulated huge number of different "laws" and "rules", concerning extremal properties of the parameters of state, which was generalized in famous Gibbs rule about extremums TP, PT in multi-phase equilibrium (see, for example, [13, 24]). Authors [31] spread this rule to the metric of incomplete Gibbs potential and extremum conditions of (d/Ji)TiP. All extremal thermodynamic "laws" and "rules" may be easily obtained from these rule - see later.
But, there is one more, important, to our opinion, case, fundamentally different from the Gibbs rule, and also characterizes passage of state parameters through extrema. This case is quite common, for example, for fusibility diagrams. Authors [32] considered some special
cases of passage through the extremum functions TP, (X{a))PT in fusibility diagrams in the conditions of two-
phase equilibrium with the participation of the phase with "unfree composition".
In this article we shall:
- demonstrate absolute topological isomorphism of the different types of diagrams of phase equilibrium, namely: liquid-vapor in the variables P(X) at T = const, liquid-vapor in the variables T(X) at P=const, liquid-solid T(X) at P = const (fusibility diagrams), liquid-solid in the variables Y - Y at P,T= const (solubility diagrams);
J J
- spread "Gibbs extremum parameters of state rule" on the phase diagrams of all types;
- spread "Unfree phase composition extremum rule" on the phase diagrams of all types.
Two-phase Equilibrium.
Generalized Differential Equation of Phase Equilibrium Shift (van der Waals Equation in Vector-Matrix Form) in the Metric of Gibbs Potential.
Let us consider two-phase {a-p) equilibrium in
n-component system. Let choose as independent variables of phases state the following: temperature - T, pressure -
P and mole components numbers - n (or molar fractions -
/« ). So, as basic thermodynamic potential we
= n/ Z nj
choose Gibbs free energy G (or metric of Gibbs potential). One can describe two-phase {a-p) equilibrium
shift by the following system of differential van der Waals equations in the most compact vector-matrix form [27, 28]:
(X{p) -Xa))G(a)dX(a) = [S(a) -S(p) + (X{p) -Xa))VS{a)]dT- (1) - [V(a)-v p + (X <« - X <a>)VV {a)]dP (X(a) -Xm)G(p)dX(p) = [S(p) -S(a) + (X-XiP))VS(P)]dT-
- [v(p) - v(a) + (X(a) - X (P))vv(m)]dP
(2)
Gia)dXia) - vs (a)dT + VV(a)dp = Gm)dXm) - vs m)dT + VV{p)dp,
(3)
Both equations (1) and (2), written in the variables of phases (a) and (p), are equivalent. The last equation (3) is so named additional condition of phase equilibrium shift. Here: V(r) and S(r) - molar phase volumes and entropies (r = a or p); VV (r) and VS(r) - concentration gradients of V and S(r) with the elements
(dv^/dX?XPr^r(I> and (qsr)/dXrx^^- corre-
T ,P, X'f * x(Z Hz)
spondingly; X(r) - vector of the state of figurative point of r phase in concentration space with elements X(r);
dX(r) - vectorX(r) shift at (a-p) equilibrium shift with elements dX(r); G(r) - operator, corresponding to the
matrix of second derivatives G
(z) .
(
QJ)
2rjz)
d G
d X {t)d X(t) ' J /
(4)
where G(t) - molar Gibbs free energy of r phase. Note, that, according to the conditions of diffusional stability of r
phase, matrices of operators G(r) are nondegenerate and corresponding quadratic bilinear forms are determined positively, according Sylvester's criterion [33].
Physical sense of the parameters
S(a^p) = [S(a) - S(p) + (X(p) - X(a))VS(a^ and
v= [V(a) -V(p) + (X(p) - X(a) )VV(a) ] are entropy and volume changes in the process of isothermal-isobaric formation of 1 mole of p phase from infinitely large mass of a phase.
Two-phase Equilibrium
Generalized Differential Equation of Phase Equilibrium Shift (van der Waals Equation in Vector-Matrix Form) in the Metric of Incomplete Gibbs Potential (Korjinskiy Potential). Together with the previous equations authors [27, 28] got differential equations system in the metric of incomplete Gibbs potential, which is introduced similarly with thermodynamic Korjinskiy potential [29, 30] for the description of the systems with "quite mobile components":
gv ] = g - 1Ln<u = Znvi' (1,1)
i=k+1 i=1
characteristic in the following variables: T, P, part of components mole numbers (^1,n2,.,nk)and chemical potentials of the rest components (fdk+1 /uk+2,... ): (Ya - Y(myG$dY(P) = [V(p) - Sa + (Ya - Y (P))Vs(P)]dT - [v(p) -v(a) + + (Y(a) - Y (P))VV(m)]dP + Z [n(qp) - n(qa) + (Y(a) - Y (p))vn(qp%q
q=k+1
(1.2)
J[t]
+ (Y rn _y ^ )V ~(a) ]dP [«^ - nf + (Y m - f^ )Vn(') }iMi
q d^q
Q$df(a) _VS(a)dT + VV{a)dP_ ^Vn^d^ =
q=k+1
(1.3)
(1.4)
= Qfdf(P) _ VS (p)dT + VV(P)dF _ £ Vn^d^
q=k+1
Designations here are similar to the equations (1), (2) and (3), but functions: G[k], S u V are calculated relative to one mole of first k components, and vectors Y(z) characterize position of figurative point of z phase (a or p) abridged (k_1 )-dimensional concentration space:
" " ' (1.5)
J=1
£f'(t) = i
In this case matrices G^ are also nondegenerate and corresponding quadratic bilinear forms are determined positively, according Sylvester's criterion [33].
(1.6) parameters
Q(Z) = Q[k ]j =
g 2Q(z) \ ° Q[k]
g y (z)g y (z) ' j
T .P. y* Jk
Physical sense of the
~(a^P) = [~(a ) _ ~(P) + (J (P) _ f (a))VS>(a )]
v{a^p) = [V(a) _ v(P) + (f(P) _ f(a))V~(a)] and
n(a-+P) = [n(a) _n(P) + (f(P) _ f (a))Vn(a) 1 are entropy,
q L q q qJ
volume and number of moles of the last (k+1, k+2,..n) components changes in the process of isothermal-isobaric formation of 1 mole of p phase from infinitely large mass
q=k +1
i=1
T .P.x
of a phase, moreover, taken into account only the masses of the first k components.
Let us choose concrete Gibbs incomplete potential; n refers to the solvent - single volatile component in the system:
(1.7),
G
[w]
= G - nnHn = 2
(Y (2) - Y «)(«dY(1) = V(1^2)dT - V^2)dP +
(Y(3) - Y m)G^dY(1) = V(1^3)dT - K(1^3)dP + n(1^3)d/uw
(Y(r) - Y (1))C?<i)dY(1) = S(1^r)dT - V^dP + n
r) Ф„
characteristic in the following variables: T, P, components mole numbers (nltn2,...,nn_ j) and chemical potentials of the n-th component (juw), "w" - index of n-th component or solvent symbolize, for example "water".
Multi-phase Equilibrium
Generalized Differential Equations Systems of Phase Equilibrium Shift. Let us consider /--phase equilibrium shift (r > 2).
Choose one phase with the maximal components number and put it number "1". As a rule, it is conveniently to choose one of the liquid phases, because solid and vapor phases often have constant composition (for example, consist of one-component), or have some limits on the composition. For example, in the systems A3B5: aA-1 - A2 -...-A- - B5-1-B2-...- B5-M (where: A3 (III group of Periodic Law)= A, Ga, In..; B5-1 = P, As, Sb, Bi...(V group)), in the liquid phase - melt there are N+M components, but in the solid solutions (s) there are only N + M -1 components, because: (s) = ^x w =1/
/=i /=i
Absolutely similar situation is observed in the other classes of solid solutions: AB6: A2-1 - A?'2 -...-AN - B61- B62-...-B6-M (where. A-1 (II group)= Cdr Hg, Ba...; B6-1 = Sr Ser Te...(VI group)), A1B7: A1-1 - A1-2 -...-A- - B7-1 - B7-2-.-B7-M (where. A1-1 (I group)= Na, K Rb...; B71 = C, Br, I...(VII group)) A4-B, where. A(B) (IV group) = C Si, Ge...) etc.
Then we write differential van der Waals equations system for the equilibria of the phases pairs (1-2; 13; 1-4... 1-r) in the variables of 1 phase in the metric of Gibbs potential:
(X(2) - X (1))G(1) dX(1) = ^ (1^2)dT -v (1^2) dP
(X(3) - X (1))G (1)dX(1) = ^ (1^3)dT -v (1^3)dP
( X( r) - X (1))G mdX(1) = S >dT -V (l^r)dP
(2.1),
or in the metric of incomplete Gibbs potential:
n
(Y (2) - y o^M dY(1) = S?(1^2) dT - V(l^dP + jn^dp,
q=k+1 n
(Y <3) - Y «X^dY(1) = ~(1^3)dT - V(1^3)dP + jjn
q=k+1
(2.3).
These systems are really the systems of (1-2-3...r)phase equilibrium shift in different potential metrics. Let us consider concrete conditions, limiting the number of thermodynamic degrees of freedom - f (number of state parameters, which one can choose independently without change of the number or nature of equilibrium phases).
Isobaric Conditions dP=0 (Metric of Complete Gibbs Potential).
(X<2) - X ®)G(1) dX(1) = S M dT (X<3) - X ®)G mdX(1) = S <M) dT
(X(r) -X(1))GmdX(1) = S(^r)dT
(2.4),
Isothermal Conditions dT=0 (Metric of Complete Gibbs Potential).
( X(2) - X (1))G(1) dX(1) = - V (1-2)dP ( X(3) - X(1) )G (1)dX(1) =- V (1^3)dP
(X(r) -X(1))G(1)dX(1) = - V{^r)dP
(2.5),
Isothermal-Isobaric Conditions dP=dT=0 (Metric of Incomplete Gibbs Potential).
(Y(2) - Y«)G«dY(1) = n™dK
(Y (3) - Y ®)£®dY (1) = nWr^d^
(Y(r ) - Y (1))(G(1>dY(1) = nW^r )d^w
(2.6).
Topological Isomorphism of the Different Types Phase Diagrams.
Let us consider the following diagrams:
1-phase is liquid (l), 2-phases is vapor (v) at P=const (l->v)P (eq.(2.4));
1-phase is liquid (l), 2-phase is vapor (v) at T=const (l-v)T (eq.(2.5));
1-phase is solid (s), 2-phase is liquid (l) at P=const (s-)P (eq.(2.4));
1-phase is solid (s), 2-phase is liquid (l) at P,T=const(s—)PT (eq.(2.6));
1-phase is solid (s), 2-phases is vapor (v) at P=const (s-v)P (eq.(2.4));
1-phase is solid (s), 2-phases is vapor (v) at T=const (s-v)T (eq.(2.5)).
One can see, that systems of differential equations are absolutely equivalent each other, because biline-
(Y(r) - Y(1) )G([l\dY(1) = S^dT-V^rdP + 2 n^djUq ar forms (X(2) - X (1))G(1)dX(1) or (Y - Y (1))(G(:)1df(1)
q=k+1
(2.2). are equivalent, matrix (corresponds to operators Gmor Or in the metrics of concrete incomplete potential G® ) are nondegenerate and determined positively, ac-
G ] = g-n n = j-1 n p one can get following expression:
cording Sylvester's criterion; and, so properties of first full
differentials of the intensive parameters: dT, dP, dpw are
determined by the signs of the effects of phase transitions only:
i=1
i=1
S(l-v)(P = const) > 0; S(s-v)(P = const) > 0; V(l-v)(T = const) > 0; V(s-v)(T = const) > 0;
S(s-)(P = const) > 0;n(s—)(P,T = const) > 0
Inequalities (3.1) are absolutely fair for phase equilibria (s—v)P, (s—v)T and (s—)P (in these cases phases can not form critical phase); for phase equilibria (l—v)P, (l—v)T inequalities (3.1) are fair far away of critical state liquid-vapor (where additionally is lost the positive definiteness of matrixes, corresponds to operators Gmor G® ); for phase equilibria (s—l), t inequalities
(3.1) are fair, when liquid solution has content of solvent more than solid phase, i.e. in solubility diagrams almost always. In exclusively rare cases occurs as named "forbidden types of solubility diagrams".
So, one can see, that all types of diagrams must have topological analogues, and (s—v)P, (l—v)P (s—)P, (s—l)PiT phase diagrams should be mirror symmetric to (——v)T, (s—v)T phase diagrams [20].
Linear Dependence of Phases Composition Condition.
Let us consider r-phase equilibrium (r>1). Suppose, that vectors of phase composition are linearly dependent in full concentration space:
(3.1).
2aX(i ) = 0
(4.1).
Let us sum equation (4.1) for all molar fractions of n components:
n r r
X X a,Xf = o or X a, = 0 j=1 1=1 ¡=1
(4.2).
Substitute (4.2) into (4.1) and get final expression:
X a (X(i) - X(1)) = 0 (4.3).
i=2
Absolutely similarly in reduced (without component w) concentration space we get:
r
X a (Y(,) - r(1)) = 0 (4.4).
Gibbs Rule about Extremum of State Parameters and its Analogs.
Multiply 1 equation in the systems (2.4), (2.5), (2.6) by a1, 2 equation - by a2... r equation - by ar, and sum all r equations in the systems (2.4), (2.5), (2.6). So, we get:
r r
Xa(X<J) -X(1))G<X)dX(1) = [X s(1^r)]dT
(5.1),
2«,(X(i) -X(i))G(i)dX(1) = -[2 v(i^)]dP
(5.2),
r r
2 a, (Y « - Y (1) = [jj K^Ww (5.3). i=2 i=2 So, if (4.3), (4.4) are valid:
dTP = 0, dPT = 0, (dw), t = 0 (5.4),
or TP, PT, (/Jw)Pj are in extrema. So, we formulate Gibbs rule and its analogs: Rule I.
I.I. If phase composition of r phases are iin-eariy dependent in full concentration space, then tempera-
ture of phase equiiibrium (at P=const), pressure of phase equiiibrium (at T=const) passes through the extremum.
I.II. If phase composition of r phases are lin-eary dependent in reduced (without component w) concentration space, then chemical potential of component w (at T, P=const) passes through the extremum.
This rule is realized in some special cases.
Two-phase Equilibrium
If we have only two phases, linear dependence of its composition symbolizes the fact of full coincidence of its compositions in full or reduced concentration range. It realizes in some cases:
A) In n-component azeotropes in liquid-vapor phase diagrams without miscibility gap in liquid (II Gibbs-Konovalov law about extrema of T, PT).
B) In n-component azeotropes in solid-vapor phase diagrams without miscibility gap in solid (spread of II Gibbs-Konovalov law about extrema of TP, PT on sublimation).
C) In n-component solid solution melts in fusibility diagrams without miscibility gap in solid (II Gibbs-Roseboom rule about extrema of TP). By analogy with azeotropes (boil without change - greek) we offer to name such points atixitrops (melt "nity" without change).
D) In (n+1)-component solid solutions in solubility diagrams without miscibility gap in solid (analog of II Gibbs-Roseboom rule about extrema of (/Jw)PiT ). By analogy with azeotropes we also offer to name such points alysotrops (solute ""ôiàÀuoq" without change). In this case figurative point of the solvent (w), liquid solution and solid phase - all belong to one straight line in the full concentration space or figurative points of liquid solution and solid phase coincide in reduced (without w) concentration space [30].
The examples of different type isomorph diagrams for the cases (A, B, C, D) in different coordinates are represented in Figures 1.
E) In n-component congruent melting compound with constant composition in fusibility diagrams (extrema of TP in distectics not widey used term - authors).
F) In (n+1)-component congruent melting compound with constant composition in solubility diagrams (extrema of ('Jw)P/T in distonics authorsterm).
G) In n-component congruent evaporating compound with constant composition in liquid-vapor phase diagrams (extrema of TP, PT in pseudo-distectics authors
term^
H) In n-component congruent sublimating compound with constant composition in solid-vapor phase diagrams (extrema of TP, PT in pseudo-distectics).
We introduce some new terms in paragraphs E) -H) similarly to the more common pairs of the terms: eu-tectic-eutonic, peritectic-peritonic.
We also can prove that for the absolutely vast majority of phase equilibrium, where are valid inequalities (3.1), is valid following addition to Gibbs rule and its analogs:
In the cases of the formation of congruent compounds with constant composition in the points of phase transformation of the last ones (cases (E, F, G, H)):
TP, (Pw)p,t passes through the maximum, and PT passes through the minimum.
r
i=1
i=2
i=2
i =2
r
r
i=2
i=2
Let us differentiate first differential equations from the system (3.4), taking into account, that com-
_ dX (1)Q <X)dX<X) + (X(2) _ X (1))Q (1)d 2X(1) + (X(2)
where: DQ(1) corresponds to matrix of second derivatives of Gibbs potential of the 1 phase, where one line is
changed by differentials
dGf>
d
д Gm
д X (Г)д X1
' j Jt ,Г,Хы
All
members in (5.5) are equal zero, except the first and the last ones, because in our case:
X(2) -X(1) = 0,dT = 0 (5.6),
So:
- dXmömdXm/ S= d2T < 0 (5.7), because bilinear form is dX(1)((mdX(1) >0, according to Sylvester criterion, and function is S> 0, according to physical sense (inequalities (3.1)). Absolutely similarly one can get:
dXmGmdXm / V™ = d2P > 0 (5.8), -dYmdYm/n™ = d2Mw< 0 (5.9). d2T < 0, d2^w< 0 is common criterion of maximum, d2P > 0 is criterion of minimum.
Fig.1.1
Fig.1.2. j
Fig. 1. Schemes of topological equivalent phase equilibrium diagrams in binary and ternary (type II in top figure) in different variables; description of the phases (here and later (v)-vapor, (l)-liquid, (s)-solid)).
T uw
lp и
T It"
■p. и ■
( л \ f Y к \
E2
e,
Fig.2.1.
a
E,
A C=compound AxBy XB,YB
Fig.2.2.
Fig.2. Schemes of topological equivalent phase equilibrium diagrams in binary and ternary (type II in Fig.2.1) in different variables.
pound has constant composition, i.e. dX(2) = 0. Number (2) is number of phase, representing compound :
-X(l))DGmdX(1) = dS(1^2)dT + S(1^2)d2T (5.5),
The schemes of different type isomorph diagrams for the cases (E, F, G, H) in different coordinates are represented in Figures 2.
K2S04
/9Li2S04'3K2S04'2CS2S04 0. >7--^ / -V0.4
li2S04 Q2 0.4 0.6 0.8 CS2SO4
Fig. 3. An example of passing through the maximum of the function (¿uw )TJ> in quaternary system Li2SO4-K2SO4-Cs2SO4-
H2O at 25 oC in the point of ternary compound (anhydrous projection of solubility diagram): thin, dash and dot lines - isopotentials of water in the crystallization field of the compound (9Li2SO4 3K2SO42Cs2SO4), where water chemical potentials or water activity are constant: (^)T p = (p.^^^ + rt aw
(aw are represented by figures in Fig.3, data is got from [30])). End of arrow symbolizes figurative point of the compound and simultaneously maximum aw = 0.938 a.u. in whole solubility diagam.
Three-phase Equilibrium.
If we have three phases, linear dependence of its composition symbolizes the fact that figurative points of all phases belong to one straight line. It may be realized in some cases:
I) In n-component hetero-azeotropes in liquid-liquid-vapor phase diagrams with miscibility gap in liquid (for example, Schreinemakers rule about extrema of TP, PT in ternary systems). Our apparatus allow to spread this rule to the systems with the arbitrary number of components.
J) In n-component pseudo-hetero-azeotropes in solid-solid-vapor phase diagrams with miscibility gap in solid (for example, analog of Schreinemakers rule about extrema of TP, PT in ternary systems). Our apparatus allow to spread this rule to the systems with the arbitrary number of components.
K) In n-component curves of solid-solidliquid equilibrium in fusibility diagrams (van Rijn points with extrema of TP in ternary systems and also in arbitrary number components systems in our case).
L) In (n+1)-component curves of solidsolid-liquid equilibrium in solubility diagrams (Type van
Rijn points with extrema of (p )PT in quaternary systems
and also in arbitrary number components systems in our case [30]).
The schemes of different type isomorph diagrams for the cases (I, J, K) in different coordinates are represented in Figures 4, 5.
Fig. 4. Schemes of topological equivalent phase equilibrium diagrams in ternary systems in different variables, dot lines -nodes, Ex - points ofextrema of TP or PT, (cr) - critical liquidliquid phase (or solid-solid solutions, if both of them are characterized by the same crystal group symmetry).
Multi-phase Equilibrium.
If we have many phases (r>3), linear dependence of its composition symbolizes the fact that: figurative points of four phases belong to one plane (r=4), figurative points of five phases belong to one three-dimensional hyper-plane (r=5), ... figurative points of n phases belong to one (n-i)-dimensional hyper-plane.
M) In n-component hetero-azeotropes in liquid-liquid-liquid-vapor phase diagrams with three-phase miscibility gap in liquid (where extrema of TP, PT in quaternary and more component systems are realized).
N) In n-component hetero-azeotropes in solid-solid-solid-vapor phase diagrams with three-phase miscibility gap in solid (where extrema of TP, PT in quaternary and more component systems are realized).
O) In n-component curves of solid-solid-solid-liquid equilibrium in fusibility diagrams (where extrema of TP in quaternary and more component systems are realized).
P) In (n+1)-component curves of solid-solid-solidliquid equilibrium in solubility diagrams (where extrema of (fdw )PiT in five- and more component systems are realized).
A (X-i =1, Y| =1)
B(X2=1,Y2=1) E[,c D=nB*mC(*pW) E^BC C(X3=1,Y3=1)
Fig. 5. Schemes of topological equivalent phase equilibrium diagrams in ternary system (fusibility) and quaternary (solubility) in different variables, dash-dot lines - nodes, E^mc) - eutectics (eutonics) points, D - distectics (distonics), R points with extrema
of TP or (¡Uw )PT - van Rijn points, dot lines - liquidus isotherms
(iso-potentials of solvent) in the fields of crystallization of single solid phase.
Extremum of State Parameters, Connected with the Decrease of Variability (Component Numbers) of One of Equilibrium Phases.
This condition is considerably less known in the comparison with the previous Gibbs rule. Only some arti-
cles in some specific cases considered this condition later [32, 34, 35, 37]. Meanwhile, this condition occurs more or less often.
Let us consider r-phase equilibrium where one of the phase (name it "1") with the variable composition is component free - i.e. has maximal component number, and it determines the component number of whole heterogeneous system - name it (n-f). The other equilibrium phase also with the variable composition is component unfree, the number of components in it is n-uf=(n-f - m), where, as a rule: m=1 or n-uf=(n-f - 1). Let us give some example of such equilibrium in different phase diagrams:
A) Fusibility diagrams in semiconductor systems A3-1 - A3-2 -...-.A3-N - B5-1- BP'2-... - 5 (where: A3 = A, Ga, In...; B5 = P, As, Sb, Bi...), in the liquid phase - melt there are N+M components, but in the solid solutions (s) there are only N + M - 1 components, because:
(*) = w =1/ 2. Similarly in semiconductor sys-/=i /=i
tems A2-1 - A2-2 -...-2 - B6-1 - B6-2-...- B6-M (where: A2 = Cd, Hg,Zn...; B = Se, Te, s ^Xw =1/2, so
/=i /=i
n-f= M+N, n-uf= M+N-1.
B) Solid-vapor or sublimation diagrams in the same semiconductor A3 - B5 or A2 - BB systems, where n-f = M+N-component vapor is in the equilibrium with the same solid solutions with n-uf = M+N-1 solid solutions. Such equilibrium occurs, for example, in the processes of MBE (Molecular Beam Epitaxy) of semiconductor layers on the substrates.
C) Solubility diagrams in salt-solvent solution with solid solutions formation at T,P=const. For example in ternary system CoCl2-NiCl2-H2O at 25oC liquid phase is really 3-component, but solid phase - solid solution Co><Nii-x*6H2O is 2-component, because XCC2 + XNC12 = 1/7, or XHO = 6/7.
We can not imagine, now anyway, nonlinear condition, limiting components number in equilibrium phase. Let us consider it linear.
Formulate theorem or later rule II.
Rule II.
II.I. If phase composition of component free phase passes through the linear condition (which is limiting component numbers in unfree component phase) in full concentration space, then:
a) temperature of phase equHibrium (at P=const and unfree component phase composition const: dX(2) = 0 ), P) pressure of phase equHibrium (at T=const and dX(2) = 0 ),
Y) chemical potentials of all components in unfree component phase (at T, P=const),
5) molar fractions of all components in unfree component phase (at T, P=const), passes through the extremum or conditional extremum. £) If phase composition of component free phase passes through the linear condition (which is limiting component numbers in unfree phase) in reduced (without component w) concentration space, then chemical potential of component w (at T, P=const) passes through the extremum or conditional extremum.
This rule is realized in some special cases.
Ternary Systems (for Fusibility and Sublimation Phase Diagrams) and Quaternary Systems (for Solubility
Diagrams).
Let as consider ternary system at the conditions dT = dP = 0 and write van der Waals equation in the variables of unfree component phase:
(X(1) - X (2))G(2)dX(2) = 0 (6.1).
Linear limiting for unfree component phase should symbolize in geometric sense, that composition vector of unfree component phase should belong to one cut in equilateral concentration Roseboom triangle. If unfree condition becomes valid for free component phase, then its composition vector should belong to this cut also. So vectors X(i\X(2),dX(2),X(1) -X(2) should be colline-ar, so one can write, that (X ^ 0):
X(1) -X(2) = XdX(2) (6.2),
XdX (2)G(2) dX(2) = 0 (6.3).
dX(2)G(2)dX(2) is positively determined bilinear form, so the equation (6.2) may be valid, only if:
dX(2) = 0, dX(2) = 0(i = 1,2....n - uf) (6.4),
Or molar fractions of all components in unfree component phase Xp should be in extremum. So guess 5) is proved.
Let us write the expression for the calculation of chemical potentials in components unfree component phase, one immediately get (using (6.4):
(6.5),
d (^')T
:£[dM^/ dX?
]
dX 2
(6.7), or: (6.8). condition:
So guess y) is also proved.
Let as consider ternary system at the conditions dP =0 and dX(2) = 0 and write van der Waals equation in the variables of free component phase:
(X(2) _ X (1))Q (1dX v = S(1-IdT (6.6).
If unfree condition becomes valid for free component phase, then its composition vector should belong to the cut in concentration triangle, whom should belong composition vector of unfree component phase. Again, in this conditions one can write, that ( X* 0 ): X(1) _ X(2) = XdX(2) XdX (2)Q(2) dX(1) = S (1-2)dT Bilinear form dX (2)Q(1) dX<X) = 0 at the dX(2) = 0, so:
Sil-2)dT = 0, dT = 0, and T is in extremum, and guess a) is proved.
Let as consider ternary system at the conditions dT =0 and dX(2) = 0. Repeat reasoning and get, that at the conditions , when unfree condition becomes valid for free component phase:
V(1-2)dP = 0, dP = 0, and P is in extremum, and guess p) is proved.
Let us consider quaternary system at the conditions dP =0, dT = 0 and df(2) = 0 and write van der Waals equation in the variables of free component phase:
(Y(2) _f^QWdf'1 = «r2) dMw (6.9).
If unfree condition becomes valid for free component phase, then its composition vector in reduced concentration space should belong to the cut in concentration
triangle, whom should belong composition vector of unfree component phase. Again, in this conditions one can write, that ( X * 0 ):
f(1) _ f(2) = Xdf(2) (6.10):
XdP2G(2) dfm = n(— d/ (6.11).
Bilinear form di(I)Q(1 df(1) = 0 at the condition: df(2> = 0,
so: «l-2) d/ = 0. d/ = 0, and /w is in extremum,
and guess £) is proved.
One can easily see, that the result is not dependent on the number of equilibrium phases, both with free and unfree composition.
Concrete examples are represented in the fragments of fusibility diagrams in A3 - B5 ternary systems in the examples of passing through the extrema composition variables (at T,P =const) (Fig.6) and liquidus temperature of solid solution (at solid solution constant composition) -(Fig.7). For the description of this theorem application we used as an experiment, as thermodynamic modeling calculations. It is quite naturally to use for this purpose semi-empirical models. In our examples we shall use following ones:
-EFLCP model (Excess Functions = Linear Combinations of standard chemical Potentials) [35-38] for the description of the excess components functions in the melt in semiconductor A3-B5 systems. Parameters of model are represented, for example in [35].
-LDM (Lennard-Jones Model) or QRSM model (Quasi Regular Solution Model) [35, 38, 39] for the description of the excess components functions in the iso-valent substitution solid solutions in semiconductor A3-B5 systems. Parameters of model are represented, for example, in [35].
c 1.0
3 0.9
o 0.8
TO "o X CM 0.7
c -O 0.6
CO X 0.5
<
o s> «> 0.4
c o X rsl 0.3
TO « 0.2
c (U c 0.1
c
o 0.0
"HI
> ........InSb
.......V-
Sum
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
a.u.)
Fig.6. The dependence of the components concentration in solid solutions on the deviation from un-freedom condition: X(2 + X^ = 0.5 at 873 K in ternary system In-As-Sb lines -
calculation, points - experimental data.
Fig.7. The dependence liquidus temperature on the deviation from un-freedom condition: x(2 + X(£ = 0.5 (at constant composition of solid solution Ga0.5As0.48Sb0.02) in ternary system Ga-As-Sb lines - calculation, points - experimental data
!=1
T .P.X
J*'.n_uf
Quaternary Systems (for Fusibility and Sublimation Phase Diagrams) and Five-component Systems (for Solubility Diagrams).
Let as consider quaternary system at the conditions dT = dP = 0 and write van der Waals equation in the variables of unfree component phase:
(X(1) - X (2))G (2)dX(2) = 0 (6.12).
Linear limiting for unfree component phase should symbolize in geometric sense, that composition vector of unfree component phase should belong to one plane in concentration tetrahedron. If unfree condition becomes valid for free component phase, then its composition vector should belong to this plane also. So vectors X(1),X(2),dX(2),X(1) -X(2) should belong to one plane, so one can write, that (constants X ^ 0 ):
XX(1) + 4 X(2) + XdX(2) = 0 (A = const) (6.13), (p X(2) + p2 dX(2))G(2) dX(2) = 0(p = const) (6.14).
Strictly speaking, we can say, that differential equation (6.14) determines conditional (not absolute) extremum of function X(2). It is enough to add additional condition, which restricts moving of the composition vector of unfree phase along some straight line (in the plane), for the real extremum realization. We shale label this additional condition as ""cond". In multidimensional geometrical (or linear algebraic) sense, it means that dependence x(2)(x<x))tpconj form positive or negative
pleats (ridges or hollows) [40].
In this case vectors X(2), dX(2) should be linearly dependent, and equation (6.14) may be rewritten:
vdX (2)G(2) dX(2) = 0(v = const) (6.15),
And one can state, that X(2) components vector passes through the conditional extremum: (dX (2))t ,P,od = 0,(dXi(2))T PPcond = 0(i = 1,2... n - uf ) (6.16),
Or molar fractions of all components in unfree component phase X{2) should be in conditional extremum. So guess 5) is proved.
Linear Conditions for Conditional Extremum Realization. Linear conditions of conditional extremum realization, which should be superimposed on the composition of unfree phase (cond), may be arbitrary, but always should be linear. For example, it maybe:
A) Plane trim, or X,(2) = constt(i) (6.17),
B) Plane section, or x,(2) / Xf = const) (6.18),
C) Condition of consistency of lattice constant of solid solution (where it is unfree phase, very common case). In this case, according to Vegard's law [35] we can get following condition:
3(4) tr i™
2 = aSut (6.19),
i=1
where: summing is carried out on the components of solid solution, number of which is 3 for ternary solid solution of 4 for mutual ternary solid solution, ai - lattice of i-th pure component of solid solution, asub - lattice period of the substrate, selected later.
D) Maybe chosen also any other linear form of the condition.
Let us write the expression for the calculation of chemical potentials in components unfree component phase, one immediately get (using (6.16):
d to = "IW' / dX(2 = 0 (6-20)'
1=1 T,P,XJ„"_uJ ,co"d
So guess y) is also proved.
Absolutely similarly one can prove, that, if unfree condition becomes valid for free component phase in the condition of consistency of composition vector of unfree phase:
d(T )px»c»d = 0 d(p)T,x<-w> d(Mw )p,TX(,cod = °>('' = U." - uf )
(6.21),
i.e.:functions
(T )
P,X( s), cond
(P)
T,X(s) ,cond 'w ^P,T,Y(s\cond
are in the con-
ditional extrema and guesses a), P), y) are valid.
One can easily see, that the result is not dependent on the number of equilibrium phases, both with free and unfree composition. The schemes of the behavior of phase composition in the points of extrema and nearby them are represented in Figure 8. Concrete examples in are represented in the fragments of fusibility diagrams in A3 - B5 ternary systems in the examples of passing through the extrema composition variables (at T,P =const) and liquidus temperature of solid solution (at solid solution constant composition). Semi-empirical thermodynamic model calculation were based on EFLCP and LDM model with the parameters from [35].
i concentration <
Fig. 8. The dependence of the components concentration in solid solutions on the deviation from un-freedom condition: X(+ X(2 = 0.5 at 873 K in quaternary system In-Ga-As-Sb
lines - calculation, points - experimental data (solid solutions are lattice matched to substrate GaSb).
Five- and More-component Systems (for Fusibility and Sublimation Phase Diagrams) and Six- and More-component Systems (for Solubility Diagrams).
Linear limiting for unfree component phase should symbolize in geometric sense, that
composition vector of unfree component phase should belong to one hyper-plane in
concentration hyper-simplex. If unfree condition becomes valid for free component phase, then its composition vector should belong to this hyper-plane also. So vectors X(1),X(2),dX(2\X(1) -X(2) should belong to one hyper-plane, so equations (6.13), (6.14), (6.15) are also valid. The dependencies x(2)(x(1))tpcod , (pi)tpcond '
(T )px (s) ,cond ' (P)T X <*W ,(pw ) PT ,Y ( s ),cond passes through
conditional extrema - positive or negative hyper-pleats (hyper-ridge or hyper-hollow).
Linear Conditions for Conditional Extremum
Realization. In this case number of the conditions for
conditional extrema realization depends on the number of components in the system (n). Number of independent conditions in n-component system is:
Ncon = n - 3 (Fusibility and sublimation phase diagrams at Tor P=const) (6.22), INcon = n - 4 (Solubiiity phase diagrams at T, P=const) (6.23).
So for ternary fusibility and sublimation phase diagrams and quaternary solubility phase diagrams there are no conditions and extrema should be unconditional Ncon = 0. If Ncon > 1, one must choose Ncon = n - 3 (n -4) from the conditions, for example (6.17)-(6.19) to realize conditional extremum.
Conclusions.
Thus, according to the authors opinion, two independent and, according to the authors opinion, the most general and comprehensive conditions of passing of the parameters of state in heterogeneous equilibrium systems are displayed in the article, regardless to the type of phase equilibrium, number of components and number of equilibrium phases. Both conditions are directly connected with the linear interdependence between phases composition in the cases of component free and component unfree phases, correspondingly. Now we can not even imagine nonlinear variant of the dependence between equilibrium phases composition.
Заключение
Таким образом, в статье отображены два независимых и, по мнению авторов, наиболее общих и всесторонних условия прохождения параметров состояния в гетерогенных равновесных системах через экстремумы, независимо от типа фазового равновесия, числа компонентов и числа равновесных фаз. Оба условия напрямую связаны с линейной взаимозависимостью между фазовым составом в случаях, когда концентрации компонентов в фазах изменяются независимо и линейно зависимо (в соответствии с природой фаз), соответственно. В настоящий момент авторы, не могут представить себе вариант нелинейной зависимости состава равновесных фаз, ограничивающий компо-нентность последних.
Funding
This work was supported by Russian Foundation for Basic Research (RFBR) (Projects Nos.18-08-00143 A, 19-015-00469 A, and 19-016-00003 A).
Research was performed using the equipment of the Resource Centers "GeoModel", Center for Chemical Analysis and Materials Research of Research park of St. Petersburg State University.
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