The condition for the conservation of momentum at the interface under phase transitions of solutions

Abstract. The present work considers a change in the momentum under the transfer of a solution through the interface. It is shown that pressure related to the partial volumes of components arises in a solution under diffusion. As a result, the distribution of the concentration of solution components differs qualitatively from the known solutions. In contrast to the description of one-dimensional interphase mass transfer using the convective diffusion problem, the proposed model provides the stationary exponential distribution of components in both phases. The model describes component segregation by the interface, observed in the experiments.

Here is the concentration distribution of the component В, z is the spatial variable, is the diffusion coefficient. It is assumed that a solution moves in the positive z direction at the specified constant velocity w. It is also assumed that the system under study is heterogeneous, The condition for the conservation of a component mass flow at the interface is The problem provides the concentration distribution Let us make transformations of this system. We substitute the expression of a component mass flow, obtained from Eq. (5), into conservation equation (6). As a result, we obtain Eq. (1). Then, we express the concentration gradient from Eq. (5) and substitute it into Eq. (1). As a result of this transformation, the linear equation is obtained Here the notation of the mass flow of the component , where ni j const  since the mass flow of any component does not depend on a coordinate in any section of the system. The solution to Eq. (7) is a sum of two exponents.
The appearance of the new solution is natural when transforming systems of differential equations. However, as well as in the considered case, such solutions often do not satisfy an initial system of equations. In this case, solution (9) does not satisfy Fick's law (5). In the general case, Fick's law (5) and the diffusion equation are a part of the general system of mass, energy, and momentum transfer. The diffusion equation is an equation of the conservation of mass flow. That is, it is a part of the general system of transfer equations. It is reasonable to assume that the analysis of the process of solution phase transition by the equation of mass flow conservation only is incorrect. It is well-known that in the general case a pressure gradient arises in a multicomponent system due to a difference in component particle masses under isothermal diffusion. Under the action of this pressure gradient, forces affect the component particles. Let us introduce pressure into Fick's law (6). We will schematically derive generalized Fick's law [11] to formulate all the assumptions used in the statement of the diffusion problem with regard to pressure.

Introduction of pressure into Fick's law.
According to the Onsager relations, the diffusion flow of the component n in a twocomponent homogeneous system is described by the equation here is the diffusion flow of the component n, are the phenomenological coefficients, are the thermodynamic forces of a diffusion process, is the thermodynamic force of a thermal conductivity process. The thermodynamic forces have the expressions c z   . By substituting the thermodynamic forces of (11) to (10), we obtain the following expression for a two-component The present work aims at analyzing a change in the component mass flows in homogeneous regions of a heterogeneous system as a result of a phase transition. During such analysis, difficulties arise in the physical interpretation of the condition of temperature constancy in phases. According to thermodynamics, the deviation of chemical potential from its equilibrium value at the interface is the motive force of the process of a phase transition. In experiments, the deviation of chemical potential from equilibrium is related to the specified velocity of interface motion through the field of a temperature gradient. In the theory of the growth of a solid phase from a single-component melt [3], the velocity of interface motion is a unique dependence on the kinetic undercooling k T  . Kinetic undercooling is the difference between the equilibrium temperature of the interface and its current temperature In the present work, the velocity of solution transfer is related to the partial velocities of the components. It is assumed that the partial velocities of the components depend on kinetic undercooling. However, in the isothermal problem 0 k T   . Therefore, we will assume that temperature gradients in phases are negligibly small to simplify the problem and consider it as the isothermal one. In this case, the isothermal problem is solved in each phase, but the temperature in the phases differs by k T  . On the one hand, this condition allows taking into account the deviation of the interface from equilibrium, i.e., taking into account the motive force of the process of a phase transition. On the other hand, it enables assuming temperature gradients to be zero in each phase. In this case, Eq. (12) takes the form We will not consider the cross-diffusion effects between the solution components in the problem under study to obtain the simplest solutions. Therefore, we assume that . In this case, component mass flows have the expressions By differentiating, we find that The coefficients before derivative concentrations are diffusion coefficients [11]  The derivative chemical potential with respect to pressure at constant temperature and component concentration is the partial specific (molar) volume.
Specific values are used in the present work. As a result, we come to the system of equations here the notations , In linear non-equilibrium thermodynamics [11], it is assumed that the coefficients of component diffusion are equal and the condition of such equality is found. However, the conditions of equality of diffusion coefficients for the isothermal, isobaric diffusion problem require the constancy of temperature and pressure. The condition of pressure constancy is not satisfied in the problem under consideration. We assume that in the problem under study the coefficients of component diffusion also are equal and find conditions under which this equality holds.
Let us equate the diffusion coefficients in system of equation (13) A sum of diffusion flows is zero, which follows from their definition. Consequently, the necessary condition of the equality of component diffusion coefficients is the equality Let us find the relation between the diffusion coefficient and phenomenological coefficients. For this to be done, we use the Gibbs-Duhem equation. At constant temperature we We write the expressions of the differentials of chemical potentials as . As a result, we come to the equation The physical meaning of the Gibbs-Duhem equation is the relation between the intensive parameters of a solution at a small deviation from equilibrium. Consequently, in this equation volume is equal to the initial equilibrium unit volume . The value of chemical potential is inf V equal to its value of an initial equilibrium solution, i.e., a solution at the concentration Cinf. Let us replace the notations and take into account relation (14) From here we find the relation between the diffusion coefficient and the phenomenological The volume depends on pressure and partial volume. We write out the volume differential Here  is the isothermal expansion coefficient, and the relation set a condition of volume constancy, 0 dV  , and we obtain the relation between the potentials of concentration and pressure The expression for the phenomenological coefficient takes the form where is the partial hydrodynamic velocity of the component B i . This equation differs from Eq. (5) in an additional term. The additional term has simple physical meaning. It is a force that affects the particles of solution components due to a change in their partial velocities. Eqs.
(16) and (6)  It is necessary to make a significant addition. Eq. (7) was obtained by the transformation of Fick's equation (5) and equation of mass flow conservation (6). If the system of Eqs. (16) and (6) is transformed in the same way, it is easy to obtain the same Eq. (7). Moreover, in special cases two more general solutions are obtained.
If one knows the distribution of the concentration , it is easy to find pressure distribution from Eq. (16) The integration constant pBi A does not depend on z. The velocity of any component n is   In addition to these four equations, solutions provide the relations between concentrations and velocities at the interface. In total, we have six equations.
One more equation is the relation between boundary concentrations and a phase diagram.
As the simplest example of an equilibrium phase diagram, we consider a eutectic phase diagram with linear equilibrium lines. The present work takes into account the deviation of chemical potential from its equilibrium value at the interface. Consequently, within the temperatureconcentration diagram under a phase transition, the temperature of the interface will differ from the equilibrium value by the value of the kinetic undercooling   Let us consider the results of the numerical calculation of concentration and pressure distribution in the system the parameters of which are shown in Table 1. The values of calculation parameters are selected so that it is convenient to interpret the physical relation between velocities, concentration, and pressure in the graphs. The concentration dependencies on spatial coordinate and kinetic undercooling are presented in Fig. 1

Discussion and conclusions.
The description of a stationary phase transition of solutions, provided in the present work, apparently is the simplest model of phase transitions. The main conclusion from the considered problem is that even in the simplest case the problem of component distribution during the phase transition of a solution should contain all the variables of chemical potential -temperature, pressure, and density of components. Indeed, if we do not take into account a change in the temperature at the interface, we come to the quasi-equilibrium solution. In this form, the solutions do not provide the limiting transition to equilibrium since the relation between the velocities and the kinetics of the addition of component particles to a new phase is lost. If the pressure is not taken into account, the problem provides a solution only which formally agrees with the solution to quasi-equilibrium problem (4). This solution corresponds to the value , i.e., trajectories 3 and 4 in Fig. 1 and trajectory 3 in Fig. 3. Second, as is well-known, it is difficult to obtain a stable plane interface in experiments.
Therefore, the question arises on the stability boundary of the obtained solutions.
The results of the present work change significantly the description of the processes of mass transfer at the interface. The kinetics of the addition of solution components to a new phase leads to a change in the momentum of components under their transfer through the interface.
Additional pressure arises in phases as a result of a change in the momentum. This additional pressure crucially affects the distribution of component concentration.