Abstract

In this paper, the Kedem–Katchalsky equations in matrix form for nonhomogeneous ternary nonelectrolyte solutions were applied for interpretation of transport through the membrane mounted in horizontal plane. Coefficients , , and (for nonhomogeneous solutions), Hij and (for homogeneous solutions) (i, j ∈ {1, 2, 3}, r = A, B), , and were calculated on the basis of experimentally determined coefficients (Lp, σ1, σ2ω11, ω22, ω21, ω12, , and ) for glucose in aqueous ethanol solutions and two configurations of the membrane system. From the calculations, it results that the values of coefficients , , , , , , and depend nonlinearly on solution concentration as well as on a configuration of membrane system. Besides, the values of coefficients , , , , , and depend linearly on solution concentration. The value of coefficients H13, H23, and H33 do not depend on solution concentration. The coefficients ψ12, ψ13, ψ22 = ψ23, ψ32 = ψ33, and ψdet depend nonlinearly on solution concentration and for  ≈ 9.24 mol m−3 are equal to zero. For  < 9.24 mol m−3, the values of coefficients ψ12 and ψ13 are negative and for  > 9.23 mol m−3, positive. In contrast, the values of coefficients ψ22 = ψ23, ψ32 = ψ33, and ψdet for  < 9.24 mol m−3 are positive and for  > 9.24 mol m−3, negative. For  = 0, we can observe nonconvective state, in which concentration Rayleigh number reaches the critical value RC = 1691.09, for is convective state with convection directed straight down and for is convective state with convection directed straight up.

1. Introduction

The membrane transport study is still a scientific challenge fulfilling the cognitive and application criteria in many processes occurring in living organisms (in membranes and biological cells, kidneys, etc.) observed in laboratory conditions and used in industry (hemodialysis, desalination of water, concentration of juices, etc.) [1, 2]. Moreover, in recent years, experimental techniques using microfluidic methods and nanoscale interface engineering to study interalia colloidal flows have been developed [3, 4]. The exchange of fluids and substances dissolved in it by biological membranes facilitates the transport of substances needed by living organisms to maintain their metabolic activity and regulation of pressure equilibrium by membranes in order to maintain the structural integrity of biological systems [5]. Passive and active transport processes control permeation of these substances. Passive transport mechanisms allow the flows of water and/or solutes to reduce their concentration gradients without energy using. In turn, active transport mechanisms allow penetration of dissolved substances against their concentration gradients, at the expense of energy supplied from metabolic reactions. The interaction between these mechanisms determines the hydrostatic fluid pressure and osmotic pressure differences through biological membranes, which are important features of biological systems [6, 7].

To describe transport phenomena in biological and technical systems, models developed in the framework of nonequilibrium thermodynamics [812], diffusion models [13, 14], and friction [1517]; models developed in the framework of statistical physics [1820]; and models developed as part of the network thermodynamics [2126] are used. In order to characterize the relationship between generalized streams of liquid and solutes and their generalized driving forces, which are derived from the electrochemical and/or chemical affinity gradient, the thermodynamics of Onsager are used [9, 11, 27, 28]. This model is based on the existence of the dissipation function, which describes the total change in the entropy of the system. In near-equilibrium systems, i.e., those in which the dissipation rate of free energy is small, linear dependencies between the flow and the driving forces (thermodynamic stimuli) can be assumed [10, 11]. The classic Kedem–Katchalsky model equations were developed in accordance with these principles [9]. This model takes into account the interaction between solvents and nonionic solutes. This gives a set of phenomenological membrane coefficients that can be easily determined experimentally in a series of independent experiments. Moreover, this formalism provides the theoretical basis for the analysis of the volume and solute fluxes in various membrane systems [10, 11]. Therefore, the KK equations are one of the basic research tools for membrane transport in both biological and artificial systems. Many versions of these equations are used: classical [7], Kargol’s [29], Chang and Pinsky [6], and network thermodynamics [2226]. These versions of KK equations for nonelectrolytes show the relationship between volume flow () and dissolved matter (Js) and thermodynamic forces: osmotic (Δπ) and/or hydrostatic (ΔP). The network form of KK equations is obtained by symmetrical and/or hybrid transformation of classical KK equations using Peusner network thermodynamics (Peusner’s Network Thermodynamics, Peusner’s NT) [22]. For homogeneous and nonhomogeneous binary solutions of nonelectrolytes, two symmetrical and two hybrid forms of KK equations are known. The symmetrical form of these equations contain Peusner coefficients: and (for homogeneous solutions) and hybrid forms- and (for inhomogeneous solutions) (i, j ∈ {1, 2}, r = A, B) [22, 24].

Therefore, the concepts of L, R, H, and P have been introduced in the form of Kedem–Katchalsky network equations for homogeneity conditions and , , , and in the form of Kedem–Katchalsky network equations for polarization concentration conditions of solutions separated by the membrane [25]. For the homogeneity conditions of nonelectrolytic ternary solutions, there are two symmetrical and six hybrid forms of network KK equations. The symmetric forms of these equations, similarly to homogeneous ternary solutions, contain and coefficients and are derived directly from Onsager’s thermodynamics and hybrid forms- , , , , , and (i, j ∈ {1, 2, 3}), which are a consequence of the application of network thermodynamics techniques [30]. For concentration polarization conditions, these coefficients should be written in the form , , , , , , , and (i, j ∈ {1, 2, 3}, r = A, B). Therefore, the concept of the form L, R, H, K, P, N, S, and W can be introduced in the network equations KK for the conditions of homogeneity and the form , , , , , , , and of the KK equations for the concentration polarization conditions of ternary solutions separated by the membrane.

The aim of the next series of papers is to present the form , , , , , , , and of the KK equations for concentration polarization conditions of ternary solutions. The aim of this work is to develop the form of of the KK equations, containing the Peusner coefficients (i, j ∈ {1, 2, 3}, r = A, B). Besides, we compare and coefficients and matrix coefficients  = det [] and  = det []. We will present the results of calculations of coefficients and matrix coefficients  = det [] and  = det [] and the quotients ij= ()/ and = ( − )/.

2. Materials and Methods

2.1. Membrane System

Similarly as in previous papers [31, 32], we will consider transport of nonhomogeneous ternary nonelectrolyte solutions with concentrations at the initial moment (t = 0) Ckh and Ckl (Ckh > Ckl, k = 1, 2) through the membrane (M) in the single-membrane system (Figure 1). This membrane separates compartments l and h and is isotropic, symmetric, electroneutral, and selective for solvent and nonionized dissolved substances. In the case of membrane located in horizontal plane that is perpendicularly to the gravity vector, there are configurations A or B of arrangement of solutions in relation to the membrane (r= A or B). In configuration A, the solution with concentration Ckl is in the compartment over the membrane while the solution with concentration Ckh is in compartment under the membrane. In configuration B of the membrane system, location of solutions is reversed. We will consider only isothermal and stationary processes of membrane transport, for which the measure is the volume fluxes () and solutes fluxes () (k = 1, 2 and r = A, B). These fluxes can be described by the KK equations for nonhomogeneous ternary nonelectrolyte solutions [33].

Under such conditions, solutes, which diffuse through the membrane, create the concentration boundary layers (CBLs) on both sides of the membrane signed by and [32, 33]. The CBL has a thickness marked as and CBL has a thickness marked as .

The mean concentrations of solutes “1” and “2” in membrane (, ) can be calculated using expressions  = (CkhCkl) [ln (CkhCkl−1)]−1 (k = 1, 2). Appearance of CBLs causes those concentrations at the interfaces of the membrane and solutions, respectively, to decrease from Ckh to and increase from Ckl to . For steady state, the following relation is fulfilled:  > ,  > Ckl, Ckh >  (k= 1, 2) [31, 34]. In addition, ρer and ρir denote the densities at interfaces and , respectively, while ρl and ρh (ρl < ρh or ρl > ρh) denote the densities of solutions outside the layers. Moreover,  > ρl or  < ,  >  or  < , and  > ρh or  < ρh [34]. If the solution with lower density is under the membrane the system, /M/ loses its hydrodynamic stability, and convective instabilities in the near membrane area are observed [33, 34].

The measure of the concentration polarization is the concentration polarization factor (). Its value depends on both the concentration of solutions separated by the membrane () and the configuration of the membrane system (r = A, B). More specifically for this case, the thicknesses of CBLs and exceed values ()crit and ()crit, and coefficient of concentration polarization () exceeds its critical value ()crit suitably [3436]. For diluted ternary nonelectrolyte solutions, the concentration polarization factor () and the thickness of concentration the boundary layers ( and ) can be described by the expression [31].

2.2. Matrix Form of the Kedem–Katchalsky Equations

According to the Kedem–Katchalsky formalism [7, 10, 33], transport properties of the membrane are determined for solutions containing a solvent and dissolved two substances (ternary solution) by practical coefficients: hydraulic permeability (Lp), reflection (σk, k = 1, 2), and permeability of solute (ωkf, k, f ∈ {1, 2}). In turn, the transport properties of the complex /M/ are characterized by hydraulic permeability (), reflection (), and permeability coefficients of solute (). The coefficients of hydraulic, osmotic, advective, and diffusive concentration polarization are defined by expressions , , , and [23]. The decrease of the value of volume and solute fluxes from and Jk (in conditions of homogeneous solutions) to and (in condition of concentration polarization) is caused by formation of the concentration boundary layers and , respectively [33].

The classical form of Kedem–Katchalsky equations for concentration polarization conditions can be written aswhere , , and are volume and solutes “1” and “2” fluxes, respectively; Lp is the hydraulic permeability coefficient; σ1 and σ2 are reflection coefficients suitably for solutes “1” and “2”; ω11 and ω22 are the solute permeability coefficients for solutes “1” and “2” generated by forces with indexes “1” and “2” and ω12 and ω21 are the cross coefficients of permeability for substances “1” and “2” generated by forces with indexes “2” and “1” respectively. is the hydrostatic pressure difference (Ph, Pl are higher and lower values of hydrostatic pressure suitably). Δπk = RT(Ckh − Ckl) is the difference of osmotic pressure (RT is the product of gas constant and thermodynamic temperature whereas Ckh and Ckl are solutes concentrations, k = 1, 2). is the mean solute concentration in membrane and is expressed by  = (Ckh − Ckl) [ln (CkhCkl−1)]−1 (k = 1, 2).

Relatively simple algebraic transformations allow transforming equations (1)–(3) into the form

The above equations are one of the forms of Kedem–Katchalsky equations obtained by the hybrid transformation of Peusner’s thermodynamic networks.

Equations (4) and (5) can be transformed by simple algebraic transformations to the matrix form:where , , , , , , , , , and [] is the matrix of Peusner’s coefficients (i, j ∈ {1, 2, 3}) for ternary nonelectrolyte solutions in conditions of concentration polarization.

It results from equation (7) for the nondiagonal coefficients  ≠ ,  ≠ , and  ≠ . On the basis of equation (7), we get , , and . Moreover, the determinant of the matrix is equal towhere , , .

Index “r” in equations (3)–(10) indicates that the fluxes , , and , coefficients (i, j ∈ {1, 2, 3} and r = A, B) and matrix [] of these coefficients, depend on configuration of the membrane system. For homogeneous conditions ( =  =  =  =  =  =  =  =  = 1), we getwhere , , , , , , , and [H] is the matrix of Peusner’s coefficients Hij (i, j∈ {1, 2, 3}) for ternary nonelectrolyte homogeneity solutions.

On the basis of equation (9), we get . The determinant of the matrix [H] is equal towhere , , and .

In order to show the relationship between coefficients and Hij and between and for A and B configurations of the membrane system (r = A, B), we can calculate using equations (6), (10), (11), and (12), the coefficients ψij = ( − )/Hij and ψdet = ( − )/. The expressions for the coefficients ψij and ψdet are given as follows:where , , , , , , , , and .

The values of coefficients ψij and ψdet show the influence of concentration polarization and natural convection on the membrane transport.

3. Results and Discussion

The values of coefficients , , , ψij = ( − )/, and ψdet = ( − )/, (i, j ∈ {1, 2, 3}, r = A, B), which describe equations (7)–(20), are calculated for polymer membrane Nephrophan and glucose solutions in aqueous solution of ethanol. The glucose concentration was marked by index “1” and the ethanol concentration by index “2”. The concentration of substance “1” in the compartment (h) take values from C1h = 1 mol·m−3 to C1h = 101 mol·m−3. In turn, concentration of a substance “2” in the compartment (h) was constant and amounted to C2h = 201 mol·m−3. The concentrations of both components in the compartment (l) were established and amounted to C1l = C2l = 1 mol·m−3. In expressions under equations (7)–(20), there are coefficients that describe transport properties of membrane (Lp, σ1, σ2ω11, ω22, ω21 and ω12), average concentrations of solutions “1” and “2” in the membrane (, ), and coefficients of concentration polarization (, , , , , , , , and ). The values of these coefficients are calculated, using conditions  =  = ζa2r = 1,  =  =  = , and  =  =  =  [33, 37].

In order to calculate Hij, , , ψij = ( − )/, and ψdet = ( − )/, (i, j ∈ {1, 2, 3}, r = A, B) on the basis of equations (7)–(20), we used the characteristics  = f(, ) and  = f(, ) presented in Figure 2 and following data: Lp = 4.9 × 10−12 m3·N−1·s−1, σ1 = 0.068, σ2 = 0.025, ω11 = 0.8 × 10−9 mol·N−1·s−1, ω12 = 0.81 × 10−13 mol·N−1·s−1, ω22 = 1.43 × 10−9 mol·N−1·s−1, ω21 = 1.63 × 10−12 mol·N−1·s−1,  = 2.79 ÷ 21.67 mol·m−3 and  = 37.71mol·m−3.

The calculations based on equations (7) and (9) show that  =  = H11 = 0.204 × 1012 N·s·m−3, and  =  = H31 = 36.77 mol·m−3 are independent of both concentration of solution and configuration of the membrane system.

Dependencies of coefficients  = f(,  = const.),  = f(,  = const.), H12 = f(,  = const.), and  =  = H21 = f(,  = const.) are shown in Figure 3. Graphs 1A, 1B, and 1 show that values of coefficients , , and H12 are decrease almost linearly with the increase in for  = 37.71 mol·m−3. These coefficients fulfill the following conditions: for  = 9.24 mol·m−3,  =  = −8.98 mol·m−3, for  < 9.24 mol·m−3, H12 >  >  < 0, and for  > 9.24 mol·m−3, H12 >  >  < 0.

Graph 2 shows that values of coefficients , , and H21 increase linearly with the increase in for  = 37.71 mol·m−3. These coefficients fulfill the condition  =  = H21 > 0. Dependencies of coefficients  = f(,  = const.),  = f(,  = const.), and H13 = f(,  = const.) are presented in Figure 4. Graphs 1A and 1B show that values of coefficients and decrease and increase nonlinearly with the increase in for  = 37.71 mol·m−3, respectively. The value of coefficient is equal to H13 = −36.77 mol·m−3 and is independent of both concentration of solution and configuration of the membrane system. These coefficients fulfill the following conditions: for  = 9.24 mol m−3,  =  = −37.49 mol·m−3, for  < 9.24 mol·m−3, H13 <  <  < 0, and for  > 9.24 mol·m−3, H13 <  <  < 0.

The graphs 1A, 1B, and 1 shown in Figure 5 illustrate the dependencies  = f(,  = const.),  = f(,  = const.), and H22 = f(,  = const.). Graph 1A shows that the values of coefficient first increase in a nonlinear way and then decrease in a nonlinear manner with the increase of value and graph 1B shows that the values increase nonlinearly with the increase of the of value (for  = 37.71 mol·m−3). In turn, the course of graph 1 shows that H22 is a linear function of (for  = 37.71 mol·m−3). These coefficients fulfill the following conditions: for  = 9.24 mol·m−3,  =  = 1.81 × 10−9 mol2N−1·s−1·m−3, for  < 9.24 mol·m−3, H22 >  >  > 0, and  > 9.24 mol·m−3, H22 >  >  > 0.

Dependencies of coefficients  = f(,  = const.),  = f(,  = const.), and H23 = f(,  = const.) are presented in Figure 6. Graphs 1A and 1B shows that values of coefficients and decrease and increase almost nonlinearly with the increase in for  = 37.71 mol·m−3, respectively. The value of coefficient is equal to H23 = 30.54 mol·m−3 and is independent of both concentration of solution and configuration of the membrane system. These coefficients fulfill the following conditions: for  = 9.24 mol·m−3,  =  = 6.98 × 10−13 mol2·N−1·s−1·m−3, for  < 9.24 mol·m−3, H23 >  >  > 0, and  > 9.24 mol·m−3, H23 >  >  > 0.

The graphs 1A, 1B, and 1 shown in Figure 7 illustrate the dependencies  = f(,  = const.),  = f(,  = const.), and H32 = f(,  = const.). Graph 1A shows that the values of coefficient first increase in a nonlinear way and then decrease in a nonlinear manner with the increase of value, and graph 1B shows that the values increase nonlinearly with the increase of the of value (for  = 37.71 mol·m−3). In turn, the course of graph 1 shows that H32 is a linear function of (for  = 37.71 mol·m−3). These coefficients fulfill the following conditions: for  = 9.24 mol·m−3,  =  = 3.35 × 10−12 mol2·N−1·s−1·m−3, for  < 9.24 mol·m−3, H32 >  >  > 0, and for  > 9.24 mol·m−3, H32 >  >  > 0.

Dependencies of coefficients  = f(,  = const.),  = f(,  = const.), and H33 = f(,  = const.) are presented in Figure 8. Graphs 1A and 1B show that values of coefficients and decrease and increase almost nonlinearly with the increase in for  = 37.71 mol·m−3, respectively. The value of coefficient is equal to H33 = 53.92 × 10−9 mol2·N−1·s−1·m−3 and is independent of both concentration of solution and configuration of the membrane system. These coefficients fulfill the following conditions: for  = 9.24 mol m−3,  =   = 12.64 × 10−9 mol2·N−1·s−1·m−3, for  < 9.24 mol·m−3, H33 <  <  > 0, and for  > 9.24 mol·m−3, H33 <  <  > 0.

The graphs 1A, 1B, and 1 shown in Figure 9 illustrate the dependencies  = f(,  = const.),  = f(,  = const.), and  = f(,  = const.). Graph 1A shows that the values of coefficient first increase in a nonlinear way and then decrease in a nonlinear manner with the increase of value, and graph 1B shows that the values increase nonlinearly with the increase of the of value (for  = 37.71 mol·m−3). In turn, the course of graph 1 shows that det [H] is a linear function of (for  = 37.71 mol·m−3). These coefficients fulfill the following conditions: for  = 9.24 mol·m−3,  =  = 6.74 × 10−6 mol2·N−1·s−1·m−3, for  < 9.24 mol·m−3,  >  >  > 0, and for  > 9.24 mol·m−3,  >  >  > 0.

Figures 39 show that there are three groups of characteristics  = f(,  = const.),  = f(,  = const.), and  = f(,  = const.), (i, j ∈ {1, 2, 3} and r = A, B). The first group includes the characteristics presented in Figure 3, the second Figures 4, 6, and 8, and the third the characteristics shown in Figures 5, 7, and 9. In the case of group 1, which includes concentration characteristics of coefficients , , , and (r = A, B), these coefficients are expressed in the same units, and their values are in the range −21.62 mol·m−3 ÷ 20.19 mol·m−3.

In the case of the second group of characteristics, the shape of the concentration characteristics of the coefficients , , , , , and (r = A, B) is very similar. However, the values of these coefficients differ by up to several orders of magnitude: −37.68 mol·m−3 ≤ ,  ≤ −37.23 mol·m−3, 0.95 × 10−13 mol2·N−1·s−1·m−3 ≤ ,  ≤ 30.54 × 10−13 mol2·N−1·s−1·m−3 and 1.51 × 10−9 mol2·N−1·s−1·m−3 ≤ ,  ≤ 53.92 × 10−9 mol2·N−1·s−1·m−3.

Moreover, the values of coefficients and are expressed in different units than coefficients , , , and .

The third group of characteristics includes the concentration characteristics of coefficients , , , , , and . Values of coefficients and are expressed in units other than , , , and . In addition, the values of these coefficients differ from each other by several orders of magnitude: 0.02 × 10−9 mol2·N−1·s−1·m−3 ≤ ,  ≤ 17.33 × 10−9 mol2·N−1·s−1·m−3, 0.03 × 10−12 mol2·N−1·s−1·m−3 ≤ ,  ≤ 35.32 × 10−12 mol2·N−1·s−1·m−3, and 0.03 × 10−6 mol4·N−1·s−1·m−9 ≤ ,  ≤ 236.09 × 10−6 mol4·N−1·s−1·m−9.

The graphs 1–5 shown in Figure 10 illustrate the dependencies  = f(,  = const.),  =  f(,  = const.),  = f(,  = const.), f(,  = const.), and  = f(,  = const.), respectively. These coefficients fulfill the following conditions: for  = 9.24 mol·m−3,  =  =  =  =  = 0, for  < 9.24 mol·m−3,  =  ≈  =  >  > 0 and  >  < 0, and for  > 9.24 mol·m−3,  =  ≈  =  <  < 0 and,  >  > 0. Besides, throughout the range of solution concentrations  =  =  = 0.

For example, we will consider equation and dependencies  = f(, ) (r = A, B) presented in Figure 2. It is drawn from the equation and Figure 2 that if ψ22 = 0, then  =  = 0.234. Moreover, by taking expressions  = D1 (D1 + 2RTω11δA)−1 and  = D1 (D1 + 2RTω11δB)−1 into account in equation (15), one can show that . From the equation, it becomes apparent that if ψ22 = 0, then δA = δB. Moreover, using equations and ρh ρl = (∂ρ/∂C1)(C1h C1l) + (∂ρ/∂C2)(C2h C2l), where (∂ρ/∂C1) = 0.06 kg·mol−1, (∂ρ/∂C2) = −0.0095 kg·mol−1, we can show that, if ψ22 = 0, it is for  = 9.24 mol·m−3 (C1h = 33.44 mol·m−3 and C1l = 1 mol·m−3) and  = 37.71 m−3 (C2h = 201 mol·m−3 and C2l = 1 mol·m−3) we get ρh ρl = 0.046 kg·m−3 and δA = δB ≈ 1.4 × 10−3 m.

Taking these data into consideration as well as D11 = 0.69 × 10−9 m2·s−1,  = 9.81 m·s−2, ω11 = 0.8 × 10−9 mol·N−1 s−1, ν = 1.063 × 10−6 m2·s−1, ρl = 998.3 kg·m−3, ζ1A = ζ1B = ζ = 0.234, and δA = δB = δ = 1.4 × 10−3 m in the expression for the concentration Rayleigh number RC = [](ρhνhD11)−1 = [] []−1 [31, 34], we get RC = 1691.09. The value is similar to the classical Rayleigh number critical value for Bernard problem RC = 1707.76 [38].

4. Conclusions

(1)The Kedem–Katchalsky equations in matrix form for nonhomogeneous ternary nonelectrolyte solutions were applied for interpretation of transport through the membrane mounted in horizontal plane. Coefficients and  = det [] (for nonhomogeneous solutions), Hij and  = det [H] (for homogeneous solutions) (i, j ∈ {1, 2, 3}, r = A, B), ψij = ()/, and ψdet = (])/ were calculated on the basis of experimentally determined coefficients (Lp, σ1, σ2ω11, ω22, ω21, ω12, , and ) for glucose in aqueous ethanol solutions and two configurations of the membrane system.(2)We can conclude that Peusner’s Network Thermodynamics (PNT) is an alternative manner of description of membrane transport both for homogeneity of solutions separated by a membrane and in conditions of concentration polarization.(3)The values of coefficients , , , , , , and depend nonlinearly on solution concentration as well as on a configuration of membrane system. The values of these coefficients in the convective state are greater then their values in the nonconvective state.(4)The values of coefficients , , , , , and depend linearly on solution concentration. The value of coefficients H13, H23, and H33 does not depend on solution concentration.(5)We can distinguish three groups of characteristics  = f(,  = const.),  = f(,  = const.), and  = f(,  = const.), (i, j ∈ {1, 2, 3} and r = A, B). In the case of group 1, which includes concentration characteristics of coefficients , , , and (r = A, B), these coefficients are expressed in the same units and their values are in the range −21.62 mol·m−3 ÷ 20.19 mol·m−3. In the case of the second group of characteristics, the shape of the concentration characteristics of the coefficients , , , , , and (r = A, B) is very similar. The values of coefficients and are expressed in different units than coefficients , , , and . The third group of characteristics includes the concentration characteristics of coefficients , , , , , and . Values of coefficients and are expressed in units other than , , , and .(6)There is a threshold value of concentration  ≈ 9.24 mol·m−3 above which the values of coefficients ψ12, ψ13, ψ22 = ψ23, ψ32 = ψ33, and ψdet are equal to zero. For  < 9.24 mol·m−3, the values of coefficients ψ12 and ψ13 are negative and fulfill the conditions  =  ≈  =  >  > 0 and  >  < 0. For  > 9.23 mol·m−3 values of coefficients ψ12 and ψ13 are positive and fulfill the conditions  =  ≈  =  <  < 0 and,  >  > 0. In contrast, the values of coefficients ψ22 = ψ23, ψ32 = ψ33, and ψdet for  < 9.24 mol m−3 are positive and for  > 9.24 mol·m−3 negative. Besides, throughout the range of solution concentrations,  =  =  = 0.(7)If ψ22 = 0, then  =  = 0.234 and it is conjugated with critical value of concentration Rayleigh number RC = [](ρhνhD11)−1 = [] [8ρhνhD14(ζRTω11)3]−1 = 1691.09. The value is similar to the classical Rayleigh number critical value for Bernard problem RC = 1707.76 [38].

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

Acknowledgments

This research was fully funded as statutory activity—subsidy of Ministry of Science and Higher Education granted for maintaining research potential in 2018 (research number BS/PB–622/3020/2014/P).