Simulation of S-Entropy Production during the Transport of Non-Electrolyte Solutions in the Double-Membrane System

Abstract

Using the classical Kedem–Katchalsky’ membrane transport theory, a mathematical model was developed and the original concentration volume flux (J_v), solute flux (J_s) characteristics, and S-entropy production by J_v, $\left({\left({\psi }_S\right)}_{J_v}\right)$ and by J_s $({\left({\psi }_S\right)}_{J_s})$ in a double-membrane system were simulated. In this system, M₁ and M_r membranes separated the l, m, and r compartments containing homogeneous solutions of one non-electrolytic substance. The compartment m consists of the infinitesimal layer of solution and its volume fulfills the condition V_m → 0. The volume of compartments l and r fulfills the condition V_l = V_r → ∞. At the initial moment, the concentrations of the solution in the cell satisfy the condition C_l < C_m < C_r. Based on this model, for fixed values of transport parameters of membranes (i.e., the reflection (σ_l, σ_r), hydraulic permeability (L_pl, L_pr), and solute permeability (ω_l, ω_r) coefficients), the original dependencies C_m = f(C_l − C_r), J_v = f(C_l − C_r), J_s = f(C_l − C_r), ${\left({\Psi }_S\right)}_{J_v}$ = f(C_l − C_r), ${\left({\Psi }_S\right)}_{J_s}$ = f(C_l − C_r), R_v = f(C_l − C_r), and R_s = f(C_l − C_r) were calculated. Each of the obtained features was specially arranged as a pair of parabola, hyperbola, or other complex curves.

1. Introduction

One of the most important properties of each non-equilibrium thermodynamic system is the continuous production of S-entropy [1,2]. The temporal change in S-entropy is a consequence of the entropy exchange with the external environment (φ_S) and the entropy production in the system ($\psi$_S). This means that for irreversible processes occurring in open systems, the S-entropy rate of change (dS/dt) is the sum of the rate of entropy exchange with the external environment (φ_S = d_eS/dt < 0 or d_eS/dt > 0) and the rate of entropy production in the system as a result of irreversible processes ($\psi$ _S = d_iS/dt > 0) [1,2,3]. The rate of entropy production can be expressed using the expression ${\Psi }_S={{\displaystyle \sum }}_i{J}_i{X}_i$, where $J_i{X}_i$ is the product of the conjugate forces ($X_i$) and fluxes ($J_i$). Prigogine [4] showed that for systems far from equilibrium, in the area of applicability of the extended non-equilibrium thermodynamics (ENET), $d{\psi }_s/dt=d_J{\psi }_S/dt+d_X{\psi }_S/dt=$ where $d_J{\psi }_S/dt\equiv {{\displaystyle \sum }}_i{J}_i\left(d{X}_i/dt\right)$, $d_X{\psi }_S/dt\equiv {{\displaystyle \sum }}_i{X}_i\left(d{J}_i/dt\right),$ and $d_X{\psi }_S/dt$ ≤ 0. Furthermore, $d_J{\psi }_S/dt$ ≤ 0, in the regime of applicability of linear non-equilibrium thermodynamics (LNET).

Membrane transport processes on the nano-, micro-, and macro-scale are the subject of interest in different areas of human activity in science, technology, and medicine [5,6,7,8,9]. One of the most important scientific achievements in this area is the double-membrane model proposed by Curran and McIntosh. This model requires the existence of two membranes (M_l, M_r) with various hydraulic permeability (L_pl, L_pr), reflection (σ_l, σ_r), and solute permeability (ω_l, ω_r) coefficients, arranged in series and separating the solutions with different concentrations (C_l, C_m, C_r) [10]. Papers published over several years have been dedicated to the analysis of transport in the double-membrane cell in order to clarify certain biophysical aspects of the membrane transport of water and dissolved substances, both in biological and artificial systems [11,12,13,14,15,16,17,18,19]. Recently, a double-membrane transducer protector [8] and double-membrane triple-electrolyte redox flow battery design [9] have been developed based on the concept of the two-membrane system.

On the basis of the characteristics illustrating the impact of concentration, pressure, and voltage dependencies on the volume flow, solute flow, and electric current, it has been shown that transport, in compliance with the concentration gradient, iso-osmotic transport, and the passive transport (against the concentration gradient), is possible in this system [11,12]. In addition, it has been shown that the double-membrane system is characterized by rectifying and amplification properties (asymmetry current-voltage characteristics [13,14,15,16,17,18,19]) and asymmetry and amplification of the volume and solute fluxes and hydromechanics pressure, which is characteristic for biological systems [20,21,22,23].

In the present paper, with the use of the Curran–Kedem–Katchalsky method utilized in the following papers [10,11,12,13,14,15,16,17,20,21,22,23], a non-linear mathematical model of transport in the double-membrane osmotic-diffusive cell was developed. This cell contains two membranes (M_l, M_r) arranged in series and separating the compartments (l), (m), and (r), which contain the solutions of various concentrations, respectively, C_l, C_m, and C_r (at the initial moment C_l > C_m > C_r or C_l < C_m < C_r). The volume of these compartments satisfies the conditions: V_m → 0 and V_l = V_r → ∞. Transport properties of the membranes M_l and M_r are characterized by coefficients of hydraulic permeability (L_pl, L_pr), reflection (σ_l, σ_r), and solute permeability (ω_l, ω_r). In order to search for new transport properties of the double-membrane system on the basis of the mathematical model, the calculations of the concentration (C_m), volume flux (J_v), solute flux (J_s), S-entropy produced by J_v, $\left({\left({\psi }_S\right)}_{J_v}\right)$ and by J_s $({\left({\psi }_S\right)}_{J_s})$, and osmotic and diffusion resistances (R_v, R_s).

2. Theory

2.1. Membrane System

Like in [7,8], let us consider the membrane system represented schematically in Figure 1. In this system, the compartments (l), (m), and (r), containing binary solutions of the same substance with the concentrations C_l, C_m, and C_r (C_l > C_m > C_r) are separated by electroneutral and selective membranes (M_l, M_r). The hydrostatic pressures in these compartments are denoted by P_l, P_m, and P_r (P_l > P_m > P_r) The membranes are characterized by the hydraulic permeability (L_pl, L_pr), reflections (σ_l, σ_r), and solute permeability (ω_l, ω_r). Compartment (m) consists of the infinitesimal layer of solution with the concentration C_m. The volume of this compartment fulfills the condition V_m → 0. The volumes of the compartments (l) and (r), containing solutions with the concentration C_l and C_r, fulfill the condition V_l = V_r → ∞.

The analysis of transport processes in this membrane system was based on the classical Kedem–Katchalsky model equations [15] using the Curran–Kedem–Katchalsky method [10,14,15]. Our starting point was the classical Kedem–Katchalsky model equations in binary and non-ionic solutions

$$J_s=\omega \mathsf{\Delta }\pi +J_v\left(1-\sigma \right)\overline{C}$$

(1)

$$J_v=L_p\left(\mathsf{\Delta }P-\sigma \mathsf{\Delta }\pi \right)$$

(2)

where J_s and J_v are the solute and volume fluxes; ω is the solute permeability coefficient; σ is the reflection coefficient; L_p is the hydraulic permeability coefficient; Δπ = RTΔC is the osmotic pressure difference (RT is the product of the gas constant and absolute temperature); $\overline{C}$ is the average solution concentration in the membrane; and ΔP is the hydrostatic pressure difference. Equation (1) describes the solute flux, the first component of this equation, J_sd = ωΔπ, describes the diffusive flux, and the second, J_sa = J_v(1 − σ) $\overline{C}$, is the advective flux. Then, Equation (2) describes the volume flux. It serves to note that the first component of this equation, J_vh = L_pΔP, is the hydraulic volume flux, and the second one, J_vo = L_pσΔπ, is the osmotic volume flux.

2.2. Model Equations

In order to describe the stationary volume flux in the membrane system shown in Figure 1, we considered Equation (1). The equations for membranes M_l and M_r are written in the following forms:

$$J_{sl}={\omega }_{l}RT\left(C_l-C_m\right)+J_{vl}\left(1-{\sigma }_l\right){\overline{C}}_l$$

(3)

$$J_{sr}={\omega }_{r}RT\left(C_m-C_r\right)+J_{vr}\left(1-{\sigma }_r\right){\overline{C}}_r$$

(4)

where ${\overline{C}}_l$ = (C_l − C_m)[ln(C_lC_m⁻¹)]⁻¹ ≈ $\frac{1}{2}$ (C_l + C_m), and ${\overline{C}}_r$ = (C_m − C_r)[ln(C_mC_r⁻¹)]⁻¹ ≈ $\frac{1}{2}$ (C_m + C_r).

In the steady state, the following conditions are fulfilled:

$$J_{sl}=J_{sh}=J_s$$

(5)

$$J_{vl}=J_{vr}=J_v$$

(6)

On the basis of Equations (3)–(6), we obtain

$$C_m=\frac{RT\left({\omega }_l{C}_l+{\omega }_r{C}_r\right)+\frac{1}{2}{J}_v\left[\left(1-{\sigma }_l\right)C_l-\left(1-{\sigma }_r\right)C_r\right]}{RT\left({\omega }_l+{\omega }_r\right)+\frac{1}{2}{J}_v\left({\sigma }_l-{\sigma }_r\right)}$$

(7)

In order to calculate J_v, on the basis of Equation (2) for the membrane system presented in Figure 1, we can write

$$J_{vl}=L_{pl}\left(P_l-P_m\right)-L_{pl}{\sigma }_{l}RT\left(C_l-C_m\right)$$

(8)

$$J_{vr}=L_{pr}\left(P_m-P_r\right)-L_{pr}{\sigma }_{r}RT\left(C_m-C_r\right)$$

(9)

Combining Equations (6), (8), and (9), we obtain the equation describing hydrostatic pressure in the intermembrane compartment of the double-membrane system

$$P_m=\frac{L_{pl}{P}_l+L_{pr}{P}_r-RT\left(L_{pl}{\sigma }_l{C}_l+L_{pr}{\sigma }_r{C}_r\right)+C_{m}RT\left(L_{pl}{\sigma }_l+L_{pr}{\sigma }_r\right)}{L_{pl}+L_{pr}}$$

(10)

Taking into consideration Equations (8) and (10), we obtain

$$J_v=\frac{L_{pl}{L}_{pr}}{L_{pl}+L_{pr}}\left[\left(P_l-P_r\right)+RT\left({\sigma }_r{C}_r-{\sigma }_l{C}_l\right)+RT{C}_m\left({\sigma }_l-{\sigma }_r\right)\right]$$

(11)

Including Equation (11) into Equation (7), we derive

$${\alpha }_1{C_m}^2+{\alpha }_2{C}_m+{\alpha }_3=0$$

(12)

where α₁ = L_plL_prRT(σ_l − σ_r)²; α₂ = 2RT(L_pl + L_pr)(ω_l + ω_r) + L_plL_pr(σ_l − σ_r)[(P_l − P_r) − RT(C_l − C_r)]; and α₃ = −2RT(L_pl + L_pr)(ω_lC_l + ω_rC_r) − L_plL_pr[(1 − σ_l)C_l − (1 − σ_r)C_r][(P_l − P_r) + RT(σ_rC_r − σ_lC_l)].

Hereby, we obtain the equation describing the solution concentration in the intermembrane compartment of the double-membrane system. Taking into consideration Equations (7) and (11), we obtain

$${\beta }_1{J_v}^2+{\beta }_2{J}_v+{\beta }_3=0$$

(13)

where β₁ = 0.5(σ_l − σ_r); β₂ = RT(ω_l + ω_r) − 0.5 L_plL_pr(L_pl + L_pr)⁻¹(σ_l − σ_r){(P_l − P_r) + RT[C_l(1 − 2σ_l) − C_r(1 − 2σ_r)]}, β₃ = −L_plL_pr(L_pl + L_pr)⁻¹RT{(P_l − P_r)(ω_l + ω_r) − RT(C_l − C_r)(σ_rω_l + σ_lω_r)}.

Taking into consideration Equations (3), (5), and (7), we obtain

$$J_s=\frac{\left({\gamma }_1+{\gamma }_2{J_v}^2\right)\left(C_l-C_r\right)+{\gamma }_3{J}_v\left(C_l+C_r\right)}{{\gamma }_4+{\gamma }_5{J}_v}$$

(14)

where γ₁ = (RT)²ω_lω_r; γ₂ = 0.5(1 − σ_l)(1 − σ_r); γ₃ = 0.5RT[ω_l(1 − σ_r) + ω_r(1 − σ_l)]; γ₄ = RT(ω_l + ω_r); and γ₅ = 0.5(σ_l − σ_r).

On the basis of Equations (12)–(14), C_m, J_v, and J_s can be calculated.

J_v and J_s can be used to calculate entropy production (${\psi }_S$) in the double-membrane system, using the expression presented in a previous paper [24]. If there is only an osmotic pressure difference Δπ = RTΔC in the double-membrane system, this expression can be written as

$${\psi }_S={\left({\psi }_S\right)}_{J_v}+{\left({\psi }_S\right)}_{J_s}=R{J}_v\mathsf{\Delta }C+R{J}_s\frac{\mathsf{\Delta }C}{\overline{C}}$$

(15)

where ${\left({\psi }_S\right)}_{J_v}$ is the S-entropy produced by J_v and ${\left({\psi }_S\right)}_{J_s}$ is the S-entropy produced by J_s.

3. Results and Discussion

The calculations of C_m = f(C_l − C_r), J_v = f(C_l − C_r), J_s= f(C_l − C_r), ${\left({\psi }_S\right)}_{J_v}$ = f(C_l − C_r), and ${\left({\psi }_S\right)}_{J_s}$ = f(C_l − C_r) were obtained on the basis of Equations (12)–(15), respectively, for the fixed hydrostatic pressure difference in the double-membrane system ΔP = P_l − P_r = 13 kPa and for two cases: C_l/M_l/C_m/M_r/C_r and C_l/M_r/C_m/M_l/C_r. The concentration difference (ΔC = C_l − C_h) has changed in the range from −700 to +700 mol m⁻³. For the M_l and M_r membranes, the following values of transport parameters were used: σ_l = 0.2, σ_r = 0.1, ω_l = 2 ×10⁻⁸ mol N⁻¹s⁻¹, ω_r = 4 × 10⁻⁸ mol N⁻¹s⁻¹, L_pl = 2 × 10⁻⁹ m³N⁻¹s⁻¹, and L_pr = 4 × 10⁻⁹ m³N⁻¹s⁻¹. The obtained results of these calculations are shown in Figure 2, Figure 3, Figure 4, Figures 7 and 8.

In Figure 2, the results of the calculations C_m = f(C_l − C_r) are shown. Parabolas 1 and 1’ were obtained for Case 1, and parabolas 2 and 2’ were obtained for Case 2. Figure 2 shows that parabola 1 crosses the concentration axis at the points C_l − C_r = −617.6 mol m⁻³ and C_l − C_r = 0, parabola 1’ at the points C_l − C_r = 0 and C_l − C_r = 213.2 mol m⁻³. The vertices of these parabolas have the following coordinates: C_m = 71.7 mol m⁻³ and C_l − C_r = −275.7 mol m⁻³ (parabola 1) and C_m = 22.8 mol m⁻³ and C_l − C_r = 110.3 mol m⁻³ (parabola 1′). In turn, parabola 2 intersected the concentration axis at the points C_l − C_r = −165.4 mol m⁻³ and C_l − C_r = 0; parabola 2′ at the points C_l − C_r = 0 and C_l − C_r = 720.6 mol m⁻³. The vertices of these parabolas had the following coordinates: C_m = 13.1 mol m⁻³ and C_l − C_r = −84.5 mol m⁻³ (parabola 2) and C_m = 90.3 mol m⁻³ and C_l − C_r = 305.1 mol m⁻³ (parabola 2′). The dotted lines illustrate the dependence of C_m = f(C_l − C_r) for C_m = 0.5(C_l + C_h). The results of the studies presented in Figure 2 indicate that in the double-membrane system, the solution accumulation effect of the intermembrane compartment of this system occurs for parabola 1 if σ_l > σ_r, ω_l < ω_r, L_pl < L_pr and −35.9 mol m⁻³ ≥ C_l − C_r < 0). For parabola 2′, if σ_l < σ_r, ω_l > ω_r, L_pl > L_pr and 0 < C_l − C_r ≤ 110.3 mol m⁻³). For C_l − C_r < −35.9 mol m⁻³ (parabola 1) and C_l − C_r > 110.3 mol m⁻³ (parabola 2′). For parabola 2 and 1’, the solution depletion effect of the intermembrane compartment of the double-membrane system occurs.

Figure 3 illustrates the results of the calculations of J_v =f(C_l − C_r) based on Equation (13) for Cases 1 and 2. The results of the calculations are shown as the parabola with branches 1a and 1b and the parabola with branches 2a and 2b. The vertex of the first parabola (C) had the coordinates: J_v = −0.92 × 10⁻³ m s⁻¹ and C_l − C_r = 562.1 mol m⁻³. The vertex of the second parabola (B) had the coordinates: J_v = 1 ×10⁻³ m s⁻¹ and C_l − C_r = −546.4 mol m⁻³. The results of the studies presented in this figure show that the solution of Equation (13) is a pair of parabolas with the common point J_v = 0 and C_l − C_r = 0.

Figure 4 presents the results of the calculation of J_s = f(C_l − C_r) based on Equation (14) for the two cases. The obtained results of the calculations are presented as parabolas with branches 1a and 1b and parabola with branches 2a and 2b. The vertex of the first parabola (C) had the coordinates: J_s = 2.4 mol m⁻²s⁻¹ and C_l − C_r = 558.6 mol m⁻³ while the vertex of the second parabola (B) − J_s = −2.4 mol m⁻²s⁻¹ and C_l − C_r = −535.1 mol m⁻³. The results of the studies presented in Figure 4 show that the solution of Equation (14), similar to Equation (13), is a pair of parabolas with a common point, J_s = 0 and C_l − C_r = 0.

We performed the procedure involving the omission of the fragments of the parabolas, which are shown in Figure 3 and Figure 4. If we leave branches 1b and 2a, section Cb of branch 1a of parabola 1 and section Bb of branch 2b, which are shown in Figure 3, we obtain the characteristic J_v = f(C_l − C_r) of the S type. Following this procedure for the relation J_s = f(C_l − C_r) (i.e., if we leave branches 1b and 2a, section Cb of branch 2b, and section Bb of branch 1a), which is shown in Figure 4, we obtain the characteristic J_s = f(C_l − C_r) of the reversed letter S type. The curve, which is shown in Figure 3, illustrates the dependence of J_v on the value of the control parameter ΔC = C_l − C_r, when the set value of the parameter ΔC₀ = 0 corresponds to three stationary states of J_va = 3.17 × 10⁻³ m s⁻¹, J_vb = 0, and J_vc = 3.0 × 10⁻³ m s⁻¹, respectively. Stable states located on the AB and CD sections of the curve were stable and the states located on the CB section were unstable. When the bifurcating values ΔC₁ = −546.4 mol m⁻³ and ΔC₂ = 562.1 mol m⁻³ were reached, the step transitions CA and BD appeared at the extreme points C and B of the curve, so the unstable states in the BC section never actually occur in real systems [25].

From the curves shown in Figure 3, it follows that for branch 1a in the area ΔC = C_l − C_r < 0, J_v > 0 and for branch 2b in the area ΔC = C_l − C_r > 0, J_v < 0. Similarly, for segments Bb of curve 2b and Ba of curve 2a, ΔC = C_l − C_r < 0, J_v > 0 and for segments Cb of curve 1a and Ca of branch 1b: ΔC = C_l − C_r > 0, J_v < 0. This means that, in these ranges of ΔC = C_l − C_r, osmotic transport occurs against the concentration gradient, furthermore, in areas where osmotic transport occurs, despite the concentration gradient R_v < 0 (see Figure 7).

In turn, the curve presented in Figure 4 illustrates the dependence of J_s on the value of the control parameter ΔC = C_l − C_r, when the set value of parameter ΔC₀ = 0 corresponds to the three stationary states of J_sc = 23 mol m⁻²s⁻¹, J_sb = 0, and J_sa = −23 mol m⁻²s⁻¹, respectively. Stationary states located on the AB and CD sections of the curve were stable and the stationary states located on the BC section were unstable. When the bifurcating values of ΔC₁ = −535.1 mol m⁻³ and ΔC₂ = 558.6 mol m⁻³ were reached, the step transitions of CD and BA appeared at the extreme points C and B of the curve, which is shown, so that unstable states on the BC section never actually occur in real systems [26]. In addition, from the curves that are shown in the Figure 4 results, for branch 1b in the area ΔC = C_l − C_r < 0, J_s > 0 and for branch 2a in the area ΔC = C_l − C_r > 0, J_s < 0. This means that in these ranges of ΔC = C_l − C_r, the diffusion transport takes place against the concentration gradient and in areas where diffusion transport occurs against the concentration gradient R_s < 0 (see Figure 8).

The presented analysis shows that the double-membrane system, which is capable of functioning in one of two stable states, has the properties of a trigger. This means that there is a change from one stable state to another as a result of the change in the value of ΔC = C_l − C_r and the change in the triad value of the membrane parameters M_l (L_pl, σ_l, ω_l) and M_r (L_pr, σ_r, ω_r). Trigger properties play an important role in biological systems, defining the directional and stepping transition from one state to another (e.g., in the process of electrical impulse along the nerve fiber transmission or in cell differentiation processes) [25].

Figure 5 presents the results of the calculations ${\left({\psi }_S\right)}_{J_v}$ (S-entropy produced by J_v) based on Equation (15). This equation shows that in order to calculate ${\left({\psi }_S\right)}_{J_v}$ we need to create the product of the universal gas constant (R = 8.31 J mol⁻¹K⁻¹), the results of the calculations J_v (presented in Figure 3) and ΔC = C_l − C_r. Figure 5 shows that ${\left({\psi }_S\right)}_{J_v}$ = f(C_l − C_r) is a combination of two curves 1a1b and 2a2b (two crossed bows in the shape of an inverted V), which intersect at the point with the coordinates ${\left({\psi }_S\right)}_{J_v}$ = 0 and ΔC = 0. It should be noted that the colors of the elements of these curves correspond to the elements of the curves shown in Figure 3: the AB segment in Figure 4 corresponds to the AB segment in Figure 3, the BC segment in Figure 4 corresponds to the BC segment in Figure 3, and the CD segment in Figure 4 corresponds to the CD segment in Figure 3. Simultaneously, the sign ${\left({\psi }_S\right)}_{J_v}$ depends on the sign J_v and the sign ΔC = C_l − C_r. The comparison of Figure 5 and Figure 3 shows that for segment AE, the relations ΔC > 0 and J_v > 0 were met; for segments EB and BE, ΔC < 0 and J_v > 0; for sections EC and CE, ΔC > 0 and J_v < 0; and for the segment ED, ΔC < 0 and J_v < 0.

Figure 6 shows the results of the calculations ${\left({\psi }_S\right)}_{J_s}$ (S-entropy produced by J_s) based on Equation (15). This equation shows that, in order to calculate ${\left({\psi }_S\right)}_{J_s}$, we need to create the product of the universal gas constant (R = 8.31 J mol⁻¹K⁻¹), the results of the calculations J_s presented in Figure 4, ΔC = C_l − C_r and $\overline{C}$. Figure 6 shows that ${\left({\psi }_S\right)}_{J_s}$ = f(C_l − C_r) is a combination of two curves 1a1b and 2a2b (bow in the shape of a jellyfish), which intersect at the point with the coordinates ${\left({\psi }_S\right)}_{J_s}$ = 0 and ΔC = 0. It should be noted that the colors of the elements of these curves correspond to the elements of the curves shown in Figure 6: the AC segment in Figure 6 corresponds to the AC segment in Figure 4, the BD segment in Figure 6 corresponds to the BD segment in Figure 4, and the BC segment in Figure 6 corresponds to the BC segment in Figure 4. Simultaneously, the sign ${\left({\psi }_S\right)}_{J_s}$ depends on the sign J_s and the sign ΔC = C_l − C_r. The comparison of Figure 6 and Figure 4 shows that for segment AE, the relations ΔC > 0 and J_s < 0 were met; for segments EC and CE, ΔC < 0 and J_s < 0; for segments EB and BE, ΔC > 0 and J_s > 0; and for the segment ED, ΔC < 0 and J_s > 0.

Moreover, from Equation (15), it follows that ${\left({\psi }_S\right)}_{J_v}$ > 0 when simultaneously J_v > 0 and ΔC > 0 or when simultaneously J_v < 0 and ΔC < 0. In turn ${\left({\psi }_S\right)}_{J_s}$ > 0, when J_s > 0 and ΔC > 0 or J_s < 0 and ΔC < 0. When J_v < 0 and ΔC > 0 or when J_v < 0 and ΔC > 0, then ${\left({\psi }_S\right)}_{J_v}$ < 0. In the case when J_s < 0 and ΔC > 0 or when J_s < 0 and ΔC > 0 simultaneously, then ${\left({\psi }_S\right)}_{J_s}$ < 0. The relations ${\left({\psi }_S\right)}_{J_v}$ < 0 and ${\left({\psi }_S\right)}_{J_s}$ < 0 illustrate a deviation from the second law of thermodynamics for the membrane system. From this law, it follows that in the single-membrane system, thermodynamic fluxes reduce the value of stimuli (to which they are induced) and cause an equilibrium state. These thermodynamic fluxes are non-zero until ${\left({\psi }_S\right)}_{J_v}$ > 0 and ${\left({\psi }_S\right)}_{J_s}$ < 0. It seems that, in the double-membrane system, due to the occurrence of the phenomenon of accumulation or depletion of the substance in the inter-membrane compartment, cases where ${\left({\psi }_S\right)}_{J_v}$ < 0 and ${\left({\psi }_S\right)}_{J_s}$ < 0 are possible.

Applying the results shown in Figure 7 and Figure 8, osmotic resistance (R_v) and diffusion resistance (R_s) were calculated using the following expressions:

$$R_v=\frac{\Delta J_v}{\Delta \left(C_l-C_r\right)}$$

(16)

$$R_s=\frac{\Delta J_s}{\Delta \left(C_l-C_r\right)}$$

(17)

The results of these calculations are shown in Figure 7 and Figure 8. It should be noted that curves 1a, 1b, 2a, and 2b (shown in Figure 7) were obtained from curves 1a, 1b, 2a, and 2b (presented in Figure 3). In turn, curves 1a, 1b, 2a, and 2b (shown in Figure 8) were obtained from curves 1a, 1b, 2a, and 2b (presented in Figure 4). Figure 5 shows that R_v > 0 for curves 2a and 1b, furthermore, R_v < 0 for curves 2b and 1a. Figure 8 shows that R_s > 0 for curves 2b and 1a, furthermore, R_s < 0 for curves 2a and 1b.

Negative resistance often determines the possibilities of their use of semiconductor components in electronics [26] and membrane systems in physiochemistry [27,28] and biophysics [29]. In membrane systems, negative resistance can be controlled by means of ion currents [29]. Therefore, the mechanism of negative resistance is the basis for the excitation of bio membranes [30].

All data were entered and calculated in Microsoft Excel 2016 and Origin Pro 2020.

4. Conclusions

These investigations showed that:

• We created nonlinear model equations of the concentration in the inter-membrane compartment (C_m), volume flux (J_v), solute flux (J_s), and S-entropy produced by J_v, $\left({\left({\psi }_S\right)}_{J_v}\right)$ and by J_s $({\left({\psi }_S\right)}_{J_s})$ for binary homogeneous, non-electrolyte solutions. The created model equations, illustrated by Equations (12)–(15), consist of quadratic equations describing the concentration in the inter-membrane compartment (C_m), volume flux (J_v), and solute flux (J_s) through the double-membrane system.
• The double-membrane system, composed of two membranes (M_l, M_r), separates the compartments l, m, and r containing the homogeneous, non-electrolyte binary solutions. The compartment m consists of the infinitesimal layer of the solution and its volume fulfills the condition V_m → 0. The volume of the compartments l and r fulfills the condition V_l = V_r → ∞. At the initial moment, the solution concentrations in the cell satisfy the condition C_l < C_m < C_r.
• Based on this model, for the fixed values of the reflection (σ_l, σ_r), hydraulic permeability (L_pl, L_pr), and solute permeability (ω_l, ω_r) coefficients, the dependencies C_m = f(C_l − C_r), J_v = f(C_l − C_r) and J_s = f(C_l − C_r) were calculated. Each of the obtained characteristics was specifically arranged as a pair of parabolas.
• The relationship ${\left({\psi }_S\right)}_{J_v}$ = f(C_l − C_r) was a combination of two curves, 1a1b and 2a2b (two crossed bows in the shape of an inverted V), which intersected at the point with the coordinates ${\left({\psi }_S\right)}_{J_v}$ = 0 and ΔC = 0. The sign ${\left({\psi }_S\right)}_{J_v}$ was the consequence of the sign J_v and ΔC: ${\left({\psi }_S\right)}_{J_v}$ > 0 when simultaneously J_v > 0 and ΔC > 0, or when simultaneously J_v < 0 and ΔC < 0. If simultaneously J_v < 0 and ΔC > 0 or when simultaneously J_v < 0 and ΔC > 0, then ${\left({\psi }_S\right)}_{J_v}$ < 0. In turn, the relationship ${\left({\psi }_S\right)}_{J_s}$ = f(C_l − C_r) is a bow in the shape of a jellyfish. The sign ${\left({\psi }_S\right)}_{J_s}$ was the consequence of the sign J_s and ΔC: ${\left({\psi }_S\right)}_{J_s}$ > 0 when simultaneously J_s > 0 and ΔC > 0 or when simultaneously J_s < 0 and ΔC < 0. If simultaneously J_s < 0 and ΔC > 0 or when simultaneously J_s < 0 and ΔC > 0, then ${\left({\psi }_S\right)}_{J_s}$ < 0. The cases ${\left({\psi }_S\right)}_{J_v}$ < 0 and ${\left({\psi }_S\right)}_{J_s}$ < 0 indicate a deviation from the second law of thermodynamics caused by the phenomenon of the accumulation or depletion of the dissolved substance in the inter-membrane compartment of the double-membrane system.
• In the solution concentration areas, where the relations were ΔC < 0, J_v > 0 and J_s > 0, ΔC > 0, J_v < 0 and J_s < 0, osmotic and diffusion transport (against the concentration gradient) occurred. In addition, in the areas where osmotic and diffusive transport took place (against the concentration gradient), osmotic and diffusion resistances (R_v, R_s) satisfied the conditions R_v < 0 and R_s < 0.