Energy conversion in Textus Bioactiv Ag membrane dressings using Peusner’s network thermodynamic descriptions Konwersja energii w opatrunku membranowym Textus Bioactive Ag w opisie termodynamiki sieciowej Peusnera

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Background
In recent years, due to the development of synthetic polymer technology, many types of synthetic membranes have found applications in modern medical therapy and diagnostics. These membranes act as selectively permeable barriers between biological tissues and the environment (e.g., hemodialyzers and active membrane dressings) or between an encapsulated drug and the internal environment of a living organism (controlled release), etc. [1][2][3][4][5] When treating difficult-to-heal wounds, such as venous stasis ulcers or severe burns, selecting a dressing that is designed to maintain a moist environment, suitable temperature and pH of the wound is important. Various types of active dressings have been used in the treatment of chronic wounds. 6 Their purpose is to protect nerve fibers from excessive stimulation during application and dressing changes, as well as protect delicate tissues from mechanical stimuli and external environmental influences to limit infections and bacterial contamination. 7 One type of these active dressings is the Textus Bioactiv Ag membrane dressing. 8,9 It is a composite/mixed polymer dressing containing thermoplastic polyethylene fibers and Ag zeolites.
According to the Kedem-Katchalsky (KK) model, the Textus Bioactiv Ag membrane has certain transport properties measured with the coefficients of hydraulic conductivity (of the solvent), reflection and diffusive conductivity (of the solution). 9 This means that the Textus Bioactiv Ag membrane can separate appropriate solutions of different concentrations and has the property of free energy conversion, similarly to Nephrophan or Bioprocess membranes. 10,11 The total energy of a thermodynamic system is the sum of its internal energy (nuclear, chemical and thermal) and external energy (connected with gravitational, electromagnetic, electrical field, etc.). 12 In both biological and physicochemical membrane systems, internal energy (U-energy) is mainly chemical energy. The internal energy consists of free energy (F-energy) and degraded energy (S-energy). The F-energy, also called exergy, defines the "quality of energy" and the part of the energy that can be practically used to do work. The S-energy, also called anergy, is the passive part of the energy that cannot be used practically. 13 The conversion of chemical energy occurs in 2 stages. In the 1 st stage, the F-energy is separated from the S-energy. In the 2 nd stage, the F-energy is converted into useful work.
The biological cell, which functions like a chemodynamic machine, is the most efficient converter of chemical energy (U-energy) into useful energy (F-energy). The most important activity of this process is converting the F-energy contained within the chemical bonds of nutrients into high-energy compounds such as adenosine triphosphate (ATP). 14 At the expense of ATP, mechanical work, osmotic work, electrical work, and biosynthesis work are performed. The production of S-energy depends on the rate of the biochemical process: the slower the process, the greater the efficiency of the cell. 15 Transport processes, including membrane transport, constitute a group of fundamental phenomena occurring at all levels of physicochemical systems. 16 The driving forces of these transporters are the physical quantities of scalar, vector and/or tensor nature, which participate in the creation of various types of physical fields that shape the field properties of nature. An example of a scalar field Streszczenie Wprowadzenie. Membrana Textus Bioactive Ag to aktywny opatrunek do leczenia ran przewlekłych, takich jak owrzodzenia żylne podudzi i oparzenia.
Słowa kluczowe: transport membranowy, membrana polimerowa, Textus Bioactiv Ag, równania Kedem-Katchalskiego-Peusnera, konwersja energii is the field of concentrations, a vector field is the gravitational field, and a tensor field is the field of internal stresses. For a scalar field to formally participate in the creation of a thermodynamic force that generates thermodynamic flows, the appropriate gradients of scalar quantities such as concentration or temperature must be created. The generation of force results in the performance of work from the energy stored in the system.
Mathematical models developed using the framework of Onsager's linear nonequilibrium thermodynamics (LNET) and Peusner's network thermodynamics (NT) 13,[17][18][19][20] are convenient tools to study membrane transport properties. One of the more important ones is the Kedem-Katchalsky-Peusner (KKP) model, [21][22][23][24] which is an extension of the KK model. 25 The classical KK equations contained transport coefficients that characterized the permeation of solvents and solutes through a membrane. 18,25 These coefficients included hydraulic permeability (L p ), reflection (σ) and solute permeability (ω). Peusner introduced L, R, H, and P versions of the KKP equations containing Peusner coefficients L ij , R ij , H ij , and P ij (i, j ∈ {1,2, …, n}) into the science of membranes and membrane techniques. 20 The Peusner coefficients are a combination of L p , σ and ω coefficients and the average concentration of solutions in the membrane (C). The coefficients of coupling (l ij , r ij , h ij , p ij (i ≠ j)) and energy conversion efficiency ((e ij ) l , (e ij ) r , (e ij ) r , (e ij ) p (i ≠ j)) can be calculated using the L ij , R ij , H ij , or P ij coefficients, respectively. In addition, the so-called coupling coefficient "Super Q", which is also a coupling measurement, can also be calculated.
In this study, the L model of the KKP form was used to evaluate the transport properties of the Textus Bioactiv Ag membrane dressing. The relationships L p = f(C), σ = f(Δπ) and ω = f(Δπ) for KCl aqueous solutions were determined experimentally according to the procedures described in previous papers. 9,18 The dependencies of Peusner (L 11 , L 12 , L 21 , L 22 ), coupling (l 12 , l 21 , Q L ) and energy conversion efficiency ((e 12 ) l , (e 21 ) l ) coefficients on osmotic pressure differences (Δπ) were calculated experimentally from the measured characteristics of L p , σ and ω as functions of osmotic pressure. Besides, dependencies (Φ S ) L = f(Δπ), (Φ F ) L = f(Δπ) and (Φ U ) L = f(Δπ) were calculated. The (Φ S ) L is the dissipated energy flux (S-energy), (Φ F ) L is the free energy flux (F-energy) and (Φ U ) L is the internal energy flux (U-energy).

Membrane system
Images of the membranous Textus Bioactiv Ag (Biocell Gesellschaft fur Biotechnologie GmbH, Engelskirchen, Germany) dressing, obtained using a scanning electron microscope (SEM; Zeiss Supra 35; Carl Zeiss AG, Jena, Germany) are shown in Fig. 1A,B. These images reveal 2 types of fibers and the mesh that prevents the membrane dressing from adhering to the wound.
Textus Bioactiv Ag is a double-layer membrane dressing used to treat wounds of various etiologies. 9 It is made of 3 types of heterogeneous thermoplastic polymer fibers. The 1 st layer contains polyethylene fibers, the core of which is hydrophobic and zeolites with silver ions are located on the hydrophilic surfaces. The task of the zeolites is to keep inactivated Ag + and/or micronized Ag inside a negatively charged polymer cage. This layer also contains hydrophilic polyethylene Super Absorbing Polymers (SAP) absorption fibers. The task of Ag + ions and Ag particles is to provide permanent and effective bactericidal protection of treated wounds against secondary infections. The 3 rd type of polymer fibers is made of polyethylene and arranged paralelly to the skin surface, as in the 2 nd layer Fig. 1. Images of membrane surfaces obtained using a scanning electron microscope (SEM). A. The surface of the Textus Bioactiv Ag membrane from the side of the polymer fibers (×1740 magnification). The cross section of the fibers is visible, whose core is hydrophobic and on the hydrophilic surface of the fibers zeolites with silver ions next to the SAP fiber are visible; B. The surface of the Textus Bioactiv Ag membrane from the mesh side (×94 magnification) with polymer fibers visible in the meshes of the net; C. The single-membrane system of the membrane these create a special mesh that prevents the dressing from sticking to the wound. The polymer fibers used in this type of membrane are thermoplastic and able to attach to structures (zeolites, AgION™) containing silver ions. Depending on the manufacturer, membrane dressings are made of various types of fibers (polyethylene, polyamide, polypropylene, polyester, polystyrene, etc.). For example, in the case of the Textus Bioactiv Ag dressing, there are various types of polyethylene fibers, in Atrauman Ag dressing-polyamide fibers, and as in the Aquacel Ag dressing -sodium carboxymethyl cellulose fibers. The type of fibers used is important for determining the properties of the dressing. They absorb exudate and increase its volume, and prevent it from sticking to the wound (polyethylene). They ensure the correct pH (polyethylene, polyamide) or, as in the case of carboxymethyl cellulose, turn into a gel when absorbing exudate. The mesh showed in Fig. 1B has the characteristics of a non-selective membrane (σ = 0). The activation process of the dressing begins after wetting the dressing with Ringer's solution, which contains Na + , K + , Ca 2+ , and Cl − ions in various concentrations, and occurs abruptly from zeolites to SAP fibers. Due to the absorption properties of SAP fibers, the dressing has a very high absorption capability (4.2 kg/m 2 ).
It should be mentioned that the surface area available to the solution on the grid side (Fig. 1B) is 60.7% smaller compared to the opposite side of the membrane (Fig. 1A). If we denote the actual membrane surface area by A h and the membrane surface area on the grid side by A l , then A l ≈ 0.61 A h . Suppose that the A h is in contact with a solution of concentration C h , and the A l is in contact with a solution of concentration C l , then we will denote the volume flux induced by Δπ by J vh . In this case, Δπ will increase solution volume (ΔV h ). If we reverse the location of the membrane, we will denote the flux for the same ∆π, through the A l by J vl . In this case, Δπ will result in a solution volume increase of ΔV l = 0.61 ΔV h . Given this, J vh = (ΔV h )A h −1 (Δt) −1 and J vl = (0.61ΔV h )(0.61A h ) −1 (Δt) −1 . This means that J vh = J vl = J v . This last relationship follows the flux continuity law. Similarly, it can be shown that for the solute flux, J sh = J sl = J s . Figure 1C shows a model of the membrane system in which the membrane (M) separates 2 homogeneous electrolyte solutions with C r and C l concentrations (C r ≥ C l ) with hydrostatic pressures of P r and P l (P r > P l , P r = P l or P r < P l ). For binary electrolyte solutions, the KK equations are as follows 18,26 (Equation 1-3):

Mathematical model
where J v -volume flux; J s -solute flux; I m -electric ion current; L p , σ, P E and ω -coefficients of hydraulic permeability, reflection, electroosmotic permeability, and solute permeability, respectively; ΔP = P r -P l -difference of hydrostatic pressure; γ -Van 't Hoff coefficient; Δπ = RT(C r -C l ) -differences in osmotic pressures (RT -the product of the gas constant and the absolute temperature; Δπ = RT(C r -C l ) are solution concentrations, C h > C l ); γ -Van 't Hoff coefficient (1 ≤ γ ≤ 2); κ -electrical conductivity; τ j , z j , ν j -transfer number, valence and ion number, respectively; and C = (C h − C l )(lnC h C l −1 ) −1 ≈ 0.5 (C h + C l ) -average concentration of the solution.
If we assume that I m = 0 in the system, we obtain equations analogous to those for non-electrolyte transport The value of the coefficients l 12 = l 21 = l is limited by the relation that −1 ≤ l ≤ +1. When l = ±1, the system is fully coupled and the processes become single processes. When l = 0, the 2 processes are completely unconjugated and no energy conversion occurs. The definition proposed by Kedem The coefficient Q L is connected with (e 12 ) l coefficient by using the Equation 14 15 : According to the first law of thermodynamics, in a membrane system, when the membrane separates 2 solutions of different concentrations and the transport processes have an isothermal-isochoric character, the following equation is fulfilled (Equation 15):  17): and using Equation 16 we get (Equation 18,19):

Methodology for measuring the volume and solute fluxes and transport parameters
The studies on osmotic volume (J v ) and solute fluxes (J s ) were carried out using the measuring set described in a previous paper and presented in Fig. 2. 28 It consisted of 2 cylindrical vessels (l and h) with each containing a volume of 200 cm 3 of aqueous KCl solution, one with a concentration in the range of 1÷16 mol/m 3 and the other with a constant concentration of 1 mol/m 3 . The solutions in the vessels were separated using a Textus Bioactiv Ag membrane dressing with an area of A = 1.15 cm 2 , located in the horizontal plane.
A pipette graduated every 1 mm 3 (KP) positioned in a plane parallel to the plane of the membrane was connected to the vessel (h) containing KCl at concentration C h . The pipette was used to measure the change in volume (ΔV) of the solution in the measuring chamber (h). The vessel (l) was connected to a reservoir containing an aqueous solution of KCl with a concentration of C l = 1 mol/m 3 , adjustable in height relative to the pipette. The measurement procedure for J v and J s was previously described. 18,29 Briefly, increases in the ΔV were measured under conditions of intensive mechanical stirring of the solutions at 500 rpm. The volume flux was directed from the vessel with the lower concentration to the vessel with the higher concentration of solutions, and the flow of dissolved substances was in the opposite direction. The measurements were carried out at isothermal conditions (T = 295 K).
The volume flux through the surface (A) of the membrane was calculated based on the volume changes (ΔV) over time (ΔV) measured in the pipette using the formula J v = (ΔV)A -1 (Δt) -1 . The fluxes of dissolved substances were calculated based on the formula J s = (V u • dC)A −1 (Δt) −1 , where V u was the volume of the measuring vessel and dC was the change in concentration of the solution measured with electrochemical methods. 30 The relative error in determining J v and J s was less than 10%. The values of coefficients coefficients L pT , σ T , and ω T were calculated based on the formulas L pT = (J v ⁄ ΔP) Ch = Cl , σ T = (ΔP⁄ Δπ) Jv = 0 , and ω T = (J s ⁄ Δπ) Jv = 0 . Based on the characteristics L pT = f(C), σ T = f(Δπ) and ω T = f(Δπ) presented in Fig. 3   show the dependencies L pT = f(C), ω T = f(Δπ) and σ T = f(Δπ) were suitable. In the case of the L pT = f(C) characteristic, it was assumed that C ≈ 0.5(C r + C l ) (C r = C l ). The characteristics shown in these figures are nonlinear. From the characteristics presented in Fig. 3, it follows that L pT increases from L pT = 5 × 10 −8 m 3 /Ns (for C = 0, pure water) to L pT = 68.5 × 10 −8 m 3 /Ns (C = 8 mol/m 3 ) -which is more than a 13-fold increase when concentrations of C changes from 0 mol/m 3 to 8 mol/m 3 ).

Determination of membrane transport parameters
The hydraulic permeability coefficient for most membranes is constant over a wide range of membrane concentrations. As can be seen in Fig. 3, this coefficient for the Textus Bioactiv    Ag membrane is not constant. The coefficient strongly depends on the KCl concentration in the membrane with increasing concentrations in the membrane, especially for concentrations above 3 mol/m 3 (in the range of 3-6 mol/m 3 ). Above 6 mol/m 3 , this coefficient does not change significantly. This causes a problem with the free use of the KK formalism over a wide range of electrolyte concentrations for this membrane. The relationship L pT = f(C) clearly shows 3 concentration ranges: an almost constant hydraulic permeability coefficient (ranges of low and high concentrations of KCl) and a transitional range of concentrations with a strong dependence of the L pT coefficient on the concentration in the membrane. This indicates the complex nature of the interaction of the Textus Bioactiv Ag membrane with electrolyte solutions (aqueous KCl solutions). The strong dependence of the hydraulic permeability coefficient on the range of concentrations in the membrane may indicate possible dynamic structural changes in the membrane itself due to the interaction of its structure with electrolyte ions within the solution being transported through the membrane.
The value of coefficient ω T increases from 2.7 × 10 −10 mol/Ns (for Δπ = ±0.86 kPa) to 3.0 × 10 −10 mol/Ns (for Δπ = ±39.22 kPa), which is an 11% increase when the concentration of Δπ changes from ±0.86 kPa to ±39.22 kPa (Fig. 4). Using the dependencies Δπ = CRTln(C r /C l ), it can be shown that Δπ = ±0.86 kPa corresponds to C = 0.1 mol/m 3 , while Δπ = ±39.22 kPa corresponds to C = 8 mol/m 3 . This indicates a significant change in the conditions of the solution transported through the membrane at this concentration range. This means that an increase in KCl concentrations in the Textus Bioactiv Ag membrane significantly improves the transport of KCl solutions through the membrane. Similarly to L pT coefficient, the osmotic pressure ranges of almost constant value of the coefficient ω can be distinguished (low and high osmotic pressures) and the osmotic pressure range (10-20 kPa) in which increase of osmotic pressure causes an increase in the coefficient ω.
The curve in Fig. 4A shows that σ T decreases from 0.09 (for Δπ = ±0.86 kPa) to 0.005 (for Δπ = ±39.22 kPa) which is related to a 18-fold reduction in the reflection coefficient when the concentration in the membrane Δπ changes from 0.86 kPa to 39.22 kPa. This indicates a significant reduction in membrane selectivity for KCl solutions with increasing KCl concentrations in the membrane. This nonlinear relationship of the transport coefficients is caused by swelling of the hydrophilic fibers within the membrane and by the hydration of K + ions. These water coatings facilitate membrane transport by reducing the friction between the membrane and the substances penetrating it, and are dependent on the concentration of the solutions. For this reason, they increase the value of L pT and ω T and decrease the value of σ T .

The Peusner coefficients (L ij ) T
The values L 11 , L 12 = L 21 and L 22 were calculated using Equation 9,10. Figures 6A-C show the nonlinear dependencies of (L ij ) T = f(Δπ) for the Textus Bioactiv Ag membrane when (a) i = j = 1, (b) i = j = 2, and (c) i ≠ j.
The value of (L 12 ) T = (L 21 ) T increases from 0.04 × 10 −7 m 3 /Ns (for Δπ = ±0.86 kPa) to 54.5 × 10 −7 m 3 /Ns (for Δπ = ±39.22 kPa). The dependence of (L 22 ) T = f(Δπ) for the Textus Bioactiv Ag membrane is nonlinear and the value of (L 22 ) T increases from 0.04 × 10 −6 mol 2 /m 3 Ns For the (L ij ) T coefficients for the Textus Bioactiv Ag membrane, 2 ranges of Δπ can be determined. For Δπ smaller than 10 kPa, the (L ij ) T coefficients do not change much and are close to 0. Above Δπ = 10 kPa, increasing the osmotic pressure causes a gradual increase in the (L ij ) T coefficients.
The coefficients l ij , (e max ) l , and Q L and fluxes (Φ S ) L and (Φ U ) L The dependencies l 12 = f(Δπ), (e max ) l = f(Δπ) and Q L = f(Δπ) for Textus Bioactiv Ag membranes were calculated based on Equation 11-13 and are presented in Fig. 7. As seen in Fig. 7, as the value of |Δπ| increases, the l 12 coefficient for Textus Bioactiv Ag membrane fulfills the condition l 12 → 1 when |Δπ| → 40 kPa. In turn, as the value of |Δπ| increases, the value of Q L also increases and fulfills the conditions for Textus Bioactiv Ag membrane Q L → 1 when |Δπ| → 40 kPa. This means that the solvent and solute transport processes are coupled to different degrees. Therefore, they act as energy converters. The measurement of energy conversion efficiency is performed using the coefficients (e max ) l and Q L . The curve (3) in Fig. 7 shows that the dependence (e max ) l = f(Δπ) has an identical maximum and minimum. For Δπ = -7.9 kPa or +7.9 kPa, the coefficient [(e max ) l ] max = 0.83 and for Δπ = -15.35 kPa or +15.35 kPa the coefficient [(e max ) l ] min = 0.53, respectively. For Δπ > 15.35 kPa and Δπ < −15.35 kPa, the dependence (e max ) l = f(Δπ) is of the saturation type. When Δπ → -40 kPa or +40 kPa, (e max ) l → 0.7. As can be seen from Fig. 7, for low KCl osmotic pressures, the coefficient values are small and close to 0, which indicates a lack of process coupling in the Textus Bioactiv Ag membrane. Increasing the osmotic pressure Δπ on the membrane causes a fast increase in coupling coefficients, which may indicate the increasing mutual influence of the membrane structure and the electrolyte f lux through the membrane. This coupling, greater for the higher applied electrolyte osmotic pressure, causes the coefficients to establish at a high level for osmotic pressures Δπ greater than 10 kPa, which may indicate a strong coupling between membrane structure and electrolyte flux through the membrane for high KCl osmotic pressure values on the membrane.
The dependencies (Φ S ) L = f(Δπ), (Φ F ) L = f(Δπ) and (Φ U ) L = f(Δπ) calculated based on Equation 16,18,19 for the Textus Bioactiv Ag membrane are presented in Fig. 8. The calculations were performed for a fixed difference of hydrostatic pressures ΔP = 40 kPa and different Δπ. As for the coupling coefficients, the energy fluxes, to a small extent, depend on Δπ in the range of small values of osmotic pressures on the membrane (Δπ < 12 kPa). An increase in the osmotic pressure on the membrane above this range causes an increase in energy fluxes on the membrane. In contrast to the coupling coefficients, which remain nearly constant at the maximum level of osmotic pressures on the membrane, the energy fluxes initially increase strongly (in the range 12 kPa ≤ Δπ ≤ 25 kPa), but then increase slower and slower with increases in osmotic pressures on the membrane. Moreover, in the same Δπ intervals, the largest values are reached by (Φ U ) L and the smallest by (Φ S ) L .
It should be emphasized that the coupling coefficients and the energy fluxes through the Textus Bioactiv Ag membrane do not depend on the direction of applied Δπ on the membrane. This may indicate a different reason for the dependence of these coefficients and fluxes on the osmotic pressures of electrolytes than the 2-layer structure of the membrane. Rather, the reason for these effects may be the changes in the structure of the basic layer of the membrane itself than in the supporting layer.