ABSTRACT A STUDY ON THE EXISTENCE OF OSCILLATIONS IN HEMBRANE TRANSPORT PHENOHERA FROM THE STANDPOINT OF IRREVERSIBLE THERMODINAHICS by Pandeli Durbetaki The object of this thesis is to derive criteria for the undamped oscillations in an ideal membrane transport system. Two investigators have reported that oscillatory phenomena were observed in systems or transport through.mem- ‘branes. The systems that supported undamped oscillations liad a concentration gradient acress the membrane in the presence of an electric field. In the present analytical work an idealized model is conceived which has a two component flow system of un- charged but polarised species. The formalism of irrever- sible thermodynamics is employed to study the possible existance of undamped oscillations in this system. Starting from the equation for the conservation of mass, the equation of motion, the equation of conservation of energy, a rede- fined Gibb's free energy, Gibb's Law and Maxwell's equations for an electric field, the internal energy and entropy balance equations are derived. Using then the irreverdible thermodynamics formalisms, the entropy flow, entropy production, and the phenomenological equations are derived Pandeli Durbetaki for the system. It becomes apparent that the conjugate forces to the flows are represented by the overall electrochemical potential gradients. The phenomenological equations are then used to derive a set of flow equations for an inert and an active membrane. Two cases are considered for both types of membranes. In the first case it is assumed that the con- centration difference across the membrane remains constant. With this assumption, a set of two differential equations are derived with the pressure and the electrical potential differences as the variables with respect to time. The analysis produces criteria for undamped oscillations and shows that such oscillations are possible when the cross phenomenological coefficient is negative for both the inert and the active membrane. The second case considered for both types of membranes assumes a concentration difference across the membrane which is variable with respect to time. The resultant three differential equations have the pressure, the electrical potential and the concentration differences as the variables with respect to time. The analysis produces generalized criteria for undamped oscillations for'both the inert and the active membrane. A STUDY ON THE EXISTENCE OF OSCILLATIONS IN MEMBRANE TRANSPORT PHENOMENA FROM THE STANDPOINT OF IRREVERSIBLE THERMODYNAMICS by Pandeli Durbetaki A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mechanical Engineering 1964 V- ‘2.”‘(‘-‘ T9 ELISABETH -11- ACKNOWLEDGMENT To my mentor and major professor, Dr. Joachim E. Lay, I wish to express my gratitude for his guidance, advice and encouragement at all times. I wish to express my appreciation to Dr. James L. Dye, a member of my guidance committee, for his valuable suggestions concerning this thesis. My appreciation goes also to the other members of my guidance committee, Dr. Rolland T. Hinkle and Dr. Charles P. Wells. I am also grateful for the friendship of my office mate, Joab J. Blech, for his suggestions and his willingness to share with me the many frustrations encountered during the research. -111- TABLE OF CONTENTS LIST OF FIGURES O O I O O O O O O O O O O O O O O SYMBOLS . CHAPTER 1 ae be 0. d. e. CHAPTER 2 CHAPTER 5 b. c. CHAPTER 4 a. b. c. CHAPTER 5 a. b. O. - IntrOd‘UCtion e e e e e s e e e e e 0 Historical . . . Stationary States md Principles . . . . General Applications Biological Systems . Non-linear Systems and Cyclic Processes Ha riational . O . - membrane Transport Phenomena . . . . Development of the Phenomenological Equations for a System with a Fixed Lattice in the Presence of an Electric Field 0 O O O O O O O O O O O O O O 0 Elements of Irreversible Thermodynamics Presence of an Electric Field . . . . . Presence of a Fixed Lattice and the Phenomenological Equations . . . . . - Inert Henlbrane e e e e e e e e e e e Deviation of the Flow Equations . . . . Case 1 - cI and cII are Both Constant . Case 2 - cI + cII is Constant . .'. . . - Aetive membrane e e e e e e e e e e e Derivation of the Flow Equations . . . Case 1 - cI and cII are Both Constant . Case 2 - cI + cII is Constant . . . . . -17... 16 16 23 36 45 45 65 69 69 77 80 CHAPTER 6 - Criteria for Oscillations a. Case 1 . . . . . . . . . b. case 2 O O O O O O O 0 O : c. Summary and Concl sio . . BIBLIOGRAPHY . . . . . . . . . . . . 102 LIST OF FIGURES Schematic Diagram of Membrane Transfer System . . . . . . D: p. 3190 bi bi bib txfl' eq 'ij SYMBOLS a constant, equation (6-48) (i=1,2,3,'°',n) constants membrane surface area a constant, equation (6-49) (i-O,1,2,"‘,n) constant coefficients of a polynomial magnetic flux density concentration in grams per liter concentration of component k given as a mass fraction membrane thickness external change of a quantity internal change of a quantity substantial (barycentric) time derivative dielectric displacement energy per unit mass electric field strength electrical potential difference at tsO electric field strength in a reversible process total external force per unit volume on the system -vii- mi RF‘I Q? m H? cu cu an c. cu an C—Ii external force per unit mass on component k specific Gibbs free energy specific Gibbs free energy in a polarized medium total Gibbs free energy total Gibbs free energy in a polarized medium total Gibbs free energy in a polarized medium and inclusive of the electrical energy partial specific enthalpy of component k in a polarized medium magnetic field strength electric current due to conduction fl total electric current density flow of component k inside the membrane per unit area and unit time total flow per unit area and unit time energy flux per unit surface area and unit time flux or flow in an irreversible process flow of component k per unit area and unit time polarization flux per unit area and unit time heat flux per unit surface area and unit time entropy flux per unit area and unit time -viii- 3k flow of component k with respect to the movement of the center of mass per unit area Lik overall phenomenological coefficient M total mass 3 mass of all components excluding the mass of component k k mass of component k Hk molecular weight of component k Nk mole fraction of component k F polarization per unit mass. ‘Fk partial specific electric polarization of component k p pressure Po pressure difference at tsO P electric polarization per unit volume q heat transferred per unit mass R the gas constant 3 specific entropy 8k partial specific entropy of component k S total entropy t time T absolute temperature -11.... id K1: specific internal energy specific energy specific energy of component k total internal energy total internal energy in a polarized medium specific volume partial specific volume of component k total volume center of mass velocity velocity of component k ' thermodynamic force in an irreversible process conjugated force for'Jk ! conjugated force for Jp conjugated force for?q total specific electric charge specific electric charge of component k (i-l,2,'°°,m; j=l,2,'°°,n) constants constant, equation (5-4) activity coefficient of component k G? C) 60 UV 0’ solubility factor (measure of deviation from inert membrane), equation (5-18) "discriminant" for a transformed third degree polynomial, equation (6-51) the vector differential operator called "del" or "nabla" inductive capacity or dielectric constant variable in a third degree polynomial, equation (6-47) a constant, equation (6-55) a constant, equations (6-15) and (6-25) magnetic permeability chemical potential of component k purely chemical potential of component k purely chemical potential of component k at the standard state electrochemical potential of component k in a polarized medium chemical potential of component k in a polarized medium the variable in a second or third degree polynomial total density density of component k entrapy production per unit volume and unit time a constant, equation (6-56) a constant, equations (6-15) and (6-23) -€— é» '8 electrical potential Qak local phenomenological coefficient Subscripts & Superscripts o ‘ subscript for membrane 1 subscript for solvent 2 subscript for solute I superscript for the solution to the left of the membrane II superscript for the solution to the right of the membrane -xii- CHAPTER 1 - Introduction a. Historical The classical macroscopic view in thermodynamics has been well established for more than a century. The principles for this approach were initiated and formalized by N. L. S. Carnot, R. Clausius and U. Thomson and imple- mented later by H. Planck, J. V. Gibbs and others. The axiomatic development of this approach is based on the principles presently known as the Zeroth, First and Third Laws of Thermodynamics. A recent publication by G. N. Hatsopoulos and J. H. Keenan [H5] proposes to unite the principles of classical thermodynamics under a single axiom. Nevertheless, the governing idea in classical thermodynamics is the simplifying condition of equilibri- um. Also, it provides a framework for the description of thermal and mechanical properties of matter and contains reciprocity rules (i.e. Maxwell relations) that couple measurable thermodynamic coefficients of physical systems. Classical Thermodynamics (sometimes aptly re- ferred to as "Thermostatics") has lacked the theory necessary to describe coupled phenomena and time depen- dent processes. After all, the condition of equilibrium is the rarely encountered case, and phenomena due to coupled processes are frequently in existance in most -1- -2- systems. Irreversible processes, such as thermoelectric phenomena, transference phenomena in electrolytes and heat conduction in an anisotropic medium, became the subject of study of many investigators. This and the desire to provide a better description of processes in an open sys- tem, such as those taking place in the living organisms, incited the development of the area known as Irreversible Thermodynamics or Nonequilibrium Thermodynamics. As early as the middle of the nineteenth century attempts were made, among others, by W. Thompson (Lord Kelvin) and R. Clausius to treat nonequilibrium processes on the basis of thermodynamic considerations. However, it was not until the second quarter of the twentieth cen- tury that a consistent phenomenological theory for irre- versible processes came into being. The reciprocal rela- tions established by Onsager [02, 03] in 1931 became'a foundation for the formalism of irreversible thermodynamics developed later by such workers in the field as J. Heixner [116, 117,118, 119], I. Prigogine [P7] , K. c. Denbigh [139], s. R. DeGroot [D5, D6] and P. Hazur [D6]. 'It should be mentioned at this point that a reformulation of the reciprocal relations by H. B. G. Casimir‘[C§] made possible the analysis of processes tak- ing place in the presence of an external magnetic field or in the presence of a uniform rotation [D1]. Thus the Casimir reciprocal relations widened the range of -3- applicability of the irreversible thermodynamics theory. Analytical and experimental verification of both the Onsager and the Casimir reciprocal relations [02, D1, D5, D6, G4, L5, H10, 1112] has been widespread. A review of the literature cited above shows that although the seeds for the development of the theory of irreversible thermodynamics were planted more than a century ago, only during the last twenty years this de- velopment has found vigor and diversity. More recently the knowledge and theories of statistical mechanics have been used to provide a statistical basis to the processes in nonequilibrium thermodynamics [11, 1321, D6, D8, El, F6, K11, K12, 1:14, r15, 01, P15, P14, zu, 25]. b. Stationary States and Variational Principles The extensive use of irreversible thermodynamics formalism in the analysis of specific systems, that is to say, applications, has justified the need for and augmented the development in this field. An important part of this analysis concentrated with systems whose time invariant condition classifies them as stationary states. Some of the stationary states dwell with the condition of equilibrium. Such systems are best dealt with the principles of classical thermodynamics. Irrever- sible thermodynamics, on the other hand, finds use in the analysis of the systems whose states are described as -4- nonequilibrium stationary states, also known as steady states. The systems thus described can be considered to represent the general case which in the limit becomes the equilibrium case. Examples of stationary states are ever- present in real biological and non-biological systems. An important characteristic extremum or vari- ational principle underlies the stationary states. It has been shown [D5, D6, D9, Gl, P7] and stated [33, 34, B6, G5, G6, K11, x12, 01, P11] that under certain condi- tions the nonequilibrium stationary state coincides with the state of the system where the rate of entropy produc- tion is a minimum. The system must be functioning under the external constraints of established mechanical equili- brium or at least a mechanical stationary state [D6, G2, G5]. Irreversible processes that primarily involve ther- mal conduction, diffusion and chemical reactions qualify to be governed by the principle of minimum entropy pro- duction [D5, D6, G5, P7]. However, the inclusion of processes of the type described here are far from general [D6, D10, F5, G5, P11]. For example, as shown by K. G. Denbigh [D10] in the case of open reaction systems the principle of minimum entropy production is valid only when the state of the nonequilibrium stationary system is very close to equilibrium. Therefore, restrictive con- ditions must be imposed, which are: the existance of’ (i) constant phenomenological coefficients, (ii) linear -5- phenomenological equations (relations between the fluxes and forces) and (iii) Onsager's original reciprocal re- lations or the modified relations in the presence of a magnetic field or uniform rotation [D6]. From the discussion above it is apparent that some of the stationary nonequilibrium process cannot comply with the restrictive conditions required for the application of the minimum entropy production principle. The analysis of the irreversible phenomena for these systems is based on other variational properties or prin- ciples [325, D6, c5, x12, L5, P11, 25]. Using both the thermodynamics and the statistical mechanics point of view, S. Ono [01] presents an extensive treatment and discussion on such variational principles for the nonequilibrium stationary state. c. General Applications I In physics, chemistry and lately in modern engineering and biology, the theory of irreversible ther- modynamics is being used to analyse a great variety of problems. The arm in this case is threefold. First of all, with the aid of existing laws, empirical or other- wise, as well as experimental data, the theory, validity of assumptions made, and formalism of nonequilibrium thermodynamics is substantiated. Secondly, nonequilibrium thermodynamics enables one to obtain a unified point of -6- view in the study of different as well as coupled irre- versible process phenomena and augments the attainment of analytical and numerical results. Finally, this new field makes it possible to improve over existing methods of analysis and calculation as well as serves to obtain en- tirely new solutions for some important problems. The various treatises on the theory of nonequili- brium thermodynamics [D5, D6, Fl, Hl, P7] have included an extensive fundamental analysis and application of this theory to processes with chemical reactions, structural relaxation phenomena, gravitational fields, electromag- netic fields and their coupled effects. In addition to this, an abundance of literature has been published presen- ting the use of irreversible thermodynamic theory and for- malism in the treatment of nonequilibrium process pheno- mena. Some of the general fields involved in the appli- cations are: diffusion and osmosis [88, B11, 02, K15, Ll, N5, ea, 31, S8, S9, 512, 815, T1, v1, v2, V5], heat and mass transfer [D7, L7], two-phase systems [B9, B10, 315, 314, 315], surface systems [312, 316, 317, B18, 319], chmmical reactions [B5, B6, B15, H4, T18], electrochemis- try [D5, F7, K10], electrochemical corrosion [F8, F9], electrical networks [01, M10], viscoelasticity and rheology [B325, 21,122], fractional distillation [T14], gas absorp- tion in liquids [1315, 1117, T18], and humidification of air [T16]. -7- Although most of the work published to date gives strictly an analytical development and discussion on the applications, some of the investigators supplemented their reporting with numerical results [02, F1, L7, H12, 812, 815, T16, 517, V1, V5]. d. Biological Systems The progressive growth of the theOry of irrever- sible thermodynamics and especially the area of nonequili- brium.stationary states has inspired the use of thermo- dynamic analysis for the description of coupled and overall processes in living systems. L. von Bertalanffy [322], U. F. Franck [F4] and others have asserted that from the thermodynamic point of view living organisms are open sys- tems in a state of constant flux. A. J. Lctka [L6] in his discussions on physical biology defines evolution as ”the history of a system undergoing irreversible changes” and points out that "the direction of evolution is identi- fied with the direction of the unfolding of irreversible processes, the direction of increase of entropy (in thermodynamics) or of increasing probability (in statis- tical mechanics)." These actions have instigated the use Of irreversible thermodynamics formalism in the description of processes in biological systems. I. Prigogine and J. n. Wiame [P7, P8] forwarded, “one [322, D5] supported and some [D9, D10, F5] had -3- reservations about the use of the theory of nonequili- brium states, such as the principle of minimum entropy production, in the analysis and better understanding of organic evolution. At present this phase of irreversible thermodynamics application remains in the state of the original proposal. The lack of development may very well bear proof to the statement of A. J. Lotka [L6] that although thermodynamic principles regulate the pro- cesses during biological evolution, "the systems under consideration are far too complicated to yield fruitfully to thermodynamic reasoning." If the focus is diverted from the analysis of the global behavior of living organisms to that of specific processes in biological systems a variety of applications come to view where the methods of nonequilibrium thermo- dynamics have been utilized. I. Prigogine, in his trea- tise on irreversible thermodynamics [P7] suggests the use of his formulations for coupled reactions, many step re- actions, and coupled diffusion-reaction phenomena in the analysis of biological processes. The bulk of the appli- cations, however, come under the heading of membrane permeability and biological excitations. A common approach in the study of processes in living organisms has been the use of artificial models. These models serve to test analyses based on classical laws, support and modify hypotheses and theories, as well -9- as bear further understanding about the behavior of biolo- gical systems. Among others, T. Teorell’ [T2, T5, T5, T6, 19, T10, 115] has contributed substantially to this approach. During the last decade the formalism of irre- versible thermodynamics was employed by L. F. Nims [N1, 32, 35, N4], J. c. Kirkwood [K8], 1. Katchalsky and 0. Kedem [K2, K5, K6], I. Tasaki [T1], and others in the study of flow processes, solvent and ion transport, and tracer movements across membranes in biological models. U. F. Franck [F4], on the other hand, used the irrever- sible thermodynamics theory in the study of models for biological excitation processes. All in all this new de- velopment has been proposed and employed as a supplemen- tary means to aid in the gain of knowledge and understanding of some of the processes and coupled phenomena in living organisms. e. Non-linear Systems and Cyclic Processes Two conditions that warrant the use of the pheno- menological formalism of irreversible thermodynamics are: (i) the processes must proceed slowly and (ii) the pro- cesses must be close to the state of equilibrium. Pro- cesses far removed from the state of equilibrium and/or proceeding st'a rapid rate will yield, in general a ‘The experiments on and analysis of oscillatory ransport phenomena in artificial membranes by T. Teorell T5, T4, T5. T7, T8, T11, T12] instigated this thesis. -10- nonlinear dependence between the fluxes and the forces and the coefficients will have an unsymmetric matrix. Thus there exist nonlinear irreversible processes. Some studies have been made concerning the extension of irreversible thermodynamics theory to non- linear phenomena. I. Prigogine [P7] discusses some of the preliminary work in this direction, including the varia- tion of the entropy production. Variational principles in the thermodynamics of nonlinear irreversible processes are also presented by P. Glansdorff [G2] and B. One [01]. Other workers in this field have forwarded and analyzed some nonlinear theory and discussed nonlinear problems in the thermodynamics of irreversible processes [B6, B7, 65, L4, 811, z1, me]. The literature points to the fact that much remains to be done in the treatment of nonlinear phenomena. In their fundamental studies of nonlinear systems I. Prigogine and R. Balescu [P7, P12] have shown that it is possible for a system to reach a stable cyclic process around a stationary state without ever reaching the stationary state. The limiting cycle can be reached either by a system far removed from equilibrium and ap- proaching the stationary state or by a system whose stationary state is disturbed by a fluctuation. Further- more, these irreversible cyclic processes will have entropy production during every cycle. -11- The periodicity of certain phenomena in living organisms is well known. The developments in the theory and methods of nonlinear irreversible thermodynamics have' been viewed as a possible means to help shed more light on understanding these cyclic biological phenomena. It has been prOposed that in some cases the cyclic processes may be analysed on the basis of oscillating chemical re- actions. Although no strong supporting evidence has been found to this effect, this spurred the interest in the study of periodic chemical reactions. Among others, J. z. Hearon [as] and '1. A. Bak [35, 36] discuss the possible existance of oscillating reactions. In many in- stances [B5, B6, P7, P12] the equations of the Volterra model of interacting biological species have been used as a possible example of periodic reactions. At this time further thorough investigations are needed both analy- tically and experimentally to substantiate or disprove clause of possible periodic reactions with this or other models of chemical equation systems. During experimental studies of permeability in systems with charged membranes T. Teorell [T5, T7, T12] and C. Forgacs [F2] reported observing some oscillatory phenomena. Forgacs, in his publication presents his find- ings as a statement of fact without analysis. Teorell, on the other hand, using the classical laws developed for the analysis of a membrane-electrolyte system, produces -12- a mathematical model of his oscillating system and presents a quantitative discussion [T5, T8, T12, T15]. The systems presented by these two investigators may very well serve as a path in the development and application of irrever- sible thermodynamic theory for periodic processes and as a tool for substantiating through experimentation some of the analytical results. It is apparent from the discussion presented in this chapter that the theory of irreversible thermodynamics must undergo further refinements and expansion, especially with regard to variational principles and nonlinear sys- tems. At the same time, the existing theory presents a challenging opportunity in the diverse area of applications. CHAPTER 2 - Membrane Transport Phenomena A membrane can be defined as a solid or liquid interphase separating two systems, with a small finite thickness compared to its surface and a function to re- tard the establishment of equilibrium between the two systems [H5, 810, 811]. It is usually regarded as a body that furnishes the path for the unidirectional or bidi- rectional transport of some or all the matter that con- stitute each system. This transfer of matter through a barrier is of great importance in both physical and biological systems. Extensive studies have been made with regards to mechanisms and the parameters that influence the membrane permeability phenomena. No attempt will be made to review or even enumerate all this work, but only to present briefly those areas that are pertinent to the presentation in subsequent chapters and as applications of the theory of irreversible thermodynamics. Permeability phenomena in artificial and bio- logical membranes were until recently studied through the use of the conventional flow equations. Several model theories were introduced aiming to describe some of these flow processes. This approach has lacked vigour, usually ignored all the secondary interaction effects and often led to contradictions [K6, 810]. However, the -13- -14- value of the contributions arising from all these studies cannot be denied. T. Teorell, in an article published about a decade ago, presented an extensive analysis of permeability properties in "electrically charged membranes" [T2] based on the conventional description of membrane transport phenomena. This work can serve also as an excellent guide to the abundant literature pertaining to membrane permeability research. In the early fifties a. J. Staverman [88, $10] applied the formalism of irreversible thermodynamics to describe membrane processes. He treated coupled effects of diffusion, electrokinetics, and osmosis and he derived relations between phenomenological and thermodynamic constants. The unifying approach of irreversible thermo- dynamics was demmstrated in the comparison of this development against several of the model theories. J. G. Kirkwood in 1955 [K8] presented a treatment for the trans- port of ions through membranes using also the phenomeno- logical methods of irreversible thermodynamics. The approach of Staverman and Kirkwood has been used later by A. Katchalsky, 0. Kedem, I. Michaeli and others [K1, K2, K5, K6, Mll, N1, N2, N5, N4, 81, s7] in the analyses and discussions of flow processes in biolo- gical systems, permeability phenomena and ion transport. Similarly, the work of H. Vink [V1, V5] on osmometry and solute-permeable membranes is influenced by these -15- developments. Vink's work is of interest in that it treats time dependence of osmotic pressure and substanti- ates his analytical treatment through experimental measurements. Mechanisms of active transport, which involve chemical energy exchange for the membrane during transport of matter through it, have also been under study in re- lation to biological systems [see for example P1, P2, P5, P4, P5]. 0. Kedem [K7] used the formalism of irreversible thermodynamics to describe active transport phenomena and presented his formulation as a possible means of quanti- tative correlations of results from different measurements. The problem under consideration in subsequent chapters is also concerned with membrane transport pheno- mena. It relates to the oscillations observed by Teorell and Forgacs and cited in Chapter 1. Although the model under consideration structurally resembles the systems of the two investigators, the conditions were idealized to permit adequate develOpment using the methods of irreversible thermodynamics. CHAPTER 5 - Development of the Phenomenological Equations for a System with a Fixed Lattice in the Presence of an Electric Field a. Elements of Irreversible Thermodynamics The formalism of irreversible thermodynamics supplies a methodical approach in describing phenomeno- logically the transport of matter and energy in a system that has departed from thermodynamic equilibrium. In such a system it is necessary, however, that at each point in the interior the thermodynamic state is defined by local thermodynamic properties. Thus, according to the prin- ciples of thermodynamics we can introduce a function 8, the entropy, which is a function of the state. The change of entropy with time in a volume element is expressed as d8 a des + dis (5-1) where dos is the entropy flow due to interplay with the surroundings and dis denotes entropy production in the system due to irreversible phenomena in the volume element. While the quantity deS may be positive, zero, or negative depending on the type of interaction of the system with the surroundings, the entropy change dis is always non- negative, i.e. d S 2 0 e (5-2) -l6- -17- The equality sign in equation (5-2) holds when the sys- tem undergoes only reversible changes. Thus, locally the second law of thermodynamics is formulated. From the discussion above it becomes apparent that the local time derivative of the entropy production dis/dt is selected as a basic quantity to characterize the irreversible processes. The macroscopic conservation laws of mass, momentum and energy are then used in the differential form along with the thermodynamic Gibbs re- lation, if the irreversible processes are sufficiently slow to justify such usage. It has been shown [D5, D6, P7] that this procedure results in the simple relationship where the local time derivative of the entropy production equals to the sum of terms, each consisting of a product Of a flux of an irreversible phenomenon and its conjugated thermodynamic force, i.e. d.S L :2 J1 XL (5-3 a Z ’ where J1 represents a flux and X1 its conjugated force. We will now consider a system in the presence of an electric field and with consideration of the polari- zation of the molecules. The formalism to be used is that presented by Smith-white [S5, S4, SS, S6], Prigogine, Mazur and Defay [P9], Defay and Mazur [D4], DeGroot [D5], and DeGroot and.Masur [D6]. The system is to consist of a18- a fluid with n+1 components, i.e. O,l,2,°'°,k,°°°,n. Since our eventual aim is to study transport phenomena in membranes, we will let one of the components, such as 0, represent the membrane lattice. The references cited have considered conditions for free diffusion. Here, a fixed lattice will be considered and prOper modifications to this effect will be made. Hans Vink [Y2]presented a de- velopment of equations for a fixed lattice in simple dif- fusion systems in the absence of an electric field and polarization. Using the (barycentric) substantial time deriva- tive we can write the following fundamental equations in the absence of electric fields, polarization and charged particles: (1) The equation for the conservation of mass for the component k is p at = —- d]V (Bk-WK (5—4) in the absence of chemical production. In equation (3-4) Gk represents the mass of component k per unit volume, that is to say Gk = Mk/V 1 (5-5) and'Pi is the velocity of component k. Summing up for all components, equation (5-4) becomes -19- 32- = —dIV(cW (3-6) at where [a , the total density, is e=zek=(zkmk)/V= M/V= I/v (5-7) if M is the total mass and v the specific volume, and 13, the velocity of the center of mass, is 1 ‘60:.(2 k (3579/6 . (3-8) If we let Jk define the flow of the substance k with respect to the movement of the center of mass, that is to at: J; = (BE - W) (5-9) and using the (barycentric) substantial time derivative 8 .- —— = —— + w Brod . (5.10) then equation (5-4) becomes i9}. = —div 5‘: -(Dkd]VW. (3-11) d’c Now we define the concentration of component k as -20- p H k (Gk/(Q = Mk/M then $20,333.» edik as “as dt but from (5-6) and (5-10) we get cio __ ___ =.—- d? dt (3 VW thus dek dck .dT:— (Jkdlvw-PQ— clkt. Substitute (5-15) into (5-11) and we get dc (D———- CH: kz—dWJIZ' We also note that from (5-7) __(°_=__|__C_LV_ dt. v2 dt thus conbinine (5-14) and (5-17) we have de _ __ | ——-——edww— ——3 dt 9... dt ° (3-12) (3—13) (3-14) (3-15) (5-16) (5—17) (5-18) -21- Also from equations (5-8) and (5-9) it follows that Z 3,1: 0. 0-19) k (ii) The equation of motion is AW _ "‘ (,3. _ _Smap “‘2‘:qu (5-20) where F> is the pressure and ii is the external force per unit mass on component k. In this equation viscous forces have been neglected. (iii) The equation of conservation of energy is (3 d(—é-::+ UL) = _div (PW + 3(1) + 2?]:ka (3'21) where u is the internal energy per unit mass (specific energy) exclusive of the barycentric kinetic energy, and Jé is the heat flow per unit area. (iv) The entropy equation we can write for the center of mass movement as as _ a_ F.__v__ Tdt ‘ dt Ptd Zhéfl at (3’22) where S is the total entropy, U is the total internal energy and}:k is the chemical potential of component k such that -22.. ['Lk=( 5U ) . (3-23) aMk S,V,MJ The Gibbs free energy function G is given as G .. U - ms + PV (3-24) thus from (5-24) and (5-25) we get as = —SdT + VdP 4'kade {5—25) and from this we see that the chemical potential can be defined as be h. = [W] ° “'26) k T,D,M .5 Euler's theorem states that: If f(x1, x2, x3,”') is homogeneous of degree n, i.e. if f(kx1, kxa, kx5,'”) - 1:11 f(x]_, x2, x3,”') 9 then nf s E 56%) 1:5. Gibbs free L energy G, at constant temperature and pressure, is homo- genous in M1, M2,”‘,Mk,"‘ and of the first degree. Thus from Euler's theorem we can write .2 5‘39. G — 5M1. Mk = Erika/1k. (5-27) k P,T,MJ k -23- Substituting (3-27) and (5-12) into (5—24) we get g = Z )1,ka = u, — Ts + Pv (3'28) k and (5-22) can now be written in terms of specific properties as follows T-ds _ du +Fyé1 dck d+. 8+. at - k Vkfi' (3-29) b. Presence of an Electric Field Let us now proceed to correct the above equations for a system where an electric field is present and polari- sation is present. Maxwell's equations for an electro- magnetic field are Vxfi =I + % (5-50) V“; = “5-,;- {5-51) where V is the vector-operator. H is the magnetic field strength, I is the electric current density and'E is the ale J Thus dG*= —saT + VdP —v(§-di) + 2):;de (3-44) k and -27- i g = Erick: u. —Ts + Pv —F-E . (5-45) k Finally, let us consider charged components in existance in the system. The total electric current T in such a system is partly due to convection and partly due to conduction. Thus if = Z (33ka = (32W + z ijl: (34-6) k making use of (3-9). In equation (3-46) ”h is the charge of component h per unit mass of k and s is the total charge per unit mass of the mixture, that is to say 2 = 71,—; (3ka = Z kak . (3.47) k If we let 3 represent the conduction current, then :- _ ‘W L _ szJk (3-48) k and (3-46) becomes T - (3:? + 3 (3-49) where Q2? represents the electric current arising from convection. We also note that from (3-16), (3-47) and (5-48) we can write the law of conservation of the charge as follows -28- (33%- : —d'\v": . (3'50) Define the specific energy of component k to include the internal energy and the electrical energy, that is to say 3k - uk + zk4J , (3-51) then using (3-47) the specific energy for the mixture of the system inclusive of the electrical energy becomes 11: LL+L|JZkaCk=LL+ 241. (3-52) Correspondingly, we define the electrochemical potential of component k as Elk:= r3; + zk41 (5-53) or N az: )‘Lk — ka p3T,"E-,Mj (3-54) where‘a is a redefined Gibbs free energy to include the electrical energy. To derive the energy conservation and entropy equations let us rewrite equation (5-20) as follows C)..— = _SrodP+E (3-55) -29- where f is the external force per unit volume acting on the system, and i3“ = 2953'“ . 0-554) k The external force per unit mass on component k, ii for our multicomponent system is given as .- Fk . .. ngradLP + (div?) Fk (5-56) where F]! is the partial specific electric polarization, that is to say . (5-5‘7) T,P,E,MJ __ _ av? k' amk From (3-55.l), (5-47), (5-37) and (3-32) we see that f on the system is given as follows § = 92?) + (divi)? . (5-58)‘ Multiply now equation (3-55) by'fi and we get W'G'j—f = 4.3er + Em (3-59) a(-12-w2) ._ GT:_Wo9rodP+F-W, (3'60) also -50- 4129*) = W \ at 2 a .. 2% _w.3map + F'W. (5-61) Substitute equation {5-18) into (3—61) and we get = __JZ—erdivw —W-3mo\P + 3°33. (3-62) Applying the (barycentric) substantial time derivative, equation (5-10), on the left hand side of equation (5-62) we get \ -2 6(Ebiw) = _medQ—PWj—é—Waedivw —W.3mdP + E-W. (3-63) But div P? = Pdiv ‘i’ + 'i-srad P (5'64) also dav(-;-ew2w) = swag—(3w?) + lewzdw. (3-65) Substitute (3-64) and {5-65) into (3-65) and we get a l *2) .. (253W -_- —div(-%- QWZVV + PW) + PdivW + {74/71. (3-66) -51- To evaluate the force-velocity product in equation (5-66) multiply equation (3-58) by'? and we get :4? = (32:47? + W-(dw-gfi , (3-67) but div(§°§W) = wfwi)? + E-Smd(w 7?) -, (3-68) substitute (3-40) into (3-68) and the result into (3-67) and we get f.w = ezE-W + div(P‘EW) + E393 - E'ifi . (3‘59) 9 dt Using {5-56) and (3-35) we see that __ g * _ _ a 4-52) AP = —Ie.E _Eo—bf- = —I"E - (2 ' (5.70) g. at t at ° substitute (3-70) into (5-69), the result into equation (3-66) and we get a | *2 ‘ -2 _ . ‘ -Z-- -- ..- -. 3.6-6“! +72“: ) — —dw[—Pw w + Pw-(P {)w] 5.5-9555 +(JzE-W + Pacvw. (3—71) Insert next equation (3-49) into (5-71) and we get l rh} ’Pifi' + Pastime-72) dt To obtain the balance equation for the internal energy we have to make use of the local form of the law of the conservation of energy. If we let e represent the energy per unit mass and 3e the energy flux per unit surface area and unit time, then %E((°e) = _dee (3-75) and Ge =(Dw- -‘2-GW2+‘\2‘:2‘ (5‘74) Similarly we have 38: jq+_\2_(3w2'\p+ (out + w —(§-§)W. (3-75) Using the (barycentric) substantial time derivative, equation (5-10), we note that for the specific internal energy u we get EL”: = .955 + W . grad u,. (3-‘75.l) di: 5+. Multiply equation (3-75.1) by (a and we get -33- (Jim (GEE — + cw gradu. (545.2) but 2 = 6_ “3:9 - (a At wok”) 5t (5 75.5) and substituting equation (5-6) into (5-75.3) we get 95% = g—tkeu.) + u. d\v(eW) . (3’7504) Thus from equation (5-75.2) and (5-75.4) we note that 9.3:“- : gage) + udiv(efi) + €W~3md u. (54755) or (4.3—: = %€(QUL) + div———-E; --J3 _ , - di: dt+ at “1 dt grkdt (3 82) flow we introduce equations (3-78), {5-18) and (3-16) into (5-82) after multiplying the latter by G) and we get ds o'“ 7"” ‘ ‘~ di- * '-*’ find—4: = —dNJ°L + L-E ‘Q(Eeq- )3}: + 2 TH. deSk (3'83) k and substituting {5-48) into (3-83) we have -35- €T_ = —de<1+:k:Zle't-- €(%-E)-:—§ + grids/3|: , (3-84) or writing (3-84) in a.more general form we have n n d5 . t v “ '* - " PT‘ZTE = — “NJ-(1+ sz'JL + zrkdIVJ-k (3-85) where n - n at -I— -’.-- — -- _a. 241- k_ 22.. k 1;. deal a)? . {5-86) However, using (3-56) we can write (5-86) as follows n :m; k =sz£° J‘: +2: (div-E.)W . {5.87) Thus from (3-86) and (3-87) and using (3-9) we have that n _G(E .. E)? = a (ex-WE); (5437.1) _9(Eq_E)£=:(a:v‘)Fk-Wkek- 2% ’)]S- we, (3437.2) and as. (ii—Zed“ E)“ - ke(div ENB'W (3—88) '51: - 99?]: (3-89) is the mass flow rate of component k per unit area. c. Presence of a Fixed Lattice and the Phenomenological L “as-.2!” ‘ Let us now adjust the equations above for a fixed lattice, in which case we have that 370 - 0 (5-90) and introducing (35-90) into (3-9) we note that (so? . (3-91) For a steady state condition we have that an __ = 3- 2 dt ( 9) and thus from equation (5-20) we get '1 zfiek _ 3"“ D = O - (3-93) Separating the first term in equation (3-93) into the -37- fixed lattice component and the components in motion we have foe. = 3de - Ema.- 0-94) I Similarly n n .. » 4- ‘- - "v ZFk-J; = Ell-31: + {50' 0 {5-95) 0 I and substituting {5-91). (3-94), (3-9) and (3-89) into (5-95) we set n n ; Ek-J; .-= Z Ek-Jk - (a grad P. (3-95) I Substitute {5-56) into the first term of the right hand side of equation (3-96) and also substitute (3-88) after using the condition of equation (5-90). Thus we get n Z113; = 27.5.; _ (.(Eeq—EEE I _P(d;VE)F.W —W.8radp. (5'97) Substitute (5-97) into (5-85) and we get n n w d8 — ' ;v.~. _L .m.-- rk . -I O * _+[(3(Eei—E)EE + (3(divE)F-W + W-Srod P] . (3-98) at -38- If we note that div-gg- = dervlil— :13- 3":°'T {5-99) and i "I * n * n ILKJL . ._ * div Q T = Z g—kdivJ‘: + ZJtSr-od); (5-100) 0 O we can rearrange equation {5-98) as follows 3 — h *j/ «Ashes. ‘1 Zr“ _lj“.3"°” dt T' 1 'T n n I I -- *- dd. .ng Tamar. -225. Jud €1-£) _r O +e(divE)F-W + 71.3er p] . (3-101) If we define 3 as the total mass flow per unit area, that is to say _- J . (.3? (3-102) then using (3-102), (3-89), (3-12) and (3-9) we note that J, . -— Jk . Jk - ckJ . (3-103) Using (3-105) we can write the third term on the right -39- side of equation (3-101) as follows h n ’k n * *- ZJL'TSrdd-th = 23k.T3er-;L -3'::CkTaro c2 (4-1) where c2 is the concentration of the solvent given as a mass fraction. The solutions on each side of the membrane are kept homogeneous throughout by means of stirrers. The membrane is assumed to be uniform and made of compo- nents that are insoluble in either the solvent or the solute and these components also do not interact with the solvent or solute. The membrane pores are assumed to be large enough to permit bulk flow but also narrow enough to support diffusion. A cathode is placed in solution (I) and an anode in solution (II). Both electrodes are considered reversible and are connected externally through a D.G. source that supplies a constant current to the system. The membrane is bound by two planes normal to the .47- D.C.JSource h v 1|ll|ll} (Cathode:////flstirror H'H t§§§8§8§8§§8§§§§§1§§§§§S§RN§R§S§S _ _ __ j _ —___——— _'_—__—’-—'—’ ‘ from H —;§/_nembrane: _ é Reservoir — ————.—»§«— g E __ <3 __ E __ <5 _ é I‘T and E represent the pressure and electrical potential difference respectively across the membrane, and .. _fl ‘1 _m‘ cg, c2, and c2 are the mean values of the solute con- centrations in the membrane. These mean values of solute concentrations can be calculated as follows: Let d d l K = C2 die dX . (4-36) 0 Assuming ideality for the solvent we note that 0 RT' )‘LTc': )‘Tc + E QMNT (4'37) where il: is the purely chemical potential for the sol- vent at the standard state, R is the gas constant, 2 is -57.. the absolute temperature, Hi is the molecular weight of the solute andl1 is its mole fraction. However, n1 - 1 - N2 {4-58) and substituting {4-38) into equation (4-37) we get __ 0 RT )"'Tc — ['ch + a Q’“(" Na) ' (4‘39) In the first approximation we note that Qm.(l - Ne) w — N2 (440) and thus from {4-39) and (4-40) we get 0 RT VWC FTC :12 2 To find the gradient of ,HWc we differentiate equation (4-41) with respect to x and noting that the process is isothermal we get = _—_—-—- {4-42) Substitute (4-42) into equation (4-36) and we note that since we have an inert membrane the concentration in the pore at the boundaries of the membrane is equal to those in the respective outside solutions [T2, T5]. Thus at -53- I x - 0 c2 - c2 (4-43) _ _ II x d c2 c2 and with these limits of integration we get . n Ca RT dC-z RT (1)2 (12 = "—_— C —d = - ._ C. '- C )1 Ma]; 2d, at 2M2[ a a) (4'44) C2 and _ RT I I I I I - “ mice“ C2)(°a *2) . (“'45) but from {4-42) we see that = .51. -_— _. 31 I _ I Arm mzAca {Aloe Ca) (4.45) thus substituting (#-46) into equation (4-45) we get )1 = é—(C2 + C?) Arm. (4-4'7) Comparing now (4-47) with the first term in equation (4-55) we note that a; . %(c§ + cgl) . (4-48) next, let -59- jl if” 2.de (4.49) and assume that‘v1 and v2 are constant. Also, now we assume constant gradients for the pressure and the solute concentrations in the membrane. These assumptions are Justified by the fact that changes in the membrane are gradual as indicated by the long period of the oscilla- tions [T5, T7, T11]. Thus we can write equation {4-49) as d K =fTH¢Z+ %x)dx (4-50) 0 using again equations {4-45). Integrating {4-50) we find that I = (C: + iZ-Ace) P = é—(c: + (:3) P- (4-51) and now comparing {4-51) with the second term in equation {4-35) we note that '3; - %(cg + oil) . {4-52) Finally, let )1 =fc2+(2L‘2- T.”)52]A,L\C _.[l_uvI + (LTZVZ — an‘)Ee:|P _ [LHF' + (L12 pe— T.H ,)Ea]£ . (4-53) The resultant equations of flow, {4-35) and {4-58), can also be Justified from another point of view. Equations {4-25) and {4-34) as they stand represent non- linear equations. A coarse method of linearising these equations will be to take 3: to represent the average solute concentration in the membrane and replace with this average value the c2 terms that contribute to the nonlinearity. Since we have assumed the membrane to be homogeneous throughout, if we let J1 and J2 represent the flows of the entire membrane of surface A, that is to 8a: Jk ' JkA (4-59) then we have that J‘ = - fl— {[(Lu- L|a)+(2L‘2- L032] Arm + [Lm + (Lav2 - LH ‘) 232] P +[LHP‘+(T_,2P2-Lu |)z2]£ {4-60) -62.. Jra= " 3' [(L\2+ L22>52 _ L22]A)*Tc +0..2 v + L22v v2)c.2P + (H230. + L22P2)EZ£} . (4-51) In order to adapt equations (4-60) and {4-61) to the conditions of our system we will have to consider the solute concentrations in the containers I and II at each side of the membrane. If we were to simulate the condi- tions in the Teorell experiments [T5, T7, rs, r12] then 0; and cgl are to be held constant. In the Forgacs experi- I II ment [F2], however, we find that while c2 and c2 are not constant, 0% + ch is constant. For convenience first let us define c as the solute concentration expressed in conventional units, i.e. grams per liter. If we assume that we have dilute solutions the difference in the purely chemical component of the chemical potential can be approximated in terms of the concentration difference. Thus from equation {4-42) we have Arm“ -RT —Ac2 ~ — -=-v Ac {4-62) M2 NH where Ac - cII - cI . {4-65) -55- I II‘ b. Case 1 - c and c are Both Constant Teorell in his experiments [T5, T7, T8, T12] has connected the solutions on each side of the membrane to a corresponding reservoir. The solutions are con- tinuously circulated between the containers and their respective reservoirs, thus maintaining a constant con- centration on each side of the membrane. is our first I case, therefore, we will assume that c is constant and cII is constant. Thus, not only cI + cII - constant (4-64) but also from equation (4-62) we see that AFT?- constant . {4-65) On this basis we can write the following equations for our system adapting the equations of flow derived above %§ - a1{le1 + J2v2) {4-66) %% ' a2(J1P1 * J2F2) (4'67) gés . a3(31 + J2) - 0 (4-68) where a1, a2, a3 are constants. In our system, unless J1 and J2 are identical to zero, we will find that J1 + J2 / 0 (4-69) -64- therefore, in equation {4-68) we must have From our equations {4-60), {4-61), {4-62), {4-66) and {4-67) we produce a system of differential equations in two variables, namely the pressure P and the electrical potential E, as follows dP __dt : GIMP + 0(‘2£ + «'5 (4-71) dE wherecxid (i-l,2; J-l,2,5) are constants. To evaluate these constants we substitute {4-60) and {4-61) into equation {4-66) and we get g_:.=_ TA-{[._.W( W) :2} +(.. HLZZVZ)CV}p _ 325i {[LI'P'J' (Harg- L“P')52]V‘ + (L‘ZPT+L22‘°2)62V:}E +-°;-"-{[(L..— LT ( W — ‘RT +[(L‘2+ L22)C2 — L22]V2} E V‘ AC. . (4-73) Therefore comparing {4-71) with {4-75) we see that A 3 ,_ - a": - 0.6V {LN +\2[(2L +L22 v—VT)-VT —LH]C2} (4-74) -65— alAvtszAc V2 _ «'5: mad L" —L'2 —L22':—‘+[2 Ll2_ L|l+(L|2+L22)-W:|C2 . (4-76) Now substitute (4-60) and (4—61) into equation (4-67) and we get ——{[< > M) }P a A {[L..l°n+(Luaa°2"—HF')EZ]1°”(L'QF' +L22P2)Eab;£ — {Ra- LWH M +[(L‘2+ L22)'62 _ Lee] F2} —W\_ v. Q - (4477) Therefore, comparing (4-72) with (4-77) we note that '0‘2l = _ 92—:M{LH+ [La2(_£'£+ 2%) + L22 if: _ L452} (4'78) “22: _ a2: ‘2{Lu +[(2La + L22‘E)?‘ 1.46:2} (4.79) and ~66- : “ZAW’I RTAC 35 Rad _ 2 P2 .. 0t L‘r-L‘2 L22L+[2L‘Z~LH+(L‘2+L22)-—]c2 . (4-80) bi PI c. Case 2 - cI + cII is Constant Unlike Teorell's experiments, if there are no reservoirs attached to the solutions on each side of the porous membrane the concentration in each solution will vary with time. However, since the total solute is con— stant we note that cI + cII - constant . (4-81) For this case the condition of equation (4-65) is no longer valid, i.e. Arm! constant . (4-82) On this basis we can write the following equations for our system, adapting the equations of flow derived earlier a"? " 8L1(51"1 * J2V2) (“-830 g " a2(Jfl°1 * 32302) (4'84) - a3(J1 + J2) . (4-85) -57- From equations (4-60), (4-61), (4-62), (4-83), (4-84) and (4-85) we produce a system of differential equations in three variables, namely the pressure F3, the electrical potential E, and the concentration difference Ac as followsi dP —d_t = oaHP + «.5; + uwAc (4'86) dE -—cx P E CX A '3: " m + o‘22 *‘ 25 C (4-87) dAc E = ub‘P + «52E + «5513c . (4'88) The constants «13 (i=1,2,3; 3:1,2,3) are evaluated in the sane manner as in Case 1 and these are 2 A °‘u= “3%{Lu WKZL J“ L221. v ‘2‘):‘2T " Luke} (4'89) deVI I a 2 _ “‘2: .. __JL{LH+[L‘2 (1‘:— + __)+ +L22 I: ‘T— Rica} (4-90) G'AV|2RT V2 _ (X‘s: wt. Ln-L 2’—L 22...“..[2L 2~—LH+(L'2+ L237] C2 (4‘91) 2 I A . , J v am: —M {L||+[Lue(fi + 112’) J’ L (“Z—vi _ Lnjaa} (4'92) -68~ L22--_2Jaf{L ”Us. + sage L452} 0‘23.: GZ'A P‘V' RT raga 05AV|RT Mad Lu+[L‘2(—£—?—+ |)+ L22 :T— -L”]aa} [WWW (“we Legs] . {LHLV -L —L£2;—T+ +2.L[ WL+(L +L )‘H52} (“-95) (4-94) (4-95) (4-96) (4-97) CHAPTER - Active Membrane a. Derivation of the Flow Equations In Chapter 4 it was assumed that the membrane was inert. This assumption will now be dropped. However, we will still consider that the membrane is uniform and made of components that are insoluble in either the sol- vent or the solute. Also the membrane pore size is once again assumed to be large enough to permit bulk flow but narrow enough to support diffusion. _ The elmination of the inertness assumption will affect the derivation of the flow equations. Let us now proceed to incorporate these changes. For the solute, from equation (4-2) we derived (4-17) which is . _- d)“Ic dhgc d D J2 " " Llacncajx— " Leeca‘gx— " (L'ac‘v' + L22V2)C237 - (saw Largej—f- <54) All the assumptions made in the derivation of this equa— tion are still valid. For the purely chemical term of the electro- chemical potential the Gibbs-Duhem equation as expressed by equation (4-18) still holds. However, we can no longer use this equation to conveniently simplify our derivations -59- -70- since we now have that dfbc # 0 . '(5-2) We therefore require a different line of attack and for this we see that we may write [V3] the following equation for the purely chemical component of the electrochemical potential of the solute __ 0 RT‘ rec- me + T22 fluke. (s-a) where FL; is the purely chemical component for the solute at the standard state, Ki is the activity coefficient for the solute and.fié is the molecular weight of the solute. Here the standard state is chosen so that {£—+—l as ca—a—O. Again we assume dilute solutions and thus we can write for the solute activity coefficient where fl is a constant and c2 the concentration for the solute given as a mass fraction. Substitute equation (5-4) into equation (5-3) and we get RT Fae: r§c+ "fi- Q’“ (C2+ (3C2) ° (5-5) a Row differentiate equation (5-5) with respect to x and we find the gradient for the purely chemical component of the solute -71- dc dCTZC = RVT | (‘+2€3C2) 2 (5-6) x X a (1+ch or _Ld 2C=_R_-_r_ --‘—+ @ fig 0 (5'7) dx Va Ca ”(5C2 dx For the concentrations in the membrane we may now write co + el + c2 - l . (5-8) Solve equation (5-8) for c1 and substitute into equation (5-1) along with equation (5-7) and we get ' at dc '==- |— C2 —c m RT' ‘ f3 2 J2 L‘2( °)C2 ax 52°? m1 02 4' H-(See dx -l:L‘2v‘(- ”C2 C’0)+Lfa?.\12] C23: [L12FI(|—C2-Co)+L22P£]C2§% (5‘9) 01' d _ _ F"- 31 (5‘12 dC-z Ja— L'FZ[( C°—)Ca c:]—— dx _L22 fiz(-‘ + \+{5ca)-c-i-x— 1.121 1cm1}:—:W111«1 1 1.11:; In equation {5-10) neglect the second powers of the solute concentration since we have dilute solutions and thus we get -72- Ja=—L 20(1—c)c—fl9— L2 2LT(1 + We )ch 2 dX M2 ‘ +pC2 dX —1—l:1_‘2v,( COW vecd] 2d: p—L-[QM co+) were]? (121—2 (5—11) For the solvent we derived equation (4-29) which is . L d L . J1: - [LIICI— Ciel (L12C1Ce):l df:|+ L: CIJe ° (5'12) Substitute equations (4-14), (4-15) and (5-9) into (5-12) and we get ‘ 3m- R_T_ <1. {3 )d‘2 -LC [‘12— CC + ‘5‘: ” 'dx 2M2 '2 C2 1+(3c2 dx dP dE. “(L11V1+ L12V2C2)C13',T - (L11P1 + L12P2C2)01E ‘ (5-13) Solve equation (5-8) for c1 and substitute into equation (5—15). neglect the second powers of solute concentration and - _ dMIC RT (5C2 dc J‘— —LII(‘ _CO-Ca) dx — L‘z—h—A:[(‘ _CO)(I + 1+{5c2)-C2]d—: cip —l:LnV1(_C C"o ("2) + L12V2( _Co) C2] '2'; -73- _[Lur‘p - c211 L404. -1194 35. (5.11.) Thus we now have the flow equations for the solute and solvent in the active but fixed lattice represented by equations {5-11) and (5-14) respectively. . Following the sane procedure as in Chapter 4 we integrate equations (5—11) and (5-1#) for the total membrane thickness d. Thus W="L2( ‘50)52 Afllc-L 22 - RVZC(\+{5C2)AC “L2 2%% —1-[L‘2v|( co)+ L22V2]52p -L[ 12P1(“'C%)+L22P2]52E (5-15) _ [L11 Mpg—Ea) 1» L‘2P2(1-€o)ie]£ , (5-16) where?o and?2 represent the average concentrations in the fixed lattice. Although the average concentrations are not equal to the mean concentrations, the assumption that they are of the same order of magnitude is -74- Justified. This approach can be considered as a coarse method of linearising the nonlinear equations. Again it should be mentioned here that we have incompressible flow and thus the quantities 31 and 32 are constant with res- jpect to position in the membrane. In equations (5-15) and (5-16) the limits applied were those corresponding to the inner surface of 'the membrane boundaries. It is desirable that we refer our solvent chemical potential, solute concentration, pressure and electrical potential difference to these existing between the two solutions on each side of the membrane. In the case of the chemical potential of the solvent, the pressure and the electrical potential the differentials between the inner surface of the membrane boundaries and between the two solutions must be equal [52, V5], as a result of the condition of local equilibri- um. We should note again that P’ and E in equations (5-15) and (5-16) represent the total pressure differen- tial and the total electrical potential difference across the membrane.‘ Due to the fact that we now have an active membrane there exists a difference between the solute concentration in the inner surface of the membrane and the solute concentration in the solution. To convert Ac2 in equations {5-15) and (5-16) to -75- Ach'I - cg — cg (5—1‘7) where cgl and cg are the solute concentrations in the solutions outside the membrane, we use the approach of Vin]: [113]. He defines a solubility factor, which we will call <5, which measures the deviation from the inert membrane. Thus we may write Cm Ca(“'°°) where c; is the solute concentration in the membrane, cg is the solute concentration in the solution outside the membrane and co is the membrane concentration. Using equation (5-18) we get AC2: 8(1 - EO)AC§_I {5-19) and converting to conventional units of grams per liter equation (5-19) becomes AC2: 8(1— EO)V‘AC (5.20) where Ac is defined by equation (4-65). Substitute equation (4-62) and (5—20) into the equations {5-15) and {5-16) and we get -75- azE‘2v1(1-E)+L22v]l> -’2[L121ol(1-E°)+L22112] -Leepg (‘5 {5-22) The membrane was considered to be homogeneous throughout, thus we substitute equations (5-21) and {5-22) into equation (4-59) and we get respectively f J,=_$%v‘{1_ e111- Queen-2.1 1.. EM151 11,,(.-e—e1.}11c+11_"v1-eo-a-21+L.2V2(._eo1e,1p {1.111010% c—o Eel-—)+L12Pa( CEO) 2}. + 1. 2%%5- > (5-25) -77- and II b. Case 1 - cI and c are Both Constant I II Using the condition that c is constant and c is constant and following the procedure in Section b of Chapter 4, the resulting equations will be in the form of (4-71) and (4-72), thus ( dp CIA _ _ _ .EE.=:_-:i—<[}MW(h-§5—CJJ+Lm€%0-—ng%v1 b _ [11m] -73- 1 +64:an P10 ’50) + L22P2JV2 1.21811«1211-6211—6121 + Luv|(\— 60—62)} V‘Ac +11. 501112231. 1.1521 _ 112221112111: +1.12}; +1.22? . (5-25) F I Therefore, comparing equation (4-71) with equation (5-25) we note that and -79- +1111+12++1+ +11 M](‘+PE)} W12" Correspondingly we have that dE_ da‘A - - -— - 31 _ - 7.1 [1.11 - 10-121 (1.. 10112] 1,. if—-)} (5-28) ’ (5-29) ~80- and comparing (4-72) with (5-29) we note that 0‘21: _ G2:""°‘{L11("Eo-52)+[L12('- 250162 .3152 1% 15} (5-50) a22=_ 02:1 12 {L”(I—Eo—Ea)+l:ZL_|a(l E)+L22'1§T]‘1;E52} (5-31) and 0:23: —%¥_:< AC{L"(l-Eo-Ez) —L‘2<\- EO)C Ca(:—|§- + 8) +|:Lla(\-‘é:,)2+L2 2(\- Cor) £3] (111(31c2)}+(5I 7(L12+ Lea-E) >. (5-32) c. Case 2 - cI + cII is Constant In this case we have the condition given by equation (4-81). We therefore follow the procedure of Section c, Chapter 4 and our resultant equations will be as follows dP E = «up + «12E + «1513c, + «14 (5-33) d5 (5-54) Ei? = malp + «2243 + ueaAc + 0‘24 -31- dAcza 9+1)! E+u [Xe-10¢ d1 51 52 55 54 ' (5’35) Evaluating then the constants 1113 (i-l,2,3; 3:43.394) in the same manner, we have that an: —1‘gi‘Z-{L"(u- EC. 59.)" [2L 2(1- '60) + 1.22%]... VT 52.} (5-36) 0115- "P-—-'{°AVL. .1-( -‘o- 63+[L,2(1-50)({f—+ “4111.22 13]} (5—37) + [L‘2(1_ 5012+ L220 - Co) %]3(1+(saa)} (5—38) v «‘4: —W(Lm+ L227?) (5-59) «2‘: — %_ELV.' {L11G‘Eo‘32)+[l—12('” COX—:3 + {7172) + Lag/1152} (5-40) 0122: — 3% {LNG-Eo- Ea)+|E2L|2(\- 64+ Leaf—2‘] 15::— 52} (5-41) (5J4) (51-45) (5-46) (5-47) CHAPTER 6 - Criteria for Oscillations a. Case 1 In Case 1 for both the inert and the active nenbrane we have two first degree differential equations in P and E as shown by equations (4-71) and (4-72). In as Inch as the constantscx13 and 0 or 2 (122 + (1120(21 < O. , (5'13) Let us first use the criteria given by equations (6-11) and (6-12) for the inert membrane. Thus substi- tute (4-79), (4-75) and (4-78) into (6-13) and we get after simplifying 1.1.1 -1 12.. 11+- 1 ”11111111211[1.1111111“2212.551 < 0 11-111 oebfie + °1V12 @2 < 0 (6-15) or where Q and Q are constants representing the quantities in the first and second large brackets respectively. Since in equation (6-15)-P1, v1, and 6) are squared the order of the inequality depends on the signs of a1 and a2. If a1 and a2 are both positive, then equation (6-15) will not held and consequently it will indicate that undamped oscillations cannot exist. 0n the other hand, if a1 and a2 have opposite signs, the direction of the inequality sign in equation {6-15) will depend whether the ratio of the two terms is greater than or smaller than one. -86- Finally, if both a1 and a2 have negative signs the ins- quality sign will stand as shown in equation (6-15). The constant a1 is a physical constant. In investigating equation (4-66) for the possible sign in a1 we note that the specific volume is positive. In our system when both J1 and J2 flow in the positive direction dPth is negative; thus a1 must be negative also under these circumstances. In as much as a1 is a constant, it must be negative at all the times. In investigating equation (4-67) for the possible sign in a2 we note that the specific polarization is positive. Following the same reasoning we presented above, a2 must be negative also. Therefore, the condition represented by equation (6-12) is satisfied. Next we substitute equations (4-74) and (4-79) into equation (6-11) and after simplifying we get L”(| - ‘52)(O‘v‘2 + (map?) + E2[2L12<01V1V2“' 02111112) ~1-L22<“o £1 + oc‘alioe + 01|5Acoe Et_ £4: £4: £1; €503— — “apes + 01224809. 1 aesAcoe Et 1; E4: EAcoe =015‘geg +o<52£0e -1 ussAcoe and simplifying it becomes (an—QB) + o(1.2Eo + “15Acoz O aflpo 1 (11(22— §)£o + 0125Ac0= 0 «$130 1- “size + (“53—€)AC0= O . Et (6-34) {6-55) {6-56) (6-37) {6-38) (6-39) Again we note that for a non-trivial solution the deter- minant of the characteristic equations must be zero, i.e. 0‘21 (“22" a) 0‘22. = 0- ((3-40) 0‘51 0‘52. (1x 55— E) Expanding the determinant we get 3 2 E "' (0‘11+ “22+ “55): + (“110‘22 + «“0159- «220(35— 0(‘2012‘ - “mum— o(230(32): - ( \Iu22u53_ M'Io‘ESuZQ — NQQMISO‘SI —cxcxo(+1x 0 (6-52) we will have one real and two conjugate complex roots as follows C, = (3 + 11> (6-53) C2,?) = - Jé—(,9+ LP) 2'2 -2LVE-<8 -cp) {6-54) where 5 3 = V— -3— HQ— {6-55) 3 19 = V—-§- -‘\fX (6-56) and 1 ' V- . {6-57) For a pair of purely imaginary roots we observe that it is necessary to have .9“? = 0 (668) which will also make the real root zero. Thus in equation (6-50) we have b - 0 (6-59) and equation (6-51) becomes 3 A = ({3}) - (6—60) Therefore, the conditions for undamped oscillations are equation (6—59) and from (6-60) and (6-52) a.>10 . {6-51) Substitute equations (6-43) and {6-44) into (6-48) and we get _ _ 1 2 2 2 1 ° " a (“11+“22 + 0155) + 5 (0‘11“22+°‘11°‘55+°(22°‘55) —(o< 01 +01 11: +01 01 ) {6-62) 21 12 1:1 51 23 52 ° Next, for the inert membrane insert equation (4-89) through (11-97) into equation {6-62) and simplifying we get _ )2 2 2 RT 0 = —L2 (a‘v‘2+an?—L) M2 2 _ 2 ‘ - 2 2 °5V1RT - L22 :5 [C2(a1"e + 02%) + ——m2 ““12 I a v RT 2 2 — _ -— Llaé 3|:aca(o|v‘v2+ 02hr?) + 5m; (I -5C2):] _ '52 O‘1".2("1Pz — V2 we dvRT Amt—[520 (v 1212122241,-1212 1 _ eangT _ <31| (v.2 — 5W2) - 020°? - 5P1 ‘50)] “L11L12(‘—5EZ)@C MUM. 1.221111)— ‘3‘“ {if T 42222;”. 1112+ v.11) +Y1lv11°2(~”"1‘va) 1111211111] +1221??? 2 v RT 2 2 c1 _ fv“ Cea‘(v‘-Vz) +52Q2(P1-P2) 2 1(av,f—2+o1211X1 4%) +<> 211(°”2 V + “2? it)” ‘L22L12510 (6-68) (2) b - 0 f {6-69) where a is described by equation (6-62) and b is des- cribed by equation (6—64). In this case it has not been possible to produce a simple set of equations. The investigation of the relative magnitude of the constants necessary to satisfy the conditions (6-68) and (6-69) will therefore be very cumbersome. However, at the present it is safe to assume that this system also will sustain undamped oscillations, since this system can be assumed to represent an idealized model of Forgac's experimental system. -101- The systems discussed here and the phenomena of undamped oscillations represent a challenge in the field of irreversible thermodynamics. The necessity for study of such systems has been forwarded both by experimental and analytical investigators. The present work is offered as a small contribution to the analysis and understanding of these systems and such phenomena. W. .‘2 VHS“ “' . B1. B2. B}. BS. 36. B7. BB. B9. BIBLIOGRAPHY F. C. 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