J. Phys. Chem. 1992, 96, 2721-2724 polar species, either in bulk water or at an aqueous interface, and (2) those which are dependent on the electrostatic self-energy of a polar or charged group. As the simplest example of the first class, consider the interaction of two spherical charges well separated in bulk liquid. From continuum electrostatics, this interaction is given by Coulomb’s law as W = (q1q2)/trlz where ql and q2 are the charge magnitudes, t is the intervening dielectric, and rI2is the separation of the charge centers. Assuming that charges q1 and q2 were not derived to explicitly take into account the dielectric constant of the water used in the simulation, the error introduced by using a TIP4P with a dielectric constant of 60 in the simulation will be 33%. Examples of problems of biological interest which are governed predominantly by electrostatic interaction energies include shifts in pK’s of surface residues on mutation and association rates of substrates with enzymes. The simplest example of the second class of properties is the free energy of transfer of an ion from water to a nonaqueous environment. From continuum electrostatics the electrostatic component of this free energy is given by the Born expression W = q 2 / ( 2 a ) l[/ t i l/t,] where a is the radius of the cavity formed by the ion in the two media and where ti and e, refer to the dielectric constants of the nonaqueous environment and of water, respectively. For the transfer of such an ion from water to a uniform dielectric with ti = 4, the error in free energy of transfer by using a water with a dielectric constant of 60 is only 1.7%. Examples of properties of biological or chemical interest which can be largely governed by electrostatic self-energies of charged or polar groups include transfer free energies, desolvation of charged or polar substrates on binding to enzymes, and protein folding. The example given above used ti = 4 to suggest an ideal protein interior. Thus interaction-energy properties but not solvation free energy properties will be sensitive to the use of a water model with a dielectric constant which is 60. It is worth making a final point concerning biological or chemical simulations which employ explicit water models. To obtain the results presented here and in other dielectric simulations, a reaction field has been introduced to account for
2721
the continuum beyond the cutoff used in the simulation. The dielectric properties of the model under study change considerably if such a convention is not introduced. In simulations of proteins in water, reaction fields have not to date been introduced. As such, the dielectric constant of the waters in such simulations is undetermined. In conclusion, our results indicated that the free energy method can give results comparable to the induced polarization method, though it produces 2-3 times the standard error when both are analyzed using block averages. Values for the dielectric constant of TIP4P determined by the free energy method and the polarization method are slightly higher than those reported for comparable conditions but fall short of the experimental value of 78. The results obtained from the two methods in a single simulation are of course correlated, in that the samplings used are identical and the fundamental physical quantity monitored in both methods is the induced polarization of the unit cell projected onto the direction of the applied electric field. The method of induced polarization has been used here as a method of monitoring the behavior of the new method. The use of the free energy perturbation method to derive dielectric properties casts the calculation of a fundamental property into a form which can be handled by many standard programs designed to examine free energy differences. Acknowledgment. We acknowledge support from NIH Grant 3 P41 RR00442-21 and express gratitude for useful discussions with Barry Honig. The computational work was carried out in the Computer Research Resource in the Biology Department at Columbia University, directed by the late Cyrus Levinthal. We would like to acknowledge Cy’s support and interest during the last year of his life. Analysis was completed using facilities at the Biosym Corporation. We also thank Kim Sharp and Howard Alper for useful comments and Alex Rashin for critical reading of the manuscript. Registry No. Water, 7732-18-5.
Asymmetry Potential in Inhomogeneous Membranes Javier Garrido* Departamento de Termodincimica, Universitat de ValPncia, 461 00 Burjassot, Valencia, Spain
and Vicente Compaii Departamento de Ciencias Experimentales, Universitat Jaume I , 12071 CastellGn, Spain (Received: August 20, 1991) In this work the asymmetry potential of ultrafiltration membranes is studied. The membranes used are made of polysulfone with a support of polyester. The theoretical analysis is developed within the frame of the thermodynamics of irreversible processes. The transient character of the asymmetry potential is emphasized. An experiment where the asymmetry potential changes following a nonstationary process has been designed and a new model for this potential is given. Membranes exhibit asymmetry potential when the concentration profile is not uniform and the profile of the apparent transference number of the cation is asymmetric. We also conclude that changes in the asymmetry potential are due to diffusion processes into the membrane.
Introduction Electrochemical properties of asymmetric membranes have recently been discussed in several papers.ld In these systems we can find peculiar effects that apparently contradict the first ~~
(1) Manzanares, J. A.; MaE, S.; Pellicer, J. J . Phys. Chem. 1991,95,5620. (2) Maf€, S.; Manzanares, J. A.; Compafi, V. An. Fis., in press. (3) Asaka, A. J . Membr. Sci. 1990, 52, 57. (4) Ssrensen, T. S.; Jensen, J. B.; Malmgren-Hansen, B. J . Non-Equilib. Thermodyn. 1988, 13, 57. (5) Higuchi, A.; Nakagawa, T. J . Chem. SOC.,Faraday Trans. 1 1989,85, 3609. (6) Takagi, R.; Nakagaki, M. J . Membr. Sci. 1986, 27, 285.
0022-3654/92/2096-2721$03.00/0
and the second law of thermodynamics.’ The asymmetry potential and the observed fluxes are not easily explained. Something similar can be said about the facilitated and reverse fluxes that these membranes exhibk6 The asymmetry potential is the electric potential difference across a membrane placed between two identical electrolyte solutions. The electric potential difference is measured by reversible electrodes dipped into the solutions; the observation must be made under potentiometric conditions. ( 7 ) Kbrasy, F. de J . Phys. Chem. 1968, 72, 2591.
0 1992 American Chemical Society
2722 The Journal of Physical Chemistry, Vol. 96, No. 6, 1992
Garrido and CompaR
The different questions which we intend to consider in this work are related to the nature and interpretation of the asymmetry potential. Liquori et a1.* explained this effect by means of two different ion-selective membranes. The TMS theory explains this potential in terms of asymmetries in the fixed membrane charge density and the ionic partition coefficients.6 Our analysis is based on a formulation of the thermodynamics of irreversible processes; in this formulation, fluxes and forces are observable variable^.^ We will discuss the temporal character of this effect and the need for a certain inhomogeneity in the membrane. Tbeary
We start with an analysis of the thermodynamic state that carresponds to the membranes which exhibit asymmetry potential. This analysis is very suitable if we want to give a proper interpretation of this effect. The apparent contradictions that one can find in the literature vanish if one makes a correct review of this basic question. Let us consider a simple membrane system: a solution with two components, solute s and solvent w, bathing the membrane. The solute is completely dissociated into vlzl-valentand v2z2-valent ions. The electrodes are reversible to the ionic species 1. Equilibrium of permeants at the membrane/solution interfaces is assumed. The dissipation function, expressed in terms of observable variables, takes the form9 \k = q A P
+ CApS + IA$
(1)
where q is the apparent volume rate measured in a subsystem, is the solute flux relative to the solvent, I is the electric current, AP is the pressure difference between the subsystems, & is the concentration-dependent part of the chemical potential of the solute between the subsystems, and A$ is the difference in electrical potential between the electrodes. We know that each one of the stationary processes which can occur in a membrane system is determined for whatever three variables among the six that appear in the phenomenological equations, i.e., AP,hc, A$, q, or I. The asymmetry potential is observed when AP = Akc = I = 0; these conditions determine a stationary process. Let us review this process. From eq 1 we obtain the surprising result that the dissipation function is zero. But this is not possible!! The second law of thermodynamics asserts that, in any system, the dissipation function must be positive if there is any nonzero flux. Therefore, we conclude that the formulation given by eq 1 is not correct. These systems evolve following a nonstationary process. Therefore, a membrane system in the conditions AP = & = I = 0 and with A$ # 0 is in a nonequilibrium state and evolves following a transient process. Having in mind this conclusion we have designed an experiment to observe the asymmetry potential effect. First, the membrane was soaked in a solution for a long time, till we were sure the equilibrium was reached. Then, the membrane was carried to a cell with a solution of different concentration and the asymmetry potential could be observed, following a transient process.
c,
Experimental Section A cell made of Pyrex glass was used. The membrane was clamped tightly between two compartments of 71 cm3 capacity. The surface area of the membrane was 4.15 cm2. Electrolyte solutions of NaCl (guaranteed reagent grade) were used. Two Ag/AgCl electrodes were dipped into the compartments. Electric potential difference was measured by means of a LI-1000 Data Logger from Li-Cor. The input impedance of this programmable voltmeter is lo5Q. This device takes measurements at intervals of 5 or 60 s; integrated values were recorded. The polysulfonic membranes were prepared at the Department of Nuclear and Chemical Engineering of the Universidad Politknica de Valencia. A solution of 15% of polysulfone U1(8) Liquori, A. M.; BotrC, C. J . Phys. Chem. 1967, 71, 3765. (9) Gamdo, J.; MafE. S.; Aguilella, V. M. Electrochim. Acta 1988, 33, 1151.
Figure 1. Scanning electron photomicrographs of the two faces of the membrane A. (i, top) Face of the polyester support; scale unity: 100 pm, (ii, bottom) Face where polysulfone was poured; scale unity: 10 pm.
tralon-s from BASF in dimethylacetamide (DMA) was poured onto a polyester support (Freundemberg and Hollytex). The film formed was immersed in distilled water at 10 "C for 10 min. Afterwards the film was allowed to dry at ambient temperature. Figure 1 shows the two faces of the membrane A. The permeabilities to air of the support were 600 f 100 dm3 m-2 s-I at 2 mbar for membrane A and 800 f 200 dm3 m-2 s'' at 2 mbar for membrane B. The thickness of the membranes are 60 pm for membrane A and 76 pm for membrane B.
Results In order to observe the asymmetry potential we soaked the membranes in a solution of NaCl 1 M for 2 weeks. Then they were placed in the cell; the two compartments were filled with a NaCl solution of lo4 M. At the same time the Data Logger started electric potential measurements. After 48 h we replaced bulk solutions by new solutions of lo4 M. Figures 2 and 3 show these measurements in two membranes. We can observe that asymmetry potential follows a relaxation process for 20 h. The last variations can be considered changes in electric potential of the electrodes. This is shown in Figure 4 where the same electrodes are dipped in lo4 M solutions without any membrane between them; we observe that the electrodes were greatly affected by the daylight.I0 To explain the results given in Figures 2 and 3 we will discuss if they might be an effect from one of the three following causes: (IO) Plsterberg, N. 0.; Jensen, J. 1978, A32, 721.
B.;Ssrensen, T. S. Acta Chem. Scad.
The Journal of Physical Chemistry, Vol. 96, No. 6, 1992 2123
Asymmetry Potential in Inhomogeneous Membranes
g
8
.
:i.
....
0 20
0
60
40
80
100
t(hour)
Figure 2. Asymmetry potential vs time for membrane A. Datta Logger is programmed to take measurements at intervals of 5 s.
lo
1 .
0 :
4'
0.6
-
0.4
-
0.2
-
.,,..'.. ..... . ''. .,,..... , . , . , . , .... .... "m
0,o
0
10
20
30
40
1 (hour)
Figure 4. Asymmetry potential of the electrodes vs time (without membrane).
(i) reverse or facilitated flux; (ii) perturbations produced by the voltmeter; (iii) diffusion processes. First we assume that the relaxation of the asymmetry potential might be due to a reverse or facilitated flux. Takagi and Nakagaki6 assert that a certain tendency toward a reverse transport of electrolyte can be recognized in some experiments. Then a concentration difference between the compartments would arise. The membrane system leaves one of the conditions of asymmetry potential, Le., AM; = 0, and therefore the electric potential difference would decrease to zero due to this cause. In order to verify that the reverse flux does not affect the relaxation process we renewed the bulk solution when 48 h had elapsed, so that the new solutions had the same composition as that in the initial stage. As we can see in Figures 2 and 3 the asymmetry potential does not reach its initial value; therefore, the reverse flux, if any, does not affect the asymmetry potential. Now, we suppose that the relaxation process is a perturbation due to the voltmeter. Then the relaxation process would exclusively depend on the number of measurements. In order to check this possibility we carried out two experiments; the Data Logger was programmed to take measurements at intervals of 5 or 60 s. The relaxation process of Figure 2 would have to be 12 times faster than that of Figure 3. Therefore, as the two processes are very
similar, it can be concluded that the perturbations of the voltmeter do not have an appreciable effect. The next step is to examine the last possibility: a diffusion process occurring into the membrane. As we have said, initially the membrane was equilibrated in a solution of 1 M for 2 weeks. When the experiment started the membrane was placed in the cell with solutions of M. Doubtless a diffusion process is taking place. Once the membrane is equilibrated with the bulk solution M), the process will have finished. The time constants of the relaxations shown in Figure 2 and 3 have an order of magnitude typical of diffusion processes. Therefore, we think that this cause can explain the process. Later we will go deeper in this thesis. It is evident that the time constant of these relaxation processes will depend on the porosity of the membranes. The membrane studied in this work is of the type used in ultrafiltration processes.
Discussion In this section we are going to show a model of membrane with asymmetry potential. This model will be congruent with the results obtained in the last section. We will assume that a concentration profile has arisen due to the treatment which the membrane has been submitted to. This study will be developed within the frame of thermodynamics of irreversible processes. The variables of the formulation will be observable variable^.^ We do not consider TMS theory, where the electric potential difference is regarded as the sum of five parts: two Nernst electrode potentials, a diffusion potential in the membrane and two interfacial Donnan potentials. We assume that the membrane system meets the following conditions: (i) AP = Ap,C = Z = 0, Le., those in which asymmetry potential is observed. (ii) Pressure has a uniform value inside the membrane. This property, concerning mechanical equilibrium, is reached very fast: the relaxation time of this equilibrium is of the order of magnitude of s. (iii) The solution fills membrane pores. Due to the previous treatment in the membrane we have a nonequilibrium profile of the solution concentration c(x). This profile evolves to a distribution equilibrium: a constant value for c ( x ) . As the relaxation time of this process is of the order of magnitude of los s, a long period of time must be allowed for. (iv) The mean activity, a, depends exclusively on concentration. Then the activity of the solution in the membrane is a function of x . (v) The electrical potential difference in the membrane system AJ, is linearly related to the electrical potential difference in the elemental layers.I1J2 (vi) The difference in electrical potential between two microelectrodes placed at the boundaries of an elemental layer dx is9
where T~ is the apparent transference number of the ionic species 2 (Hittorfs reference). The value of r2 depends on solution concentration and membranesolution interaction. For inhomogeneous membranes in equilibrium with a fixed solution, T~ can vary with x in each layer, Le., 7 2 = 7 2 ( X ) . Nevertheless, for homogeneous membranes, r 2 has a uniform value. Now we have to evaluate the asymmetry potential for a membrane system. This quantity will be
(3) where Y = v l + u2. Kamo et aL13 predict that asymmetry potential given by eq 3 must be always zero because we are integrating between two identical solutions. Nevertheless, such a result is only valid for certain systems, for example, in the case of homogeneous membrane systems. In these systems the electric potential takes the ( 1 1) Staverman, A. J. Trans. Faraday SOC.1952, 48, 176. (12) Siegel, R. A. J . Phys. Chem. 1991, 95, 2556. (13) Kamo, N.; Kobatake, Y . J . Colloid Interface Sei. 1974, 46, 8 5
2124 The Journal of Physical Chemistry, Vol. 96, No. 6, 1992
Garrido and Compah
TABLE I: Asymmetry Potential for Different Profiles" b A&. mV b A$. mV 0 0.011 -0.7 740 -0.5 530 -0.8 850 -0.6 640 Oc(x) and Q ( X ) are given by IO-' M; ~ ~ (=0 0.513. )
eqs 4 and 5; n = 2 X
T 2 (X) 0.50
M;m = 5 X
0,30 0,007
TABLE II: Asymmetry Potential for Different Profiles" b Ai4 mV b A$, mV 0 8.6 X 3 7.3 1 2.4 4 9.8 2 4.9 " c ( x ) and X
T ~ ( x are )
given by eqs 4 and 5; n = 5
lo-' M; T 2 ( 0 ) = 0.098.
X
0,0021 -0,5
M;m = 4.98
T*(x)
are given by eqs 4 and 5; n = 2 X
M;b = -0.5;
~ ~ (=0 0.513. )
value of zero because 72and a exclusively depend on concentration c. But for inhomogeneous systems, as r2 also varies with the position x, the asymmetry potential can be different from zero, as we can see later. The next step is to solve eq 3 for different profiles c(x) and T ~ ( x ) to inhomogeneous membrane systems. We assume the following ad hoc profiles c(x) = m sin AX 72
fz(O)(1
+n
+ bx)
0 0.0
. 0,5
1 1,o
0 1.5
Figure 5. Concentration and apparent transference number profiles: m = 5 X lo-' M;n = 4.98 X IO-' M;T 2 ( 0 ) = 0.513; b = -0.5.
TABLE IV: Asymmetry Potential for Mfferent Profiles" m. lo-' M A&. mV 2.76 3.00 4.00 4.09 4.82 4.75
TABLE 111: Asymmetry Potential for Different Profiles" m, IO-' M A A mV 1 .oo 118 1.78 198 250 2.24 " c ( x ) and
0.012
(4) (5)
where m, n, r2(0),and 6 are parameters; n is the solute concentration in the subsystems, and rz(0)is the value of r2within the membrane in x = 0. The inhomogeneityof the membrane is given by the parameter 6. An example of these profiles is shown in Figure 5. For homogeneous membranes, b = 0. For 1:l electrolytes, eq 3 can be written as
where we have neglected the changes in mean activity coefficient. Tables I-IV show the values of A+ for different profiles. After examining these tables we can assert that (i) the asymmetry potential is zero for symmetric membranes (b = 0) and will be different from zero for asymmetric membranes (b # 0) and (ii) the asymmetry potential is also zero when the solution concentration into the membrane has a uniform value. Thus we deduce that for asymmetry potential to be different from zero (i) &/dx must be different from zero at least in a certain
" c ( x ) and T ~ ( x are ) given by eqs 4 and 5; n = 5 X 72(0) = 0.098.
IO-' M; b = 2;
interval of the membrane and (ii) the profile 72 = Q(X) must be asymmetric.
Conclusions Figures 2 and 3 show the time evolution of asymmetry potential. Now we can give some explanations of this behavior. The measurements started when the membrane was placed between 10" M solutions: a diffusion process took place. A nonuniform concentration profile a r w in the membrane and we could observe a nonzero asymmetry potential. The diffusion process smoothed down the concentration profile; then the asymmetry potential decreased. This nonequilibrium thermodynamics formulation gives a satisfactory explanation for the asymmetry potential. The temporal character of this effect is well assured: a concentration gradient is necessary. The inhomogeneity in the membrane is expressed by the profile of the apparent transference number of the cation. The basic laws of thermodynamics are not violated in this effect.I Nevertheless, we can find systems with asymmetry potential subjected to a stationary process. In fact we can build experimental devices where solution migrates within the membrane to its edges, where water is lost by evaporation.' Then a stationary concentration profile is permanently reached along the pores. In these systems, if inhomogeneous, we can observe a constant value of asymmetry potential. Acknowledgment. This work is part of a project of the DGICYT (Ministry of Education and Science of Spain) and has been partially sponsored by the Institucid Valenciana d'Estudis i Investigacid. We thank Dr. E. Soriano, Universidad Politknica de Valencia, for the preparation of the membranes.
