7830
J. Phys. Chem. B 2003, 107, 7830-7837
Role of Standard Chemical Potential in Transport through Anisotropic Media and Asymmetrical Membranes Nikolai Kocherginsky* and Yan Kun Zhang DiVision of Bioengineering and Department of Chemical and EnVironmental Engineering, National UniVersity of Singapore, 10 Kent Ridge Crescent, Singapore 119260 ReceiVed: NoVember 27, 2002; In Final Form: May 6, 2003
Description of mass transport in a media or through a membrane usually is based on the assumption that a standard chemical potential µ0 of a substance is constant and does not change with the distance in the media or membrane. This common assumption is not valid for asymmetrical or nonhomogeneous membranes. More accurate analysis of the transport, given in this paper, is based on one of the major principles of linear thermodynamics, according to which any flux is proportional to the gradient of electrochemical potential µ. It also takes care of possible changes of standard chemical potential µ0 in asymmetrical media or membrane. It is demonstrated that an asymmetrical media, but not a membrane separating two similar phases, can act as a diffusion-based diode. Still, even with the membranes, if both the standard chemical potential and activity gradients are negative, the flux is increased and the time lag is decreased. Time lag, which is usually used to calculate a diffusion coefficient, can be much less than the one on the basis of the Einstein-Barrer equation. Asymmetry also allows additional improvement of membrane selectivity. Transport through both the membrane interface and the inner part of asymmetrical membrane far from equilibrium even for nonelectrolytes can be described by equivalent circuit, which has three diodes, with one of them looking to the opposite direction. The developed model and equivalent circuit are reduced to the common concepts if the asymmetry of the membrane decreases.
where
1. Introduction It is well-known that in an ideal diluted homogeneous solution of one component the flux J (mol‚cm-2‚s-1) and concentration changes for a noncharged substance A in the x-direction are described by the first and second Fick’s laws:
J ) -D
dc dx
∂c ∂2c )D 2 ∂t ∂x
(1) (2)
The coefficient D (cm2‚s-1) is positive and is called diffusion coefficient. Diffusion is possible only from high to low concentration due to the concentration gradient in the direction x. When we are dealing with a membrane, usually we do not know concentration profiles along the membrane thickness L. The measurable values are the concentrations c1 and c2 in the bulk donor and acceptor volumes. As a result it is necessary to add boundary conditions at both membrane surfaces. The common approximation is an equilibrium distribution of molecules on both sides of the membrane, described by distribution constant K. In this case the steady-state flux is characterized by linear profile of concentration and is described by simple equation
J)-
KD (c - c1) L 2
(3)
* To whom correspondence should be addressed. E-mail: chenk@ nus.edu.sg. Phone: 65-68745083. Fax: 65-67791936.
K ) e-(∆µ0/RT)
(3a)
The characteristic time for the movement to the steady-state transport through homogeneous membrane after addition of a substance is called the time lag and is described by the EinsteinBarrer equation:1,2
τlag )
L2 6D
(4)
Development of modern membrane technology demonstrates that in many cases it is necessary to have more advanced model of transmembrane transport, taking into account the fact of membrane asymmetry. Examples of recent papers dealing with asymmetry of membranes can be found in nanofiltration,3 pervaporation,4 biomedical,5,6 and electrochemical applications.7,8 Asymmetry can be especially important for biomembranes, where both the composition and physicochemical properties are different on internal and external surfaces.9-11 There were some attempts to describe the role of membrane asymmetry with the additional assumption that either diffusion coefficient is not constant and is a function of a coordinate x or the K value is different at both surfaces.1,12,13 One of possible problems with some of these models is that, for example, due to the difference of K at the two surfaces, it is possible to create a gradient of concentration in a membrane even if the concentrations in both donor and acceptor solutions are the same. On the basis of the first Fick’s law, it means that simple difference of K should give the directed transport of a substance, which is physically impossible. In this case the work of directed
10.1021/jp027572l CCC: $25.00 © 2003 American Chemical Society Published on Web 07/17/2003
Role of Standard Chemical Potential in Transport
J. Phys. Chem. B, Vol. 107, No. 31, 2003 7831
transport is conducted without any external source of energy, which is equivalent to the “perpetual motion” machine. This problem was mentioned in our early paper14 and then by others in a paper.15 To our deep surprise, we did not find in the literature a simple but still rigorous thermodynamic description of the transmembrane transport of one noncharged species, which would take care of the membrane asymmetry. The membrane asymmetry in this paper means that at least some properties, including the standard electrochemical potential (µ0), are different at the first (x ) 0) and the second (x ) L) membrane surfaces. The membrane with a thin and selective skin layer is one of the extreme cases of this asymmetry. The description below is based on common assumptions of linear thermodynamics and the fact that in a nonhomogeneous media with properties depending on the spatial coordinate, the standard chemical potential µ0 of a substance is also not constant anymore. These changes of µ0 play a role of additional driving factor, leading to a directed transport in the media, but usually are not taken into account. A general description of the transport in the anisotropic media and then through the asymmetrical membrane will be given. A special situation when the standard chemical potential µ0 is a linear function of the distance in the asymmetrical membrane will be considered. Both the steady and nonsteady state conditions will be analyzed, and the time lag that is generally used to characterize the transition period will be considered. Finally it will be demonstrated that both the asymmetrical membrane and interface can be described on the basis of the equation similar to the equation for a junction diode. This results in new “rectifying” properties of transport of noncharged substances in an asymmetrical media but not for the transmembrane transport. 2. Model Development General Description and the Fick’s First Law. If the system center of gravity does not move, the flux J of a substance A in the direction x can be expressed as changes of activity by volume per area per time, and according to linear nonequilibrium thermodynamics, it is proportional to the product of activity and electrochemical potential gradient:
J ) -Ua
dµ dx
(5)
Usually this equation is used for diluted solutions, and in this case, it has concentrations instead of activity.16 The flux in this case has common units, mol‚cm-2‚s-1. In our model we neglect the effects of other components in the solution; i.e., we assume that we are dealing with monocomponent solution. The driving force is the gradient of µ, J mol-1 cm-1, which has the meaning of force/mol. U is called mobility and has the units cm mol s-1 N-1. The electrochemical potential of the substance A in isobaric and isothermal conditions is dependent on x and is given by
µ(x) ) µ0(x) + RT ln a(x) + zFψ
(6)
where µ0 is the standard chemical potential, z is the charge number, F is Faraday’s constant, a is activity, and ψ is electric potential. In the case of noncharged species, the last term is not important. For simplicity we will consider this case. Activity a at the point x can be expressed in terms of activity coefficient γ and concentration c:
a ) γc
(7)
It is easy to find that in a common situation of homogeneous phase, where µ0 is constant for all x and the activity coefficient is equal to unity (dilute solution) or at least constant, eq 5 can be reduced to the Fick’s first law. Equation 5 in combination with the eq 6 at constant µ0 and ψ gives known relationship of U with the diffusion coefficient D, which is often called Fokker-Einstein relation:
U ) D/RT
(8)
In a general case of nonhomogeneous media, µ0 is not constant, is dependent on the local media properties, and is a function of x. The reason for these changes could be, for example, the change in polarity and intermolecular interactions. The value of µ0 for each point could be calculated on the basis of, for example, experimental “local” distribution constants between this point in a media and homogeneous surroundings. In all papers known to us, especially for a gas-phase transport processes, µ0 is assumed constant and intermolecular interactions are taken care of by activity coefficients. In the case of liquid or solid nonhomogeneous media, intermolecular interactions are different from point to point even for very diluted solutions, when activity coefficient should be equal to unity, and these effects should be described by changes of µ0. Substituting eq 6 into the eq 5, we have
J(x) ) -D(x)
[
]
da(x) a(x) dµo(x) + dx RT dx
(9)
i.e., transport of nonelectrolyte in nonhomogeneous media is possible as the result of both the activity gradient (diffusion) and the variation of standard chemical potential from one point to another. Equation 9 is similar to the situation for transport of ions in the presence of external electric field in homogeneous media,17 or diffusion in atmosphere in the presence of gravity field.18 As it is often the case in linear thermodynamics, the solute flux is a linear combination of the two terms proportional to different driving factors. In this case we prefer to use the term driving factor instead of driving force because, for example, the gradient of activity does not have units of force. The second process is a drift in the field of potential energy, and its rate is proportional to the activity. This process becomes dominant if the activity gradient in the system is small:
-
∆µ0 a1 - a2 > RT a2
(10)
Diffusion Resistance in Anisotropic Media. Let us consider the diffusion in a plane slab of anisotropic media. Activity in the slab a(x) is a function of x. J is the same for all points for a steady-state transport though the slab. Solution of the eq 9 in this case should have the form19
a(x) ) A(x) exp[-µ0(x)/RT]
(11)
Evidently for the two surfaces of the slab, we have
A(0) ) a(0) exp[µ0(0)/RT]
(12)
A(L) ) a(L) exp[µ0(L)/RT]
(13)
After substitution and integration of eq 9 from 0 to L, we can
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get the activity profile and then the equation for Js at the steady state.14,17 Omitting the intermediate steps, we have
Js ) -
a(L) exp[µ0(L)/RT] - a(0) exp[µ0(0)/RT]
∫0
L
exp[µ0(x)/RT] D(x)
(14)
dx
At equilibrium Js is equal to zero, and we have that the equilibrium activities ae in the media are described by the Gibbs law:
ae(L) ∆µ0 ln )RT ae(0)
(15)
On the basis of eqs 6 and 14, we have that the nominator for transport in any nonhomogeneous media is determined by the difference exp[µ(0)/RT] - exp[µ(L)/RT]. As usual it depends only on the properties in the initial or final points and is not dependent on the way µ0 changes with x. Still the exact value of Js will be sensitive to the detailed properties of the media due to their influence on the denominator, which represents the resistance to the transport. It is important that if we swap the sides 0 and L, the flux will change not only the sign but also the value. In other words the same difference of activities gives different value of Js, depending on orientation, and in this sense we have an effect that can be called diffusion rectification. Steady-State Transport in the Media, when µ0 is a Linear Function of x. As a specific and simple but important example of transport through nonhomogeneous media, we can consider the case when µ0 is a linear function of x:
µ0(x) ) µ0(0)(1 + βx)
(16)
where µ0 (0) and β (cm-1) are constants. From a mathematical point of view, this case is similar to the Goldman model for the ion transport in constant gradient of electric field.17 It is also similar to the simultaneous steady-state diffusion and pressure-driven transport through pores.20 For simplicity we assume that viscosity or D is also constant for different x. This factor was discussed earlier in the literature1,2 and is not considered here. Using dimensionless parameter
∆µ0 ω) RT
(17)
instead of eq 9, we have
Js ) -D
da ωD a dx L
(18)
Equation 18 differs from usual Fickian relation by the second term. After integration from x ) 0 to x ) L, we have the expression for the activity profile and also the expression for the steady-state flux Js, which are
as(x/L) ) [a(0) - a(L) exp(ω)] - [a(0) - a(L)] exp[ω(1 - x/L)] 1 - exp(ω) (19) and
ωD a(0) - a(L) exp(ω) Js ) L 1 - exp(ω)
Figure 1. Dimensionless activity profile inside anisotropic media for different values of ω under the boundary condition of R(1) ) a(L)/ a(0) ) 0.5. Linear activity profile corresponds to the homogeneous membrane and results in the Fick’s law.
(20)
Figure 2. Dimensionless steady-state flux Φs inside anisotropic slab as a function of ω with different ratio of concentrations at both ends, R(1) ) a(L)/a(0). 1, R(1) ) 0; 2, R(1) ) 1; 3, R(1) ) 2; 4, R(1) ) 5.
It is easy to derive this solution also directly from eq 14. Steadystate activity profile can be either convex or concave, depending on the sign of ω. As before, if ω is small (almost homogeneous media), the concentration profile is reduced to a linear function of x (Figure 1), and the flux is described by the Fick’s first law. In the opposite situation, the asymmetry is playing the dominant role. If ω is a large negative value, the flux is much higher than in the case of simple diffusion and is determined by the species movement from a point with high to low standard chemical potential (Figure 2):
Js ) -
∆µ0 D a(0) RT L
(21)
If ω is a large positive value, the flux has the opposite direction, i.e., from low to high concentration and is
Js )
∆µ0 D a(L) RT L
(22)
These two situations correspond to the downhill and uphill mechanical movement. On the basis of the eq 20, the steadystate flux can be described as the difference between unidirec-
Role of Standard Chemical Potential in Transport
J. Phys. Chem. B, Vol. 107, No. 31, 2003 7833 membrane are the same (same solvent) and using them as a reference state, where µ0 is equal to 0 (Figure 4), we have
[ [
] ]
a(0) ) a1 exp -
µ0(0) RT
a(L) ) a2 exp -
µ0(L) RT
(26)
On the basis of the eq 20 and assuming that D is constant, we have
Js ) Figure 3. Dimensionless steady-state flux Φs as a function of a total driving factor W in the dimensionless form. R(1) is 0.0001, 1, 2, and 10 for curves 1-4, respectively.
tional fluxes Js ) B J - A J, and their ratio is [a(0)]/[a(L)] exp(-ω). Using this equation and radioactive indicators, it is possible to determine the value of ∆µ0 in experiments and thus to characterize the anisotropy of a media. To simplify the analysis, we could use dimensionless forms: R(δ) ) a(δ)/a(0) R(1) ) a(L)/a(0) δ ) x/L ω ) βµ0(0)L/(RT) ) ∆µ0/(RT) Φs ) JsL/[Da(0)]
dimensionless activity at δ position dimensionless activity at x ) L dimensionless distance dimensionless chemical potential difference dimensionless steady-state flux
In this case the dimensionless equations for the flux and activity are
Φs ) R(δ) )
ω(φeω - 1) 1-e
ω
1 - φeω + (φ - 1)eω(1-δ) 1 - eω
(23)
(24)
It makes sense to discuss the steady-state flux as a function of total driving factor, which is electrochemical potential gradient. Dimensionless parameter W can be determined as
W)
∆µ ) ω + ln R RT
(25)
Though the value of electrochemical potential difference is dependent on activities, still there is a possibility to change µ by changing only µ0 without changing local activities. The dimensionless steady-state flux as a function of W is plotted in the Figure 3 for different R(1). All dependences go through the origin of coordinates. The flux is proportional to the driving factor if a(L) ) a(0) (curve 2) but behaves as a diode if activities are different on the opposite sides of the slab. In the case when a substance was added from one side of a slab (R(1) is small, curve 1), the process is relatively slow, and it is determined by diffusion if W is near 0. High positive W values are not efficient because there is no substance on opposite side, but high negative values of W can accelerate the process by many times. Steady-State Transport through a Membrane, when µ0 Is a Linear Function of x. In the case with transmembrane transport, we should concede also the processes on the surfaces. We can use the common approximation that there is pseudoequilibrium at both surfaces. Assuming also that the standard chemical potentials of the bulk phases on both sides of the
D (a - a1) L(1/K)LM 2
(27)
where the logarithmic mean of 1/K is
(1/K)LM )
1/K0 - 1/KL
(28)
ln(KL/K0)
Equation 27 demonstrates that exponential terms and parameter ω disappear and the direct flux through an asymmetric membrane is proportional to the simple activity difference in the donor and acceptor phases and is not direction dependent; i.e., it does not change the value after the changes of position of the first and second bulk solutions. Evidently if activity in both solutions is the same, there is no directed transport. Instead of one distribution coefficient (eq 3) in this case, we have to use the reciprocal of the logarithmic mean of 1/K for the two membrane surfaces. For symmetrical membrane, when both sides are characterized by the same K and µ0, eq 27 is reduced to the well-known expression
J)-
KD(a2 - a1) L
(29)
If KL is bigger than K0, we have an increase of Js in comparison to the membrane with one value of distribution coefficient, equal to K0. Increase of K even at the acceptor side with constant K at the donor side results in the increase of transmembrane flux. As long as two different substances in the same asymmetrical membrane have different values of µ0 at both surfaces, we have an additional way to regulate membrane selectivity. Similar mechanisms can be important in pervaporation, where a high concentration of one penetrating substance changes the local properties of a membrane, including local activity coefficients. This is an additional factor, similar to the changes of standard chemical potential and determining the flux of a second substance and also the membrane selectivity. More general equation when µ0 is an arbitrary function of x can be derived on the basis of the eqs 14 and 26. In this case for constant D we have
Js ) -
D(a2 - a1)
(30)
∫0 exp[µ0(x)/RT] dx L
Unsteady-State Transport and the Fick’s Second Law. In the unsteady-state conditions, if U is constant based on eq 5, we have
( ) [
]
∂a ∂µ ∂J ∂ ∂µ ∂2µ ∂a )- )U +a 2 a )U ∂t ∂x ∂x ∂x ∂x ∂x ∂x
(31)
This is more general equation than the Fick’s second law, described by the eq 2.
7834 J. Phys. Chem. B, Vol. 107, No. 31, 2003
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Figure 4. Linear profile of the standard chemical potential µ0 (a) and corresponding activity profile (b) in a plain membrane, separating two bulk phases with the same µ0 ) 0.
Unsteady-State Transport and Time Lag when µ0 is a Linear Function of x. This case, together with eq 6, results in a drift of nonelectrolyte simultaneously with diffusion and is again similar to the well-known equation for diffusion and advection:
∂2a ωD ∂a ∂a - )0 + L ∂x ∂t ∂x2
(32)
D
Evidently this equation is reduced to the Fick’s second law if ω ) 0. Using the dimensionless time θ ) tD/L2 and distance δ, we can write the above equation in a dimensionless form
∂R(δ,θ) ∂R(δ,θ) ∂2R(δ,θ) )ω + ∂θ ∂δ ∂δ2
(33)
If the membrane is initially empty, R(δ,0) ) 0 and δ ∈ (0,1). The boundary conditions with the substance on both sides can be expressed as
R(0,θ) ) 1, R(1,θ) ) R(1)
R(δ,θ) ) Rs(δ) + e
∞
∑BnCn
(35)
n)1
where the first term on the right-hand side, as(δ), is the steadystate solution for the given boundary conditions (see the eq 19). In this case -n2π2θ
Cn ) e
sin(nπδ)
(36)
and Bn is the Fourier coefficient, which does not depend on time or position:
Bn )
8nπ[(-1)nR(1)eω/2 - 1] 4n2π2 + ω2
The total amount of the substance, which achieved the opposite side for unit surface area, can be calculated as
M(t) )
(34)
Using Fourier series expansion, we have the solution of eqs 33 and 34 [21]: -{[(ωδ)/2]+[(ω2θ)/4]}
Figure 5. Dimensionless time lag θlag as a function of the dimensionless standard chemical potential difference ω when R(1) ) 0.
(37)
To find the time lag, we must calculate the flux J through the surface at x ) L and then the amount of a substance that has gone into the acceptor phase as a function of time.
∫0t J(L,t˜) dt˜
(38)
where J is the flux through the surface L. As in the simple case of transport through homogeneous membrane,1 this function can be extrapolated by a straight line at high values of time ,and its intercept with the x-axis gives the time lag. As a result, the dimensionless time lag θlag is a function of ω and also the ratio of concentrations on both sides R(1):
64π2 sinh θlag )
() ω 2
∞
∑
ω[R(1)eω - 1]n)1
n2[(-1)n - R(1)eω/2] (4n2π2 + ω2)2
(39)
The plot for the situation when a substances is added only on one side of the membrane (R(1) ) 0) is shown in the Figure 5. In this case if ω ) 0 (membrane is symmetrical), θlag ) 1/6, which agrees with the classical eq 4. In opposite situation when the membrane is asymmetrical and is described by the eq 16, the time lag is proportional to L2, but the proportionality coefficient is smaller due to the action of a second factor in addition to diffusion. In general it can be not only the gradient of chemical potential, but also the gradient of electrical field, of pressure, etc. The plot is symmetrical with respect to ω ) 0,
Role of Standard Chemical Potential in Transport
Figure 6. Activity distribution inside asymmetric slab during transition period when R(1) ) 0.5, ω ) -5. The corresponding times for the lines from left to right were θ ) 1/64, 1/32, 1/16, 1/12, 1/8, 1/6, 0.3, 1, respectively.
Figure 7. Dimensionless mass transfer flux at δ ) 1 as a function of time during transition period when R(1) ) 0.5, ω ) -5.
meaning that θlag depends only on the absolute value of ∆µ0/RT, and the change of its sign does not change the value of time lag. A special effect is observed for asymmetrical membrane if the substance is added on both sides and R(1) is not zero anymore. Figures 6, 7, and 8 show as an example the changes in activity, flux, and total mass transferred inside asymmetric membrane during transition period when R(1) ) 0.5 and ω ) -5. Results in Figure 6 demonstrate that when R(1) is not zero, at the very beginning of transition period mass transfer flux through the surface x ) L is negative (Figure 7). It changes from negative to positive (Figure 8) only with time. The effect of chemical potential on the time lag under different boundary conditions is shown in Figure 9. In a symmetric membrane (ω ) 0), when a substance is added only in one phase (curve 1, R(1) ) 0), as usual we have θlag ) 1/6 (same as Figure 5 and eq 4). If the substance is present in both phases outside the membrane, R(1) > 0 and the value of the time lag is increased. In an asymmetric membrane, the time lag decreases and approaches to zero when the absolute value of w becomes high enough. In the special situation when R(1)eω ) 1, the time lag is infinitely high and the steady-state flux is zero (see also Figure 1). 3. Discussion We have discussed the direct transport of nonelectrolyte in an anisotropic media and then through the asymmetrical membrane, where the standard chemical potential is not constant
J. Phys. Chem. B, Vol. 107, No. 31, 2003 7835
Figure 8. Total dimensionless mass transferred out of the interface at δ ) 1 as a function of time when R(1) ) 0.5, ω ) -5.
Figure 9. Dimensionless time lag θlag as a function of the dimensionless standard chemical potentials difference ω in the asymmetric membrane: 1, R(1) ) 0; 2, R(1) ) 0.25; 3, R(1) ) 1.
from point to point. The asymmetry can be created not only in the solid, but also in a liquid media. One of the well-known examples is so-called gradient mixer, where two different liquid phases are connected by a narrow channel, which allows formation of pH, density, and some other gradients and is widely used in bioseparations.22 The same method can be used to make a gradient of polarity, which could result in separation of molecules based on their hydrophobicity. One of the ways to regulate the membrane asymmetry is, for example, to add a substance, interacting with the membrane components, only into one of the solutions. It is known that in the cell biology different compositions of the solutions surrounded by a membrane and outside of the membrane will create and maintain the membrane asymmetry, thus participating in the biogenesis. Asymmetrical distribution of Ca ions and the membrane curvature are playing an essential role in regulation of biomembrane transport.11 The essential difference between the asymmetrical and symmetrical membrane is the resistance to the mass transport, which is not equal to L/KD anymore. In this case it is necessary to use the logarithmic average of the two values 1/K at both surfaces. As a result the increase of the distribution coefficient even at the surface in contact with the acceptor phase makes the asymmetrical membrane with the linear profile of µ0 more permeable than usual. In the typical situation, the substance is present only on one side of the membrane and is removed from another side. In this case R(1) ) 0, and based on Figure 2, the decrease of ω from 0 to -2, which corresponds to ∆µ0, only 2RT and easy to do, will give the increase of the transmembrane flux by 2.3 times. It means that the asymmetrical membranes are more
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attractive in the separation processes and could be used in the bioseparations, where the surface in contact with the biological donor media should be first of all biocompatible, while the properties of another surface could be easily changed. Another specific aspect of the asymmetrical membranes is the change of the coefficient necessary for calculation of the time lag, which characterizes the rate of approaching to the steady state. In the case of asymmetrical membrane, this time is shorter due to the influence of an additional factor, which is the gradient of standard chemical potential. One of the reasons that the experimental value of diffusion coefficient determined in the steady-state measurements is sometimes not equal to the value determined in the nonsteady-state conditions23 could be the asymmetry of the membrane. It is important to remember that τlag for transport through asymmetric membranes without knowledge of asymmetry parameter ω does not give correct value of D. Total flux in the anisotropic slab can be characterized by nonlinear behavior similar to the one observed in the p-n junction diodes. Addition of the two interfaces in the case of membranes results in simpler behavior of the flux. For direct diffusion through an asymmetrical membrane, the flux is proportional to the difference of activities between the donor and acceptor phases, which is similar to the first Fick’s law. In the case of direct diffusion due to the linearity of transport equations and pseudoequilibrium boundary conditions on a membrane, it is impossible to make a “diffusion-based” diode, where the rate of transmembrane transport is dependent on the orientation of applied difference of activity. In a more complex situation, where penetration into the membrane is accompanied by chemical reactions, it is possible to have an active transport of a substance. One of the wellknown examples is the so-called coupled facilitated transport of ions. As an illustration we can mention liquid membranes with hydrophobic fatty acids as monovalent cation carriers.24 In this case it is possible to observe an active transport of K+ into more concentrated salt solution, if this solution has higher H+ concentration than the feed. The H+ concentration difference in this case serves as a source of energy and induces the asymmetrical distribution of protonated and K+-form of the carrier in a membrane. As a result it seems that potassium permeability through the membrane is different in different directions. If the boundary conditions are different in comparison to the eq 26, i.e., for example, there is no equilibrium through the interface, initially we would have transport even though the activities are the same in both solutions, but this process will stop as soon as local equilibrium for each point in the membrane is reached. If the membrane separates two different phases, for example aqueous and organic phases, where the transported species have different µ0, the process will stop when electrochemical potential of the substance is the same in both phases. Evidently concentrations and activities of the penetrating substance will be different in these media, and the driving factor is not a simple activity difference but the difference of electrochemical potentials, ∆µ. Similarity of the electrophysical behavior of the nerve cell membrane and diodes was demonstrated in experiments and is well-known.20 Now we would like to discuss the electric analogue for the transport in anisotropic media. We can write the eq 20 for J as
[
a(L) J s ) J0 1 exp(ω) a(0)
]
(40)
Figure 10. Diode-based equivalent circuit for nonelectrolyte transport through a thin asymmetric membrane with two interfaces.
where
J0 ) -
a(0) ωD L 1 - exp(ω)
(41)
The role of changes of standard chemical potential µ0 is similar to the effects of electric field (see the eq 6), and the denominator of eq 11 is similar to the nonlinear resistance in the electric circuits. In the extreme case when ω is negligibly small, asymmetric (rectifying) properties disappear, which is similar to the behavior of Schottky diode.25 The same asymmetry (decrease of the standard chemical potential) directed from the feed to the strip (“forward biased” diode) will increase the flux, while orientation in the opposite direction (“reverse biased diode”) will practically stop it. Similarity of the effects due to gradients of electric voltage and µ0 is not surprising. The electrochemical potential is a linear function of these, and other intensive parameters such as pressure and temperature and application of a gradient of any of these factors together with concentration gradient should result in rectification effects. Nevertheless the case of nonelectrolyte transport through asymmetrical membranes is slightly different because the gradient of µ0 is available only in the membrane, while electric field, pressure, and temperature gradients are applied through the membrane to the donor and acceptor phases. Because of this difference and the effect of boundary conditions, the asymmetrical membrane loses its rectifying properties when we talk about diffusion. Finally we would like to note that the interface of a membrane can be considered not as an idealized flat surface but as a thin transitional layer, and it means its behavior also could be described as an analogue of a junction diode. In this case the whole membrane system can be presented as a series of three diodes, one of them connected in opposite direction to the two others (Figure 10). If the membrane is symmetrical, the internal part of it behaves like a simple resistance, which is commonly used in the equivalent electric circuits for description of membrane transport.20 Usually in the experiments with artificial membranes, the interface resistance to mass transport is much lower than that of the membrane and can be neglected.2 Still, this assumption cannot be always correct. Recently we have demonstrated that the interface resistance plays essential role for exit of nitroxide radicals from biomimetic membrane26 and for transport of H+ and electrons through electroconductive polyaniline membranes.27 If the membrane is very thin (common situation in biology) and transport through the membrane is fast, the assumption of low interface resistance cannot be valid anymore, and more accurate description should be based on the equivalent circuit with diodes, describing the changes of standard chemical potential through the interface. Acknowledgment. Help of N.K.’s former student Feng Ji and other students who had taken my Membrane Science course at NUS is highly acknowledged.
Role of Standard Chemical Potential in Transport List of Notations J D L K τlag U µ µ0 ψ R z F T γ a ae c c1, c2 β ω R(x) ) a(x)/a(0) R(1) ) a (L)/a(0) δ ) x/L ω ) βµ0(0)L/(RT) ) ∆µ0/(RT) W ) ∆µ/(RT) ) ω + ln R θlag Φs ) JsL/[Da(0)] Cn ) e-n2π2θsin (nπδ) Bn ) {8nπ[(- 1)nφeω/2 - 1]}/[4n2π2 + ω2]
flux (mol‚cm ‚s ) diffusion coefficient (cm2‚s-1) membrane thickness distribution constant time lag mobility (cm/s)/(N/mol). electrochemical potential standard chemical potential electric potential gas constant charge number Faraday’s constant absolute temperature, K activity coefficient activity activity at equilibrium concentration concentrations in the bulk donor (1) and acceptor (2) volumes constant (cm-1), characteristic of the gradient of standard potential dimensionless standard chemical potential dimensionless concentration at x position dimensionless concentration at x ) L dimensionless distance dimensionless standard chemical potential difference dimensionless chemical potential difference -2
-1
dimensionless time lag dimensionless steady-state flux coefficient Fourier coefficient
References and Notes (1) Crank J. The Mathematics of Diffusion, 2nd ed.; Oxford University Press: Oxford, 1975.
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