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A mechanistic model of transport processes in porous membranes generated by osmotic and hydrostatic pressure
Journal of Membrane Science 191 (2001) 61–69
A mechanistic model of transport processes in porous membranes generated by osmotic and hydrostatic pressure Armin Kargol∗ Department of Physics, Tulane University, New Orleans, LA 70118, USA Received 27 October 2000; received in revised form 11 April 2001; accepted 11 April 2001
Abstract This work presents a simple molecular model of membrane transport processes generated by hydrostatic (P ) and osmotic (Π ) pressure. We postulate that the porous membrane permeability is determined by the size and distribution of its pores. From this postulate and a mechanistic interpretation of the flows we obtain equations for the volume flux Jv , the solute volume flux Jvs , and the solvent volume flux Jvw : Jv = Lp P − Lp σ Π Jvs = (1 − σ )c¯v¯s Lp (Π + P ) Jvw = Jv − (1 − σ )c¯v¯s Lp (Π + P ) We also observe that the porous membrane properties are described by two phenomenological parameters (Lp and σ ), as opposed to three parameters (Lp , σ , and ω) in the standard Kedem–Katchalsky formalism. The extra parameter was eliminated using the correlation relation for the transport parameters found previously and modified in this work. It has a form: ω = (1 − σ )cL ¯ p © 2001 Elsevier Science B.V. All rights reserved. Keywords: Membrane transport; Theory; Porous membranes
1. Introduction Passive transport of nonelectrolytic solutions in biological and artificial membranes includes several phenomena, most notably osmosis, filtration, diffusion, and convection. For the purpose of this work, we use a somewhat restricted definition of osmo∗
´ etokrzyska Academy, On leave from: Physics Institute, Swi¸ 25-406 Kielce, Poland. Fax: +1-504-862-8702. E-mail address: [email protected] (A. Kargol).
sis: the solvent diffusion through a semi-permeable membrane, i.e. one that forms an ideal barrier to the solute molecules. These transport phenomena can be driven by hydrostatic pressure, osmotic pressure, or both. In a general case, a quantitative technique has been proposed four decades ago by Kedem and Katchalsky [1–4]. The cornerstone of their method are the so-called Kedem–Katchalsky practical transport equations derived by means of thermodynamics of irreversible processes. In their standard form the Kedem–Katchalsky equations represent the linearized
0376-7388/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 6 - 7 3 8 8 ( 0 1 ) 0 0 4 5 0 - 1
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A. Kargol / Journal of Membrane Science 191 (2001) 61–69
permeation coefficient depends on the solute concentration c¯ according to the following [1,3]:
Nomenclature c1 , c2 jd jk js Jv , Jva , Jvb Jvsb Jvwa , Jvwb Lp P1 , P2
concentrations (mol/m3 ) diffusive solute flux (mol/m2 s) convective solute flux (mol/m2 s) solute flux (mol/m2 s) volume fluxes (m/s) solute volume flux (m/s) solvent volume fluxes (m/s) filtration coefficient (m3 /N s) hydrostatic pressures (Pa)
ω=
Greek symbols Π osmotic pressure (Pa) σ reflection coefficient ω, ω permeation coefficient (mol/N s) force-flow relationships, i.e. express the volume and solute fluxes in terms of the driving forces: Jv = Lp P − Lp σ Π
(1)
js = ω Π + (1 − σ )cL ¯ p (P − σ Π )
(2)
where Jv , js are the fluxes, and c¯ is the average solute concentration. The volume flux Jv is defined as Jv = jw v¯w + js v¯s = Jvw + Jvs
(3)
where jw , js are the solvent and solute fluxes, v¯w , v¯s the partial molar volumes of the solvent and the solute, and Jvw , Jvs the solvent and solute volume fluxes. The membrane properties are described by the phenomenological coefficients, Lp , σ , and ω, called the coefficients of filtration, reflection and permeation, respectively. They are defined as Jv (4) Lp = P Π=0 P σ = (5) Π Jv =0 js ω= (6) Π Jv =0 The above equations besides specifying the physical meaning of each parameter also to some extent determine their measurement techniques [1,4,5]. The
(Lp LD − L2pD )c¯ Lp
(7)
where LpD = (Jv /Π )P =0 and LD = (JD / Π )P =0 ; JD is called the diffusion flux. The Kedem–Katchalsky technique has become one of the main tools in both scientific and technological studies involving transport across biological and artificial membranes. However, since the early days there have been many attempts at generalization and modification of the formalism to address both the question of phenomena not covered by the standard method (e.g. the effect of boundary layers) and certain questions concerning physical principles governing membrane transport [6–17]. This is also the subject of the present work. Our technique in this paper is however different from Kedem and Katchalsky’s. Rather than using thermodynamics tools we are interested in a molecular, mechanistic approach. We present a model of a porous membrane and describe the transport processes in such a membrane starting from few fairly natural assumptions and a mechanistic interpretation of the flows. Of course, such a model has its limitations, however we believe it adequately describes porous membranes. The method is based on a postulate that in a porous membrane the size of pores determines permeability to a given solute. Among all pores present in a membrane, a certain number na has radii sufficiently small to become totally impermeable to the solute molecules while the remaining nb pores (na +nb = n) are ideally permeable, i.e. allow movement of both solute and solvent molecules. Extending the notion of the reflection coefficient to a patch of a membrane, or even a single pore, we postulate that a single membrane pore has the reflection coefficient for a given solute equal to either 1 or 0. Inspired by the Kedem–Katchalsky formalism we derive equations for: the volume flux Jv , the solute flux js , and the solvent volume flux Jvw . The three phenomenological transport parameters introduced by Kedem and Katchalsky (cf. Eqs. (4)–(6)), i.e. the parameters of filtration, reflection, and permeation are redefined to reflect a somewhat different form of these transport equations.
A. Kargol / Journal of Membrane Science 191 (2001) 61–69
In the standard Kedem–Katchalsky method three independent membrane transport parameters (Lp , σ and ω) are introduced, however as previously noted [18] the available experimental data strongly suggest some form of correlation among them. In [18] a correlation relation has been obtained within the framework of the Kedem–Katchalsky equations using standard curve fitting techniques albeit some simplifications had to be made (we argued that certain terms are typically small and can be neglected). Here, we modify this correlation relation based on the mechanistic equations and as we show no approximations are needed. The significance of this observed correlation (which we derive from the mechanistic equations for porous membranes but as we mentioned earlier the experimental data available to us strongly suggest that it is a more general phenomenon) is that it allows elimination of one of the parameters, preferably the one that is the hardest to measure, i.e. ω, from the description of the flows. Accordingly, the equations for the volume, solute and solvent fluxes are written in a reduced form expressing this correlation. We emphasize that the correlation relation in a general case is a phenomenological observation and it relies on curve fitting to experimental data, but it can be derived within our mechanistic model for porous membranes. Overall, in this paper we obtain equations that are in many aspects similar to the KK equations but also differ at certain points. We believe that our approach which is a continuation of earlier work [19] gives new insight into membrane transport mechanisms and is also quite intuitive.
2. Analysis and interpretation — transport equations 2.1. Porous membrane — an equivalent membrane model We consider transport processes across porous membranes driven by hydrostatic and osmotic pressure gradients. A typical sieve-type membrane has some number n of pores with random radii, from very small to a certain maximal value. The pores are distributed randomly in the entire membrane area. Examples of such membranes are: cellophane, colloid,
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and nephrophane membranes. As we mentioned in the introduction we extend the concept of the reflection coefficient for a membrane to a small patch of the membrane or even individual pores. We postulate that pore properties are determined by their sizes. For a given solute a fraction of pores allow free movement of solute molecules while the others form an ideal barrier due to their smallness compared to the solute molecule size. In terms of the reflection coefficient we postulate that its value for a single pore is either 0 or 1, and not fractional. This reflects the fact that a single pore can be either permeable, meaning equally permeable to both the solvent and the solute, or semipermeable that is permeable to the solvent while totally impermeable to the solute. The reflection coefficient of the entire membrane (a complex of n pores) depends on the proportion of permeable to impermeable pores. In other words, it equals zero if all of its pores are permeable and 1 if all the pores are impermeable to the solute. If there are both types of pores (solute permeable and impermeable) present, the reflection coefficient of the entire membrane has a fractional value 0 < σ < 1. Having this in mind, we consider an equivalent model membrane, such as in Fig. 1, where all the pores are spatially separated according to size. For a given solute a part of the membrane (part (a)) has pores that are impermeable to the solute, while the other part of the membrane (part (b)) has only permeable pores. The two parts of the membrane have reflection coefficients σa = 1 and σb = 0. The filtration coefficients for both parts of the membrane are denoted Lpa and Lpb . In the following sections such an equivalent membrane is used to analyze hydrostatically and osmotically driven transport. 2.2. Filtration and reflection coefficients — volume transport equation The membrane in Fig. 1 separates well-mixed solutions of concentrations c1 and c2 (c1 < c2 ) subject to hydrostatic pressures P1 and P2 (P1 < P2 ). This represents a generic situation with both the osmotic Π = RT c = RT(c2 − c1 ) (where R is the gas constant and T is the absolute temperature), and hydrostatic P = P2 − P1 pressures (driving forces) across the membrane. The pressure gradients generate volume fluxes Jva and Jvb in parts (a) and (b),
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A. Kargol / Journal of Membrane Science 191 (2001) 61–69
Fig. 1. The equivalent membrane model system (M–membrane with the semipermeable (a) and permeable (b) parts). Volume (Jva , Jvb , Jvwb ) and solute (jd , jk , js ) fluxes as described in text. The concentration and pressure gradient directions marked with inequality signs.
respectively, of the membrane. The fluxes can be described as Jva = Lpa P − Lpa Π
(8)
Jvb = Lpb P
(9)
since σb = 0. Given the pressure gradients the flux Jvb flows to the right, while the direction of Jva depends on the relative strength of the driving forces. In Fig. 1, we assumed that the osmotic pressure dominates. The total volume flux across the membrane is the net of the two: Jv = Jva + Jvb
(10)
We use a sign convention where fluxes directed to the right are positive. Given osmotic pressure Π the total volume flux vanishes if the hydrostatic pressure P = Pn equals Lpa Π Pn = Lpa + Lpb Lpa Π Lp
Lpa Lpa = Lpa + Lpb Lp
(12)
This equation can also be written as P σ = Π Jv =0
(13)
which is the same as the standard definition of the reflection coefficient (Eq. (5)). On the other hand according to Eq. (11) the coefficient σ is a ratio of the filtration coefficient of the part of the membrane containing impermeable pores to the total filtration coefficient Lp . That gives the reflection coefficient a new mechanistic interpretation. For instance, if Lpa = 0 then σ = 0, or if Lpb = 0 then σ = 1. Eq. (11) can also be written as Lpa = σ Lp
(14)
Lpb = (1 − σ )Lp
(15)
Jv = Jva + Jvb
where Lp = Lpa + Lpb is the total filtration coefficient of membrane M. Denoting: σ =
Pn = σ Π
When the hydrostatic pressure is different from the reversal value Pn then there is a nonzero net flux across the membrane and given the driving forces P and Π it can be written as
or Pn =
we get the relation:
= (Lpa + Lpb ) P −
Lpa (Lpa + Lpb ) Π Lpa + Lpb
or, taking Eq. (11) into account: (11)
Jv = Lp P − Lp σ Π
(16)
A. Kargol / Journal of Membrane Science 191 (2001) 61–69
i.e. in a form identical with Eq. (1), the first Kedem–Katchalsky equation.
This volume flux has two components: the solvent volume flux (Jvwb ) and the solute volume flux (Jvsb ):
In the previous section the equivalent membrane with various size pores was used to derive the first of the Kedem–Katchalsky equations for the volume transport. As one might expect the pore size has even greater effect on the solute transport since in parts of the equivalent membrane such transport does not occur at all due to the pore smallness. Starting from similar microscopic assumptions a macroscopic equation for the solute transport is derived in this section. Let us recall the second Kedem–Katchalsky equation (Eq. (2)) which shows the total solute flux as a sum of diffusive flux (diffusion due to concentration gradient) and the convective flux (i.e. the solute convection in the volume flux): (17)
The Kedem–Katchalsky formulas for the two fluxes are as follows: jd = ω Π
(18)
Jvb = Jvwb + Jvsb > 0
(20)
These two fluxes represent the two phenomena generically occurring on the membrane: the filtration (forced by hydrostatic pressure P ) and diffusion (since Π = 0). For the moment we consider a simplified case where P = 0. Due to concentration difference between both sides of the membrane we observe solute and solvent diffusion in opposite directions. In such case Eq. (20) reads: Π Π Π Jvb = Jvwb + Jvsb =0 Π is the diffusive solvent volume flux and where Jvwb Π Jvsb the diffusive solute volume flux. The superscript Π emphasizes the fact that we assumed the case of fluxes driven by the osmotic pressure only. With the concentration gradient directed as in Fig. 1, both Π < 0, J Π > 0) and can be fluxes are nonzero (Jvwb vsb expressed as Π Jvwb = −Lpb c¯w v¯w Π
and jk = (1 − σ )c(L ¯ p P − Lp σ Π )
In part (b) of the membrane, the situation is more complex. Since the reflection coefficient is 0, we have: Jvb = Lpb P − Lpb σb Π = Lpb P
2.3. Solute transport equation — permeation coefficient ω
js = jd + jk
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(19)
All these fluxes are marked in Fig. 1. Again, we use a sign convention where all fluxes directed to the right (Jvb , jd , jk , js and Jvwb ) are positive. The former four have been defined in the previous paragraphs and the latter (Jvwb ) is the solvent volume flux permeating across part (b) of the equivalent membrane. The total volume flux Jv is positive if |Jva | < |Jvb |. Let us consider Eqs. (8) and (9) again. The former refers to the part of the membrane that is impermeable to the solute. Henceforth, even if there are both osmotic and hydrostatic pressure gradients across this part of the membrane no solute transport, neither diffusive nor convective, takes place. In other words, the solute flux is zero (jsa = 0) and the volume flux consists of the solvent flux only (Jva = Jvwa ). The subscript “w” stands for “water” (the solvent).
and Π Jvsb = Lpb c¯s v¯s Π
where c¯w denotes the average solvent concentration (for diluted solutions c¯w ≈ 1) and c¯s = c¯ = (c1 + c2 )/2 is the mean solute concentration, both measured in the pores of part (b) of the membrane. We assume that the phenomena of filtration and diffusion can be in this formalism decoupled, i.e. the diffusive solute volume flux in a generic case (Π = 0 and P = 0) is the same as just described in a special case (Π = 0 and P = 0). In other words, we can drop the superscript Π and express the diffusive solute volume flux in a generic case as Jvd = Lpb c¯v¯s Π
(21)
Inserting relation (15) to (21) we get: jd = ω Π
(22)
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2.4. Solvent volume flux Jvw
where
¯ p (1 − σ ) ω = cL
(23)
The latter equation expresses correlation among the membrane transport parameters postulated in our earlier work [18] and modified here. We return to this issue later in the paper. Here, let us only add that it can be also written in a form: ω =
(Lpa Lpb + L2pb )c¯ Lp
analogous to Eq. (7) known from the work of Kedem and Katchalsky [1,3]. Solute convection is obviously restricted to the membrane part (b) as well. The corresponding flux can be written as jk = cJ ¯ vb
(24)
and since Jvb = Lpb P = (1 − σ )Lp P as we determined in the previous section, we get: ¯ p P jk = (1 − σ )cL
(25)
The form of this equation requires a more detailed physical explanation. It is obvious that, although jk denotes the convective solute flux in the entire membrane, the phenomenon takes place in this part of the membrane that has pores permeable to the solute (part (b)). Hence, properties of part (a) should not affect the flux. On the other hand in part (b), the reflection coefficient equals zero σb = 0 and so does the effective osmotic pressure σb Π = 0. We conclude that the solvent convection is driven exclusively by the hydrostatic pressure P and is not affected by the osmotic pressure on the membrane. Combining Eqs. (17), (22) and (25) we get: ¯ p P js = ω Π + (1 − σ )cL
(26)
This is the second mechanistic equation (the first one being the same as the Kedem–Katchalsky equation for the volume flux) for the solute flux. As can be seen it differs from the corresponding the Kedem–Katchalsky equation (2). As argued in the previous paragraph, the term in the convective flux involving the osmotic pressure is absent. The last equation can also be written as Jvs = js v¯s = ω v¯s Π + (1 − σ )c¯ v¯s Lp P where Jvs is the solute volume flux.
Let us recall that Kedem and Katchalsky define the total volume flux (Eq. (3) as Jv = jw v¯w + js v¯s = Jvw + Jvs . This equation states an obvious fact that both solute and solvent permeate across a generic membrane. From the equivalent membrane viewpoint we separated the volume flow into two components Jv = Jva + Jvb (see Eq. (10)). As explained in previous sections the flux Jva equals the solvent flux Jvwa generated in this part of the equivalent membrane by both hydrostatic and osmotic pressures and the volume flux in part (b), given by Eq. (9), consists of the solvent flux Jvwb and the solute flux Jvs . The latter is explicitely given by Eq. (27). Eqs. (3) and (10) are simply different groupings of all the volume fluxes in the membrane. In particular:
(27)
Jvw
Jv = Jva + Jvb = Jvw + Jvs
= (Jvwa ) + (Jvwb +Jvs ) (28)
Substituting (16) to (27) and (28) yields an equation for the solvent volume flux: ωv¯s Jvw = Jv − Jvs = − σ + Lp Π Lp (29) +[1 − (1 − σ )c¯v¯s ]Lp P 3. Transport parameters Lp , σ, and ω and their correlation — reduced transport equations In previous sections we introduced, in analogy to the Kedem–Katchalsky formalism [1,3,4], three membrane transport parameters Lp , σ , and ω . The former is defined as Jv Lp = (30) P Π=0 according to Eq. (16). This is the same as definition (3); the parameter Lp in the Kedem–Katchalsky formalism and in our approach presented here has the same interpretation and the same methods of its experimental measurements are suggested. The second parameter (σ ) we interpret somewhat differently from Kedem and Katchalsky (recall the Stavermann relation σ = (LpD /Lp ) [1]). According
A. Kargol / Journal of Membrane Science 191 (2001) 61–69
to Eq. (11) σ equals the ratio of the filtration coefficient Lpa of part (a) of the equivalent membrane to the total filtration coefficient Lp of the entire membrane (σ = Lpa /Lp ). From Eq. (16) we see however that the basic definition of the coefficient σ is: P σ = (31) Π Jv =0 that is the same as the Kedem–Katchalsky definition (Eq. (5)). Consequently, similar experimental techniques are suggested to determine this coefficient as well. The situation is quite different with the latter coefficient (ω ). Kedem and Katchalsky define it as (Eq. (6)): js ω= Π Jv =0 while from Eq. (26) we arrived at the following alternative measure of the membrane permeability: jd ω = (32) Π P =0 In both the cases, the idea is to measure the solute flux under the condition of no solute convection. However, when Jv = 0 there must be a certain nonzero hydrostatic pressure P = Pn = 0 on the membrane. Then there is a nonzero flux Jvb and hence the solute is transported in part (b) of the equivalent membrane both diffusively (flux jd ) and convectively (flux jk ). Accordingly, the parameter ω ought to be measured with P = 0 (and hence Jvb = 0) and not Jv = 0 as required by the Kedem–Katchalsky formalism. Unfortunately, standard techniques meet considerable difficulties here. When P = 0 we would observe a nonzero osmotic volume flux Jv = Jva = Jvwa = 0 and the solution concentration would increase on one and decrease on the other side of the membrane. This effect occurs in addition to the concentration change caused by solute diffusion. It is then very difficult, if not impossible, to determine the diffusive flux jd from the time dependence of the solution concentrations. As can be seen, the coefficient ω is significantly more difficult to determine than the other two (Lp and σ ). In biophysical studies it is often virtually impossible. This difficulty can be eliminated using the corelation relation of the parameters Lp , σ , and ω given by Eq. (23). For more discussion of this relation we refer
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to an appendix. Let us only mention that it eliminates one of the parameters from the transport Eqs. (27) and (29), that is the equation for the solute flux (Jvs ) and for the water flux (Jvw ), which can be rewritten in the following reduced form: Jvs = (1 − σ )c¯v¯s Lp (Π + P )
(33)
Jvw = Jv − (1 − σ )c¯v¯s Lp (Π + P )
(34)
where the flux Jv is given by Eq. (16)): Jv = Lp P − Lp σ Π In view of our previous comments it is natural to eliminate the coefficient that is the most difficult to determine experimentally, i.e. ω . Eqs. (33) and (A.1) are written in terms of the parameters that are the easiest to measure, i.e. Lp and σ .
4. Conclusions Membrane transport equations for binary solutions, originally derived by Kedem and Katchalsky [2], result from thermodynamic analysis of membrane transport. As such they are fairly general and apply to a variety of membranes and transport mechanisms, independently of the underlying molecular principles. In this paper, we offer an alternative approach based on a molecular, mechanistic model of a membrane and flows within. Our model seems to adequately describe transport phenomena in porous membranes. We obtained transport equations that are in many aspects similar or identical with the Kedem and Katchalsky approach, however certain parameters and terms require a new physical interpretation. 1. The equations apply to sieve-type porous membranes, as we postulate that the selective properties of the membrane result from the pore size distribution. We make a simple assumption that a certain fraction of membrane pores are too small to allow the solute flow while still being permeable to the solvent. The rest of the pores are bigger thus equally permeable to both solvent and the solute. The reflection coefficient of the entire membrane has a fractional value depending on the proportion of permeable to impermeable pores. If σ = 1 then all membrane pores are impermeable to the solute
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A. Kargol / Journal of Membrane Science 191 (2001) 61–69
Table 1 Measured membrane transport parameters Membrane/solution
σ
Lp (m3 /N s)
Nephrophane/ethanol Nephrophane/ethanol Nephrophane/glucose Nephrophane/glucose Cellophane/ethanol Cellophane/glucose Dialysis membrane/glucose
0.025 0.016 0.068 0.065 0.02 0.1 0.13
5.0 × 10−12 5.0 × 10−12 5.0 × 10−12 5.0 × 10−12 2.23 × 10−12 2.23 × 10−12 1.09 × 10−12
and only the solvent flows across driven by osmotic (Π ) and/or hydrostatic (P ) pressure. In the case σ = 0 the magnitude of the volume flux depends on the hydrostatic pressure only. The osmotically driven solute and solvent flows balance out and the net flow equals the hydrostatic component of the solvent volume flux. In other words if P = 0 and Π = 0 then there are nonzero solvent and solute diffusive fluxes while the total flux Jv equals 0. 2. In a generic case (0 ≤ σ ≤ 1) osmotically and hydrostatically generated transport can be described by the reduced equations: Jv = Lp P − Lp σ Π Jvs = (1 − σ )c¯v¯s Lp (Π + P ) Jvw = Jv − (1 − σ )c¯v¯s Lp (Π + P ) involving only two membrane transport parameters (Lp and σ ), and not three as in the Kedem–Katchalsky equations. 3. The above equations we derived using the correlation relation among the membrane transport parameters:
ω = cL ¯ p (1 − σ ) Details of this relation are given in the Appendix A.
Appendix A. Correlation relation for the parameters Lp , σ, and ω Our previous papers [18] present a derivation of the correlation relation for the membrane transport parameters based on the Kedem–Katchalsky formalism. It has been written in a form:
ω (mol/N s) 20.0 × 10−10 18.9 × 10−10 5.5 × 10−10 8.0 × 10−10 6.3 × 10−10 2.3 × 10−10 2.95 × 10−10
ωt = ω = K
c¯ (mol/m3 ) 400 400 100 200 300 150 300
Lp (1 − σ ) v¯s
ωt (mol/N s) 19.5 × 10−10 19.7 × 10−10 4.7 × 10−10 9.35 × 10−10 6.56 × 10−10 3.0 × 10−10 2.8 × 10−10
(A.1)
where K has been interpreted as a universal constant. We obtained the relation (A.1)) by standard curve fitting techniques to the plot of the normalized solvent volume flux, i.e. the magnitude of the flux per unit hydrostatic conductivity and unit osmotic pressure versus σ for 50 transport parameter sets collected from the available literature. The details of the procedure are presented in [18]. The value of constant K was found to be 0.021 ± 0.015. We also computed the values of ωt from Eq. (A.1) using the experimental data for Lp and σ and the above value for constant K. For most data sets, we found a good agreement between the computed and experimental values of coefficient ω. Significant errors for some data sets were attributed to large relative error in constant K and possible systematic experimental errors. Our analysis of the transport processes presented here led us to a new form of the correlation relation, i.e. Eq. (23): ωt = ω = cL ¯ p (1 − σ ) This form agrees with Eq. (A.1) if we put K = c¯v¯s , however the K can no longer be interpreted as a universal constant. As a preliminary verification of Eq. (2) we measured parameters (Lp , σ , and ω ) for seven different membranes, solutes and their concentrations shown in Table 1. Values of ωt computed from Eq. (23) are shown in the last column in Table 1 and show a very good agreement with experimental data. References [1] A. Katchalsky, P.F. Curran, Nonequilibrium Thermodynamics in Biophysics, Harvard University Press, Cambridge, 1965.
A. Kargol / Journal of Membrane Science 191 (2001) 61–69 [2] O. Kedem, A. Katchalsky, Permeability of composite membranes (Part I–III), Trans. Faraday Soc. 59 (1963) 1918– 1953. [3] O. Kedem, A. Katchalsky, Thermodynamic analysis of the permeability of biological membranes to non-electrolytes, Biochim. Biophys. Acta 27 (1958) 229–246. [4] O. Kedem, A. Katchalsky, A physical interpretation of the phenomenological coefficients of membrane permeability, J. Gen. Physiol. 45 (1961) 143–179. [5] C. Grygorczyk, Measurement of practical phenomenological coefficients of membranes, Curr. Top. Biophys. 3 (1978) 5– 19. [6] A. Zelman, Membrane permeability. Generalization of the reflection coefficient method of describing volume and solute flows, Biophys. J. 12 (1972) 414–419. [7] K.S. Spiegler, O. Kedem, Thermodynamics of hyperfiltration (reverse osmosis): criteria for efficient membranes, Desalination 1 (1966) 311–326. [8] Yusuke Imai, Membrane transport system modeled by network thermodynamics, J. Membr. Sci. 41 (1989) 3–21. [9] L.F. Del Castillo, E.A. Mason, Generalization of membrane reflection coefficients for nonideal, nonisothermal, multicomponent systems with external forces and viscous flow, J. Membr. Sci. 28 (1986) 229–267. [10] M. Kargol, A more general form of Kedem and Katchalsky’s practical equations, J. Biol. Phys. 22 (1996) 15–26.
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[11] M. Kargol, Full analytical description of graviosmotic volume flows, Gen. Physiol. Biophys. 13 (1994) 109–126. [12] D.G. Levitt, A new theory of transport for cell membrane pores. I. General theory and application to red cell, Biochim. Biophys. Acta 373 (1973) 115–131. ´ ezak, Modification of Kedem–Katchalsky [13] M. Kargol, A. Sl¸ practical equations, Technical Univ. Kielce, Zesz. Nauk. 16 (1985) 5–12 (in Polish). ´ ezak, B. Turczy´nski, Modification of the Kedem– [14] A. Sl¸ Katchalsky equations, Biophys. Chem. 24 (1986) 173– 178. ´ ezak, B. Turczy´nski, Generalization of the Spiegler– [15] A. Sl¸ Kedem–Katchalsky frictional model equations of the transmembrane transport for multicomponent non-electrolyte solutions, Biophys. Chem. 44 (1992) 139–142. [16] A. Kargol, Effect of boundary layers on reverse osmosis through a horizontal membrane, J. Membr. Sci. 159 (1999) 177–184. [17] A. Kargol, Modified Kedem–Katchalsky equations and their applications, J. Membr. Sci. 174 (2000) 43–53. [18] A. Kargol, M. Kargol, S. Przestalski, Correlation relation for the membrane transport parameters Lp , σ , and ω, J. Biol. Phys. 23 (1997) 233–238. [19] M. Kargol, A. Kargol, Membrane transport generated by the osmotic and hydraulic pressure. Correlation relation for parameters Lp , σ , and ω, J. Biol. Phys., in press.
A mechanistic model of transport processes in porous membranes generated by osmotic and hydrostatic pressure Armin Kargol∗ Department of Physics, Tulane University, New Orleans, LA 70118, USA Received 27 October 2000; received in revised form 11 April 2001; accepted 11 April 2001
Abstract This work presents a simple molecular model of membrane transport processes generated by hydrostatic (P ) and osmotic (Π ) pressure. We postulate that the porous membrane permeability is determined by the size and distribution of its pores. From this postulate and a mechanistic interpretation of the flows we obtain equations for the volume flux Jv , the solute volume flux Jvs , and the solvent volume flux Jvw : Jv = Lp P − Lp σ Π Jvs = (1 − σ )c¯v¯s Lp (Π + P ) Jvw = Jv − (1 − σ )c¯v¯s Lp (Π + P ) We also observe that the porous membrane properties are described by two phenomenological parameters (Lp and σ ), as opposed to three parameters (Lp , σ , and ω) in the standard Kedem–Katchalsky formalism. The extra parameter was eliminated using the correlation relation for the transport parameters found previously and modified in this work. It has a form: ω = (1 − σ )cL ¯ p © 2001 Elsevier Science B.V. All rights reserved. Keywords: Membrane transport; Theory; Porous membranes
1. Introduction Passive transport of nonelectrolytic solutions in biological and artificial membranes includes several phenomena, most notably osmosis, filtration, diffusion, and convection. For the purpose of this work, we use a somewhat restricted definition of osmo∗
´ etokrzyska Academy, On leave from: Physics Institute, Swi¸ 25-406 Kielce, Poland. Fax: +1-504-862-8702. E-mail address: [email protected] (A. Kargol).
sis: the solvent diffusion through a semi-permeable membrane, i.e. one that forms an ideal barrier to the solute molecules. These transport phenomena can be driven by hydrostatic pressure, osmotic pressure, or both. In a general case, a quantitative technique has been proposed four decades ago by Kedem and Katchalsky [1–4]. The cornerstone of their method are the so-called Kedem–Katchalsky practical transport equations derived by means of thermodynamics of irreversible processes. In their standard form the Kedem–Katchalsky equations represent the linearized
0376-7388/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 6 - 7 3 8 8 ( 0 1 ) 0 0 4 5 0 - 1
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A. Kargol / Journal of Membrane Science 191 (2001) 61–69
permeation coefficient depends on the solute concentration c¯ according to the following [1,3]:
Nomenclature c1 , c2 jd jk js Jv , Jva , Jvb Jvsb Jvwa , Jvwb Lp P1 , P2
concentrations (mol/m3 ) diffusive solute flux (mol/m2 s) convective solute flux (mol/m2 s) solute flux (mol/m2 s) volume fluxes (m/s) solute volume flux (m/s) solvent volume fluxes (m/s) filtration coefficient (m3 /N s) hydrostatic pressures (Pa)
ω=
Greek symbols Π osmotic pressure (Pa) σ reflection coefficient ω, ω permeation coefficient (mol/N s) force-flow relationships, i.e. express the volume and solute fluxes in terms of the driving forces: Jv = Lp P − Lp σ Π
(1)
js = ω Π + (1 − σ )cL ¯ p (P − σ Π )
(2)
where Jv , js are the fluxes, and c¯ is the average solute concentration. The volume flux Jv is defined as Jv = jw v¯w + js v¯s = Jvw + Jvs
(3)
where jw , js are the solvent and solute fluxes, v¯w , v¯s the partial molar volumes of the solvent and the solute, and Jvw , Jvs the solvent and solute volume fluxes. The membrane properties are described by the phenomenological coefficients, Lp , σ , and ω, called the coefficients of filtration, reflection and permeation, respectively. They are defined as Jv (4) Lp = P Π=0 P σ = (5) Π Jv =0 js ω= (6) Π Jv =0 The above equations besides specifying the physical meaning of each parameter also to some extent determine their measurement techniques [1,4,5]. The
(Lp LD − L2pD )c¯ Lp
(7)
where LpD = (Jv /Π )P =0 and LD = (JD / Π )P =0 ; JD is called the diffusion flux. The Kedem–Katchalsky technique has become one of the main tools in both scientific and technological studies involving transport across biological and artificial membranes. However, since the early days there have been many attempts at generalization and modification of the formalism to address both the question of phenomena not covered by the standard method (e.g. the effect of boundary layers) and certain questions concerning physical principles governing membrane transport [6–17]. This is also the subject of the present work. Our technique in this paper is however different from Kedem and Katchalsky’s. Rather than using thermodynamics tools we are interested in a molecular, mechanistic approach. We present a model of a porous membrane and describe the transport processes in such a membrane starting from few fairly natural assumptions and a mechanistic interpretation of the flows. Of course, such a model has its limitations, however we believe it adequately describes porous membranes. The method is based on a postulate that in a porous membrane the size of pores determines permeability to a given solute. Among all pores present in a membrane, a certain number na has radii sufficiently small to become totally impermeable to the solute molecules while the remaining nb pores (na +nb = n) are ideally permeable, i.e. allow movement of both solute and solvent molecules. Extending the notion of the reflection coefficient to a patch of a membrane, or even a single pore, we postulate that a single membrane pore has the reflection coefficient for a given solute equal to either 1 or 0. Inspired by the Kedem–Katchalsky formalism we derive equations for: the volume flux Jv , the solute flux js , and the solvent volume flux Jvw . The three phenomenological transport parameters introduced by Kedem and Katchalsky (cf. Eqs. (4)–(6)), i.e. the parameters of filtration, reflection, and permeation are redefined to reflect a somewhat different form of these transport equations.
A. Kargol / Journal of Membrane Science 191 (2001) 61–69
In the standard Kedem–Katchalsky method three independent membrane transport parameters (Lp , σ and ω) are introduced, however as previously noted [18] the available experimental data strongly suggest some form of correlation among them. In [18] a correlation relation has been obtained within the framework of the Kedem–Katchalsky equations using standard curve fitting techniques albeit some simplifications had to be made (we argued that certain terms are typically small and can be neglected). Here, we modify this correlation relation based on the mechanistic equations and as we show no approximations are needed. The significance of this observed correlation (which we derive from the mechanistic equations for porous membranes but as we mentioned earlier the experimental data available to us strongly suggest that it is a more general phenomenon) is that it allows elimination of one of the parameters, preferably the one that is the hardest to measure, i.e. ω, from the description of the flows. Accordingly, the equations for the volume, solute and solvent fluxes are written in a reduced form expressing this correlation. We emphasize that the correlation relation in a general case is a phenomenological observation and it relies on curve fitting to experimental data, but it can be derived within our mechanistic model for porous membranes. Overall, in this paper we obtain equations that are in many aspects similar to the KK equations but also differ at certain points. We believe that our approach which is a continuation of earlier work [19] gives new insight into membrane transport mechanisms and is also quite intuitive.
2. Analysis and interpretation — transport equations 2.1. Porous membrane — an equivalent membrane model We consider transport processes across porous membranes driven by hydrostatic and osmotic pressure gradients. A typical sieve-type membrane has some number n of pores with random radii, from very small to a certain maximal value. The pores are distributed randomly in the entire membrane area. Examples of such membranes are: cellophane, colloid,
63
and nephrophane membranes. As we mentioned in the introduction we extend the concept of the reflection coefficient for a membrane to a small patch of the membrane or even individual pores. We postulate that pore properties are determined by their sizes. For a given solute a fraction of pores allow free movement of solute molecules while the others form an ideal barrier due to their smallness compared to the solute molecule size. In terms of the reflection coefficient we postulate that its value for a single pore is either 0 or 1, and not fractional. This reflects the fact that a single pore can be either permeable, meaning equally permeable to both the solvent and the solute, or semipermeable that is permeable to the solvent while totally impermeable to the solute. The reflection coefficient of the entire membrane (a complex of n pores) depends on the proportion of permeable to impermeable pores. In other words, it equals zero if all of its pores are permeable and 1 if all the pores are impermeable to the solute. If there are both types of pores (solute permeable and impermeable) present, the reflection coefficient of the entire membrane has a fractional value 0 < σ < 1. Having this in mind, we consider an equivalent model membrane, such as in Fig. 1, where all the pores are spatially separated according to size. For a given solute a part of the membrane (part (a)) has pores that are impermeable to the solute, while the other part of the membrane (part (b)) has only permeable pores. The two parts of the membrane have reflection coefficients σa = 1 and σb = 0. The filtration coefficients for both parts of the membrane are denoted Lpa and Lpb . In the following sections such an equivalent membrane is used to analyze hydrostatically and osmotically driven transport. 2.2. Filtration and reflection coefficients — volume transport equation The membrane in Fig. 1 separates well-mixed solutions of concentrations c1 and c2 (c1 < c2 ) subject to hydrostatic pressures P1 and P2 (P1 < P2 ). This represents a generic situation with both the osmotic Π = RT c = RT(c2 − c1 ) (where R is the gas constant and T is the absolute temperature), and hydrostatic P = P2 − P1 pressures (driving forces) across the membrane. The pressure gradients generate volume fluxes Jva and Jvb in parts (a) and (b),
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A. Kargol / Journal of Membrane Science 191 (2001) 61–69
Fig. 1. The equivalent membrane model system (M–membrane with the semipermeable (a) and permeable (b) parts). Volume (Jva , Jvb , Jvwb ) and solute (jd , jk , js ) fluxes as described in text. The concentration and pressure gradient directions marked with inequality signs.
respectively, of the membrane. The fluxes can be described as Jva = Lpa P − Lpa Π
(8)
Jvb = Lpb P
(9)
since σb = 0. Given the pressure gradients the flux Jvb flows to the right, while the direction of Jva depends on the relative strength of the driving forces. In Fig. 1, we assumed that the osmotic pressure dominates. The total volume flux across the membrane is the net of the two: Jv = Jva + Jvb
(10)
We use a sign convention where fluxes directed to the right are positive. Given osmotic pressure Π the total volume flux vanishes if the hydrostatic pressure P = Pn equals Lpa Π Pn = Lpa + Lpb Lpa Π Lp
Lpa Lpa = Lpa + Lpb Lp
(12)
This equation can also be written as P σ = Π Jv =0
(13)
which is the same as the standard definition of the reflection coefficient (Eq. (5)). On the other hand according to Eq. (11) the coefficient σ is a ratio of the filtration coefficient of the part of the membrane containing impermeable pores to the total filtration coefficient Lp . That gives the reflection coefficient a new mechanistic interpretation. For instance, if Lpa = 0 then σ = 0, or if Lpb = 0 then σ = 1. Eq. (11) can also be written as Lpa = σ Lp
(14)
Lpb = (1 − σ )Lp
(15)
Jv = Jva + Jvb
where Lp = Lpa + Lpb is the total filtration coefficient of membrane M. Denoting: σ =
Pn = σ Π
When the hydrostatic pressure is different from the reversal value Pn then there is a nonzero net flux across the membrane and given the driving forces P and Π it can be written as
or Pn =
we get the relation:
= (Lpa + Lpb ) P −
Lpa (Lpa + Lpb ) Π Lpa + Lpb
or, taking Eq. (11) into account: (11)
Jv = Lp P − Lp σ Π
(16)
A. Kargol / Journal of Membrane Science 191 (2001) 61–69
i.e. in a form identical with Eq. (1), the first Kedem–Katchalsky equation.
This volume flux has two components: the solvent volume flux (Jvwb ) and the solute volume flux (Jvsb ):
In the previous section the equivalent membrane with various size pores was used to derive the first of the Kedem–Katchalsky equations for the volume transport. As one might expect the pore size has even greater effect on the solute transport since in parts of the equivalent membrane such transport does not occur at all due to the pore smallness. Starting from similar microscopic assumptions a macroscopic equation for the solute transport is derived in this section. Let us recall the second Kedem–Katchalsky equation (Eq. (2)) which shows the total solute flux as a sum of diffusive flux (diffusion due to concentration gradient) and the convective flux (i.e. the solute convection in the volume flux): (17)
The Kedem–Katchalsky formulas for the two fluxes are as follows: jd = ω Π
(18)
Jvb = Jvwb + Jvsb > 0
(20)
These two fluxes represent the two phenomena generically occurring on the membrane: the filtration (forced by hydrostatic pressure P ) and diffusion (since Π = 0). For the moment we consider a simplified case where P = 0. Due to concentration difference between both sides of the membrane we observe solute and solvent diffusion in opposite directions. In such case Eq. (20) reads: Π Π Π Jvb = Jvwb + Jvsb =0 Π is the diffusive solvent volume flux and where Jvwb Π Jvsb the diffusive solute volume flux. The superscript Π emphasizes the fact that we assumed the case of fluxes driven by the osmotic pressure only. With the concentration gradient directed as in Fig. 1, both Π < 0, J Π > 0) and can be fluxes are nonzero (Jvwb vsb expressed as Π Jvwb = −Lpb c¯w v¯w Π
and jk = (1 − σ )c(L ¯ p P − Lp σ Π )
In part (b) of the membrane, the situation is more complex. Since the reflection coefficient is 0, we have: Jvb = Lpb P − Lpb σb Π = Lpb P
2.3. Solute transport equation — permeation coefficient ω
js = jd + jk
65
(19)
All these fluxes are marked in Fig. 1. Again, we use a sign convention where all fluxes directed to the right (Jvb , jd , jk , js and Jvwb ) are positive. The former four have been defined in the previous paragraphs and the latter (Jvwb ) is the solvent volume flux permeating across part (b) of the equivalent membrane. The total volume flux Jv is positive if |Jva | < |Jvb |. Let us consider Eqs. (8) and (9) again. The former refers to the part of the membrane that is impermeable to the solute. Henceforth, even if there are both osmotic and hydrostatic pressure gradients across this part of the membrane no solute transport, neither diffusive nor convective, takes place. In other words, the solute flux is zero (jsa = 0) and the volume flux consists of the solvent flux only (Jva = Jvwa ). The subscript “w” stands for “water” (the solvent).
and Π Jvsb = Lpb c¯s v¯s Π
where c¯w denotes the average solvent concentration (for diluted solutions c¯w ≈ 1) and c¯s = c¯ = (c1 + c2 )/2 is the mean solute concentration, both measured in the pores of part (b) of the membrane. We assume that the phenomena of filtration and diffusion can be in this formalism decoupled, i.e. the diffusive solute volume flux in a generic case (Π = 0 and P = 0) is the same as just described in a special case (Π = 0 and P = 0). In other words, we can drop the superscript Π and express the diffusive solute volume flux in a generic case as Jvd = Lpb c¯v¯s Π
(21)
Inserting relation (15) to (21) we get: jd = ω Π
(22)
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2.4. Solvent volume flux Jvw
where
¯ p (1 − σ ) ω = cL
(23)
The latter equation expresses correlation among the membrane transport parameters postulated in our earlier work [18] and modified here. We return to this issue later in the paper. Here, let us only add that it can be also written in a form: ω =
(Lpa Lpb + L2pb )c¯ Lp
analogous to Eq. (7) known from the work of Kedem and Katchalsky [1,3]. Solute convection is obviously restricted to the membrane part (b) as well. The corresponding flux can be written as jk = cJ ¯ vb
(24)
and since Jvb = Lpb P = (1 − σ )Lp P as we determined in the previous section, we get: ¯ p P jk = (1 − σ )cL
(25)
The form of this equation requires a more detailed physical explanation. It is obvious that, although jk denotes the convective solute flux in the entire membrane, the phenomenon takes place in this part of the membrane that has pores permeable to the solute (part (b)). Hence, properties of part (a) should not affect the flux. On the other hand in part (b), the reflection coefficient equals zero σb = 0 and so does the effective osmotic pressure σb Π = 0. We conclude that the solvent convection is driven exclusively by the hydrostatic pressure P and is not affected by the osmotic pressure on the membrane. Combining Eqs. (17), (22) and (25) we get: ¯ p P js = ω Π + (1 − σ )cL
(26)
This is the second mechanistic equation (the first one being the same as the Kedem–Katchalsky equation for the volume flux) for the solute flux. As can be seen it differs from the corresponding the Kedem–Katchalsky equation (2). As argued in the previous paragraph, the term in the convective flux involving the osmotic pressure is absent. The last equation can also be written as Jvs = js v¯s = ω v¯s Π + (1 − σ )c¯ v¯s Lp P where Jvs is the solute volume flux.
Let us recall that Kedem and Katchalsky define the total volume flux (Eq. (3) as Jv = jw v¯w + js v¯s = Jvw + Jvs . This equation states an obvious fact that both solute and solvent permeate across a generic membrane. From the equivalent membrane viewpoint we separated the volume flow into two components Jv = Jva + Jvb (see Eq. (10)). As explained in previous sections the flux Jva equals the solvent flux Jvwa generated in this part of the equivalent membrane by both hydrostatic and osmotic pressures and the volume flux in part (b), given by Eq. (9), consists of the solvent flux Jvwb and the solute flux Jvs . The latter is explicitely given by Eq. (27). Eqs. (3) and (10) are simply different groupings of all the volume fluxes in the membrane. In particular:
(27)
Jvw
Jv = Jva + Jvb = Jvw + Jvs
= (Jvwa ) + (Jvwb +Jvs ) (28)
Substituting (16) to (27) and (28) yields an equation for the solvent volume flux: ωv¯s Jvw = Jv − Jvs = − σ + Lp Π Lp (29) +[1 − (1 − σ )c¯v¯s ]Lp P 3. Transport parameters Lp , σ, and ω and their correlation — reduced transport equations In previous sections we introduced, in analogy to the Kedem–Katchalsky formalism [1,3,4], three membrane transport parameters Lp , σ , and ω . The former is defined as Jv Lp = (30) P Π=0 according to Eq. (16). This is the same as definition (3); the parameter Lp in the Kedem–Katchalsky formalism and in our approach presented here has the same interpretation and the same methods of its experimental measurements are suggested. The second parameter (σ ) we interpret somewhat differently from Kedem and Katchalsky (recall the Stavermann relation σ = (LpD /Lp ) [1]). According
A. Kargol / Journal of Membrane Science 191 (2001) 61–69
to Eq. (11) σ equals the ratio of the filtration coefficient Lpa of part (a) of the equivalent membrane to the total filtration coefficient Lp of the entire membrane (σ = Lpa /Lp ). From Eq. (16) we see however that the basic definition of the coefficient σ is: P σ = (31) Π Jv =0 that is the same as the Kedem–Katchalsky definition (Eq. (5)). Consequently, similar experimental techniques are suggested to determine this coefficient as well. The situation is quite different with the latter coefficient (ω ). Kedem and Katchalsky define it as (Eq. (6)): js ω= Π Jv =0 while from Eq. (26) we arrived at the following alternative measure of the membrane permeability: jd ω = (32) Π P =0 In both the cases, the idea is to measure the solute flux under the condition of no solute convection. However, when Jv = 0 there must be a certain nonzero hydrostatic pressure P = Pn = 0 on the membrane. Then there is a nonzero flux Jvb and hence the solute is transported in part (b) of the equivalent membrane both diffusively (flux jd ) and convectively (flux jk ). Accordingly, the parameter ω ought to be measured with P = 0 (and hence Jvb = 0) and not Jv = 0 as required by the Kedem–Katchalsky formalism. Unfortunately, standard techniques meet considerable difficulties here. When P = 0 we would observe a nonzero osmotic volume flux Jv = Jva = Jvwa = 0 and the solution concentration would increase on one and decrease on the other side of the membrane. This effect occurs in addition to the concentration change caused by solute diffusion. It is then very difficult, if not impossible, to determine the diffusive flux jd from the time dependence of the solution concentrations. As can be seen, the coefficient ω is significantly more difficult to determine than the other two (Lp and σ ). In biophysical studies it is often virtually impossible. This difficulty can be eliminated using the corelation relation of the parameters Lp , σ , and ω given by Eq. (23). For more discussion of this relation we refer
67
to an appendix. Let us only mention that it eliminates one of the parameters from the transport Eqs. (27) and (29), that is the equation for the solute flux (Jvs ) and for the water flux (Jvw ), which can be rewritten in the following reduced form: Jvs = (1 − σ )c¯v¯s Lp (Π + P )
(33)
Jvw = Jv − (1 − σ )c¯v¯s Lp (Π + P )
(34)
where the flux Jv is given by Eq. (16)): Jv = Lp P − Lp σ Π In view of our previous comments it is natural to eliminate the coefficient that is the most difficult to determine experimentally, i.e. ω . Eqs. (33) and (A.1) are written in terms of the parameters that are the easiest to measure, i.e. Lp and σ .
4. Conclusions Membrane transport equations for binary solutions, originally derived by Kedem and Katchalsky [2], result from thermodynamic analysis of membrane transport. As such they are fairly general and apply to a variety of membranes and transport mechanisms, independently of the underlying molecular principles. In this paper, we offer an alternative approach based on a molecular, mechanistic model of a membrane and flows within. Our model seems to adequately describe transport phenomena in porous membranes. We obtained transport equations that are in many aspects similar or identical with the Kedem and Katchalsky approach, however certain parameters and terms require a new physical interpretation. 1. The equations apply to sieve-type porous membranes, as we postulate that the selective properties of the membrane result from the pore size distribution. We make a simple assumption that a certain fraction of membrane pores are too small to allow the solute flow while still being permeable to the solvent. The rest of the pores are bigger thus equally permeable to both solvent and the solute. The reflection coefficient of the entire membrane has a fractional value depending on the proportion of permeable to impermeable pores. If σ = 1 then all membrane pores are impermeable to the solute
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A. Kargol / Journal of Membrane Science 191 (2001) 61–69
Table 1 Measured membrane transport parameters Membrane/solution
σ
Lp (m3 /N s)
Nephrophane/ethanol Nephrophane/ethanol Nephrophane/glucose Nephrophane/glucose Cellophane/ethanol Cellophane/glucose Dialysis membrane/glucose
0.025 0.016 0.068 0.065 0.02 0.1 0.13
5.0 × 10−12 5.0 × 10−12 5.0 × 10−12 5.0 × 10−12 2.23 × 10−12 2.23 × 10−12 1.09 × 10−12
and only the solvent flows across driven by osmotic (Π ) and/or hydrostatic (P ) pressure. In the case σ = 0 the magnitude of the volume flux depends on the hydrostatic pressure only. The osmotically driven solute and solvent flows balance out and the net flow equals the hydrostatic component of the solvent volume flux. In other words if P = 0 and Π = 0 then there are nonzero solvent and solute diffusive fluxes while the total flux Jv equals 0. 2. In a generic case (0 ≤ σ ≤ 1) osmotically and hydrostatically generated transport can be described by the reduced equations: Jv = Lp P − Lp σ Π Jvs = (1 − σ )c¯v¯s Lp (Π + P ) Jvw = Jv − (1 − σ )c¯v¯s Lp (Π + P ) involving only two membrane transport parameters (Lp and σ ), and not three as in the Kedem–Katchalsky equations. 3. The above equations we derived using the correlation relation among the membrane transport parameters:
ω = cL ¯ p (1 − σ ) Details of this relation are given in the Appendix A.
Appendix A. Correlation relation for the parameters Lp , σ, and ω Our previous papers [18] present a derivation of the correlation relation for the membrane transport parameters based on the Kedem–Katchalsky formalism. It has been written in a form:
ω (mol/N s) 20.0 × 10−10 18.9 × 10−10 5.5 × 10−10 8.0 × 10−10 6.3 × 10−10 2.3 × 10−10 2.95 × 10−10
ωt = ω = K
c¯ (mol/m3 ) 400 400 100 200 300 150 300
Lp (1 − σ ) v¯s
ωt (mol/N s) 19.5 × 10−10 19.7 × 10−10 4.7 × 10−10 9.35 × 10−10 6.56 × 10−10 3.0 × 10−10 2.8 × 10−10
(A.1)
where K has been interpreted as a universal constant. We obtained the relation (A.1)) by standard curve fitting techniques to the plot of the normalized solvent volume flux, i.e. the magnitude of the flux per unit hydrostatic conductivity and unit osmotic pressure versus σ for 50 transport parameter sets collected from the available literature. The details of the procedure are presented in [18]. The value of constant K was found to be 0.021 ± 0.015. We also computed the values of ωt from Eq. (A.1) using the experimental data for Lp and σ and the above value for constant K. For most data sets, we found a good agreement between the computed and experimental values of coefficient ω. Significant errors for some data sets were attributed to large relative error in constant K and possible systematic experimental errors. Our analysis of the transport processes presented here led us to a new form of the correlation relation, i.e. Eq. (23): ωt = ω = cL ¯ p (1 − σ ) This form agrees with Eq. (A.1) if we put K = c¯v¯s , however the K can no longer be interpreted as a universal constant. As a preliminary verification of Eq. (2) we measured parameters (Lp , σ , and ω ) for seven different membranes, solutes and their concentrations shown in Table 1. Values of ωt computed from Eq. (23) are shown in the last column in Table 1 and show a very good agreement with experimental data. References [1] A. Katchalsky, P.F. Curran, Nonequilibrium Thermodynamics in Biophysics, Harvard University Press, Cambridge, 1965.
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