Dec., 1958
PERMEATION OF CELLOPHANE MEMBRANES BY DIFFUSION
1483
ON THE PERMEATION OF CELLOPHANE MEMBRANES BY DIFFUSION BY L. B. TICKNOR Research and Development Division, American Viscose Corporation, Marcus Hook, Pa. Received April 81, 1968
The assumption that permeation of membranes by liquids is a diffusive process leads t o equations which have the proper form; Le. the quantity of permeant transferred is directly proportional to the pressure difference and inversely proportional t o the bulk viscosity of the liquid. It is therefore necessary to consider diffusion as a possible mechanism in liquid permeation of membranes. By comparison of permeation and self-diffusion it is found that the permeability of cellophane to water and methanol is greater than would be expected from a diffusive flow mechanism; and, therefore, the permeation in these cases is largely a result of viscous flow. With other alcohols the permeation is about the same as would be calculated for a diffusive flow mechanism. An approximate calculation shows that the pore radii obtained on the assumption of Poiseuille flow are small enough to suggest that part of the transport is by a diffusion mechanism. The variation of the permeability constant with molecular size also suggests that the permeation is not simple viscous flow. These results and other considerations mentioned suggest that, in general, permeation of cellophane membranes by liquids involves both a viscous and a diffusive flow mechanism.
Introduction In permeability measurements a gas or a liquid is forced through a membrane by a pressure difference. I n liquid permeation the volume of material transferred in unit time through unit cross sectional area is found to depend linearly on the pressure difference in accordance with the equations Q = -A(AP) =
-
1IL
(1)
A is termed the “permeability”, K is the “permeability constant,” 1 is the viscosity of the liquid and L is the membrane thickness. This equation is derived readily on the assumption of Poiseuille flow through capillaries. It also may be obtained from the assumption of a diffusive flow through the membrane. Whether the permeation of cellophane membranes by liquids is a viscous or diffusive type of transport is difficult to ascertain. These two types of flow differ in whether the molecules move in groups or move individually in a random walk manner. Since it is conceivable that with a membrane of very small capillaries the molecules of the permeant would be separated from each other by the material of which the membrane structure is composed, the permeant molecules would only move as individuals and their movement could be in a random walk manner. The gaseous permeation of polymer membranes involves a real concentration gradient and has often been interpreted as a diffusion pr0cess.l In the permeation of membranes by liquids there is essentially no concentration gradient; but there is a gradient in the chemical potential because of the gradient in hydrostatic pressure. Whereas the gradient in hydrostatic pressure would lead to mass flow, the gradient in chemical potential would lead to a diffusive flow. Which type of flow predominates would depend on the sizes of the capillaries, the size of the permeating molecules and the bonding of the permeant to the film. We might expect that in some cases, as has been indicated by Mauro,2 the total permeation would be a combination of viscous flow and diffusion. A distribution in capillary sizes would lead to a combined mechanism; and even within some capillaries the molecules along the (1) R. M. Barrer, “Diffusion in and Through Solids,” Cambridge University Press, Cambridge, 1951. (2) A. Mauro, Science, 126, 252 (1967).
sides might be diffusing or moving individually while those in the center were moving as groups. The experimental determination of whether permeation is a viscous flow or diffusion process is of necessity dependent on a comparison of the permeation rate with the rate one would expect if diffusion were the only process taking place. If the molecules of permeant moved through the membrane in groups, the permeation rate would be expected to be much larger than if they moved individually. In order to make this comparison we have calculated a “coefficient of diffusion” from rates of permeation of cellophane by water and alcohols. These diffusion coefficients are compared with the self-diff usion coefficient for each permeant. To make this comparison valid the frames of reference for the calculation of the diffusion coefficients must be equivalent in the two cases and the frictional forces for the movement of the permeant molecules in the membrane must be comparable to those in the bulk liquid. In the calculations given below these two requirements are essentially satisfied; the frames of reference are equivalent because the volume in that part of the system wherein diffusion takes place remains constant,* and the frictional forces are similar because of the hydroxyl groups on the cellulose molecule and the solvation of the cellulose by the permeant. Equations for Diffusive Flow If we make the assumption that all the permeation is a diffusive flow process resulting from the difference in chemical potential of the permeant at the two sides of the membrane, we may use the procedure of Mauro2and derive the equation DeV( A P ) & = -~ RTL
where D is the coefficient of diffusion, E is the fractional void volume of the membrane, V is the molar volume of the bulk liquid, AP is the difference in hydrostatic pressure on the two sides of the membrane and L is the thickness of the swollen membrane. The concentration of the permeant in the membrane has been taken as E/V. This is equivalent to correcting the area of the membrane to that fraction of the area that is permeable to the liquid. Equation 2 is identical in form to the first part of equation 1. (3) G. 8. Hartley and J. Crank, Trans. Farodav SOC.,45, 801 (1949).
L. B. TICKNOR
1484
I n order to relate the permeation by diffusion to the second part of equation 1 we need a relation between the diffusion coefficient D and the viscosity 7. Johnson and Babb4 have shown that the Stokes-Einstein theory, the kinetic theory of Eyring, the hydrodynamic-thermodynamic approach of Hartley and Crank, and the theory of irreversible thermodynamics all lead to a relationship between diffusion and viscosity coefficients of the form D
= BRT --
N?l
TABLEI CALCULATED “DIFFUSION COEFFICIENTS” FOR THE PERMEATION OF A WATERSWOLLEN CELLOPHANE MEMBRANE BY WATER Temp., OC. A X 1011 D x 105 20 25 28 33.5 38 40
(3)
where B depends on various molecular dimensions. Substitution of equation 3 into 2 gives This equation is identical with the second part of equation 1. Thus we see that the assumption of a diffusion mechanism leads to permeation equations identical to those obtained through the assumption of viscous flow. Therefore, we must consider diffusion as a possible mechanism for the permeation of membranes by liquids. Application of Equations Madras, McIntosh and Mason5 have measured the permeability of cellophane to pure liquids with a van Campen6 type osmometer. They used a commercial type cellophane after swelling it in various media and conditioning it in the permeating liquid. Their measurements were carried out a t 30” except for water for which the temperature was varied. They measured the fractional void volume e by two methods. We will use the values derived from the thickness of the swollen membrane because that method appears to be more reliable.7 They found that the fractional void volume was nearly independent of the permeant, but it increased with stronger swelling agents. They obtained values of 0.38, 0.59 and 0.62, respectively, for swelling in 60% ethanol, water and 3% NaOH. From equations 4 and 1 we have
Vol. 62
1.95 2.17 2.28 2.65 3.08 3.30
37 42 45 53 63 67
TABLEI1 CALCULATED “DIFFUSION COEFFICIENTS” FOR THE PERMEATION OF CELLOPHANE MEMBRANES BY ALCOHOLS AT 30” Alcohol
Methanol Ethanol I-Propanol I-Butanol
-Membrane swollen inSelf-diff. 00% Ethanol Water 3% NaOH coef. X 106 D X 106 D X 106 D X IO5
2.5 1.16 0.73 0.58
2.97 0.87 0.33 0.20
7.37 1.95 0.70 0.39
10.3 2.3 1.06 0.47
fusion coefficients to be no larger than the selfdiffusion coefficients of the bulk liquid if diffusion through the membrane were the controlling process. I n the case of water permeation the values in Table I may be compared with the results of Wang, et al., who obtained self-diffusion coefficients for water ranging from about 2.8 to 4.4 X a t 25 to 45”. Since the constants in Table I are about ten times as large, it appears that a faster transfer mechanism, buch as viscous flow, is the predominant transport process. In the case of the alcohols, the calculated diffusion coefficients in Table I1 are of about the same order of magnitude as the self-diffusion coefficients. For methanol the coefficients for all three membranes are somewhat larger than the self-diffusion coefficient. This suggests a certain amount of mass flow in these cases. For t h e other alcohols it is found that for the least swollen membranes the self-diffusion coefficient of the liquid is larger than the calculated diffusion coefficient. In these cases RTLA D=(5) it appears that diffusive flow may predominate. eV It was suggested above that the size of the capiland we may calculate D from the values of A reported by Madras, et al. (their P values). I n laries would determine whether the flow was preequation 5 D has units of cma2/sec.,A is in cm.2- dominantly viscous or diffusive. The following sec./g., L is in cm., T is in OK., R is 8.314 X lo7 simple calculationlo shows under what conditions one may expect a transition from one type of flow to ergs/mole-deg., and V is in cc./mole. Application of equation 5 t o Madras’ data on the the other. Consider a system of capillaries of uniradius r and length L passing perpendicularly water permeability of cellophane (taking L equal to form through a membrane. If the fractional void vol84 p ) yields the results listed in Table I. When we use the data of Madras, et al., for the ume is e, the number of capillaries per unit surface permeation of alcohols, we obtain the calculated area is e/nr2. Then, according t o Poiseuille, the “diffusion coefficients” given in Table 11. I n addi- rate of transport &’ of a liquid in cm.8/cm.2-sec.is tion, the values of the self-diffusion coefficients for the alcohols as estimated from a paper by Partington, et aZ.,8are given. In order to compare this result with diffusive Discussion transport the Stokes-Einstein relation In general, we would expect these calculated dif_ - 6rr*q (4) P. A. Johnson and A. L. Babb, Chem. Reus., 66, 387 (1956). (5) S. Madras, R. L. McIntosh and S. E. Mason, Can. J . Research.
27B,764 (1949). ( G ) P. van Campen, Rec. trau. chim., 60, 915 (1931). (7) J. M . Kuzmak, Ph.D. Thesis, McGill University, 1953. (8) J. R. Partington, R. F. Hudson and K. W. Bagnall. Nalure, 169, 583 (1952).
where r* is the radius of the molecules, is substituted into equation 2 giving (9) J. H. Wang, C. V. Robinson and I. S. Edelman, J . A m . Chem. Soc., 76, 466 (1953).
(IO) Suggested by Professor Carl Wagner.
..
Dec., 1958
STABILITY OF HYDROGEN-BONDED POLYPEPTIDE STRUCTURES IN SOLUTION & = - - ET’( A P ) 6~r*qNL
(7)
The two transport rates may now be compared by taking the ratio of Q’ to Q For close-packed hard spheres the fraction of the total volume that is occupied by the spheres is about 0.74. For a real liquid we can only estimate this ratio and we will take the convenient value of 4/9. Then, substitution in equation 8 for the volume in terms of the molecular radius shows that the two rates of transport would be equal if r =2r*
(9)
Therefore, within the limitations of the above assumptions, viscous flow across a membrane would predominate if *the capillary radius were much larger than the molecular radius. If, on the other hand, the capillary radius were only slightly larger than the molecular radius, diffusive flow would be important. Madras, et al., calculated the effective pore radius to be about 1.5 X lo-’ cm. for water permeation of their cellophane membranes. Their method of calculation (assumption of Poiseuille flow) gives values of 6 X 10-8 cm. for permeation by ethanol of a water swollen membrane and 3.6 X cm. for the permeation by butanol of a membrane previously swollen in 60% ethan0140% water. Thus, equation 8 suggests that for water viscous flow would predominate and for the alcohols both types of flow would occur, which agrees with the results of Tables 1 and 11. It would be difficult to separate the permeation into these types of flow, although it has been attempted for gaseous permeation of membranes. l1 (11) H. L. Frisch, THIEJOURNAL, 60, 1177 (1950).
1485
If viscous flow were the only transport mechanism, the permeation of a membrane by various liquids (assuming they did not change the pore radius) should yield the same permeability constant. Madras, et al., however, found permeability constants for water swollen membranes varying from 3.6 X 10-l6 cm.-2 for permeation by methanol to for hep1.8 x 1O-la for butanol and 0.8 X tane. On the other hand if the mechanism were partly viscous flow and partly diffusive flow, it would be expected that an increase in the molecular size of the permeant would cause a transfer from a convective to a diffusive mechanism. Thus, the observed results can be explained by a combined mechanism. The increase in the calculated diffusion coefficients with increased degree of swelling suggests an increase in the size of the capillaries and a transfer to more convective and less diffusive flow. Other investigators have found evidence for a diffusion mechanism in membrane permeation. Barrer and others’ have treated the gaseous permeation of organic membranes as a diffusive process. It has been shown by McBain and Stuewer12 that an ordinary cellophane membrane when swollen in water and used as an ultrafilter will partially hold back small molecules and ions such as sucrose, dextrose and potassium chloride. C. E. Reid and E. J. Breton13have reported on the use of cellulose acetate membranes for the ultrafiltration of a 0.5% aqueous NaCl solution; salt concentration was reduced by about 95%. These results suggest that, in general, permeation of membranes by liquids may be interpreted in part by a diffusion mechanism. (12) J. W. McBain and R.F. Stuewer, ibid., 40, 1157 (1936). (13) Charles E. Reid and Ernest J. Breton, “Water and Ion Flow through Cellulose Acetate Membranes,” a paper presented a t the 130th National Colloid Symposium, Madison, Wisc., June 18-22, 1956.
T H E FACTORS AFFECTING THE STABILITY OF HYDROGEN-BONDED POLYPEPTIDE STRUCTURES I N SOLUTION BY JOHN A. SCHELLMAN* Division of Physical Chemistry, Chenzistry Department, University of Minnesota, 1cf inneapolis, Minn. Received A p r i l 21, I968
A previous discussion of the thermal transition of helices is extended t o include the effect of fluctuations, mixed organic solvent systems and pressure. It is found that though fluctuation effects are very large for helices of low stability, the average thermodynamic properties are altered in a relatively minor way. The fluctuation treatment reveals an asymmetry in transitions which has been verified experimentally. An essential difference between the transitions of helices in pure and mixed solvents is demonstrated. The form of the results gives an explanation for the inverted transitions which have been observed in proteins and polypeptides. The calculated properties of helices should have semiquantitative application t o proteins.
It is now known that the various hydrogenbonded polypeptide structures which have been proposed1-a have a sufficiently limited stability in aqueous solution that a large number of the classical denaturation reactions can be qualitatively * Department of Chemistry, University of Oregon, Eugene, Oregon. (1) M . L. Huggins. C h e n . Revs., 3 2 , 195 (1943).
(2) L. Pauling, R. B. Corey and H. R. Eranson, PTOC.Nall. Acad. S c i . ( U . S.), 37,205 (1951). (4) B. Low and H. J. Grenville-Wells. ibid., 99, 785 (1953).
explained as the disruption of these weak structures by heating, denaturants, pH changes, e t ~ . ~ J This interpretation of denaturation is not always valid since cooperative transformations are known in which the configuration of the polypeptide back(4) W. Kauzman, “Rleohanism of Enzyme Action,” W. McElroy and B. Glass, Johns Hopkins Press, Baltimore, Md., 1954. (5) J. Schellman, Compl. rend. trav. Lab. Cadsberg. Ser. chin., 29, No. 15 (1955).
