1426
E. J. ZWOLINSKI, H. EYRING, AND C. E. REESE
The validity of conclusions as to the behavior of the polyamide antifoams in a boiler, based upon observations a t 40°C. and pH 5.0, may be questioned. However, although the temperatures involved in boiler operations will be considerably higher (the temperature of saturated steam a t 250 pounds per square inch, gauge pressure, is about 208°C.) and this increase in temperature will tend to promote spreading of the films, the p H levels usually encountered in modern boiler practice are also much higher, being in the approximate range 10 to 11.5, and from the known effect of p H on hydrogen bonding and our experimental observations, it does not appear to be unreasonable to suppose that the effect of p H would compensate the effect of temperature. SUMMARY
The behavior of the monolayers of three polyamide antifoams has been studied, and the effect on the monolayer of such factors as pH, molecular configuration, and the presence of alkaline earth ions and tannic acid, is reported. The results have been interpreted to explain some of the observed operating characteristics of the polyamide antifoams in terms of a recently proposed theory of their action. REFERENCES (1) ADAMAND DYER:Proc. Roy. SOC.(London) A106, 694-709 (1924). (2) ALEXANDER: Proc. Roy. SOC.(London) A179, 470-85 (1942). (3) COCKBAIN AND SCHULMAN: Trans. Faraday SOC.36, 716-27 (1939). (4) HARKINS AND ANDERSON: J. Am. Chem. SOC.69, 2189 (1937). (5) HARKINS, CARMAN, A N D RIES: J. Am. Chem. SOC.68, 1377-83 (1936). (6) HARKINS AND FISCHER:J. Chem. Phys. 1, 858 (1933). (7) HARKINS A N D MYERS:J. Chem. Phys. 4, 716-24 (1936). (8) JACOBY: J. Phys. Colloid Chem. 62, 689-94 (1948). Proceedings of the Seventh Annual Water Conference, En(9) JACOBY A N D THOMPSON: gineers Society of Western Pennsylvania, January, 1947, pp. 31-41. (10) WILHELMY: Ann. Physik 119, 177 (1863).
DIFFUSION AND MEMBRANE PERMEABILITY. I" BRUNO J. ZWOLINSK1,a HENRY EYRING,
AND
CECIL E. REESE
Department of Chemistry, University of Utah, Salt Lake City, Utah Received March 83, 1949
One of the basic physical phenomena for sustaining the growth and development of plants and organisms is that of diffusion. It has arrested the attention of investigators from the time of Fick (10) in 1855, who, drawing the analogy between conduction of heat and the transport of matter, was responsible for giving a quantitative formulation of the basic laws of diffusion. After some hundred 1 Presented a t the 113th Meeting of the American Chemical Society, which was held in Chicago, Illinois, April, 1948. Contribution No. 82 from the Department of Chemistry of the University of Utah. a American Chemical Society Postdoctoral Fellow, 1947-48.
DIFFUSION AND MEMBRANE PERMEABILITY. I
1427
years of investigation, there are aspects of the problem of diffusion which remain unexplainable. A detailed kinetic approach to diffusion is presented in the hope that it will clarify established concepts and provide impetus to a fresh approach t o the existent problems in the field of biological diffusion. The absolute rate theory treatment of diffusion and membrane permeability provides a general unified point of view applicable t o systems of varying degrees of complexity. It is equally adaptable to the treatment of the permeabilities of membranes t o electrolytes and to non-electrolytes under the driving forces of a concentration gradient, an activity gradient, and external and internal potential gradients. Applications are made of the general equations derived to the permeabilities of egg cells of marine invertebrates and of plant cells to water and to non-electrolytes. I n Section I, Fick’s first and second laws of diffusion for a two-component system are derived by use of difference equations. The general treatment of diffusion from the point of view of rate theory is given in Section 11. For steadystate diffusion, the flux is defined by new equations which take explicit account of all types of potential barriers crossed by the migrating particles. The effect of external forces on the diffusing system is then taken up in Section 111. In Section IV the general equations of Section I1 are simplified on the basis of assumed models of diffusing systems. I n Section V a somewhat detailed analysis is presente,d for determining the mechanism of the permeation process from studies on distribution coefficients and temperature coefficients. Section VI includes calculations of the various thermodynamic functions from permeability data. I. FICK’S FIRST AND SECOND LAWS OF DIFFUSION
Molecular migration in condensed phases may be treated as point-to-point jumps of the elementary particles governed by a rate constant. The nature of the elementary jumps will show very many variations, depending on the nature of the diffusing components. It is instructive to analyze the various types of relaxations or jumps t o see how they lead to the relations usually applied. We consider first a single two-component system with molecules which are sufficiently alike so that the whole may be thought of as forming a more or less perfect lattice. In figure 1 we have a schematic potential diagram. If Ciis the concentration per cubic centimeter at the ithposition, then the amount of material in a square centimeter of cross-section and length h (the distance between equilibrium minima in figure 1) is hCi. Let k represent the number of times per second a molecule jumps. At the steady state, let Q be the amount of C passing per second through a square centimeter of surface. Then
Q = kXCi - kXCi+l The concentration gradient between the ithand the (i
+ l)tbposition is
(1)
1428
B. J. ZWOLINSKI, H. EYRINQ, AND C. E. REESE
Thus
This is Fick's first law, ordinarily written as
-D-dC
Q=
dx where D = kX2, a result derived earlier (9). Fick's second law is obtained equally easily. Thus, the rate equation is
K
'
c-
K
DISTANCE FIG.1. Schematic potential energy profile
Here we have the rate of increase in number of molecules a t the ith position equal t o the difference between the number of molecules jumping into the ith position from the (i - l ) t h and (i l)th positions and the number of molecules leaving the ith position. Now, further, equation 3 gives the concentration gradient (Ci+l - CJ/X at a point halfway between point i and i 1. Similarly (Ci Ci-l)/X is the concentration gradient halfway between the ith and the (i - l)th positions. The second derivative at the ithposition is then
+
+
Ci+i - Ci dx2
h
- Ci+l -
x
- Ci - Ci-i
- 2ci + ci-1 XZ
h
(5)
DIFFUSION AND MEMBRANE PERMEABILITY. I
1429
Rearranging equation 4, we obtain:
It might be pointed out that the approximations made are valid only in the limit where
This implies that a smooth continuous resistance is offered to the diffusing molecule, which mayoonly be approximately true in the case of the thin natural membranes (50-150 A.) whose complex structure of large protein and lipoid molecules
kCY
W
2
....
w w w
E
L
A,
+hi
A,.
FLOW COORDINATE. FIG.2. An energy profile curve in which free energy is plotted against the flow coordinate
may offer an irregular resistance to flow. The latter point is implicit in the derivation in the following section. 11. GENERAL TREATMENT O F DIFFUSION AND PERMEABILITY
By regarding the flow of molecules as a series of successive jumps from one equilibrium position t o another, diffusion in membranes and their permeabilities are readily treated. In figure 2 an energy profile curve is constructed where free energy is plotted against the flow coordinate. Although this schematic energy diagram as drawn represents diffusion through a heterogeneous three-layer membrane or transport of matter through a solution-membrane-solution system, we shall generalize and assume a diffusion system where all the rate constants for forward and reverse flow are unequal. Concentration a t the various minima Ai is defined as concentration of particles per square centimeter cross-section of thickness X in the direction of flow. The specific rate constant in the forward
1430
B. J. ZWOLINSKI, H. EYRINQ, AND C. E. REESE
direction for Ai is defined as ki. Similarly, k: is the specific rate in the (-x) direction from the equilibrium position Ai. It is to be noted that Ai = XiC;. For steady-state conditions the flux at the various maxima is given by the set of rate equations : Q = koXCo - k:XC1
Q
= klXC1
- khC2
......................
(7)
Q = knACn - k:+JCn+l Here the flux Q represents the net flow of material through a unit cross-sectional area per second. We are assuming an ideal system and also taking all the A’s, the distances between the various minima, t o be equal. Solving the second equation for CI and substituting in the first equation one obtains:
+
koACo - - [Q kiACzl ki Then using the third equation to eliminate C2 one obtains:
Q
=
Continuing the process and solving for Q, we arrive at the expression that ko Q = -
,
When the distances hi between equilibrium positions are not the same, we have the more general expression:
Q=
(9)
From the theory of absolute rate processes, a specific reaction rate constant for any process is given by k ’ = ~ k T- -eA F ~ I R T h where K is the transmission coefficient, and k T / h is a frequency factor involving the Boltzmann constant, IC, the absolute temperature, T , and h, Planck’s constant. If we assume the transmission coefficients t o be unity, which appears to be true for most processes, equation 8 simplifies to Q = -h
e-AF:IRTcc
1
- e exp
( A F ;~ AF;)
+AFL;~
+ 2 e exp (- 2 ( A F P - AFT)/ R T ) r-1
Z-1
DIFFUSION AND MEMBRANE PERMEABILITY. I
1431
The rate of movement of matter is thus governed by the relative heights of the potential barriers, as given explicitly in equation 10. Since the specific rate constants enter as ratios, it is the difference in the free energies of activation for the individual unit processes of flow which will determine the flux. 111. EFFECT OF EXTERNAL FORCES
How external forces act on a kinetic system can readily be incorporated into our general expression for the transport of material under steady-state conditions. The procedure to be followed has been successfully employed in interpreting rate phenomena such as plastic flow, creep, viscosity, relaxation of dielectrics, conductance in solution, and other related phenomena (11).
L I
A
d
,
DISTANCE FIG.3
For simplicity of treatment, consider a two-component diffusing system governed by a single unimolecular rate constant. The basic assumption is that the external force acting on a single unit process simply provides an additional amount of work W , which will tend to aid or hinder the process by increasing or decreasing the free energy of the initial and final positions. Assuming a symmetrical potential barrier as diagrammed in figure 3, the work done is given by the force f acting only through the distance X/2 between the initial and the activated states. The net linear velocity in terms of the specific rates for the forward and reverse steps is
v
=
x(/c,
- ICJ = /Cox(ew'kT - e-w'kT)
(11)
This equation can also be written in the form
V
=
Xko 2 sinh (WIkT)
(12)
1432
B . J. ZWOLINSKI, H.. EYRING, AND C. E. R E E S E
With the exception of plastic flow and creep in high-polymeric materials, it is usually found that W > 1, respectively, the flux is
21zoXCo sinh a
(21) When the forces are large, the reverse flow over the potential barrier becomes negligible and the net transport is thus independent of the thickness of the membrane and the final concentration. The second case of small external forces and thin films is of greater practical interest. Here, since CY k,, 111. k , M k,, >> k,, The fourth possibility that 12, z k,,
