SOME FUNDAMENTAL DEFINITIONS AND CONCEPTS IN DIFFUSION PROCESSES

BY G. S. HARTLEY AND J. CRANK

Received I 3th January, I 949

The definition of a diffusion coefficient is examined critically, with particular reference to the necessity of defining clearly the section across which the rate of transfer of the diffusing substance is measured, and the units of concentration and space co-ordinate adopted. It is shown that, by suitable choice of these three factors, the one-dimensional diffusion behaviour of a two-component system can always be represented in terms of a single variable diffusion co- efficient even if the partial molal volumes of the two components vary with composition.

The relations between the diffusion coefficients defined with respect to three different frames of reference of practical importance are established, and several standard methods of measuring diffusion coefficients are examined in order to see which coefficient is observed in each method. Some applications of the different frames of reference are discussed.

The physical nature of a pure diffusion process is discussed in terms of random molecular motions. It is suggested that, in a binary solution, the net rate of transfer of either component is the result of a transfer by pure diffusion coupled with a transfer of that component due to a mass-flow of the whole solution. The mass-flow is due to the fact that, in general, the intrinsic rates of pure diffusion of the two components will differ and hence there is a tendency to set up a hydrostatic pressure in the solution which is relieved by a mass-flow. Reference is made to an experimental demonstration of the mass-flow in a particular system, in which one component is a high polymer.

A tentative quantitative theory is proposed, incorporating the effect of mass-flow, and an expression for the mutual diffusion coefficient in a thermo- dynamically non-ideal binary solution is derived.

1. Definition of a Diffusion Coefficient.-Quantitative measurements of the rate at which a diffusion process occurs are usually expressed in terms of a diffusion coefficient. The present discussion is confined to diffusion in one dimension only, in which case the diffusion coefficient is defined as the rate of transfer of the diffusing substance across unit area of a section, divided by the space gradient of concentration of the substance at the section. Thus, if the rate of transfer is S, C the con- centration of diffusing substance and if x denotes the space co-ordinate, then .-

and ( I ) is a definition of the diffusion coefficient D. In order that this definition shall be unambiguous it is necessary to specify carefully the section used. and the units in which S , C and x are measured. Only the simplest system of practical importance is considered, which is a two- component system, since it is not possible to set up and observe a con- centration gradient of a single substance in itself without introducing complicating features such as pressure gradients, etc. The diffusion of isotopes is best regarded as a special case of a two-component system.

2. A Frame of Reference when the Total Volume of the System remains Constant.-Consider the inter-diffusion of two liquids A and B in a closed vessel and assume that there is no overall change of volume

29 801

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802 DIFFUSION PROCESSES

of the two liquids on mixing. for each liquid, may be defined by the relations

Two diffusion coefficients DI, DZ, one

( 2 )

C, and CB are the concentrations of A and B respectively, each expressed in the usual way in any convenient unit of amount (e.g. g. or, in the case of simple molecular substances, g. mole) per unit overall volume. S, and SB are the rates of transfer of A and B measured in the same units of amount per unit time, across a section which is defined by the condition that the total volume on either side of it remains constant as diffusion proceeds. In the particular case under consideration it is therefore fixed with respect to the containing vessel. The origin from which x is measured is such that the x-co-ordinate of the section is constant. x is measured in normal units of length, e.g. cm., and the same unit of length is used in measuring the volume which appears in the definition of concentration. If the unit of time adopted is the second it follows that the units of DI and DZ are each cm.2 sec.-l. These somewhat obvious statements are made here in full because it will be seen later in 3 4 below that other scales of length and alternative ways of measuring concentration are more suitable in some circumstances.

Let V, and V, denote the constant volumes of the unit amounts used in defining the concentrations of A and B. Thus if C, is expressed in g. per unit volume, V, is the volume of I g. of A. In dilute solutions, where the volume changes in the range of concentration concerned can be considered negligible, V , and V, will be the partial specific or molar volumes. That of the solute may be very different from the specific volume in the pure state. The volume transfer of A per unit time across unit area of the section defined is therefore

3CA -DVV -, A A 3x

3C 3 X

and that of B is

- DZVB >. By definition of the section as one across which there is no net transfer of volume we have immediately,

The volume of A per unit overall volume of solution is VACA and of B is VBCB, so that, since only molecules of A and B are present, we have

which, following differentiation with respect to x, becomes

In order that (4) and (6) shall both be satisfied i t follows that

or else that

If V, = o and VB .t. 0, it follows from (6) that

01 E DZ, . - (7)

VA = 0, or V , = 0. . ' (8)

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G. S. HARTLEY AND J. CRANK 803 and further reference is made to this case in 5 3 ( c ) below. In either case the behaviour of a two-component system, satisfying the condition of zero volume change on mixing, may be described in terms of a single diffusion coefficient, which may vary with composition. It is convenient to refer to the single coefficient as the mutual diffusion coefficient, denoted henceforth by DV. This coefficient is familiar in the interdiffusion of gases.I

3. Some Standard Methods of Measuring Diffusion Coefficients .- An examination of the standard methods of measuring diffusion CO- efficients reveals that for the most part they have been applied to systems in which the total volume remains constant, or can be assumed to do SO with sufficiently good approximation, and that the diffusion coefficient measured is the coefficient DV introduced above.

(a) GRAHAM’S SECOND METHOD.-The so-called second method of Graham 2 consists essentially in placing a solution of the diffusing sub- stance, e.g. NaCl solution, beneath a long column of water and examining the concentrations of the diffusing substance in a number of different layers of the water column at subsequent times. This method has been elaborated by numerous workers and Stefan’s tables are commonly used to facilitate the calculation of a diffusion coefficient from the experi- mental data. Detailed reference to the various forms of apparatus based on this general method are to be found in a review article by Williams and Cady.4 In all cases the origin from which the space CO- ordinate is measured is the base of the containing vessel and so, provided there is no overall change of volume of the system, the diffusion coefficient DV is measured.

(b) STEADY-STATE METHoDs.--If a heavier-than-solvent solid is placed in the bottom of a vertical vessel and the whole vessel immersed in a larger one filled with solvent, diffusion occurs from the saturated solution at the base of the inner vessel to a concentration effectively zero at its open upper end. The dilute solution here “ spills ” over the rim and, if the outer vessel is large enough, the conditions in the inner vessel will remain effectively steady over a long period. This first, or “ bath ” method of Graham5 has been elaborated by later workers and, by arrangement for upward as well as downward “ spillage,” an approxi- mately steady-state diffusion layer can be maintained between any two concentrations (see e.g. Griffiths ‘j). Clack 7 j has described a formal correction for displacement of solvent to bring into line experiments where a solid is dissolving with those in which neither reservoir is saturated. The coefficient obtained by this correction is in effect our D: (see below).

When the two solutions have only very small difference of density, a case of most practical interest in the study of dilute solutions, the method is very sensitive to disturbance by convection. This has largely been overcome in the porous disc method. Since this is normally applied to very dilute solutions, the ambiguities in definition due to volume change are small compared with purely experimental errors and the theoretical treatment as for constant volumes is adequate. The method was introduced by Northrop and Anson and subsequently modified by McBain and Liu.lo The diffusion cell consists essentially of a bell- shaped vessel, closed at the narrow top end by a stopcock and at the wide

Its physical significance is discussed in 3 6 below.

Jeans, Kinetic Theory 0.f Gases (Cambridge, 1940). Graham, Annalen, 1862, 121, I. Stefan, Akad. Wiss . Wien . Math. naturw., Klnsse 11, 1899, 179, 161. Williams and Cady, Chem. Rev., 1934, 14, 171. Graham, Phil. Tyans., 1850, 140, I. Griffiths, Proc. Physic. SOL, 1915, 28, 21. Clack, ibid., 1916, 29, 49. Northrop and Anson, J . Gen. Physiol., 1929, 12, 543.

lo McBain and Liu, J . Amer. Chem. Soc., 1931. 53, 59.

Clack, ibid., 1921, 33, 259.

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804 DIFFUSION PROCESSES bottom end by a sealed-in sintered-glass disc. The cell is filled with the solution, the diffusion rate of which is to be measured, and is immersed in the solvent. The function of the porous disc is to allow diffusion to proceed through i t without interference from convection currents. The quantity observed is the rate of passage of solute through the disc in the steady state condition. The condition satisfied, because of the design of the apparatus, is that the volume of solution in the diffusion cell remains constant and therefore measurements of the rate of transfer relative to the disc are in effect relative to a section fixed so that the total volume on the cell side of it is constant during diffus'on.

(c) OVERALL ABSORPTION BY A PLANE SHEET.-Suppose an infinite plane sheet of material B (in practice a sheet the area of whose plane surfaces is much greater than the area of its edge) is immersed in a bath of solution or in a vapour A. A comparatively simple observation to make is the rate of uptake of solution or vapour A by the sheet B. The simplest case is one in which the dimensions of the sheet do not change appreciably, i.e. the sheet does not swell, as A diffuses into it. This implies that V , = o in which case eqn. (8) and (9) hold. This state of affairs holds approximately for non-zero values of VA if the concentration of A is very small. A method of dealing with the case where volume change is important is dealt with in 0 4. In the former case the rate of diffusion of A is dependent on the coefficient DV and is governed by the familiar equation of diffusion in one dimension,

If DV be assumed constant, (10) takes the simpler foim

A convenient ditions

and the initial

solution of this simpler equation for the boundary con-

C, = C, = constant, x = f I, t > 0, - (11)

c,=o, - z < x x z , t = o , . * (12)

condition

takes the form

11.1,

M t is the total amount of A absorbed by the sheet of thickness 22 at time t, and Moo the equilibrium absorption attained theoretically after infinite time. By comparing the experimentally measured time a t which LWt/hfm achieves a certain value, say 0.5, with the value of DVt/4Z2 a t which M t / M o o calculated from (13) has the same value, DV is readily determined if the thickness of the sheet is known. This method has been used to determine the rate of diffusion of direct dyes into cellulose sheet (Neale and Stringfellow 11), and in studying the absorption of oxygen by muscle (Hill l 2 ) .

(d) OPTICAL METHODS-A number of optical methods have been devised for measuring the refractive index-distance relation or some more complicated relation, from which the diffusion coefficient can be calculated if the relation between concentration and refractive index is known. In some cases, following Wiener l3 and Thovert," observations

l1 Neale and Stringfellow, Trans. Faraday SOC., 1933, 29, 1167. l2 Hill, Proc. Roy. SOC. B, 1928, 104, 39. l 3 Wiener, Wied. Ann. , 1893, 49, 105. l P Thovert, Ann. Physik, 1914, 2, 369.

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G. S. HARTLEY AND J. CRANK 805

are made of the interference bands produced by light passing through different layers of the diffusion cell, or of the deviation of a narrow beam of light, while in others the blurring of an initially sharp boundary be- tween two solutions in contact is observed (Svedberg l5 ; Furthls). A recent development in optical methods is reported by Coulson and others.1' From the point of view of the present paper, however, the optical methods of measurement involve no new principle. In all cases the condition of zero volume change is considered to hold and measurements are made with reference to an initial boundary or fringe on both sides of which the volume remains constant, so that in these methods too, DV is measured.

4. Alternative Frames of Reference. (a) GENERAL.-The definition of the volume-fixed section used in 0 z above is unambiguous only as long as the total volume of the diffusion system remains constant. If there is an overall change of volume of the two components on mixing, the side of the section on which the volume is to remain constant must be chosen arbitrarily, and the diffusion coefficient becomes equally arbit- rary. In such a case some alternative frame of reference must be used in defining the section across which transfer of diffusing substance is to be measured. There are clearly several possible alternatives. Thus, for example, the total mass of the system will always be conserved even though volume is not, and a section can be defined consistently such that the mass of the system on either side of the section remains constant during diffusion.

Where a convention other than that of constant volume on either side is used in defining a section, the second-order differential equation de- scribing diffusion may not take the standard form of eqn. (10). It is clearly convenient if it can be made to do so since the standaxd form has frequently been used as the starting point in calculations of diffusion behaviour. This can always be arranged by departing from the orthodox linear scale, e.g. cm., for measurement of the spacial co-ordinate so far denoted by x, and by measuring concentration in a certain way. Let some modified scale of length be denoted by [, and consider two sections, fixed on the same convention at 5, and 5 + dc. The rate of entry of A into the volume enclosed between these sections is S, and that of departure

is SA + ($)df.

The rate of accumulation is therefore

and this is always true independently of how S, and 5 are measured. It can only be equated to (3CA/3t)d5, however, when C, and 5 are measured in certain consistent units. Thus, if the sections are fixed with respect to total mass, then [must be measured so that equal increments of 5 always include equal increments of total mass, and C, must be defined as the amount of A per unit total mass. Similarly, if the sections are fixed with respect to volume or mass of component B, equal increments of 5 must include equal increments of amount of B, and C, must be ex- pressed as the amount of A per unit amount of B. I n general, for all values of 6 and t, the element of unit length in terms of 6, and of unit cross-sectional area, is that which contains an amount of A equal to the unit used in defining the concentration C,. When the quantities C, and 5 satisfy this condition the usual relation

lS Svedberg, Colloid Chemistry (Chemical Catalog.

16 Furth et al., Kolloid-Z., 1927, 41, 300. l7 Coulson, et al., Proc. Roy. SOC. A , 1948, 192, 382.

2nd ed.

* (14)

Co., New York, 1928),

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806 DIFFUSION PROCESSES

follows at once and by substituting for S, from the relation

we derive the familiar form of the diffusion equation

which reduces to the form ( Ioa) when D is constant. It is convenient that 6 should have the dimension of length and D

the usual dimensions of (length)2 (time)-l. This can be arranged without interfering with the generality or simplicity of eqn. (16), by multiplying the mass of component B (or the total mass of A and B together if this is the reference system being used) by an arbitrary constant specific volume. The volume represented by the product of a mass B, for example, and this arbitrary specific volume will be referred to for convenience as the basic volume of that mass of B.

The concentration of A was defined above as the amount of A per unit amount of B. We now re-define the concentration of A as the amount of A per unit basic volume of B, and unit 6 to contain unit basic volume of B per unit area. A convenient arbitrary specific volume is that of the pure component B, so that the basic volume of a certain mass of B is the volume that mass of B would occupy in the pure state.

The same arbitrary specific volume is used for concentrations expressed in the original definition per unit mass of A and B together, i.e. the basic volume of a mass of A and B together is obtained by multiplying the mass by the same arbitrary specific volume. This is true also for deriving the basic volume of A alone, so that the basic volume has a simple physical significance only in the case of the basic volume of B. Nevertheless the use of this particular basic volume has the convenience that all the concentrations measured in the different frames of reference tend to the same value in dilute solutions.

Concentration is, of course, frequently expressed in a number of different ways and so the symbol C, is retained, but the appropriate index V, B or M as superscript is added, so that the concentration of A is written CI, C: or C: according as to whether the amount of A is contained in unit volume of solution, or in unit basic volume of R or in unit basic volume of total mass. According as to whether unit 6 contains] per unit area, unit basic volume of B or of A and B together the symbol tB or tBf is used. The diffusion coefficients 01, Di, DF also carry an index to indicate the frame of reference to which they refer. The arbitrary specific volume may be denoted by Yi, and then tB and & are defined formally by the respective relations

dtn = VgCE dx, . - (17) dtM = V i ( C I f Ci)dx. . - ( 1 7 4

( b ) SECTIONS FIXED WITH RESPECT TO TOTAL MASS AND MASS OF ONE CoMPoNEm.-It was found in $ 2 that the behaviour of a two-com- ponent system satisfying the condition of zero volume change on mixing, can be represented in terms of a single diffusion coefficient DV. A similar result follows readily for a system in which volume changes occur, pro- vided the diffusion coefficients are defined with respect to a mass-fixed section. Thus thz equation defining such a section is

and the definitions of Cf and C z lead immediately to the equation

Cf + c; = I / V O B' ' - (19)

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G. S. HARTLEY AND J. CRANK 807

D f = Df. . - (20)

On differentiating (19) with respect to tM and comparing with (IS) we find

If a section fixed with respect to one component, say B, is used, then clearly DE z= o and only the coefficient D; is needed to describe the dif- fusion behaviour. Thus the statement that the diffusion behaviour of a two-component system can be described in terms of a single diffusion coefficient, is valid whether there is a change of volume of the whole system or not, provided the appropriate irame of reference is used in defining the diffusion coefficient. Frames of reference could be so chosen that the two coefficients are not identical and neither is zero, but they would be related through some function of the partial volumes and would not be independent measures of two separate diffusion processes. The possi- bility of measuring the diffusion of the two molecular species independently is discussed in 3 7.

(c) RELATIONS BETWEEN THE DIFFUSION COEFFICIENTS : 01, DZ, D:.- The rate of transfer of A through a B-fixed section is greater than that through a total-mass-fixed section by an amount given by the con- centration of A per unit mass of B multiplied by the flux of B across the mass-fixed section. Thus the flux of A across a B-fixed section in the direction of .$ increasing is

using (19).

that we have But the rate of transfer across a B-fixed section is - D;3C”,3SB SO

From the definitions of C: and C t it is easy to show that

so that finally, by substituting (25) and (23) in (zz), we find,

Dt = Dl“r(V;CZ)2 . - (26) since rearrangement of the partial derivatives in (22) is permissible. For a system in which there is zero volume change on mixing, so that V A and V B are constant, the relation between D: and DV can be similarly established. Thus the flux of A. across a B-fixed section in the diregtion of 5 increasing is

using ( 5 ) and ( 6 ) . is - D: 3C:/3&,, so that we have

But the rate of transfer of A across a B-fixed section

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808 DIFFUSION PROCESSES

From the definition of CI and C: i t follows that

and from (24) we have immediately,

* (30) - - VBC,', .

0; = Dv( T/'BC:)'. . - (31)

3 X so that on substituting (29) and (30) in (28) we find

Since (31) applies only when VB is constant, and therefore when VB = Vg, comparison of (31) and (26) shows that when there is no overall volume change accompanying diffusion

D M = DV(C,V/C,M)2 = DV (Basic total volume/true total volume)2. (32) 5. Applications of the New Scales of Length and Concentration.-

It is proposed to discuss briefly two examples which arise in practice where i t is desirable to adopt one of the alternative frames of reference just discussed even though there is not necessarily a change of volume of the whole system.

study the rate of diffusion of a gas or a vapour A through a polymer B, a modified form of the porous disc method may be used, in which the disc is replaced by a membrane of the polymer and the steady rate of flow of A through it is observed. If the membrane swells as the steady state is established, and this is most likely to occur, then the thickness of the membrane in ordinary units of length, e.g. cm., will vary with time. In such a case the modified scale of length tB may usefully be introduced since the thickness on this scale, being the basic volume of polymer per unit area of the membrane, remains constant and equal to its unswollen value. We are dealing here with transfer across a section fixed with respect to one component and the rate of transfer, S,, of A through unit area of the membrane is given by

(a) STEADY-STATE DIFFUSION THROUGH A MEMBRANE.- -h order to

Integrating (33) with respect to SB, remembering that for steady state flow S, is independent of fB, we have

(34)

where a is the thickness of the unswollen membrane, and the con- centrations of A a t the two surfaces of the membrane are C, and zero. By determining S, experimentally for different values of C o and differ- entiating the curve relating Sb to Co, the diffusion coefficient Dz is ob- tained for various concentrations of A. This method has been applied by King18 to the diffusion of water through a horn keratin membrane, though the swelling of the membrane and its effect on the diffusion co- efficient measured was not discussed. If the swelling is in accordance with the condition of zero overall change of volume of the system, the coefficient DV may be derived by use of eqn. (31).

tion of a vapour or liquid A by a sheet B has already been stated in 9 3 ( c ) for the case in which no swelling of the sheet occurs. Thus in the boundary condition (11) and the mathematical solution (13) the thickness 21 of the sheet is assumed to remain constant. If swelling occurs, however, the condition (11) is to be applied on moving boundaries. Furthermore,

(b) ABSORPTION BY A SWELLING SHEET.-The problem Of the absorp-

King, Trans. Faraday SOC., 1945, 41, 479.

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G. S. HARTLEY AND J. CRANK 8% the motion of the boundaries is not known till the solution of the diffusion equation has been evaluated and this requires knowledge of the motion of the boundaries. The problem is therefore awkward and would, in this form, involve some method of successive approximations. When re-formulated in terms of the variable of length tB, however, these diffi- culties are removed and the boundaries of the sheet correspond to constant values of f B . The solution (13) then applies with 21 denoting the un- swollen thickness of the sheet and with DV replaced by the coefficient Di provided this is constant. In the likely event of D: being dependent on the concentration of A, the comparison of experimental and calculated absorption-time curves suggested in Q 3(c) yields some mean value of Dt. A method of deriving the concentration dependence of the diffusion coefficient from a number of mean values of this kind is described else- where (Crank and Park In).

(c) DIFFUSION OF ONE SUBSTANCE THROUGH A SECOND SUBSTANCE WHICH IS CONFINED BETWEEN MEMBRANES.-A case of considerable theoretical interest is the steady state of diffusion of one substance through another when the second substance is confined between membranes. Let the membranes be impermeable to B and let the concentrations of A in contact with the membrane be maintained by supply and removal of A through the membranes from and to reservoirs of vapour or of solu- tion of A in B or other impermeant substance.

At the steady state, sections at fixed distances from the membranes are fixed with respect to amount of component B, and the rates of transfer of A across all such sections must be equal, so that

We will assume that the partial volumes are constant. By substitution of (24), (29) and (31)~ (35) becomes

DV ac: v,c; 3%

- -- - = const. .

Substituting from (6) we have DV 3Cl - - = const. . v,c; ax

(37)

If, instead, we maintain the steady state by substituting membranes permeable only to B and supplying and removing B in the reverse direc- tion we find, by interchange of A and B in (36) and reversal of signs to allow for reversal of direction

DV -- = const. . v,cI ax

It is evident that eqn. (37) and (38) require different concentration- distance functions. The steady state is therefore different, even between the same concentration limits, according as to which component is con- strained and which is free to diffuse.

We shall assume here, in the interests of further simplicity, that DV is constant. The concentration of A being very low, VBCi can be assumed to be unity. Eqn. (36) now becomes

This difference is particularly evident in dilute solutions.

D V E: = constant rate of transfer of A. . * (364 3%

This is valid for A diffusing and B restrained. l9 Crank and Park, Trans. Faradny sot., 1949, 45, 240.

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810 DIFFUSION PROCESSES Substituting from (6) we may modify (38) to

DV acz DV 3 - - _ _ - - (log Cx) = constant rate of transfer of B. (38a) VBC: ax VB3x

This is valid for A restrained and B diffusing between the same concen- tration limits.

It will be seen that, when the dilute component is diffusing, its con- centration gradient, under these simple conditions, is linear. When the same limits are maintained by diffusing the " solvent " through membranes impermeable to the solute, the gradient of the logarithm of concentration of the latter is linear and hence the concentration itself varies exponentially with distance.

When the total fall of concentration ACI of the dilute is small compared with its mean concentration, (36a) becomes

DV rate of transfer of A = - ACI, .

and (38a) becomes Dv ACI

a C,'VB equivalent rate of transfer of B = - - .

where a is the distance between the membranes. The rate

constituent

* ( 3 8 4

of diffusion of water in such a system down a given small mean vapour pressure gradient is therefore expected not to be constant but to be inversely proportional to the mean concentration of the solute. It will further be proportional to the diffusion velocity of the solute given by (36b). Thus if water diffuses from 99.1 to 99.0 :h relative humidity through a layer containing hydrogen chloride, it will do so about twice as rapidly as when the layer contains sodium chloride. If the diffusion occurs from 90.1 to 90-0 yo relative humidity, each rate will fall to one-tenth of its former value.

These conclusions refer to the case where the membranes are separ- ated by a fixed distance. If, as is more likely to be true in practice, the membranes confine a given amount of component A so that the volume between them will vary inversely as the mean concentration of A, the distance a in (38b) will be more nearly inversely proportional to C, than constant. In this case the rate of transfeI of B will be, to a first approxi- mation, dependent on ACA only and not on the mean value of C, This result would be obtained diiectly of course from equations in 5 as in the treatment already given of the swelling membrane. Diffusion of solvent through a constant amount (per unit area) of solute is thus simpler than diffusion through a constant thickness. This conclusion is not a t once obvious and may have some important applications in physiological processes.

A second conclusion of interest may be drawn if we consider what happens in such a steady state system, if the membranes are suddenly rendered completely impermeable. Sections fixed with respect to the membranes are now fixed with respect to volume of solution. The change of concentration with time will therefore from now on be governed by

(39)

but, a t the instant of change of membranes, eqn. (36) still holds if A has been the diffusing component, whence

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G. Combining (39) and

since

S. HARTLEY AND J. CRANK 811 (40) we obtain

(3, = - const. x v,, . ' (4')

for CI constant. Now the constant in (35) and (41) is the rate of transfer of A at the steady state in standard units of amount. Multiplied by I/, it represents the volume rate of transfer of A, or, since we are always considering transfer across unit area, the linear apparent velocj t y with which A was passing through the system.

We thus find that, at the instant of cessation of flow through of A, the whole concentration-distance distribution commences to move back- wards at the velocity with which substance A previously passed through the membrane. With increasing time, the distribution will, of course, flatten out from the low concentration upwards as the substance A diffusing down the gradient now accumulates.

It js evident, therefore, that in the steady state during the passage of substance A, there was superimposed on the statistical diffusion process a real flow of the whole system. The physical significance of compensat- ing mass-flow is dealt with in 0 6.

6. Physical Picture of Pure Diffusion and Compensating Mass- flow.-Probably a majority of students are first introduced to the subject of diffusion by way of the classical experiment in which a tall cylindrical vessel has the lower part filled with, say, iodine solution and a column of solvent is carefully placed on top so that no convection currents are set up. In this initial state there is a sharp, well-defined boundary separating the coloured solution from the pure solvent above it. On inspecting the system later it is found that the upper part has become coloured, the lower part being correspondingly less intensely coloured, and after sufficient time the whole solution appears uniformly coloured. It is therefore evident that a transfer of iodine molecules from the lower to the upper part of the vessel has occurred, this transfer taking place in the absence of convection currents. This is the process commonly re- ferred to as diffusion, the iodine being said to have diffused into the solvent.

Now if i t were possible to watch the individual molecules of iodine, and this can be done effectively by replacing them by particles small enough to share the molecular motions but just large enough to be visible under a microscope, it would be found that the motion of each molecule is a random one. In a dilute solution each molecule of iodine behaves independently of the others, which it seldom meets, and each is constantly undergoing collision with solvent molecules, as a result of which col1isioI:s it moves sometimes towards a region of higher, sometimes of lower, con- centration, having no preferred direction of motion towards one or the other. The motion of a single molecule can be descr,bed in terms of the familiar " random walk " picture, and whilst it is possible to calculate the mean square distance travelled in a given interval of time it is not possible to say in what direction the molecule will move in that time. That is, if a molecule is observed a t a given time the contours of equally probable displacements at successive times will be consecutive spheres with centre at the original position of the molecule, provided that the time between two successive observations is great compared with the time between two successive molecular collisions.

This picture of random molecular motions, in u-hich no molecule has a preferred direction of motion, has to be reconciled with the fact that a transfer of iodine molecules from the region of higher to that of lower concentration is nevertheless observed. Consider any horizont 71

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812 DIFFUSION PROCESSES section in the solution and two thin equal elements of volume one just below and one just above the section. Though it is not possible to say which way any particular molecule will move in a given interval of time, it can be said that on the average a definite fraction of the molecules in the lower element of volume will cross the section from below and the same fraction of molecules in the upper element will cross the section from above, in a given time. Thus, simply because there are more mole- cules in the lower element than in the upper one, there is a net transfer of molecules from the lower to the upper side of the section as a result of the random molecular motions.

Where the solution is thermodynamically non-ideal, the movement will not be entirely random, even apart from the effect of compensating mass flow referred to below. If the molecules, as is generally the case, find themselves in a less favourable molecular force environment in solution than in the pure state (or more concentrated solution), then, in a gradient of concentration, they will experience a real average force in the direction of higher concentration and therefore have a component of real velocity opposing the statistical flow.

Similarly, in the presence of a temperature gradient, a gradient of molecular force will arise which will not in general be the same for both components. This will give rise to a real velocity of the molecules which will cause a slight separation of the two components and a con- centration gradient will be built up. This phenomenon, the Soret effect, is usually called “ thermal diffusion”. For the reasons discussed above, Toms and one of the present authors (G. S. H.20) have suggested that “ thermal migration ” would be a more correct name. In the correspond- ing phenomenon in gases the primary effect in the thermal separation is difference in mass of molecules and thermal diffusion may be a more correct term. There is no doubt, from the highly specific nature of the effect in liquids, that the force field effect predominates.

In the simplest case, where the molecules of A and B are identical in mass and size, the rates of transfer of the two components, due to random motion across a volume fixed section, may reasonably be ex- pected to be equal and opposite. In general, however, differences of mass and size of A and B molecules result in the transfer of A by random motions being greater or less than that of B. Consequently a hydro- static pressure tends to be built up in the region of the solution which contributes least to the volume rate of transfer. This pressure is relieved by a compensating mass-flow of A and B together, that is of the whole solution (Hartley 21).

The overall rate of transfer, say of component A, across a volume- fixed section may thus be expressed as the combined effect of a mass-flow and a transfer due to pure diffusion, that is due to the random motions of non-uniformly distributed A molecules. Since the amount of A trans- ferred by the mass-flow of the whole solution is proportional to the concen- tration of A, it follows that the diffusion coefficient DI must necessarily be a function of the concentration of A, a result which was deduced above by a slightly different method.

Whilst the resolution of the overall transfer of matter into a mass- flow and a pure diffusion process, is mathematically simple, in the physical picture of such a combination it is not quite as easy to separate out diffusing molecules from those taking part in both processes simultaneously.

It is, however, possible to demonstrate the separate existence of pure diffusion and mass-flow in the case of gases,when diffusion occurs across a porous plate which offers considerable viscous resistance. In this case an increased pressure is found to arise in that part of the vessel occupied initially by the slower diffusing component. Similarly, in a

2 0 Hartley and Toms, Nature, 1946, 158, 451. Hartley, Trans. Faraday Soc., 1946. 42B, 6.

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G. S. HARTLEY AND J. CRANK 813 liquid system, the separation takes a tangible form if, between the two solutions, a membrane is inserted which is permeable to one component only. In this case diffusion occurs until a definite equilibrium pressure, familiar as the osmotic pressure, is built up and restrains further diffusion.

The interpretation for liquids and solids appears simpler if the current model of diffusion, in which it is assumed that a molecule spends most of its time oscillating within a cage or hole formed by the closelypacked surrounding molecules is accepted. The molecule performing this oscillatory motion will occasionally acquire sufficient energy to break through the surrounding potential barrier and thus form a new constella- tion of neighbours. It is then said to undergo diffusion, and calculations of the energy required by a molecule before it can jump in this way show that, in a typical case, less than I yo of the total number of molecules are undergoing a net displacement relative to the surrounding molecules a t any time. Thus in this case it is permissible to think of the diffusing molecules, i.e. those undergoing jumps, as distinct from the surrounding oscillating molecules which take part in the mass flow, if any. The theoretical diffusion coefficient, calculated in terms of the frequency of the molecular jumps, thus gives the rate of diffusion relative to surrounding molecules, i.e. mass-flow is not included.

The distinction between jumping molecules and those which merely oscillate enables us to suggest a molecular mechanism for the compensating mass-flow. Molecules do not always jump into holes which are adequate to accommodate them. They may pass through a very unstable stage of high potential energy to a new position which, although it has lower energy than the barrier just passed, has higher energy than the space which has been vacated. As a result the new constellation of neighbours is pushed slightly further apart in order to reduce potential energy. This effect will be greater in general where it results from the molecules which are intrinsically capable of more rapid diffusion. The additive effect over a large number of such processes is the microscopic increase of hydro- static pressure which is relieved by the mass-flow.

We have mentioned above that the microscopic picture of pure dif- fusion can be represented in terms of a series of contour shells about the initial position of any molecule, which represent equal probabilities of displacement after a finite time interval. In the general case, when mass-flow takes place concurrently with pure diffusion, these contours will assume an ellipsoidal form which can be resolved mathematically into the sum of a random displacement proportional to the square root of the time interval and a linear displacement proportional to the time interval.

7. Intrinsic Diffusion Coefficients.-From the point of view of interpreting diffusion coefficients in terms of molecular motions, the mutual diffusion coefficient, Dv, thus appears to be unnecessarily com- plicated by the presence of the mass-flow. It is desirable to define new diffusion coefficients, gA and gB in terms of the rate of transfer of A and B, respectively, across a section fixed so that no mass-flow occurs through it. Such a section may be in practice impossible to determine, except in special conditions mentioned below. It is fixed in a different way from any of the other sections previously dealt with, and it must follow the mass-flow althpugh this flow is not normally directly observable. These new dif- fusion coefficients will be referred to as “ intrinsic diffusion coefficients.” When the partial volumes axe constant they are related to the mutual diffusion coefficient in the following way.

On one side of a section fixed so that no mass-flow occurs through it, there is a rate of accumulation of total volume of solution, which may be denoted by 4, where

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814 DIFFUSION PROCESSES 4, as thus defined, is actually the rate of increase of volume on the side of smaller x, arid this must be equal to the rate of transfer of total volume by mass-flow across a volume-fixed section. Such a mass-flow involves a rate of transfer of A of 4C1, so that, equating two expressions for the net rate of transfer of A across the volume-fixed section, we find

On substituting for 4 from (42) and using (6) we have finally Dv = VACI (BB - 9 ~ ) + 9~ . * (44)

If the molal volumes vary with composition, the coefficient DV has no significance, but gA, gB can still be defined in terms of the rates of transfer of A and B respectively across a section which moves so that there is no mass-flow of A and B together, through it . It is convenient in this case to relate the intrinsic cbffusion coefficients to D:, Since the net rate of transfer of B through a €3-fixed section is, by definition, zero, it follows that the contributions to the transfer of B resulting from the overall mass-flow and from the true diffusion of B relative to the mass- flow, must be equal and opposite. The rate of transfer of B by true dif- fusion relative to the mass-flow is

ac; 9 B -

ax in the direction of x increasing, and hence the volume transfer of the whole solution accompanying mass-flow with respect to the B-fixed section is given by

in the direction of x increasing. This produces a fate of transfer of A through the B-fixed section of

due to the mass-flow. of A relative to the mass-flow which is given by

This is to be combined with the rate of transfer

to give the net rate of transfer of A across a B-fixed section, which is simply

When the molal volumes are not constant, the relation

still holds, but the differentiated form (6) is to be replaced by VACA + VBCB = 1,

Since

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G. S. HARTLEY AND J. CRANK 815 it follows immediately from (30), (45) and (46) that

This reduces to (44) when V , and VB are constant. It is not possible, therefore, to deduce the values of gA and gB separ-

ately, unless some other information is available or some approximation can be made. The frame of reference is not one which can normally be located experimentally. One of us (Hartley 21) has introduced a tech- nique which makes it possible to do so in certain special circumstances. We require reference points which move with the bulk of the solution as a whole and these can be provided by particles of inert solid, too large to diffuse, suspended in the solution, which must, however, be so viscous that the movement of the particles under gravity can be ignored. This is possible, in the case of a swelling polymer, over a considerable range of concentrations. The particles can be inserted into the original un- swollen polymer in the form of a scale which can then be used as an experimental 5 scale for diffusion calculations. As it has been exploited so far, it is doubtful whether this method can be used in concentrations where the value of gB (referring to the polymer molecules) is distinguish- able from zero. The method is capable of further refinement, however, and eventually it may be possible to use it for actual measurement of gB.

Where gB can be assumed zero, substitution in (44) gives

In this case, therefore, the intrinsic diffusion coefficient of A and its dependence on the concentration of A can be deduced from observations of DV.

8 . A Tentative Quantitative Theory for all Concentrations of a Binary Solution.-According to Nernst's method, the solute, in a con- centration gradient, is considered to be driven down the gradient by a force determined by the gradient of osmotic pressure. The velocity of the solute molecules under the influence of this force is obtained by dividing by a resistance coefficient which, according to Einstein, can be taken to be that of a sphere of volume equal to the molecular volume. There is some disagreement about this derivation of the coefficient, e.g. it is considered that 4"" gives a better fit for small molecules than the 6"" of Stokes' law for macroscopic spheres of radius a. There is also good evidence that, for extremely unsymmetrical molecules, the estimate requires correction. These points, however, will not concern us here. We shall simply accept the resistance coefficient as a ~ , where u is a function of molecular size and shape without necessarily assuming it to be inde- pendent of the viscosity 7). We might therefore replace or] by a single symbol but we leave it as a product to indicate that, in simple solutions, a is approximately independent of 7.

The conception of an osmotic pressure gradient as a driving force is theoretically unsatisfactory for several reasons, not least of which is one noted above, that, in pure diffusion in a simple solution, no molecule has any finite average velocity in any preferred direction. The osmotic pressure method does, however, give the correct result and there follows below an equivalent treatment free from the theoretical objections.

Let the gradient of (low) concentration C, of A be maintained in an equilibrium condition by the application of a force FA per g. mole of A in the direction of increasing x. This is purely a hypothetical operation but can be realized in the case of large molecules much different in density from the solvent, by a centrifugal field. The condition for this thermo- dynamic equilibrium is, for an ideal dilute solution,

a log c, FA = RT-- * (49) ?x * -

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816 DIFFUSION PROCESSES In the absence of the concentration gradient the force FA will produce

a real average velocity vA of the A molecules in the direction of increasing x , where

and therefore a rate of transfer SA g . mole/sec. across unit cross-section, where

and N is Avogadro’s number. Since this real migration is exactly balanced when the concentration

gradient is operating together with the force FA, we may assume that, in the absence of the force, the gradient will produce a transfer, by dif- fusion, of equal magnitude but opposite in sign (S, = - S’A). Since the diffusion transfer S, is defined by

we obtain, combining (49), (51) and ( 5 2 ) ,

The thermodynamic relation of a concentration gradient to the force necessary to maintain it, and the equality in magnitude but opposition in sign of the transfers of solute produced by the gradient and the force acting separately, is the logical essence of the classical method of cal- culating diffusion coefficients in liquid systems.

In attempting to extend the theory to concentrated solutions three complicating factors must be taken into account-

(i) The thermodynamic equilibrium equation (49) must be generalized to make allowance for the deviation from ideal solution behaviour which will usually arise when the concentration is increased.

(ii) Some correction must be made for the effect of interference between solute molecules on their velocity under the influence of a steady force, when their distribution may no longer be considered entirely random. (The “ relaxation ” and “ endosmotic ” terms of the Debye-Huckel theory of electrolyte conductivity are a special case of this.)

(iii) Allowance must be made for the displacement of one molecular species by the other.

We shall not consider (ii) further in detail in this discussion, but simply accept the possibility that, in the expression for the average resistance coefficient, oar], of A molecules in the solution, the value of u, may in- volve other factors than the effective radius of individual A molecules and may therefore vary with the composition of the solution. (i) is simply dealt with by orthodox thermodynamics, and if suitable measure- ments have been made in the system, e.g. of vapour pressure at all compositions, the correction can be made quantitatively precise. The relevant equations are interpolated below in dealing quantitatively with (iii) which has been discussed qualitatively above. In what follows we shall assume for simplicity that the partial volumes are constant.

A generalized theory has already been presented by Ole Lamm.22 This author presents the thermodynamic correction in precisely the same way but treats the viscous resistance problem in a radically different way which seems to us to be even more remote than our own from prac- tical usefulness. He does not make the distinction as we have done be- tween pure diffusion and real flow and it seemed simpler to treat the

22Lamm, J . Physic. Chem., 1947, 51, 1063.

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G. S. HARTLEY AND J. CRANK 817 subject anew rather than to attempt to relate our conclusions to his equations.

According to the accepted definition of the activity coefficient f , the thermodynamic equilibrium equation (49) becomes, in its generalized form

where N , is the mol fraction of A. Similarly

WE. may note that the total force acting, at equilibrium, on a volume containing a total of I g. mole of A and B together, is

(56) N F log N A f A 3 N A log N B f B E B )

A A -I- NBFB = RT( d log N A 3X + d log N B 3~ ' by (54) and (55) . Since the Duhem relationship gives

and, since

and therefore

it follows from (56) that this total force is zero, as is of course necessitated by the condition of equilibrium. We have now to estimate the transfer rates of A and B produced by the forces F A and F B if acting in the absence of the balancing composition gradient.

The hydrodynamics of the movement of closely contiguous spheres is obviously a subject of great complexity. It is, however, unimportant, in the derivation of the condition of thermodynamic equilibrium between a concentration gradient and a maintaining force, whether this force operates uniformly over all the molecules or is unequally distributed be- tween them, provided the distribution is entirely indiscriminate and changes with time. We shall therefore considerably simplify the hydro- dynamic part of the calculation by considering the force, F A , acting per g . mole of A, to operate at any one instant only on so small a fraction of the A molecules that the velocity of these molecules is uninfluenced by the numerous other A molecules in the vicinity.

Thus we replace the force F A I N acting on every molecule (at a given value of x ) by a force kFA/N acting on a small fraction, I / k only, of these molecules. These will have a velocity kvA where

k F A k V A = -

N U A T ' '

and the resulting rake of transfer of A is

* (59) k F ~ g. mole/sec. .

Since this is equal and opposite to the rate of transfer of A by diffusion, in absence of the force, relative to a section through which there is no mass-flow, we have on substituting for F A from (54)

* (60) 3 C A C A R T b log h T A . f A gA- -=- JX N U A T ax

and hence,

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818 HYDROGEN BONDING IN NITRONAPHTHYLAMINES By using the relations

i t can be shown that CAVA + CBV, = I, N A + N B = I, NAG, = NBCA,

so that (61) may be written

Similarly it can be shown that

On substituting from (63) and (64) in (44), recalling (57), we have finally

This is an expression for the mutual diffusion coefficient Dv in a binary solution, whose thermodynamic behaviour is non-ideal, when the effect of the overall mass-flow of the solution is included.

Courtadds Limited, Research Laboratory,

Maidenhead.

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