A serious experimental study of the diffusion of one solution into another was first under- taken by Thomas Graham (I), who devised a number of experimental techniques for studying the phenom- enon and obtained a great deal of qualitative and quantitative data. His achievements in this field have perhaps been overrated and he is more justly remembered for his distinction between crystalloids and colloids in terms of the different rates a t which they pass through membranes, and for his description of the process of dialysis. A far more important name in the early literature of the subject is that of Adolf Fick (1829-1901). "Fick's Laws of diffusion" and the "Fickian frame of reference" are terms which fre- quently appear in modern papers on transport prop- erties. Virtually all experimental papers on diffusion are concerned, in the first instance, with the deter- mination of diffusion coefficients defined in a manner similar to that proposed by Fick, namely.

J = -D dcldz (1)

H. J. V. Tyrrell The University

Sheffield, England

where J is the one-dimensional flow in moles or grams per unit area per unit time across a reference plane, D is the diffusion coefficient, and c is the concentration in moles or grams per unit volume. An equation of this Form is often termed Fick's first law, and the derived equation.

The Origin and Present Status of Fick's Diffusion Law

Fick's second law. Furthermore, attempts to obtain semiempirical equations to predict values of diffusion coefficients from other physical properties have a long history, while a priori calculations of transport coef- ficients, including diffusion coefficients, in liquids from statistical-mechanical considerations have at- tracted considerable attention from modern theore- ticians. Clearly, Fick's ideas have proved fruitful for more than a century, almost as fruitful in fact as the very similar ideas of Ohm in the field of electrical conduction, and of Fourier in that of heat conduction. Yet his paper on diffusion is usually misquoted; his name, unlike those of Fourier and Ohm does not appear in general works of reference in English, such as "Encylopaedia Rrittanica" or "Chamber's Encyclopae- dia," and the details of his career are unknown to most physical scientists, though net to physiologists. This paper is not intended as a biographical memoir but rather as a review of the ideas which led Fick to his conception of the basic diffusion law (Grundgesetz) and an assessment of its present importance.

Fick, the youngest of nine children, showed a strong mathematical bent a t school in Cassel, where he graduated from the gymnasium in 1847. He proceeded to the University of Marburg where his eldest brother

Ludwig, sixteen years older than he, was Professor of Anatomy, and another brother, Heinrich, seven years his senior, and later Professor of Commercial Law a t Ziirich, was then privatdozent. Adolf's original inten- tion was to take mathematics, but Heinrich, with great perception, suggested that he should read for a medical degree, studying mathematics as a subsidiary subject, since his aptitude for this would be of ines- timable value if he pursued a career in medical re- search. This judgement was later fully vindicated, because Adolf Fick became an outstanding figure in that small group of nineteenth century physiologist8

Ad0lf Fick

who applied the concepts and methods of physics to the study of living organisnls, and thereby laid the founda- tions of modern physiology. He was graduated as a doctor of medicine from Marburg in 1851, and early in 1852 was invited by his friend and teacher from earlier days in hlarburg, Carl Ludwig, then Pro- fessor of Anatomy and Physiology in Ziirich, to take up a post as Assistant (Prosektor) in the Anatomy Dc- partment there. The contact with Ludwig was re- sponsible for Fick's interest in diffusion, since Ludwig, in common with several other physiologists of the period, had been interested in diffusion through mem- branes for some years previously.

Fick's principal paper on diffusion entitled "Ueber Diffusion" appeared in Poggendorff's Annalen in 1856 (94, p. 59). It was clearly intended to be read by physical scientists. Following a not uncommon prac- tice, he comnlunicated the same results in a modiied form to the Zeitschrift fur rationelle Medicin (6, 28% (1955)) (2) in order to bring his ideas to the attention of physiologists and medical men generally. The paper in Annala begins with the remark that hydrodiffusion

Volume 41, Number 7, July 1964 / 397

through membranes is an important physiological problem which should also be of great interest to the physicist. After quoting several references to earlier work in this field, Fick stated that he had himself car- ried out experiments of this kind but with limited suc- cess. Smce it was not possible to carry the work fur- ther he proposed to describe the results as far as they went, and to draw attention to the analogy, which had formerly been neglected, between diffusion through membranes and the "simple spreading of a soluble sub- stance in its solvent." The quantitative aspects of Graham's work were strongly criticized and Fick went on to announce his intention of discovering the basic law governing the transfer of material from one layer of solvent to the next. To do this he began by devising a model to be used in describing the state of matter taken up by a single substance and the process of the diffusion of one substance into another. It was con- ceived in terns of attractive and repulsive forces be- tween particles. A single substance was assumed to be made up of two kinds of "atoms,"' "ponderable atoms" attracting one another by gravitational forces, and "aether atoms" which repelled one another with a force proportional to the product of their masses and to an inverse power of the distance r between them which was greater than 2, indicated by f(r). Likewise, the two types of "atoms" were assumed to attract one another with a force, also proportional to the product of their masses, which fell off less rapidly with increasing separation than f(r). This distance function was termed ~ ( r ) . According to Fick, a model of this kind leads to a system in which each "ponderable atom'' acts as the center of a sphere of "aether atoms," the density of the latter falling off as the distance from the central "atom" increases. Such aggregates he termed "molecules." Diffusion is discussed in terms of two assemblies, one of "molecules" of species A, and the other of species B, placed side by side. Assuming that the attraction between "n~olecules" of unlike species is greater than that between those of like species, "mole- cules" of species A will be drawn into the region formerly occupied by those of B and vice versa, until the "mole- cules" of each species are distributed uniformly over the whole region. The model can be used to explain why "molecules" of A and B do not coalesce entirely; the gravitational attraction of the two species of "ponder- able" atoms is balanced before this can occur by the in- creasing repulsive forces between the surrounding spherical shells of "aether atoms." The most im- portant aspects of this to modern eyes are the very clear recognition of the part played by molecular motion (using molecule in the modern sense) in diffusion proc- esses, and the use of a model in which the macroscopic properties of matter depend upon the balance of at- tractive and repulsive forces between microscopic particles, an idea which is of central importance in all modern theoretical treatments of similar problems. It is of considerable historical interest to note that, a t this date, Fick found it necessary to introduce a spe- cial hypothesis to explain the random distribution of A and B "molecules" which was the final result of the diffusion process. Graham was faced with the same

' Where the terms "atom" and "molecule" are used in a sense other than a modern one, they areput hetween quotation marks.

problem and solved it in the same way, rather than by applying the essentially correct explanation given by Dalton of the interdiiusion of two gases (that the proc- ess was akin to the expansion of a gas into a vacuum) to the case of liquid diffusion, though he mentions this possibility in afootnote. The revolution brought about by the kinetic theory of gases can be seen from the standpoint taken up by Fick in a series of six popular lectures "Die Naturkriifte in ihrer Wechselsbeziehung" published in 1869, where his views on this subject are essentially those held today (2).

Fick's reasoning can best be followed from this point by direct quotation:

The first task would now he ta establish the basic law far this mode of motion [i.e., of diffusion] from the general laws of mo- tion, and this should, I hope, d m be possible without knowing the functions f(r) and q(r). My attempts to do this have how- ever had no success. On the other hand, when first considering this hasic law, it occurred to me that a very similar supposition has, in my opinion, been fully proved hy experiment. In fact, it might be added, that from the outset, nothing is more probable than this; the spreading of a. dissolved suhstance in a solvent, provided that it is not affected by the exceptional influence of molecular farce^,^ takes place according to the same law which Fourier has suggested for the spreading of heat in a conductor, and which Ohm has already applied to the spreading of elec- tricity, where it is, of course, not quite correct.' It is only neces- sary to replace, in Fourier's law, the words "quantity of heat" with the words "quantity of dissolved substance," and the word "temperature" by "solution density." The conductivity, in our case, corresponds ta a constant dependent on the affinity (ver- wmdsehaft) of the two substances.

Fick thus obtained the "Grundgesetz" for the diffu- sion process solely by analogy with the laws of Fourier and Ohm, and could give no a priori reason for the choice of the solution density gradient as the driving force for diffusion. In applying this law to real sys- tems, he specified that the volume change on mixing the diffusing solutions must be ignored; this proviso enabled him to pass from the solution density gradient to the concentration gradient as the driving force, con- centration being expressed as weight units per unit volume. He also pointed out that the density gra- dient had to be of such a kind that the heavier layers of liquid lay below the lighter ones.

The mathematical development of the basic law pro- ceeded as follows:

A quantity nf salt will he transferred into the elementary layer between the planes z and z + d z (where the concentration is y) from that between x + d z and z + 2dz (wbere the concentration is y + d y l d z . d z ) equal to -Q k d y l d z dt , where Q is the cross- sectional area of the layer, and k s. constant dependent on the nature of the substances. Naturally, an amount of water equal in volume to this amount of salt enters from the upper into the lower layer. Exactly by the method used for the development of Fourier's law oneobtains from this basic law of diffusion flow, the differential equation,

ay/st = - k(a2 y /arz + 1 /Q d ~ / d z s y / a z ) (3) where Q is aasumed to be n function of the height of the plane

This proviso, which Fick did not discuss further, could be stretched to cover all deviations from the hssic law. It is ao general, however, that i t can he regarded on Fick's part as seien- tifie caution rather than as exceptional prescience.

=Fick gives no indication of what he has in mind here. In fact, Ohm's law is obeyed more exactly than Fourier's law in the sense that the electrical conductivity of a substance is less de- pendent on the potentid gradient than is the thermal conductivity on the temperature gradient.

398 / Journol o f Chemical Education

above the base. If this is constant, the differential equation becomes,

Fick's notation is here retained without any modifica- tion.

The equation now referred to as Fick's first law has the form (1); the "basic law" referred to by Fick as a modification of Fourier's law, is not exactly this, and was never written down explicitly in the Annalen paper. Equation (4) would now he called Fick's second law for the restricted case where the diffusion coefficient, written ask by Fick, is independent of con- centration. Further points of particular interest are that Fick was fully aware of the mutual nature of the diffusion process, migration of one component in one direction being balanced by the migration of an equal volume of the other component in the opposite direc- tion. The plane of reference, with respect to which the (one-dimensional) flows are measured, while not specifically discussed, is obviously one across which no net volume transfer occurs. If there is no change in volume on mixing as Fick assumed, this reference plane will also he fixed with respect to the containing vessel. This is probably the most useful way in which to de- fine a diffusion coefficient, and Fick's contribution in this respect can hardly he overrated. However, it will be apparent that the detailed molecular model which was to provide the basic law contributed noth- ing to its development, and the only real justification for equations (3) and (4) would he a demonstration that the quantity k (i.e., in modern terms, the diffusion coefficient) is, in fact, independent of the concentration gradient. This was the next task to which Fick turned his attention.

The first possibility considered was that of integrat- ing equation (4) subject to appropriate boundary con- ditions in order to find the concentration y as a function of position x and time, a procedure which is the basis of ahnost all methods in common use a t the present time. Fick rejected this possibility because of the computational labor involved if the law were to be tested adequately, even for those cases where an in- tegral solution could be obtained in closed form; this was the reason for not presenting any such solu- tions. A second method, found to be unsatisfactory, was to establish a diffusion boundary in a tall cylindri- cal vessel by introducing concentrated sodium chloride solution beneath a water layer, allowing the diiusion process to continue for a definite period, then sampling layers a t different levels in the liquid column, and ana- lyzing. Ay/Ax, and A2x/Ay2 could in principle be obtained from these measurements, but the experi- mental errors were such that their ratio (k) was vari- able. However, the qualitative change of concentration with the coordinate x was that expected. A more suc- cessful technique was then devised. A column was set up with sodium chloride crystals a t the bottom and pure water, constantly renewed, a t the top. The con- centration distribution along the column was allowed to become that characteristic of a time-independent state in which the flow of solute (and, in the reverse direc- tion, of solvent) was the same across any horizontal crosssection irrespective of the value of x. The sys-

tem is described more completely in the paper in the Zeitschrift jfir ratimelle Medicin; and there Fick used the terms, still in current use, "stationary state" and "dynamic equilibrium" to describe conditions within the column. The amount of solute diffusing out a t the top of the column when the stationary state had been attained was measured, and the concentration gradient taken to he the ratio of the solubility of the salt to the height of the column. This is only true if the diffusion coefficient is independent of the concentration, and Fick's "constants" are therefore averaged diffusion coefficients. He does not seem to have considered the possibility that the "constant" might vary in this way possibly because this would not strictly he consistent with his "basic law of Diffusion." Three columns of different heights were used, and the coefficients derived from experiments on each were considered to be sufficiently concordant for the diffusion law to be taken as experi- mentally established. Other experiments were done with a cone-shapedcolumninorder toestablish thevalid- ity of an equation related to equation (2). The table shows some of his results on columns of uniform cross section, recalculated to modern units; some accurate differential diffusion coefficients for sodium chloride in water are also shown. It is worth mentioning that an improved form of this method was used over fifty years later by Clack (3) who measured the local concentration gradient a t successive horizontal planes in diffusion columns which had been allowed to reach the steady state, and corrected for the solvent counterflow; for many years they provided the best available diffu- sion data for electrolytes in water.

Fick's Average Diffusion Coefficients for Aqueous Sodium Chlorides

T m w Medium Short colunk column column

T ("c) ( x 109 ( x lo5) ( x 105) 14.8-15.8 1.12 1.12 1 .07

20 1.29 . . . 1.27 2W21 1.37 . . . 1 .29

Modern differential coefficients ( 2 5 W ( X 10-4' 0.5 M 1.474 2.0 M 1.514 4.0 M 1.58a

Values are recalculated to cm' seo-lunits. A value of 1.50.X 10-6 at 25' would correspond approximately to 1.34 X lo-' at 20' and 1.20 X 10"at l 5 T .

1 STOKE% R. H., J . Am. Chem. Soc., 72, 2243 (1950).

Although Fick stated that his results had confirmed the basic law, subsequent work has shown that, as in the case of so many simple physical "hws," Fick's "Grnndgesetz" is not correct in the general sense that its author clearly hoped that it would he. His views did not escape criticism a t the t i e . Beilstein (4) sug- gested that the effect of varying the concentration had not been adequately investigated, and that there was no more reason for choosing the first power of the con- centration gradient as the driving force rather than some power of it other than this. I n particular, Beilstein proposed that the density gradient in Fick's formula- tion should be replaced by (density)"'/x, a curious sug- gestion later treated with some scorn by Fick (6). He attacked the accuracy of Fick's experimental

Volume 41, Number 7, July 1964 / 399

data, and did many quantitative messurements de- signed to support Graham's conclusion (1) that the amount of salt transferred into distilled water in unit time was proportional to the weight of salt contained in the solution from which the transfer took place. Fick's reply to these criticisms forms the substance of his third, and last, paper on diusion (6) and he claimed that since, in his own experiments, the concentration varied along the column from saturated solution a t the base to pure solvent a t the top, the effect of concentra- tion changes were automatically taken into account; Fick does not seem to have realized that his assumption about the constancy of the concentration gradient in the colun~n was the weak point in this argument. As for the suggestion that the concentration gradient was involved to some other power than the first, he claimed that the observed constancy of his experimental coefficients would not have been found if any other assumption than this had been used. Later work has shown that the diffusion coefficient usually varies ap- preciably with concentration, and Nernst (6) took the gradient of osmotic pressure as the driving force for diffusion. This is equivalent to taking this force as the gradient of chemical potential, proposed independently by Gibbs (7) (1899) and later by Schreiner (8) (1922) and by Hartley (9) (1931) to whom is due the main credit for the specific application of this conception to the diusion problem. This conclusion has been confirmed by the application of steady-state thermody- namics to the problem (10). For dilute electrolyte solutions the correction introduced into the original diusion law4 suggests that D ( l + b In y/b in m ) ~ ; should be independent of concentration and be equal to the theoretical limiting Nernst value (6 ) which can be calculated from the mobilities of the constituent ions. Experiment has shown that this quotient is not quite independent of concentration but does tend to the limiting value a t low dilutions. The observed deviation is due to neglect of certain interionic attraction terms, as can be seen clearly from the thermodynamic treat- ment (11) . These can be calculated successfully for I : l electrolytes by a kinetic thecry due to Onsager and Fuoss (10) which is, however, rather less successfnl for other classes of electrolytes and breaks down for con- centrated solutions. While detailed evidence is ra- ther scanty as yet, for certain classes of nonelectrolyte pairs insertion of the thermodynamic term does give a quotient which is constant within experimental error. For others. D (1 + b in f/b 1nN) ;$ varies much more with changing concentration than does D it- self (18). Qualitatively, such variations are often attributed to the formation of complex species in the solution, each with its own characteristic diffusion co- efficients (18), but ideas of this kind have not yet led to

7 is the activity coefficient on the molality scale, and f that on the mole fraction scale.

quantitative developments. Indeed, a fully sahfac- tory explanation of the observed values of diusion coefficient and of their variation with concentration would not appear to be an immediate prospect.

More than a century of experience has shown that the value of Fick's contribution to the study of diusion in liquids, and also in its later application to gases, lies preeminently in the stimulus i t gave, and is still giving, to accurate experimental work, and in the provision of a concise and easily appreciated form for the expression of experimental data. A glance a t Graham's extensive, and almost unreadable, descriptions of quantitative studies on diffusion, will show how great a contribution this was. The original concept has been extended to cover the phenomenon of self-diffusion, and, for a binary mixture, a full description of the transport of matter within the system requires three diffusion coefficients, one for inter-diiusion of the two compo- nents, and two self-diffusion coefficients. Fick's hope that diffusion coefficients could be calculated from con- siderations of the repulsive and attractive forces be- tween molecules of the same, and of different, species bore no fruit a t the time or for many years afterwards. Hydrodynamic theories, like the Stokes-Einstein theory held, and in many ways still hold, the field; but increas- ing attention is being paid to the statistical mechanical description of transport processes in liquids (14) which, it can reasonably he hoped, will eventually be successful in realizing the ambitious program set out by Fick in 1855. Literature Cited (1) GRAHAM. T.. Phil. Trans. Rou. Soc. L a d n . 140. 1. 805

(1850);. 141,483 (1851); is:, 183 (1861); Am. c&. 77, 58, 129 (1851); 80, 197 (1851).

FICK, A,, "Gesammelte Schriften," Stahelverlag, Wiirshurg, 1903-04, R. Fick, editor. (The papera referred to here are reprinted in volume 1 of this four volume work. The portrait here reproduced is also taken from this work.)

CLACK, B., Proc. Phys. Soc., 21, 374 (1908); 24,40 (1912); 27. 56 (1914): 29. 49 (1916): 33. 259 11921). . .

BEIL&&_ F R . : ' A ~ ~ . Chem.. lbb. 165 (1856) (5j FICK, A., ~ n n : C h a . , 102; 97 ?1857j, (also reprinted in

reference (2)). (6) NERNST, W., Z. physik. Chernie, 2 , 613 (1888). (7) "The Scientific Papers of J. Willard Gibb~," Longmars,

Green, snd Co. New York, 1906, p. 429. (Letter dated 1899, to Wilder D. Bancroft).

(8) SCHREINER, E., TidSskr. Kjemi, Bwgvesa Met., 2, 151 (1922); ef. L m , O., S v m k Ka. Tidskr., 72, l(1960).

(9) HARTLEY, GI S., Phil. Mag., (7) 12,473 (1931). (10) ONSAGER, L., AND FUOSS, R., J . Phys. Chem., 62,404 (1958). (11 ) TYRRELL, H. J . V., "Diffusion and Heat Flow in Liquids,"

Butterworth, London, 1961, chsp. 4. (12) ANDERSON, D. K., HALL, J. R., AND BABB, A. L., J . Phys.

Chem., 62, 404 (1958). (13) ADAMSON, A. W., AND IRANI, R., J. Phys. Chem., 64, 199

11960). (14) BEARMA, R. J., J. Phys. Chem., 65, 1961 (1961); RICE,

S. A,, AND FRISCH, H. L., Ann. Rev. Phys. Chem., 11, 187 (1962); COLLINS, F. C., AND RAPFEL, H., Advan. Chem. Phys., 1,135 (1958).

400 / Journal of Chemical Educaiion