t
w
I’
T,,, - Tb for condensation, A T = Tb - T,,,for evaporation = time, or exposure time of jet, sec. = mass rate of condensation or evaporation: g. cm.* sec.-I = correction factor, dimensionless
literature Cited
Alty, A., Proc. Roy. SOC.l S l A , 554-64 (1931). Hickman, K., Ind. Eng. Chem. 46, 1442-6 (1954). Hickman, K., Proceedings of First International Symposium
on Water Desalination, Washington, D. C., Vol. I, pp. 180-223, October 1965. Maa, J. R., IND.ENG.CAEM.FUNDAMENTALS 6, 504-18, (1967). Maa, J. R. IND.ENG.CHEM.FUNDAMENTALS 8, 560 (1969). Maa, J. R., Vacuum Sci. Technol., 6 , No. 5, 153-4 (1968). Schrage, R. W., “Theoretical Study of Interphase Mass Transfer,” Columbia University Press, New York, 1953. Trevoy, D. J., Znd. Eng. Chem. 46, 2366 (1953). Wyllie, G., Proc. Roy. SOC.197, 383-95 (1949). RECEIVED for review May 1, 1968 ACCEPTEDJanuary 30, 1969 Work supported by a grant from the Office of Saline Water, U. S. Department of the Interior.
SURFACE DIFFUSION I N MONOLAYERS E D W A R D K . S A K A T A ’ A N D J O H N C. B E R G Department of Chemical Engineering, University of Washington, Seattle, Wash. 98106 Surface diffusion in a monomolecular myristic acid film on a water substrate is studied using radioactive tracers. The results obtained are presented in terms of a surface diffusion coefficient, D,, employing a twodimensional form of Fick‘s law. The results are discussed in terms of the structure of the monolayer, and the values of D , obtained are compared with those predicted by several molecular models.
SURFACE diffusion-Le., two-dimensional migration of atoms or molecules in the plane of an interface-has long been regarded RS an important process on solids, but has received almost no attention in the case of liquids. In solids it has been useful to postulate surface diffusion to explain numerous phenomena, including observed patterns of crystal growth, aggregation of unstable metallic films into crystallites, variation in thermionic emission of tungsten unevenly coated with adsorbed metallic films (Blakely, 1963), and unusually rapid transport of adsorbed gases through porous media (Dacey, 1965). Parallel phenomena have not been observed in the case of liquids, however, so that until recently it has not been necessary to give serious consideration to surface diffusion on liquids. Apparent surface migration in fluid systems can usually better be explained by convective or diffusive transport in the adjacent bulk fluid phases. Although interfacial diffusion effects in fluid systems of practical interest would usually be expected to be masked by convection in the adjacent bulk phases, there is no reason to suppose that the lateral diffusion process is not occurring. Indeed, in some fluid systems, such as those involving insoluble monolayers, surface diffusion may represent the only feasible mechanism for the lateral transport of material. In studies of capillary effects in fluid mechanical problems, in particular those induced by the presence of such surfactant monolayers, the possible importance of the surface diffusion process has been suggested by Levich (1962), Whitaker (1964), Berg and Acrivos (1965), and others. In all these studies, the hydrodynamical boundary condition a t the fluidfluid interface depends upon the surface distribution of the surfactant molecules, which in turn is determined a t least in part by the surface diffusion process. Surface diffusion would also be important in studies of kinetics of chemical reactions occurring in monomolecular films, and although this ‘Present address, E. I. du Pont de Nemours & Co., Inc., Wilmington, Del. 570
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FUNDAMENTALS
has been recognized by Gaines (1966) and others, the diffusion process itself has not been studied. At fluid interfaces, one would expect lateral variations in the concentration of the diffusing species to provide the sole driving force for the migration, and Fick’s law to give an adequate representation for the surface migration process. The history of surface diffusion studies in fluid systems is meager. Crisp (1946) suggested that an upper limit might be calculated for gaseous monolayers on aqueous substrates by assuming a t least half the polar group of the surfactant molecule to be “submerged” and equating D, to the diffusivity of the submerged portion of the molecule in the substrate. The “hydrodynamical theory” (Bird et al., 1962) of diffusion in liquids then leads to the expression
D, = kT/3?rpr0
(1 1
where
k = the Boltzmann constant T = absolute temperature p = substrate viscosity ro = radius of polar group sq. cm. per which typically yields values of the order of second. Blank and Britten (1965), on the other hand, have predicted values of D, for condensed monolayers, using a model based on equilibrium fluctuations in monolayer density. Their theory leads to a D, value of sq. cm. per second for a condensed film of stearic acid. However, the same theory predicts values for surface viscosity in condensed films several orders of magnitude smaller than those obtained experimentally. The only experimentally determined surface diffusivity on a liquid substrate was obtained by Imahori (1952) for denatured horse serum albumin (a complex protein of molecular weight 70,400) on water a t surface pressures of the order of
0.5 dyne per cm. The value obtained for D, was 1.1 X 10-6 sq. cm. per second. In the present work, selfdiffusion in monomolecular films of myristic acid at the air-water interface was studied. Radioactive tracers were used to obtain unsteady-state surface concentration profiles which were analyzed successfully in terms of Fick’s law. Surface diffusion coefficients were obtained for monolayers in both the liquid-expanded and the intermediate state of the film. A full discussion of the thermodynamic phase behavior of insoluble monolayers is given by Adam (1938) and Gaines (1966). Experimentation
The surface self-diffusion coefficient of myristic acid nionomolecular films was determined by a radioactive tracer technique utilizing carbon-14-tagged myristic acid. Two films spread upon a 0.01N 13C1 substrate contained in a shallow trough and initially separated by a barrier were allowed to diffuse into each other. One film was ordinary carbon-12 myristic acid and the other was carbon-14-tagged myristic acid, both a t the same surface concentration and surface pressure. The over-all surface concentration was thus uniform, and upon removal of the barrier, there was equimolar counterdiffusion of the ordinary myristic acid and the tagged myristic acid. The extent of diffusion was determined by measuring the surface activity (the radioactivity in counts per minute) of the carbon-14-tagged myristic acid with a radiation detector as it, diffused into the carbon-12 myristic acid. The over-all initial surface concentration of myristic to 4.26 X mole per acid was varied from 7.27 X sq. cm., corresponding to initial specific areas of 23 to 39 sq. A. per molecule. The apparatus is shown schematically in Figure 1. The trough was milled from a block of 3-inch-thick polypropylene, and its inside dimensions were 5.1 X 28 X 0.35 cm. Two movable barriers (not shown in the diagrams), used t o sweep the surface before spreading the film and to confine the area of the film, were made of 4 X 33 X $ inch Teflon. The
11
REGION 11
1
REGION I
A
e
Figure 1. A. E.
L X - 4
Diagram of trough
Divided into two equal parts b y flexible barrier Detector placed over trough to measure activity a t position x
surface area of the substrate was divided into two equal parts by a flexible barrier made by cutting a 0.5-mm. strip from a piece of Teflon tape 0.003 inch thick. The surface radioactivity was measured with a thin end window GeigerMuoller (G-M) tube, Amperex Type 200 LB, connected to a Nuclear-Chicago Model 186A scalar. The G-M tube was mounted in a carriage which could be moved along the length of the trough on runners. The tube could also be moved vertically by means of a vernier, allowing the detector t o be placed at any desired height above the surface film. A slitted cap, made from a 3-mm.-thick piece of Plexiglas with a 4-mm.-wide by 4.7-cm.4ong slit, was placed at the end of the detector, so that the surface activity a t any position along the length of the trough could be measured. The carriage assembly and the trough were enclosed inside a Plexiglas cover to eliminate air currents and minimize evaporation. Water wicks-Le., beakers filled with water and containing pieces of filter paper-were also placed inside the cover t o reduce evaporation and to provide a saturated atmosphere. The temperature of the room in which the experiments were run was maintained a t 22’ f 0.5’ C. The monomolecular film of myristic acid was spread on the substrate from a dilute benzene spreading solution (0.0031M). The normal carbon-12 myristic acid, 99+% purity, was supplied by the Nutritional Biochemicals Corp., Cleveland, Ohio, and the tagged carbon-14 myristic acid was obtained from the Nuclear-Chicago Corp. in a benzene solution. All apparatus and glassware coming in contact with either the HC1 solution or the myristic acid solution were cleaned in hot chromic acid cleaning solution, then rinsed with sirupy phosphoric acid and distilled water. The flexible Teflon barrier was secured to the top inside edges a t the center of the trough with petroleum jelly, care being taken that the petroleum jelly did not touch the inside surfaces of the trough. The substrate solution was poured into the trough to a depth of 5 mm., providing a positive meniscus around the trough. The movable Teflon barriers were swept across each surface on either side of the dividing barrier to provide a clean surface, and were positioned a t the ends of the trough. The regular carbon-12 myristic acid was spread on one side of the flexible barrier with a micrometer syringe, and the tagged carbon-14 myristic acid was spread on the other side with a separate micrometer syringe. ilny difference in surface pressures between the two sides could be detected readily by the shape of the flexible barrier lying on the surface. If the surface pressure was greater on one side, the barrier “bowed out” toward the opposite side. When the surface pressures were equal, the flexible barrier lay more or less in a straight line across the center of the trough. If the surface pressures were not equal, they could be equalized by adding a fraction more to one side or moving the Teflon barrier located a t the ends of the trough. The benzene was allowed t o evaporate and the surface pressures were precisely equalized. The flexible barrier separating the two regions was then lifted, allowing the tagged and untagged myristic acid films t o diffuse into each other. The detector was placed over the surface a t a height such that the cap a t the end of the detector was 2 mm. above the surface, and the surface radioactivity along the length of the trough was measured immediately before and after the barrier was lifted. By moving the detector along the length of the trough externally from outside the cover, unsteady surface concentration profiles could be obtained a t successive times by measuring the surface activity at various distances along the length of the trough. The movable carriage was not mechanically connected t o the trough, so that its motion, or its change of position, would not induce disturbances in the liquid. Desorption of the myristic acid film into the substrate VOL.
8
NO.
3
AUGUST
1969
571
played an important part in the over-all mass transfer of the monolayer molecules, and its rate had to be determined in separate experimental runs. In these runs, the tagged carbon-14 myristic acid was spread over the entire surface without the surfacedividing barrier, and the surface activities were again measured a t various times. In this case, however, it was not necessary to scan the surface with the detector, since the surface activity remained uniform over the surface, decreasing uniformly with time. The radioactive source used, carbon-14, is an emitter of low energy beta-rays which can be detected by scintillation, proportional, or G-?\I counting. The G-M counter was chosen for this work because of the stability of the counting system and the simplicity of the electronics involved. The G-hf tube had a window thickness of approximately 5 microns (1.4 mg. per sq. cm.) and an effective window diameter of 2.4 cm., with a transmission of approximately 40% a t 50 K.e.v. (the average energy of carbon-14 beta-rays). The geometric arrangement of the counting setup used is shown schematically in Figure 2. It was assumed that the activity of the surface a t any distance z was uniform in the lateral direction-i.e. along the length of the slit-that the activity across the width of the slit was uniform, and the activity seen by the G-M tube was a t a specific location 5. Actually, the activity varied across the width, d, of the slit, so that as seen by the G-M tube it was some mean value across this finite distance. It would have been necessary to correct for such variation in surface activity if the counting rate were extremely low, the width of slit d were large, the thickness of the cap were small, or the height of the cap from the surface were large, since each of these factors would tend to increase the “view” of the surface as seen by the G-?VI tube. However, with the physical dimensions involved in this experiment, the variation of surface activity across the slit could be neglected. Experimental Model and Evaluation of D,
If Fick’s law gives an adequate representation of the surface migration, experimental surface diffusion coefficients may be
rn
calculated by solving the equation for onedimensional diffusion, assuming 0, to be independent of concentration. Utilizing Fick’s equation for diffusion in a monolayer on a liquid surface, however, requires the assumption that the monolayer molecules remain a t all times within the film and do not desorb into the liquid substrate. For a monolayer in which the molecules are even very slightly soluble in the liquid substrate, it would be necessary to account for desorption. This was the case for the myristic acid monomolecular film on water, for even though the solubility of myristic acid is very small (0.002 gram per 100 grams of water a t room temperature), it required modification of Fick’s equation. With monolayer molecules desorbing into solution, one must consider the alternate paths which a diffusing molecule may take, as shown in Figure 3. Normal surface diffusion will take place with the molecules diffusing within the monolayer (arrow 1). The monolayer molecules, in addition, may also desorb into the substrate (arrow 2), diffuse laterally in the bulk liquid (arrow 3), and eventually readsorb back to the monolayer (arrow 4). Thus, the monolayer molecules may migrate laterally by surface diffusion and desorptionbulk diffusion-readsorption. The diffusion equation employed must thus take into account the desorption-bulk diffusion-readsorption as well as surface diffusion. In separate experiments conducted to determine the desorption rate of the myristic acid, it was found that the monolayer molecules dissolved a t a rate proportional to the monolayer surface concentrat on-i.e., a rate equal to -kr, where k is the desorption rate constant and r is the surface concentration of the desorbing molecules. Once in solution, the molecules may readsorb to the monolayer a t a rate equal to ( h C ) ,where k-1 is the adsorption rate constant, C is the concentration of the molecules in the bulk liquid, and h is the depth of the substrate. The substrate is thus assumed to be sufficiently thin and the desorption rate sufficiently low that the vertical variation of surfactant concentration in the bulk may be neglected. This was borne out by experiment. Subject to the above assumptions, we may write:
with
GM TUBE
1.C.
r = ro a t t =
0
The surface and bulk concentrations will eventually attain equilibrium values, (00 ), so that we may write k
-=
K = - .hc, =-
k-1
rcn
The solution of Equation 2 is then
Figure 2.
Geometry of detection system SURFACE DIFFUSION
- 1 ___L
BULK DIFFUSION IN SUBSTRATE
Figure 3. 572
l&EC
FUNDAMENTALS
Surface migration of nonvolatile monolayer
ro- r, r m
(3 )
Normally, the measured surface activity would have to be corrected for the contribution to the surface count by the desorbed molecules in the substrate-that is,
where
Thus, desorption-readsorption is characterized by the single parameter, k. When surface diffusion occurs, the total lateral migration is described in terms of simultaneous differential equations for the surface diffusion and the horizontal bulk diffusion. These are given below with the appropriate initial and boundary equations for one-dimensional diffusion in a region of length L.
D.E. 2.
ahC
at
.-
a2hC D-
ax*
+ k r - k-l(hC)
-- 0 13.c.1 , 2 . ar 8X
a t z = 0, L, all t
n.c. 3 , 4 . ac --0 ax
a t z = 0, L , all t
1.c.1.
1'= I'o a t 1 = O , O < x < -
I.C. 2 .
C=O
(7 1
1,
2
where a b is the equivalent surface activity from the desorbed molecules in the substrate. However, because of the relatively low concentration of desorbed tagged myristic acid in the substrate, and the short range of the low-energy beta-rays, the beta-rays are sufficiently adsorbed by the liquid substrate to make this correction unnecessary. This was confirmed by sweeping the surface clean of the monolayer and measuring the activity of the monolayer-free surface. The activity, or counting rate, was substantially the same as the background count. Results and Discussion
The desorption rate of the myristic acid monolayer depends linearly upon the surface concentration under the conditions studied (Figure 4 ) . The slope of the plot is k' in Equation 4, from which the desorption rate constant, k , was 4.10 X lo-' min.-l This value of the rate constant was then used in the solution to the modified diffusion equation, Equation 6. The unsteady surface concentration profiles obtained for three different initial surface concentrations are shown in Figures 5 to 7. The solid lines represent solutions to Equation 9, using, in each case, the value of D, which produced the best fit of the data (Table I ) .
att=O,O
