Dialysis of sodium dodecyl sulfate, its activity above the critical micelle

418

MOHAMMAD ABU-HAMDIYYAH AND KAROL J. MYSELS

The Dialysis of Sodium Dodecyl Sulfate, Its Activity above the Critical Micelle Concentration, and the Phase-Separation Model of Micelle Formation1

by Mohammad Abu-Hamdiyyah and Karol J. Myslels2 Department of Chemistry, University of Southern California, Los Angeles, California (Received September 6, 1966)

90007

The phase-separation theory of micelle formation assumes that the activity of monomers becomes independent of concentration when micelles are present. Dialysis, however, continues a t all concentrations though slower when micelles are present on both sides. Hence, activity increases with concentration and this increase can be accounted for on a simplified mass action model. The measurements were performed in a new kind of dialysis cell which permitted continuous monitoring of the composition of the dialyzate and a large number of separate kinetic experiments using the same membrane. Thus, the permeability of a particular membrane to various solutes could be studied, and, in particular, its permeability to micelles could be estimated using solubilized dye as a tracer.

It is generally agreed that micelles of surfactants are formed from monomeric molecules or ions above a certain rather well-defined critical micelle concentration (cmc). Below the cmc, the activity of the solute is to a good approximation similar to that of other low molecular mass molecules or ions, but above the cmc it certainly varies much less rapidly with concentration. This has led to the point of view that micelle formation is akin to a phase separation3s4and to the proposal that the phenomenon can, and should, be treated as corresponding to zero change of the activity of monomers above the cmc. It has even been stated that the other approach which assumes a mass action equilibrium between micelles and monomers is incorrect.4 This phase-separation approach has the great merit of simplicity and has been supported by experimental results dealing with surface tension meas~rernents,~ultrafiltration,'j and dialysis.' General arguments against the validity of the phaseseparation approach and in favor of the mass action approximation have been given8-'" and will not be repeated. More recently, ultrafiltration experiments of Schottll on LZ purified nonionic surfactant showed that the monomer concentration-and therefore activityincreases significantly above the cmc. Surface tension measurements of Elworthy and Mysels'O show that if proper experimental precautions are taken, the acThe Journal of Physical Chemistry

tivity of sodium dodecyl sulfate clearly increases above the cmc band that this increase can be accounted for very well by a simplified mass action theory.12 A preliminary report on dialysis experimentsg showed that dialysis continued above the crnc through a membrane impervious to micelles, which means that the activity of monomers continued to increase with concen(1) Based on part of the Ph.D. Thesis of M. Abu-Hamdiyyah, University of Southern California, Los Angeles, Calif., 1965. (2) Research Department, R. J. Reynolds Tobacco Co., WinstonSalem, N. C. 27102. (3) (a) G . Stainsby and A. E. Alexander, Trans. Faraday Soc., 46, 527 (1950); (b) K. Shinoda, Bull. Chem. SOC.Japan, 26, 101 (1953); (c) K. Shinoda, T. Nakagawa, B. Tamamushi, and T. Isemura, "Colloidal Surfactants," Academic Press Inc., New York, N. Y., 1963, P 6. (4) E. Hutchinson, A. Inaba, and L. G. Bailey, Z . Physik. Chent. (Frankfurt), 5, 344 (1955). (5) K. Shinoda and E. Hutchinson, J . Phys. Chem., 66, 577 (1962). (6) E. Hutchinson, 2. Physik. Chem. (Frankfurt), 21, 38 (1959). (7) J. T. Yang and J. F. Foster, J . Phys. Chem., 57, 628 (1953). (8) P. Mukerjee, ibid., 66, 1375 (1962). (9) K. J. Mysels, P. Nukerjee, and hI. Abu-Hamdiyyah ibid., 67, 1943 (1963). (10) P. Elworthy and K. J. hfysels, J . Colloid Interface Sci., 21, 331 (1966). ( 1 1 ) H. Schott, J . Phys. Chem., 68, 3612 (1964). (12) (a) R. C. Murray, Trans. Faraday SOC.,31, 207 (1935); (b) X. J. Mysels, J . Colloid Sei., 10, 507 (1955); (c) P. Mukerjee, K . J. Mysels, and P. Kapauan, unpublished data.

DIALYSIS OF SODIUM DODECYL SULFATE

tration. We are now reporting details of these and additional experiments along with a fuller interpretation confirming the original conclusion. The first dialysis experiments? on a surfactant suggested that the dialyzate and retentate did not reach the same concentration. This result was criticizedL3 as being due to impurities and indeed later experim e n t ~ ' agree ~ ~ ' ~that equilibrium is reached more or less rapidly.15" The authors of these later experiments suggest, however, that this is due to a slow permeation by micelles rather than to an activity gradient of monomers. In no case, however, has the permeability of the membranes to micelles been measured directly and only in one caseL5could the same membrane be used for both pore size determinations and surfactant dialysis. The difficulty of repeatedly determining the concentration during an experiment in the conventional dialysis apparatus also limited the value of the kinetic data obtained. Our results have been obtained using an apparatus which permitted the use of the same membrane through a long series of experiments. I n fact, one of them was used intermittently for 3 years and showed no detectable change in permeability to either surfactant or dye during that time. (Another membrane was also used and gave very similar results.) The apparatus also permitted continuous monitoring of the concentration of the dialyzate during an experiment and good control over all experimental variables. The dye-tagging technique of Hoyer16was used to study any transport of micelles through the membrane and showed, along with other evidence, that it was negligible. We shall consider the theory involved in the dialysis experiments, then experimental tests of their validity, and finally the results obtained for sodium dodecyl sulfate or NaLS. Experimental Section

Apparatus. The dialysis membrane was a section of Visking cellophane sausage casing "/32 in. flat width and about 10 cm long. After thorough washing, its ends were forced along with the platinum rings upon the Teflon cones as shown in Figure 1. The cones were then mounted on the central glass rod to give a rigid assembly. One of the cones was fixed by means of the screw cap in one arm of the glass cell. A screw plug in this cone also allowed filling and emptying the retentate without removing the assembly from the cell. The other arm of the cell had an opening for handling the dialyzate. In this arm there was also a small conductivity cell whose electrode seals served as the axis around which the cell was rocked at 1 cps by means of a string attached to the larger screw cap and to a motor-driven eccentric. This position of the elec-

419

a

r V l S K l N G TUBING

Figure 1. Dialysis cell permitting repeated use of same membrane and continuous conductometric monitoring of the dialyzate.

trodes minimized the strain on the wiring, despite the energetic mixing of both compartments. A Teflon sphere acted as a one-way valve in one of the connections between the two arms and ensured continuous and quite rapid circulation of the dialyzate between the two arms of the cell during the rocking motion. Concentration gradients within the dialyzate dissipated completely in less than 1 min. The cell was mounted in an air bath maintained a t 25 i 0.05". Conductivity was determined using a precision bridge with an accuracy much exceeding our requirements. Conductance data were translated into concentration using previously obtained value^'^^'^ for the same sample and several new points obtained in the dialysis cell. Materials. Distilled water equilibrated with the laboratory air was used to minimize conductance fluctuations due to atmospheric CO,. Reagent grade chemicals, Kational Bureau of Standards standard sucrose, and a high-purity sample of SaLS previously describedL7were used. Two samples of Orange OT (1-o-tolylaso-2-naphtol) were used. One was a purified commercial sample and has been described earlier;17 the other was freshly synthesized and purified by reprecipitating twice from acetone with water (mp 131", emax 1.91 X lo4 independent of pH at X 493-494 mp). The optical density of (13) K. J. Mysels in discussion of ref 7. (14) B. S. Harrap and I. J. O'Donnel, J . Phys. Chern., 5 8 , 1097 (1954). (15) H . B. Klevens and C. W. Carr, ibid., 60, 1245 (1956). (15a) NOTE ADDED I N PROOF.However, G. Bobalek and E. G. Bell, Ofic. Dig.Federation Soc. Paint Technol., 35, 423 (1963), support ref 7. (16) H. W. Hoyer and K. J. Mysels, J. Phys. Colloid Chem., 54, 966 (1950); H. W. Hoyer, K. J. Mysels, and D. Stigter, J . Phys. Chern., 58, 385 (1954). (17) K. J. Mysels and R. J. Otter, J . Colloid Sci., 16, 462 (1961). (18) R. J. Williams, J. N. Phillips, and K . J. Xysels, Trans. Faraday Soc., 51, 728 (1955).

Volume 7 1 , Number 2 January 1967


420

saturated aqueous solutions of the two dye samples was undetectable in 1-em cells and amounted to 0.002 in a 10-em cell. Assuming that molar absorbancy is the same as in acetone solutions, this corresponds to a solubility of about 1 X lo-' M or 0.03 mg/l., which is several times lower than some earlier reportslg but in agreement with the value found previously in this laboratory. l7 A Carey Xodel 11 spectrophotometer was used in this last determination. Procedum. In experiments with solubilized dye, the concentration of the dye varied between 40 and 70% of saturation. These solutions were prepared by gently shaking part of an NaLS solution with excess dye for several hours, filtering through Whatman No. 41 filter paper, diluting with the same NaLS solution, and shaking for several hours to dissolve any remaining dye particles. 'The optical density was determined in a Beckman DU spectrophotometer a t 494 or 498 mp, which corresponds to the maximum of the new and old dye samples, respectively. Preliminary tests showed that in neither case (contrary to previous indications20) was the result dependent on pH and no acid was added prior to the measurement. The samples were withdrawn from and then returned as promptly and as completely as possible to the dialyzate compartment of the cell. In the dialysis of sucrose, very small samples were withdrawn, and their concentration was determined using a Brycez1type of differential refractometer. Except for special experiments, a standardized procedure was followed, namely, through rinsing of both compartments, draining, and then introducing 6 ml of solution into the casing and 23 ml into the other compartment of the cell. A syringe with a thin Teflon tubing attached to the needle was used to introduce and withdraw solution from the casing through the narrow opening in the Teflon cone without danger of damaging the casing. The exact values of volumes used were determined by weighing the cell and, where needed, the results corrected for small deviations.

Theory Dialysis of Monomer. Experimentally, we can measure C, the concentration of the dialyzate as a function of time t. We can also determine the initial volumes vi and vd and concentrations cp and CO of the retentate (inside) and the dialyzate (outside) the membrane. Assuming that these volumes remain constant during the experiment, conservation of mass gives the difference of concentrations AC between the two Compartments as

AC/V, =

(ciovd)

The Journal of Physical Chemistry

(Co/Vi)

- (YC

(1)

where =

+

(2) At equilibrium AC = 0, and both compartments have a concentration C,, given by (Y

(vi

vd)/viVd

(3) With the usual assumptions of irreversible thermodynamics22 about linear coupled flows, we can write for our system J , = LilFl f LJ72

+ Lid's

(4)

where i = 1, 2, and 3 for the anion, cation, and water respectively, Fi are the forces exerted on species i, and the phenomenological coefficients Li, obey Onsager's relations Ltj = Lit. We now assume that the transport of ions is not affected by the transport of water, i.e. Lia = 0; that the gradient of electrochemical potential is the only driving force, i.e., that the pressure difference between the two sides of the membrane is negligible; and that the charges on the membrane have a negligible effect; i.e., the concentration of anions and cations is equal to every point in the membrane. I n view of electroneutrality, the membrane potential can then be eliminated to give

We further assume that the solution is sufficiently dilute so that the only interaction between the flows of ions is due to electroneutrality, hence Lt2 = 0. Introducing the values of Lit = uiCi (where u is the mobility and C the concentration of our 1-1 electrolyte), of dp, = R T d In at, and of dpl dp2 = RT d In a+ gives

+

J

=

2URTC d In a+/dx = 2URT(dC/dx)(d In ak/d In C) = D dC/dx

(6)

+

where the reduced mobility U = uluz/(ul uz) and D is by definition the diffusion coefficient. Thus D = Dideald In a*/d In C where Didea, = 2RTU. Since we are dealing with a membrane whose accessible cross section A is unknown, we can only measure experimentally the permeation P = A J . Furthermore, the membrane and its stagnant layers represent a (19) I. M. Kolthoff and W. F. Graydon, J. Phys. Chem., 55, 699 (1951); M.W.Rigg and F. W. Liu, J . Am. Oil Chemists' SOC.,30, 14 (1953). (20)D.Stigter, R. J. Williams, and K. J. Mysels, J. Phys. Chem., 59, 330 (1955). (21) B. Brice and M. Halwer, J . Opt. SOC.Am., 41, 1033 (1951). (22) I. Prigogine, "Thermodynamics of Irreversible Processes," C. C Thomas, Springfield, Ill., 1955, p 40.


42 1

system of unknown thickness Ax within which concentration gradients are unknown but have to sustain a constant permeation. However, the over-all concentration difference AC can be determined. The experimentally accessible permeation corresponds therefore not to the differential diffusion coefficient D as derived above, but to :t semiintegral diffusion coefficient D’ which is a weighted average of D over AC. Hence dC/dt = P/’Vo = AJ/Vo = (AD’/Vo)AC/Ax

(7)

Introducing the expression obtained earlier for AC, this equation can be integrated if one makes the approximation that D‘ remains constant, which is often reasonable because the two concentrations change in opposite direction so that the center of the AC range involved in the averaging of D remains fixed. I n other words, there is always a point in the membrane where C = Ceq. This gives, after introducing the initial conditions (2.3/(~)log [(C,,, - C)/(Ceq - C,O)I = (A/Ax)D’t

(8)

In this equation, the left side contains only measurable quantities, the right-hand side the duration of the experiment t , the unknown ratio of membrane dimensions (A/Ax) which has been assumed to be constant, and D’. Hence, if the experiment conforms to the above reasoning and assumptions, a semilogarithmic plot of (Ceq C)/(C,, - Co) us. t should give a straight line. The slope of this line k = (aA/2.3Ax)D’ is a measure of the rate of dialysis and, as long as A/Ax and a remain constant, is proportional to D’. This should be the case for any given substance provided experimental conditions remain constant. Dialysis of T?.acer Dye. We now consider an experiment in which both compartments contain the same concentration of micelles Cm, but one side contains initially also some dye whose concentrations will be denoted by c. The dye is mainly solubilized by the micelles, but a small fraction will be present “free” in simple solution in mater. The equilibrium between these two fractions can be advantageously described in terms of a partition coefficient r between the “micellar” and aqueous “phases.” In view of the very small amount of dye present, we can make the approximation that this partition coefficient is independent of dye concentration and also that the presence of dye does not cause any net dialysis of micelles (or that if it does this has no effect upon the dialysis of the dye). The fact that the solubility of the dye increases linearly with concentrationla indicates also that the partition coefficient is independent of total concentration of micelles. Heme

+

cw/(cm/Cm) = r; c m cw = c (9) if we denote by cm and cw the amount of solubilized and of free dye per milliliter of solution. This gives

+ r/Cm)

c = cm(1

a cm

(10)

where the approximation is justified if the dye is sufficiently water-insoluble to give a very small r (=lo-6 equiv/l. in our case) and the concentration of micelles is not too low. It is then easy to show by following the reasoning of the preceeding section that if the dye has an over-all (total) diffusion coefficient D’, in the membrane, one can experimentally measured an over-all rate of dye dialysis kt by measuring dye concentration as a function of time according to the equivalent of eq 8 log [(ceq

- c)/(c,q

-

=

CP)]

(c~At/2.3A~)D’tt = ktt

(11)

As the dye may cross the membrane in one of two states, solubilized or free, we may assign two separate diffusion coefficients, Dm and Dw,to these two modes of transport. The problem is to separate the two, since Dm is an indication of the permeability of the membrane to micelles. The total flux of the dye is the sum of its fluxes in the two forms

+

Jt

=

J i

= Didci/dx

Jm

Jw

and by definition (13)

Following the above reasoning, we obtain dc/dt = (AD’t/Vo)Act/Ax = (AmD’m/Vo)Acm/AX

+ (AwD’w/Vo)Acw/Ax (14)

This equation is based on the assumption that whereas the thickness of the membrane and of its stagnant layer, Ax, is the same for free and solubilized dye, the effective cross section A may differ. In fact, because of the heterogeneity of the membrane and the great difference in size between the kinetic unit of solubilized and free dye, this difference may be expected to be much larger than that between their diffusion coefficients. Simplifying and introducing the values of cm and cw from (9) gives

AtD’t = AmD’,

+ (AwDwr)(l/G‘m)

(15)

Equation 15 shows that one cannot separate the correspondingpairs of AiD’i terms and that if these products for the solubilized and for the free dye are assumed to be constant, the product for the total transport must Volume 71 Number 8 January 1967 I


422

c

0.I

I/c-crnc, IO* I/M Figure 2. The total rate of dialysis of Orange OT as a function of concentration of sodium dodecyl sulfate a t 25'.

Hours

vary with the concentration of micelles present during the experiment. This corresponds to the fact that as the concentration of micelles increases, the fraction of dye which is free decreases and so does the importance of its transport. I n view of the assumption that Ax is the same, we can rewrite eq 15 as

kt = k m

+ kwr/Cm

(16)

which shows that a plot of the experimentally measured over-all rate of dye dialysis us. the inverse of the concentration of micelles should give a straight line. Its intercept gives the rate of dialysis of solubilized dye, its slope that of the free dye times the distribution ratio r. Figure 2 shows that the experimental points indeed define a line with a very small intercept. A leastsquares calculation of the intercept from this plot overemphasizes t8hepoints at low C m . It is, therefore, better to rearrange eq 16 into

ktCm = k m C m

+ k,r

(17)

and to calculate the slope of the straight line thus defined. We thus find k m = 3.6 (*3.3) X 10-4 hr-1 for which will be compared later to k = 1.6 X NaLS above the cmc. l./equiv The slope of the line of Figure 2 is 1.1 X hr. The value of r can be estimated from the findings of Williams, et aZ.,l* that the optical density of satucmc) rated Orange OT in NaLS is given by 6.00(C

-

Figure 3. Dialysis of solutions initially a t 6 x cmc against water showing continuous change of rate in the cmc region, continuation of dialysis when both sides are above the cmc, and reproducibility of the measurements: taut membrane, Q; slack membrane, (3; normal membrane, 0;second membrane, 0 .

g/cm 100 ml., Le., 173 equiv/cm 1. We have found the optical density of water saturated with Orange OT to be 0.002 cm-l; hence, if we assume that the molar absorbancy of the dye does not vary much between the equiv/l. Hence, k, for solvents, r = 1.15 X Orange OT is approximately 1.0 hr-1, which is the value used later in discussing the permeability of the membrane to various solutes.

Results Validity of the Method. In discussing the theory, a number of assumptions were made which can be checked experimentally. Among these were that the total effects upon the rate of dialysis of the initial volumes were properly taken into account by conservation-of-mass eq 1. I n fact, second-order effects were present, but they were small as a 25% change in Vd and a 15% change in Vi produced 1.6 and 2.8% changes in D' (eq 8), respectively. I n all experiments, these volumes were kept within 1.5% of 23.0 and 6.0 ml and were corrected for in calculating IC when needed. Hence, this assumption seems well justified. Another assumption is that hydrostatic pressure produced by osmotic flow during dialysis has no effect of

DIALYSISOF SODIUM DODECYL SULFATE

423

F

-

t

I

L

I

f

Sucrose

0

10

20

30

40

50

Ceq mole per liter x io4

Figure 4. Effect of concentration and size upon the rate of dialysis through the same membrane at 25".

the latter. As the dialysis tubing normally has a considerable slack, a certain amount of osmotic flow can occur without changing the pressure within it. To test whether this was a critical factor, the tubing was made initially as taut as possible and as slack as possible in some experiments, but as shown in Figure 3 there was no detectable effect. As a further control, the size of the air bubble within the casing was observed from time to time as an indication of the pressure prevailing within the tubing. For the experiments reported, no significant changes were noted. In these experiments the initial concentration difference does not exceed 6 X cmc and is often less. In one experiment in which a solution at 32 X cmc was dialyzed against initially pure water, the situation was different, however. The air bubble decreased in size until it disappeared and the tubing bulged conspicuously. The rate of dialysis did not seem to be affected significantly until the disappearance of the bubble, which was a very different situation from the rest of the experiments. Incidentally, the properties of the membrane were not affected measurably by this mistreatment. Figure 3 also shows the over-all reproducibility of the measurements. A further assumption is that electroneutrality is the only charge effect, i.e., that the effect of charges in the membrane is swamped by the electrolyte concentration present as far as dialysis of monomers is concerned. (This last limitation is important as these same charges

may well be decisive in preventing the dialysis of micelles which are not only larger but also carry much higher charges.) To study the effect of membrane charges upon small particles, the dialysis of KC1 and KaLS was measured as a function of concentration and that of tetra-n-butyl iodide and of sucrose was measured at one concentration. The results are shown in Figure 4. It may be seen that at higher dilutions dialysis is significantly slower but that the rate levels off at higher concentration and for XaLS becomes constant which is about one-third of the cmc. above 3 X Hence, no significant rate effect from this source is expected at the cmc or above. If the membrane were homogeneous in pore size and presented the same effective cross section A to all small particles, the ratio of dialysis rates would be equal to the ratio of free diffusion coefficients once membranecharge effects have been swamped. The free diffusion coefficients are known or can be calculated from limiting conductivities of the ions and show that this is not the case. The larger particles are slowed increasingly by the membrane. Taking KC1 as the standard, the effect is 1.3 for tetrabutyl iodide, 2.8 for NaLS, and 16.5 for sucrose. The dialysis rate of Orange OT is much more uncertain but appears to be high in comparison with sucrose, suggesting the presence of an alternative transport mechanism for more hydrophobic particles. This inhomogeneity of the membrane made it unpromising to attempt any calculation of absolute values of diffusion coefficients from our experiments by comparison with a standard. An assumption which is not always expected to be valid is that of the constancy of D' = [Didea,d In a+/d In cIav over the span of concentrations involved in an experiment and therefore in the integration leading to eq 8. The differential term can be expected a priori to change as the cmc is crossed and this should result in a curve instead of a straight line after integration with a marked change in the cmc region. Figure 3 shows that this is indeed the case. On the other hand, as shown by Figure 5, excellent straight lines are obtained when the retentate and the dialyzate both remain either above or below the cmc as equilibrium is approached. (This is also the case for the lower part of Figure 3.) Depending on the analytical precision, good to excellent straight lines could also be obtained for solubilized dye and for the other solutes as shown in Figure 6. The Behavior of Sodium Dodecyl Sulfate. As pointed out in the preliminary comm~nication,~ the curve of Figure 3 shows clearly that NaLS continues to dialyze, though at a reduced rate, after its concentration on both sides of the membrane is above the cmc, and the Volume 71,Number B January 1967

424

MOHAMMAD ABU-HAMDIYYAH AND K A R O L

Time, hours

I1

5

t t

10

I

I

I

Time, minutes 20 I

J. A'fYSELS

30 I

Dye in 6 c.m.c.

Dye in 1.9 c.m.c.

20

'I ll 40 60 80 100 120 140 160 180 200 220 240 260 280

Time, h o u r s

Figure 6. Dialysis of solubilized dye and of tetra-n-propyl iodide through the same membrane a t 25'

T i m e , hours

Figure 5. The dialysis of sodium dodecyl sulfate when both sides are either above or below the cmc and straight lines are obtained. Note large difference of time scales. The conductances are a linear function of concentration above the cmc and are plotted for this region. The numbers indicate Ceqin mmoles/l.

process continues until an equilibrium is reached. This equilibrium can be calculated from the initial conditions or measured after dialysis has proceeded long enough for the concentration not to change perceptibly further. Table I shows that the two values agree very well.

Table I: Dialysis of Sodium Dodecyl Sulfate at 25' (All Concentrations in mmoles/l.) -c

cia

1.oo

5.95 16.25 24.75 48.00 50.6 16.06 31.98 47.97 80.16

CQ

0 0 0 0 0 0

8.03 15.99 23.98 64.02

C q

Exptl

Calcd

3.27 5.01 9.82 10.05 9.70 18.40 29.14 67.72

0.20 1.26 3.32 5.02 9.82 10.14 9.75 18.64 29.40 67.72

The Journal of Physkal Chemistry

Slope, hr-1

0.18 0.26 0.50 0.50 0.0165

0.0165 0.034 0.0170 0.0164 0.027

As pointed out above, when the points lie on a straight line extending to lowest concentrations, a meaningful dialysis rate is given by its slope and this is the case when both solutions are above or below the cmc. I n the immediate neighborhood of t,he cmc, no straight line is obtained. Hence only the slopes, k, of such straight lines are listed in Table I. Figure 7 shows these slopes and the range of concentrations over which they were obtained as a function of concentration. The slow increase a t low concentrations which we attribute to swamping effects, the sudden change in t,he crnc region, and the low but clearly nonzero rates above the cmc, which increase at higher concentrations, are clearly seen. hr-l, is Even the lowest of these rates, 164 X much larger than the rate k , found above for the dialysis of dye solubilized within micelles, i.e., 3.6 (*3.3) X hr-l which may, in fact, not differ significantly from zero. If it is really zero then, of course, our membrane was truly impervious to micelles and the observed rates are due to monomers alone. If the rate is nonzero, however, this could be due to the presence of some small but macroscopic leak in our system. In that case, the transport of dye would be an exact measure of the transport of any solute including untagged micelles and this would then be negligible. This type of leak is unlikely, however, because the transport of dye was exactly the same for a second piece of cello-


425

c

c.m.c.

d 100, p.30

I

/n= /'

9-

1

10-4

1

1 1 1 1 1 1 1

1

1 I 1 1 1 1

10-3

t

I 1 1 1 1 1 1

10-2

10-1

Conc. of NoLS, M / I

Figure 7. The rate of dialysis of sodium dodecyl sulfate as a function of concentration. Horizontal segments show the range of linear behavior leading to the rate shown.

phane tubing after the apparatus had been completely dissassembled and reassembled again. It is more likely, therefore, that the transport, if any, occurs through pores of molecular dimensions in the membrane. I n this case, it is probable that untagged micelles will dialyze faster than the tagged ones and the question arises whether they could account for the observed transport. As is shown elsewhere,23the size distribution of the tagged micelles is very likely to be weighed toward large micelles according to the size; i.e., their number distribution is equal to the weight distribution of the uiitagged ones. Thus, if the niembrane permitted the passage of a micelle only '/IO as large as the average one, this size would be only '/lo as frequent among the tagged ones and the rate for the untagged ones would be ten times larger than measured by the dye technique. This is a rather extreme possibility, since a micelle of about seven monomers is highly unlikely, and still it would account for only a small part of the transport above the cmc. A strong argument against any significant contribution by the small micelle lies in the fact that a t highest concentrations the rate increases, whereas the proportion of the smaller micelles must decrease radically according to the law of mass action. Thus it is likely that the permeability of our membrane to micelles was negligible and the observed transport is due entirely or predominantly to monomers (and any dimers2*present). This is further supported by the semiquantitative explanation of the transport above the cmc asing a simple mass action approach as shown below. It should be noted, however, that whether transport by micelles is postulated or not, the complete and relatively rapid equilibration at all concentrations shows

2

3

5

4

6

7

e

cov Figure 8. Comparison of averaged experimental value of In C with those calculated for several models of micelles: experimental, 0 ; calculated, 0;p from ref 12b has same meaning as s in text and in ref 1%.

+ = d In a*/d

clearly again that the chemical potential of NaLS continues to increase above the cmc at a significant rate. Solely monomeric transport permits, however, a more complete analysis and a test of that part of the phase separation theory which assumes5 that the activity of the monomers remains practically constant above the cmc. The Activity Change above the Cmc. As shown above, the rate of dialysis is proportional to D i d e a l d In a*/d In C averaged over the membrane thickness and if this is reasonably constant, the experimental points lie on a straight line in a semilogarithmic plot according to eq 8. Thus, the rates obtained from the straight lines under comparable conditions permit an intercomparison of these averages. If we define experimentally

4 =

kabove cmc/kjust below emc

this should be also given by

4 =

(Dideal

d In a/d In

C)above/ (Dideal

d In a d d In

c)below

If we assume transport by monomers only, the Dideal term should cancel and any nonideality existing both just below and above the cmc (including contributions by dimers) will also cancel. Hence, will measure the change in the differential term due solely to the formation of micelles and to accompanying changes in ionic strength, etc. The solid points of Figure 8 show the experimental values of 4 from our measurement. (23) K. J. Mysels, unpublished data.

(24) P. Mukerjee, J. Phys. Chem., 6 2 , 1404 (1958).

Volume Y l , Number 2 January 1967

426

The lines of Figure 8 have been calculated on the assumption that the system is ideal, i.e., neglecting ionic strength effects, etc., that the micelles are monodisperse and characterized by a degree of association n and an effective charge s which is constant but takes into account12c all nonidealities of the system. Values previously computedlZbusing a simple law of mass action approach have been utilized. The computed points are A In a*/A In C taken between the tabulated values and plotted a t the midpoint of C. Thus, the averaging process is not necessarily analogous to the experimental one and may account for some of the discrepancies. It may be noted that the calculated curves all show the rise a t higher concentrations which is also shown by our experiment and corresponds to the double curvature of the activity-log C plot and of the experimental surface tension results.1° Over the range of parameters used, the curves differ markedly and the one for n = 50

The Journal of Physical Chemistry


and s/n = 0.15 comes closest to the experimental results. It does not seem likely that significantly better agreement can be obtained by other choice of the parameters and these are also reasonably close to, e.g., n = 62 and an effective ionization of 0.17 obtained from light scattering.26 Thus dialysis results are not only in disagreement with the phase separation theory, but are in fair agreement with values estimated using a highly simplified mass action approach. I n view of the difficulty of comparing the theory with the experiment, the agreement may be as good as could be expected.

Acknowledgment. This work was supported in part by fellowships sponsored by the Continental Oil Co. and awarded to Mohammad Abu-Hamdiyyah. (25)

K. J. Mysels and L.H. Princen, J . Phys. Chem., 63,1966 (1959).

Dialysis of sodium dodecyl sulfate, its activity above the critical micelle

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