THE JOURNAL OF
PHYSICAL CHEMISTRY (Registered in
U. S. Patent Offic~)
(0Copyright, 1956,by the American Chemical Society)
MARCH 20, 1956
VOLUME60
NUMBER3
FACTORS RESPONSIBLE FOR THE STABILITY OF DETERGENT MICELLES BY IRVING REICH‘ University of Southern California, Los Angeles, Calif. Received Auguat 7 , 1965
The theory of Debye, which ascribes micelle formation to the opposition between hydrocarbon-chain attraction and ionic repulsion, leads to grossly incorrect calculations of micelle size distribution. Theories which depend on ionic repulsion are also unable to account for the formation of micelles by non-ionic detergents. A general theory for the formation of micelles by non-ionic detergents is outlined, based on the Hartley picture of micelle structure. For ordinary solutes, the energy of the s stem decreases indefinitely as the degree of aggregation increases. For detergents, the energy of the system no longer dkcreases after aggregates reach a certain size. Consequent1 further aggregation does not occur, since that would involve an entropy decrease. A general equation for micelle size &tribution is developed, and the modifications required for ionic detergents are discussed.
Introduction In 1949 D e b ~ e outlined ~.~ a general theory of micelle formation. This theory was intended to explain why micelles do not grow indefinitely and separate out as an ordinary phase. In Debye’s theory, limitation of micelle size is attributed to electrostatic repulsion between the head groups of the detergent ions which make up the micelle. In the first part of the present paper, it is shown that this theory is incorrect due to two rather fundamental errors. In the rest of this paper a theory is outlined explaining the formation of micelles in terms of an entropy effect.
”
Debye’s Theory of Micelle Formation Debye assumed the micelles to be disk-shaped McBain platelets. He assumed that as the micelle grows, each entering detergent ion releases the same amount of hydrocarbon-chain adhesional energy, so that this energy is proportional to the size of the micelle. Then the total work of assembling the micelle is W
= N’lawe -
Nw,
(1)
where N is the number of detergent ions in the micelle, wm is the work of hydrocarbon-chain ad(1) Lever Brothers Research Center, Edgewater, New Jersey. (2) P. Debye, THISJOURNAL, 53, 1 (1949). (3) P. Debye. Ann. N. Y. Acad. Sci., 61, 573 (1949).
hesion per ion, and we is a constant which relates to work done against electrostatic repulsion of the ionic head groups. Plotting W against N gives a curve with a minimum. Debye considered the value of N a t which the minimum occurs to specify the stable micelle size since “work is required to increase or decrease the equilibrium number No.” In deriving equation 1, the effect of gegenions in screening the electrostatic repulsion between head groups is neglected. Hobbs4 attempted to refine the theory by taking into account the effect of the gegenion atmosphere, but mathematical difficulties prevented the development of general equations. The above theory appears to be incorrect in three ways : (1) The stable micelle size (or size distribution) must be that which results in minimum free energy for the system. This is quite different from the criterion of minimum free energy per micelle. (2) Growth of micelles will involve a decrease in the number of independent particles in the system, and hence will involve a decrease in total entropy. This must be taken into account. (3) Energy of hydrocarbon chain adhesion per ion must increase as the micelle grows. If it remained constant, as Debye assumes, growth beyond dimer size would not occur. (4) M. E. Hobbs, THIEJOURNAL, 66, 675 (1951).
257
258
IRVING REICH
Suppose C molecules of detergent are present in the system and they aggregate to form C / N micelles of N molecules each. Then, accepting equation l, the total energy of micelle formation will be (C,") (N"We - Nwm) or C(N"ZW~ - w m ) . It is this rather than N'/zw, - N w , which must be minimized to determine the stable micelle size. However if C ( N l / W e - Win) is plotted against N , it is found to increase steadily with increasing N without passing through a minimum. Consequently there would be no preferred micelle size. Instead aggregates of all sizes would be present-dimers in highest concentration and larger aggregates in steadily decreasing concentrations. This is a picture which differs radically from generally accepted ideas about micelle size distributions. More correctly we should plot free energy rather than total energy against N , which means that a temperature-entropy term should be subtracted from the total energy. The more disperse the system, the greater will be the entropy. Hence entropy will favor the smaller as against the larger aggregates without changing the trend of the curve. I n order to account for the existence of a preferred micelle size, account must be taken of the variation of wm with N . Debye's assumption that wm is constant appeared t o lead to a preferred micelle size only because of the incorrect minimization of energy per micelle instead of energy of the system. Micelles of Non-ionic Detergents.-Non-ionic detergents, such as polyoxyethylene derivatives of fatty acids or fatty alcohols, are known to form micelles in much the same way as ionic detergents.6$6 These molecules do not have ionic groups capable of exerting long-range electrostatic forces. Consequently a general theory of micelle formation cannot involve ionic-group repulsion as an essential element. Unquestionably the repulsion between head groups plays an important role in determining the critical concentrations and sizes of ionic micelles. Unfortunately a general quantitative treatment of the problem does not appear possible a t present owing to the difficulty of specifying the effect of the gegenion atmosphere. There is much to be said in favor of focussing our attention on the simpler case of non-ionic micelles in attempting t o set up any general theory of micelle formation. The remainder of this paper will consist of an outline of a theory of non-ionic micelle formation, followed by a brief discussion of the modifications required in proceeding t o ionic micelles. Thermodynamic Equations Governing Aggregation.-In any solution, solute aggregates of all sizes have finite probabilities of existence. The solute consists of an equilibrium mixture of all aggregate sizes. At equilibrium, there will be a definite aggregate-size distribution which may be shown by plotting aggregate concentration [AN] against aggregate size N . The equilibrium size distribution will be that which gives lowest free energy for the system. Of course some aggregate sizes may be present in undetectably small concen(5) J. W. McBain and E. Gonick, J.A m . Cham. Soc., 69. 334 (1947). JOURNAL, 58, 1163 (6) I,. M . Kiislinrr and W. D. Hubhard, THIS 1954).
Vol. 60
trations or even in concentrations so small that a system of ordinary size will only rarely form a single aggregate. The monomer is in equilibrium with aggregates of each size, so that the concentration of monomer determines the concentration of aggregates of each size. This is to say that the concentration of monomer determines the complete size-distribution curve. I n dealing with micelle formation, it is convenient t o consider how the size-distribution curve changes as the monomer concentration is increased to the critical concentration. I n Fig. 1, four types of size distribution are shown. Curve 1 represents an ordinary solution, Monomer is present a t appreciable concentration. Dimers and higher aggregates are present at undetectably low concentrations. Curve 2 represents a hypothetical system which would occur if the energy of aggregation specified in equation 1 were correct. Aggregates of all sizes exist in steadily decreasing concentration. Curve 3 represents ordinary phase separation. As in curve 1, monomer is present in substantial concentration, while dimers and higher aggregates are present only in very low concentrations. But here the concentration curve rises for aggregates of very large size. Finally curve 4 represents micelle formation. The left-hand portion of the curve resembles the corresponding parts of curves 1 and 3. However at some value of N , generally in the range 50-200, the concentration rises and then falls again, so that a considerable concentration of small aggregates is present. When the concentration of an ordinary solute is increased, a critical concentration is eventually reached at which the situation changes from that of curve 1 to that of curve 3. When the concentration of a detergent is increased, the change a t the critical concentration is from the situation of curve 1 to that of curve
I
%-
4. In a solution, aggregates of each size are formed reversibly from monomer N A = AN
(2)
so that the standard free energy of formation is AF" = -k2' In K
(3)
or AE"
-
[ANI l'AS' = -kT In [AI
(4)
which may be rearranged to give In [AN] =
TAS" - AEo kT -k N l n [AI
(5)
Equation 5 is a perfectly general thermodyiiamic size distribution equation. It gives aggregate concentration [AN] as a function of N . This size distribution function may be calculated a t a series of different monomer concentrations [A], to det,ermine what happens when [A] reaches its critical concentration. However AX" and AE" vary with the aggregate size N . They must be evaluated as functions of N before the equation can he employed. It will be assumed that the monomer and aggregates form ideal solutions. In that case, activities are equal to concent,rations, and these will be expressed in mole fractions. This has the advantage
I
I
STABILITY OF DETERGENT MICELLES
Mar., 1956
of defining the standard states as pure liquid monomer and pure liquid aggregates. AS” is the entropy change involved when N moles of pure liquid monomer forms one mole of pure liquid N-mer. It does not include any entropy of dilution. The important assumption will now be made that s, the standard entropy change per molecule, is the same regardless of the size of the aggregate formed. This is undoubtedly true for large aggregates. If it is true for smaller aggregates also, then AS” = N s
(6)
and, from equation 5
259
for if we set In [A] equal to (equation 10, we get
E
In [AN] = - N Z / a ( e / k T )
- sT)/kT
+NA
+ A in (12)
If A is zero or negative, In [AN]falls rapidly with N, so that the aggregate size distribution is as shown in curve 1 of Fig. 1. If A becomes infinitesimally greater than zero, the aggregate size distribution follows the same course for low values of N , but for very large values of N , concentrations become appreciable, and in fact infinite aggregates will form. This is the size distribution shown in curve 3 of Fig. 1. Thus the concentration defined in equation 11 is a critical concentration at which a new phase separates.
The remaining step is to evaluate AEO. One possibility is to employ equation 1, setting AEO equal t o N 8 ’ W e - Nwm,SO that
+ In [AI)
- Na//a+ (:)
(8)
Since w eis positive, the second term is negative. The first term also is negative, except in physically meaningless cases where [AN] is greater than 1. Hence In [AN]always decreases with increasing N . At very low monomer concentration [A], the size distribution, is that shown in curve 1 of Fig. 1. At higher monomer concentration, it is that shown in curve 2 of Fig. 1. This is the conclusion reached earlier by qualitative reasoning. The same types of size distribution obtain if the electrostatic repulsion work is eliminated by setting the second -$ term equal t o zero. Clearly the aggregation energy specified in equation 1 is not correct. As the first step toward setting up a more valid I specification of aggregation energy, it will be help/ / ful to consider an ordinary solute rather than a de/ tergent. Suppose we have an aliphatic hydrocar/ bon dissolved in water. Each molecule will show / / some tendency to fold up on itself because of the / tendency of segments of the chain to escape from the water and contact each other. To permit calculation, it will be assumed that each niolecule or aggregate is a tightly-packed sphere which does not include any water. When molecules join to form an aggregate, the total surface is reduced. If E is the total surface energy of a single molecule ’I sphere, then the maximum energy change when N molecules aggregate would be - N E . This would hold only for very large aggregates; for small ag11 L L ------- -gregates the energy change would be less, since appreciable hydrocarbon surface would remain exN , -. posed to contact with water. It is readily calcuFig. 1.-Types of aggregate size distribution: 1, ordinary lated that when N spheres unite to make a larger solute; 2, hypothetical system obeying equation 1; 3, sphere, the fraction of the original surface which is phase separation; 4, micelle formation. eliminated is 1 - 1/N1’3. Hence the energy of The case of non-ionic detergent molecules may aggregation is now be considered. The hydrocarbon portion of Ah’’ = - N e ( l - I,”‘/*) (9) the molecule is assumed to be curled up into a tightly-packed sphere. This, while not likely to be On substituting this into equation 7, we obtain an exact picture, is probably reasonably close to the NsT + Ne(1 - l/N1/a) In [AN] = N In [A] fact.’ Of the total surface of the molecule, A , a kT portion S is covered by the polar group, while the This equation predicts phase separation a t a criti- remainder, A - S, consists of hydrocarbon surface. cal concentration defined by When molecules unite to form an aggregate, the
*I
3
I_ _ _ _ I I
+
In [ A ] =
-e ~
- s?’
k 7’
(11)
(7) A . E. Alexander and P. Johnson, “Colloid Science,” Val. 11, (’larendon Press, Oxford, 1949, p. 671.
IRVING REICH
260
hydrocarbon chains are assumed to blend together to form a larger sphere, with the polar groups occupying the surface. This is essentially the Harley picture of micelle structure. Here again let B be the total energy of the hydrocarbon-water interface of a single molecule. The energy of aggregation will be - - E multiplied by the fraction of the original hydrocarbon surface area which is eliminated by aggregation. Since the original hydrocarbon surface is NA - N S and the hydrocarbon surface of the aggregate is N’/aA - N S , the fraction eliminated on aggregation is A/(1 - l/N’/a) ( A - 8 ) . If, for example, the polar group covers one fourth of the surface of the curled-up molecule, so that A/(A - S) is 4/3, the fraction of the surface eliminated will increase with N to the value 1a t N = 64. An aggregate of 64 of these molecules will be in effect a tiny oil droplet with its surface completely covered by polar groups. All of the hydrocarbon will then be withdrawn from contact with water and will be in contact with other hydrocarbon. In this case the stable micelle size should be 64 molecules. Aggregates should grow to this size for the same reason that they grew in the previous case-in order to eliminate as much as possible of their hydrocarbon surface from contact with the water and thus to minimize the energy of the system. However for the case of the dissolved hydrocarbon, the fraction of the hydrocarbon surface which was eliminated became I only for infinite aggregates, so that aggregates grew indefinitely. Here it becomes 1 for aggregates of very small size. There is no energetic reason for these to grow larger. Accordingly their further growth would not be expected, since that would involve a decrease in total number and a corresponding decrease in the entropy of the system. Aggregates of the size a t which the surface becomes completely covered with polar groups will be called “complete micelles.” On thermodynamic grounds it is certain that some aggregates larger and smaller than complete micelles will be present. I t is necessary to show that their concentrations drop away rapidly as N becomes larger or smaller than the complete micelle value. If an aggregate grew larger than the complete micelle size and still remained spherical, the surface would not accommodate all of the polar groups. Some polar groups would have to be buried in the interior. This is unlikely energetically. Such large aggregates mould presumably be flattened suficieiitly to create enwgh extra surfwe to accommodate all the polar groups. Thus as N increases beyond the complete micelle size, the aggregates 1)ecome larger and flatter. The fraction of the hydrocarbon surface eliminated remains 1. Accordingly for values of N less than that of tlie complete micelle, equation 7 becomes NsT In [AN] =
+ N e (*AE) (1 - l/N‘/s) kT + N l n [AI
(13)
while for values of N greater than that of the complete micelle equation 7 becomes In [ A N ] =
Ns‘l’
+ N e + N l n [AI
kT
(14)
Vol. 60
The value of N for the complete micelle is given by (+8)(1
- l/N’/s)
= 1
(15)
Rough calculations of aggregate size distribution can now be made, based on approximate values of A I S , B and s. Taking a polyoxyethylene ether of dodecyl alcohol as the non-ionic detergent, it will be assumed arbitrarily that the polar group covers one-fourth of the curled up hydrocarbon portion of the molecule. Then A = 4s and, from equation 15, the size of the complete micelle is given by N = 64. I n calculations of this type it has sometimes been assumed that the standard entropy change Y is approximated by the entropy of fusion of the appropriate hydrocarbon. The entropy of fusion of dodecane is 16.8 cal./deg. However this approximation does not seem satisfactory if the Hartley micelle structure is postulated, since the micelle interior is assumed to be liquid rather than crystalline solid. It seems more reasonable to view the standard entropy change as the entropy change of condensing dodecane vapor into liquid. However the vapor must be assumed to have first been compressed to the density of the liquid, so that no volume change occurs during the condensation. The quantity desired can be calculated as Satd. dodecane vapor
A
+
Compressed vapor (density 0.751)
B -3
4
r
Liquid (density 0.751)
During step A, one mole of dodecane vapor a t 25’ is compressed from its saturation pressure (0.115 mni.) to a pressure sufficient to increase its density to 0.751 g./cc. The vapor is assumed to behave as a perfect gas. The entropy change is R In PI/P2,which in this case is -26.68 cal./mole deg. During step B the compressed vapor condenses to a liquid occupying the same volume. The change in configuration and organization of the dodecane molecules involves the entropy change s’ which we wish to calculate. The over-all entropy change for steps A B is simply the ordinary entropy of condensation - AHere this is -49.16 cal./mole deg. SubH,,,/T. tracting from this the value for step A, -26.68, we have -22.5 cal./mole deg. or -11.35 IC as the value for s‘. This is the standard entropy change for the hydrocarbon portions of the molecules. However there will also be some entropy decrease due to the restriction of the movement of the polyoxyethylene chains. The micelle may he looked upon as a tiny hydrocarbon oil droplet with waving polyoxyethylene tentacles emerging from its surface. The portions of the poiyoxyethylene chains within a few atoms’ distance of the oildrop surface will be greatslyrestricted as to position and freedom of motion. This restriction involves an entropy decrease which is estimated as of the order of two-thirds s’, or about -8k. Thus the total standard entropy of micelle formation is of the order of -2Ok, and this value will be used in the calculations. The interfacial energy e may be calculated by multiplying the surface area of a molecule-sphere by the macroscopicallydetermined interfacial energy of the hydrocarbon. The total interfacial energy
+
L
N
STABILITY OF DETERGENT MICELLES
Mar., 1956
of dodecane against water is 65.3 ergs/cm.2. This leads to a value of 40kT for the interfacial energy of a curled-up molecule of dodecane. Since we are assuming that one-fourth of the hydrocarbon surface is permanently screened off by the polyoxyethylene group, the value for e of the detergent becomes 30kT. Specifying A = 4S, s = -2Ok, and e = 30kT, equations 13 and 14 become N 5 64: N 2 64:
+
In [AN] = N(20 In [A]) - 40N% (15) In [AN] = N(10 -I-In [A]) (16)
These equations may be used to calculate the entire aggregate size distribution curve at any concentration of monomer. Thus at monomer concentration [A] = 2.01 X 10" (curve 3 of Fig. 2), In A N falls away rapidly with increasing N . Although it rises again to a maximum at N = 64, the concentration of these aggregates never exceeds the undetectably small value of e-52. Hence the solution is essentially an ordinary solution of monomer. However as the concentration of monomer is increased, the maximum rises, so that a t monomer concentration [A] = 3.67 X lo", large numbers of micelles of sizes in the neighborhood of N = 64 are present.
-100-
- 120 0
I
20
40
I
I
60
I
I
80
I
I
100
N. Fig. 2.-Semilogarithrnic plot of calculated aggregate size distributions for non-ionic detergent at different monomer concentrations: 1, [A] = 3.67 X IOus; 2, [AI = 3.00 X 10-6; 3, [A] = 2.01 X 10-6.
The sharpness which micelles appear and the narrowness of their size distribution is seen more clearly in Fig. 3, where aggregate concentrations are plotted as actual weight per cent. concentrations instead of as logarithms of mole fractions. The detergent is assumed to have a molecular weight of 500. The monomer concentrations chosen here are much closer together than those in Fig. 2. It is clear that there is a sharp critical micelle concentration at a detergent concentration of about 0.097%. At a detergent concentration slightly less, 0.0947%, very few micelles are present. On the other hand to increase the monomer concentration to O.O9970j, requires the presence of over 0.5% of detergent, most of which goes to form micelles. Approximately 90% of the micelles are in the size region 64 to 74. A complete micelle in the sense defined above may not be attainable. If the polar group screens off a small enough portion of the molecule, the aggregate may grow t o twice the chain length of the molecule before its surface is completely covered with polar groups. The aggregate cannot grow
26 1
larger as a sphere without polar groups being pulled into the interior.. Hence any further growth would involve flattening of the aggregate. Such growth would not decrease the fraction of the hydrocarbon surface eliminated and hence would not be favored. I n this case, therefore, the stable micelle would be a sphere whose diameter is twice the length of the detergent molecule. 0.10 I
50
1
h
60
70
80
90
N. Fig. 3.-Direct plot of calculated aggregate size distributions for non-ionic detergent at different monomer concentrations: A, [A] = 0.0997%; B, [A] = 0.097%; C, [AI = 0.0947%.
According to the foregoing treatment, the size of micelles depends on the geometric ratio S / A . On the other hand the critical micelle concentration depends on the relative magnitudes of the standard entropy s and standard energy E . Precise calculations of critical concentrations and sizes depend on evaluation of these quantities. It would be expected that, for detergents with short polyoxyethylene chains, the critical micelle concentration would increase with polyoxyethylene chain length, due to the increasing entropy change when the motion of these chains is restricted. However for detergents with long polyoxyethylene chains, critical concentration should no longer vary with chain length, since the outer portions of the polyoxyethylene chains would not suffer much restriction in their movement. Critical concentration should always decrease with increasing hydrocarbon chain length since this would increase e. Micelle size should increase with increasing hydrocarbon chain length, since this would result in larger A . However it should not be appreciably influenced by the polyoxyethylene chain length beyond the fist few units. Dr. Marjorie J. Vold of this University has pointed outsto the writer that it may be necessary to consider the energy of dehydration of the polyoxyethylene groups. As the micelle grows, these groups will be forced closer together. If dehydration of ether oxygen occurs, the energy associated with this must be subtracted from e.
The McBain Micelle The above calculations have been based on Hartley's picture of the micelle. An alternative picture is that of McBain. The micelle is assumed to grow in two dimensions as a platelet, with the hydrocarbon chains aligned parallel to each other and the polar groups covering the flat faces. No matter how large such an aggregate may become, the hydrocarbon-water interface will never be eliminated completely, since the detergent molecules a t the edge will have their hydrocarbon chains ex-
posed to water. The usual assumption will he made that, the plat,eIet aggregates are disksl I aped. The original separstn niole~~ules, as tzfore, are assuinecl to t)o crirlecl up illto spheres. The pro+ ess of forniing :L MaHain aggregate may I)e atialyzetl ititlotwo steps. First, the tnolecules un~!url,assumiiig c:ylindric*alshape. This involves an increase in hydrocarbon surface with a corresponding energy change NeI. Then the molecules form the disk aggregate. 1 1 1 this process the flat ends of the hydrocarbon chains are completely covered. This iiivolves ai1 energy change -Nez. The sides of the hydrocarbon cylinders are covered except for the exposed portions of the cylinders a t the edge of the aggregate. If is the total interfacial energy of the cylinder side, then the energy change which occurs when the aggregate is formed is -Ney(l - 1/N1I1). Thus the standard energy of forming an aggregate is Ah’’ = Nci
- Ne2 - Ner(1 -
I/N’/t)
(17)
When this is substituted into equation 7, the aggregate size distribution obtained is that for phase separation rather than for micelle formation. The critical concentration is given by
The situation here is like that defined by equations 10-12. At monomer concentrations less than the critical concentration, very few aggregates are present. When the monomer concentration is raised to the critical value, the aggregates grow to infinite size. I t follows that, according to the theory here proposed, micelles of non-ionic detergents must have the structure proposed by Hartley rather than that proposed by McBain. Preston8 pointed out that soaps arid other colloidal electrolytes exhibit transition temperatures. Above the transition temperature micelles form at a certain critical concentration. Below the transition temperature, when the critical concentration is attained, a solid phase separates out. This may be explained if the transition temperature is considered a microscopic melting point. Below the transition temperature the hydrocarbon chains in an aggregate are aligned parallel to each other forming a crystalline plate. Accordingly aggregates grow to infinite size. Above the transition temperature the hydrocarbon chains form a liquid sphere, so that the aggregate is of the Hartley type and is limited in size. This treatment apparently ignores the effect of electrostatic repulsion energy which must be a factor for the ionic detergents which Preston considered. However Halseyg has shown that when an aggregate reaches large enough size, each entering detergent ion will be repelled only by the portions of the aggregate near its entry point. The more distant parts of the aggregate would be screened off by a cloud of gegenions. The work of electrical repulsion per entering ion would then become constant rather than increasing and would no longer limit (8) W. C. Preston. THISJOURNAL, 62, 84 (1948). (9) Q. D. Halsey, Jr., ibid., ST, 87 (1953).
the growth of the aggregate. Hence the transition from liquid t o crystalline state for the hydrocarbon chains would permit infinite growth of the aggregate and the electrical work would be merely a constant factor to be subtracted from the hydro(barbon adhesion work. Micelle Formation by Ionic Detergents.-Thr! formation of micelles by ionic detergents may 1)n accounted for in the same way as their formation by non-ionic detergents. However the additional factor of electrostatic repulsion between ionic head groups must he introduced. As a first approximation, we may adopt Debye’s function, adding the term -weNa’%/kT to the right-hand sides of equations 13 and 14. This does not change the general form of the size distribution curve. However it causes smaller aggregates to be favored as against larger ones and resuIts in a decreased average micelle size and an increased critical micelle concentration. If We is of sufficient magnitude, the stable micelle size may be considerably smaller than the “complete micelle” defined earlier. Before the aggregate had reached completion, the electrical work term would grow larger than the hydrocarbon-adhesion energy, preventing further growth, even though the aggregate still had exposed hydrocarbon. If we assume the aggregates to be crystalline platelets, we must add -weNa/*/kT to equation 17. Substituting this into equation 7 leads t o a modified equation which predicts micelle formation. Thus it appears that electrical repulsion may be capable of stabilizing a McBain-type micelle. However the electrical work term employed here does not take into account the screening of repulsion by the gegenions. This effect will become increasingly important as the aggregate grows. Indeed, as Halseyg has shown, for a sufficiently large aggregate the N8I2factor becomes simply N , so that the aggregate is not stabilized against further growth. Without exact calculation of the potential due to gegenions, it is not possible t o determine whether electrical repulsion is in fact capable of stabilizing McBain micelles. Appendix Since this paper was first submitted for publication, Ooshikalo has published a theory of micelle formation. Ooshika here points out the incorrectness of Debye’s treatment in assuming constant energy of hydrocarbon chain adhesion and in minimizing the free energy of the micelle rather than of the entire system. Ooshika’s treatment does not account for the formation of micelles by non-ionic detergents. The micelle size which he calculates is given by N = wB/we. For non-ionic detergents, where w e is zero, the aggregates would become infinite. Acknowledgment.-The author wishes to express his thanks to Dr. Marjorie J. Vold and to Dr. Karol J. Mysels of the University of Southern California for their helpful suggestions. This study was part of a program a t the University of Southern California supported by the Lever Brothers Company. (10)
Y.Ooshika, J . Colloid Sei., 9, 254 (1954).
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