Langmuir 1995,11, 2911-2919
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Micelles of Polysoaps 0. V. Borisov Institute of Physics, The Johanes Gutenburg University, 55099 Mainz, Germany
A. Halperin” CRPCSS, 24 av. d u President Kennedy, 68200 Mulhouse, France Received March 28, 1995. In Final Form: May 11, 1995@ Polysoaps, hydrophilic polymers incorporating m amphiphilic monomers, can form intrachain micelles. The structure of the intrachain micelles, the configurations of the polysoaps, and their modification in the presence of free amphiphiles are analyzed for the high salt, dilute solution limit. The important polymeric parameters are the polymerization degree of the spacer chains joining adjacent surfactants, n, and of the chain as a whole, N . The structure of spherical intrachain micelles is similar to that of free micelles up to n * 10. For larger n the aggregation number decreases because of the crowding of the loops formed upon aggregation by the spacer chains. For n 5 400 and above the intrachain micellization may be repressed. The chain dimensions are greatly reduced by the intrachain aggregation. When m 2 peq,the equilibrium aggregation number of a single micelle, the chain dimensions are those of a single micelle. In the opposite case the thermodinamically favored configuration is that of a branched structure. The addition of free surfactants results in the formation of mixed micelles once the critical association concentration (cac) is exceeded. The cac is typically much smaller than the cmc of free micelles. The formation of mixed micelles ends when a well-defined saturation concentration is attained. At this point the chain is fully unfolded and the structure of the bound, mixed micelles is essentially that of free micelles. When the polymerized surfactants tend to form cylindrical micelles, the shape of the intrachain micelles can vary with n and with the concentration of free amphiphiles.
I. Introduction Short chain amphiphiles can be incorporated into the backbone of polymer chains. The resulting macromolecules, upolysoaps’’,are capable of forming both intrachain and interchain aggregates.l Polymeric surfactants selfassemble into a variety of aggregates such as LangmuirBlodgett layers, assorted mesophases, and simple intrachain micelles. Polysoaps can also form mixed aggregates that incorporate free, monomeric surfactants.2 In the following we present a theory of the self-assembly of polysoaps in thedilute limit; i.e., chain-chain interactions are mostly ignored. The polysurfactants discussed are linear copolymers comprising altogether N monomers of which m > Rim
e
e
fFfH
fy
H x (bh)1/4n3/4b; F,,,,,,/kT
x (bh)5/8n3/8 (4)
For extended spherical brushes
H x p U5n315b;
FcoronalkTM p U 2ln(R,
+ HYR,
5
pu21nn ( 5 ) Strictly speaking, an extra term, k T In n, should be added to Fcorona to allow for the loss of configurational entropy due to loop formation. However, in the following we ignore this term because it is independent of the aggregation number and thus plays no role in setting the equilibrium structure. Accordingly, (19) Milner, S. T.; Witten, T. A.; Cates, M. E. Macromolecules 1988, 21, 1610. (20) Halperin, A,; Tirrell, M.; Lodge, T. P. Adu. Polym. Sci. 1990, 100, 31. (21) Halperin, A. In Soft Order i n Physical Systems; Bruinsma, R., Rabin, Y., Eds.; NATO AS1 series, Volume 323, Series B, Physics; Plenum: New York, 1994. (22) de Gennes, P. G. Scaling Concepts i n Polymer Physics; Come11 University Press: Ithaca, NY, 1979.
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(6) Note that the constant transfer free energy term, 6, is ignored as in (3). As in the case of free surfactants, the micellization is driven by the surface tension term, ya. However, it is now opposed by two terms: the head groups repulsion, yao2/a,and F,,,,.IkT. Consequently, the effect ofFeomna is to increase the equilibrium area per head group, aeq,aboveao, i.e.,aeq2 ao. For free surfactants in aqueous solutions, a0 depends on the parameters controlling the head group repulsion, eg.,pH and ionic strength. a0 is independent of the micellar shape. For polysoaps aeq increases with the length of the spacer chains, n. Furthermore, since the functional form of Fcomna varies with the micellar shape, aeqshould vary between micelles of different geometry even if formed by identical surfactants and with identical n. The allowed micellar forms are now determined by a modified packing parameter, vla& rather than by vlad,. Because the formation of intrachain micelles does not result in loss of translational entropy, one may no longer argue that the smallest possible aggregates attain the lowest free energy. To determine the relative stability, it is necessary to compare the minimized free energies of the different species. The variation of Fc,,,, with the geometry, for a given n and aeq,flat > cylindrical > spherical, gives a rough idea of the relative stability. The preceding discussion focused on the characteristics of a single, noninteracting intrachain micelle. It is strictly applicable when the equilibrium aggregation number pes is, at most, equal tom, i.e., m Ipq. In the opposite case, m x- pes,the chain supports many intrachain micelles. How does the large scale configurational free energy of the chain affect the characteristics of the intrachain micelles? The precise answer depends on the model adopted to describe the string of micelles. For spherical intrachain micelles one may consider a linear string or highly branched structures. In the case of cylindrical micelles both rodlike and semiflexiblechain configurations are conceivable. Yet, in all the scenarios considered in the following, the equilibrium structure of the micelles is that of a single noninteracting intrachain micelle; i.e., the effects of micelle-micelle interactions and of the elasticity of the strong of micelles are negligible. This is well illustrated by the case of a linear string of spherical intrachain micelles. The total free energy of the polysurfactant in this case, as we shall see, is of the form FlkT = (mlplpc, (m/p)avxmicelle.Here mlp is the number of micelles, vficelle% H3 is the micellar volume while a % 115 and x % 215 within the Flory approximation. The first term reflects the contribution of the mlp individual micelles while the second allows for micelle-micelle repulsive interactions and for the elasticity of the chain. The full free energy per surfactant, allowing for all contributions, Zp, is thus Zp cp ma-lp-avXmicelle. For m p, clearly Zp = cp, i.e., the equilibrium micellar structure is that of a single, isolated intrachain micelle. Having said that, it is important to stress that these contributions play a crucial role in determining the configuration of the chain as a whole. Thus far our discussion of the single chain behavior concerned two regimes: m 5 pes and m "pes. In the first regime a single intrachain micelle of p = m is expected. In the second, a string of mlpeqmicelles ofp % p e sshould obtain. It should be noted that when mlpq > 1is not too large, a string of micelles of p t pes is expected. This effect is most pronounced when 2 > m/pq > 1. The average aggregation number, (p),is easily found upon noting that cp for Ip - pesl/pes ao, vlaeqlc< 113 and the intrachain micelles are spherical irrespective of n. Because the value of aeqof mixed micelles is intermediate between the above two values, these, too, are always spherical. Initially, we focus on the structure ofintrachain micelles in the absence of free surfactants. The interaction with free amphiphiles will be considered in section V. The starting point of our % p V 2In n, as is discussion is eq 6 for cp with F,,,,,IkT appropriate for an extended spherical corona. To proceed in terms of a and ao. fprther it is helpful to express Fco,,, A spherical micelle comprisingp amphiphiles satisfies pv % Rh3 and pa % RinZ,where Ri, denotes the radius of the hydrocarbon core. Accordingly p
= v2/u3
and, defining po % v2Iao3,one obtains FcomnJkT ( ~ d aIn) n,~ thus ~ leading to
(12) %
porn-
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2916 Langmuir, Vol. 11, No. 8, 1995
Introducing u = (udal3= plpo we obtain
Here K poU2In nlyao > 0 is a dimensionless parameter measuring the relative importance of Fco,,. as compared to the second penalty term, the head group repulsion. In particular, K is the ratio of F,,,,,./kT and the head group repulsion term for a = ao. When K > 1,F,,,,,, is the dominant penalty term and, as a result, aeq > a0 andpe, < PO. To obtain the equilibrium characteristics we set ac#dau = 0, thus leading to U e q2/3
+
KU,?
b
e 1.
ueq 5 1 decreases monotonically as K increases, i.e., a,, increases whilep,, decreases. For K X,,,. In this regime polysoaps carrying saturated mixed micelles coexist with free surfactants. WhenX, is reached, the free surfactants form micelles on their own. The polysoap behavior is unmodified in this regime where XI =Z X,, and only the number of free micelles is increasing as more surfactant is added. SinceXs,t may be very near toX,, the distinction between the last two regimes may be impractical. When K is not very small, (30) is not easily solved and a more involved analysis is called for. While in the case o f K 1,(30)and (31)completely determine the equilibrium state, in the general case it is also necessary to invoke the condition for the cac, Q(1, u)= Q(a,y). First it is helpful to consider (31)in greater detail (Figure 4). The function g(y) appearing in (31) has a maximum, g,, = 1 - 23n13 at yo = 2-3/2. Equation 31 has no solution while (2yao)-l In XJX > gm,. It only has solutions for X 2 XOwhere XO is defined by
( 2 y ~ z , ) lnX,jXo -~ = 1- 23'213
(36)
Asingle solution,yo, exists forX=Xo. This is the minimal possible physical solution for y of a mixed micelle. For X > XOtwo solution, y < yo .and y > yo are found, however, only they > yo solution is physical. The nature of the equilibrium solution depends on K . In one regime yea, = ueqat Xcacwhile in the other there is a discontinuousjump in the micellar characteristics, y, * ueq. The transition between the two regimes occurs at K, = ZU4 as determined by (15) for u = UO,i.e., uoW3+ K U ~ =~ 1. / ~If K i K,, ueq2 yo and a smooth transition between the pure intrachain micelle and the mixed one is possible. On the other hand, when K > K,, ueq< yo and a discrete jump, between ueq< yo and ycac 2 yo, occurs at X,,,. To analyze the behavior atX,,,, it is helpful to use (15)and (30) in order to rewrite
4.05
-0.1
I-/ ' '
0.2
0
0.8
0.4
0.8
1
Y
1
Figure 4. A plot ofg(y) demonstratingthe graphical solution of eq 31.
the equilibrium condition Q(1, ues)= Q(G,, formz7
ycac)
in the
where
@ ( x ) = -1x 3
u3
+ - 5x
-u3
-2
3
AtX=Xcac (37)is supplemented by (30) and (31)in the form ycacW3 KCtcac3/2yca?/6 = 1 and ( 2 y ~ ~ J -Inl XJXcac= &cac)u For K i K, all three equations are satisfied by ycac = Ueq and G a c = 1. As Xincrease above the X,,,, yesvaries according to (31). This is a quadratic equation with a solution
+
(39) where the corresponding a,,as determined by (301, is
(40) In the limit of K K, and uq uq. Because of (37) this also imposes a finite jump to a, 1. The subsequent development of the system again follows (39) and (40).
VI. A Little on Cylindrical Micelles A much richer behavior is expected when the free surfactants form cylindrical micelles; i.e., the unpolymerized amphiphiles are characterized by a packing parameter 113 I vlad, i lI2. Two qualitatively new features are predicted in this case. First, when n is large enough, the intrachain micelles are spherical rather than cylindrical. This effect is due to the coronal penalty,F,,, which favors aeq > UO. Second, spherical intrachain micelles can convert to cylindrical mixed micelles upon addition of free surfactant. The free surfactants inserted into the mixed micelle decrease the loop crowding thus (27) This form is obtained by substituting K = (1 - ue 2/3)/ueq5/6,as follows from(l51, into R(1, u,) and G , ~ K = (1 -ycae2/3)/yc.~6, as follows from (30),into R(G.,, yc.,). In turn this leads to Q(1, ues)= In X, + yaocNy,,) and to Q(G,,,Y,.J = ln(XdX,,) + (~adae~~)O(y,,,). Equating the two expression leads to (37) and (38).
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lowering Fcorona and decreasing aeqto the point where cylindrical geometry is enforced. A full analysis of the cylindrical case is beyond the scope of this paper. It is nevertheless instructive to discuss some of the basic features for comparison purposes. For simplicity we limit the discussion to the case of 113
