Langmuir 1993,9, 66-70
66
Simulation of the Formation of Micelles by Diblock Copolymers under Weak Segregation Yongmei Wang and Wayne L. Mattice' Institute of Polymer Science, The University of Akron, Akron, Ohio 44325-3909
Donald H. Napper Department of Physical and Theoretical Chemistry, School of Chemistry, The University of Sydney, New South Wales 2006, Australia Received December 18,1991. In Final Form: August 21,1992
Simulation of the self-assembly of diblock copolymers, AN~BN~, into micelles has been performed on a cubic lattice. In all calculations, the pairwise energy of interaction of A with B is identical with the p+rwise interaction (EM)of A with a void (solvent). With this assignment of the energies, the critical micelle concentration (cmc) depends strongly on NAand the interaction parameter, which is denoted ae x. The cmc is weakly dependent on NBif it is expressed in terms of the volume fraction of A, and NB is not too different from NA. The cmc is determined for the conditions 3 < (X/Z)NA < 6, where z is the 8 ln vAmC/8[(X/z)NA1 lattice constant. In this range, the cmc shows an exponential dependence on XNA, = -1.2. At constant XNA,the value of V A ~ increases C slightly when N A decreases. In the micelles the insoluble block is slightly collapsed, and the soluble block becomes stretched as NBincreases, with N Aand the number of chains in the micelle held constant. The size of the micelle is close to that estimated by a simple hard-core shell model. al. extended that work to the case of asymmetric triblock
Introduction
copolymers.11 Their theoretical formulations provide numerical predictions of the cmc. An analytical form for the dependence of the cmc on the number of unita in the insolubleblock, NA,and the interaction parameter, x , was obtained under conditions of strong segregation. Recently we have developed a simulation on a cubic lattice of the formation of micelles by block copolymers.12 The relatively small size of the lattice used in the initial development did not permit study of the cmc. Here, by increasingthe volume of the lattice by factors of 2 q S ,we can follow the cmc for different values of x. This paper will document that fact, demonstratehow the cmc is easily extracted from the simulations, and present the dependence of the cmc on X N Aunder conditions of relatively weak segregation. The results suggest that some of the predictions obtained previously using analytical treatmenta that assume strongsegregationmay retain predictive power well into the region of weak Segregation. We will also discuss the shape of the micelle, on the basis of the result from the simulation. Stretchingof the soluble block is seen as the number of units in the soluble block, NB, increases.
The self-assembly of diblock copolymers. into micelles in a selective solvent comprised of small molecules, or in a matrix comprised of a homopolymer, is a well-known phenomenon. The micelles in dilute solution are easily detected by light scattering. For example, this technique has been wed to investigate the structure of the micelles formed by block copolymers of polystyrene and poly(methyl methacrylate) in selective solvents.lI2 Light scattering is less useful in the study of the critical micelle concentration (cmc) because the onset of aggregation of block copolymers can occur at extremely low concentrations. Techniques that utilize the response of a fluorescent probe to the presence of the micelles have an advantage over light scattering because the experiments can be performed at a much lower concentration of the block cop~lymer.~-~ Several groups have also studied the cmc of diblock copolymers in a matrix comprised of molecules of the homopolymer with the same repeating unit as that found in one of the blocks.6-g Both transmission electron microscopy and small-angle X-ray scattering have been employed to determine the cmc for diblock copolymers of different compositions in the matrix formed by a homopolymer.819 Leibler et al. consideredthe cmc of a symmetric diblock copolymer in the homopolymer matrix,1° and Balsara et
Method The simulation is performed on a cubic lattice of dimensions 8 8 3 with periodic boundary conditions in all three directions. The diblock copolymers, A N ~ B Ncontain ~, NAbeads of A and Ne beads of B. No lattice site can be occupied by more than one bead. Vacant sites are considered to be occupied by solvent, S. Reptation and the extended Verdier-Stockmayer moves13are used to convert one replica into another, with the Metropolis rules1' employed for acceptance of a new replica. Pairwise interactions, Em and Em, are applied whenever A has a nonbonded B, or a vacant site, a~ a nearest neighbor. All of
443, 603, or
(1) Tanaka,T.; Kotaka, T.; Inagaki, H. Polym. J. 1972,3, 338. (2) Kotaka, T.; Tanaka, T.; Inngaki, H. Macromolecules 1977,II,138. (3) Zhao, Y.L.;Winnik, M. A,; R i m , G.; Croucher, M. D. Langmuir 1990,6,514. (4) Xu, R.; Winnik, M. A.; Croucher, M. D. Macromolecules 1991,24,
AI.
(5) Wilhelm, M.;Wao, Y.;Xu, T.; Winnik, M. A,; Croucher, M. D. Macromolecules 1991,24,1033. (6) Selb, J.; Marie, P.; Duplemix, R.; Gallot, Y.Polym. Bull. 1983,10,
__-.
AAL
Roe, R. J. Macromolecules 1984,17,1778. (8)Kinning, D. J.; Winey, K. I.; Thomas, E. L. Macromolecules 1988, 21,3502. (9)Kinning, D.J.; Thomas, E. L.; Fetters, L. J. J. Chem. Phys. 1989, (7) Rigby, D.;
(11)Baleara, N. P.; Tirrell, M.; Lodge,T. P. Macromolecules 1991,24, 1975. (12) Rodrigues, K.; Mattice, W. L. J. Chem. Phys. 1991,94, 761. (13) Verdier, P. H.; Stockmayer, W. H. J. Chem. Phys. 1962,36,227. (14) Metropolis, N.; Roeenbluth, A. N.; Roeenbluth, M. N.; Teller, A. H.; Teller, E. J. Chem. Phys. 1969,21,1087.
90,5806.
(IO)Leibler, L.;Orland, H.; Wheeler, J. C. J. Chem. Phys. 1983,79,
3650.
0743-7463/93/2409-0066$04.00/ 0
Q
1993 American Chemical Society
Langmuir, Vol. 9, No.1, 1993 67
Formation of Micelles by Diblock Copolymers the simulations reported here will use Em = Em, with all remaining pairWiee interactions being zero. The equivalence of Em and EM would be moat easily obtained in an experiment if the solventwere asmall oligomer formedfrom the aame monomer that occurs in the B block. End effects, if present, are ignored. Reduced interactions will be denoted by
XU
zE',
(2)
where z is the coordination number. Since all simulations are performed on a cubic lattice, z is a constant. We shall adopt x / z aa an abbreviation for E'ABand E'm. Two diblock copolymers are considered to participate in an aggregate if a bead of A on one chain is a nearest neighbor of a bead of A on the other chain. The basis for this choice comes from a consideration of the three types of changes in nearestneighborinteractionsdepicted in eqs 3-5. With the assignments
ocPa 0
0.006
-
0.004
-
With the assignments of the pairwise interactions that are used here, the conversion from E' to the Flory-Huggins x is xm = z E ' ~
0
0.008 -
"."""
0.00
0.01
A
0
0
0.02
0 , -
0.03
0
0.04
- 0 -
0.05
c
0.06
A...S
+ A...S
-D
A..A
+ S...S
(3)
VAB Figure 1. Volume fraction of free AloBlo y VU when xlz = 0.45. Points plotted ae + are from a 609 lattice, and the other points are from a 443 lattice.
B...S
+ B...S
-D
B...B
+ S...S
(5)
N,,and the number of sites, L3, on the lattice:
of the pairwise energies that are used here, the rearrangements of beads depicted in eqs 4 and 5 involve no change in energy,but the energy can change in the rearrangement depicted in eq 3. Therefore,it is only the creationof A--Acontacta that can produce stable aggregates. The method is essentially the same ae the one described previously,'2 the differences being the increase in the size of the lattice, the change in the number of chains on the lattice, the change in the values of NA, NB,and x, and the incorporation of the extended Verdier-Stockmayer moves.
Results and Discussion Detection of the Cmc. One of the characteristics of micellization is the relationship of the concentration of free chains to the total concentrationof all chains. When the total concentration is well below the critical micelle concentration (cmc),the chains do not form micelles and the concentration of free chains is directly proportional to the totalconcentration. As the totalconcentrationrises above the cmc, the concentration of free chains becomes relatively constant. In the simulation, we can group the chains into those that are in the singly dispersed state (meaning the chain has no bead of A that has as a nearest neighbor a bead of A from another chain) and those that form aggregates or micelles (meaningthe chain has at least one bead of A that has as a nearest neighbor a bead of A from another chain). When all of the pairwise interactions are zero, and the volume fraction of the diblock copolymers is small, most of the chains do not participate in contacts with other chains.12 Movement produces occasional collisions between chains, but the absence of any attractive energy causes the small aggregates that result from these collisionsto have a short lifetime in the simulation. Hence MJMo fluctuates about values that are close to 1, and often may be equal to 1. Here M,,denotes the numberaverage molecular weight of the aggregates, and MOis the molecular weight of a single chain (or an 'aggregate" that coxmieta of a single chain). Figure 1 depicts the behavior of 23 equilibrated ensembles containing AI& when the nonzero pairwise interactions are x / z = 0.45. T h e volume fraction of the copolymer, denoted by VAB,is plotted on the abscissa. That number is lees than 0.06. It is completely specified by NA,NB, the number of chains of the diblock copolymer,
It does not depend on the arrangement of the chains on the lattice. The ordinate is the fraction of the volume occupied by chains that do not make a nearest-neighbor A--A contact with another chain. That number is determined by the organization of the chains on the lattice. It is averaged over 400 independent replicas of each system. The magnitude of the fluctuationof Vmh is around 10%. The figure contains two straight lines. T h e one line has a slope of 1 and passes through the origin. The other line has a slope of 0 and a value of 0.008 at ita intersectionwith the ordinate. The former line approximates the behavior of the pointa evaluated for the most dilute systems, and the latter line describes the "moreconcBlltfBted systems. We identify the cmc with the total concentration at the point defined by the intersectionof the two straight lines. Thus,the cmc is a volume fraction of 0.008 f 0.001 when the chainsare A&o and the nonzero pairwiae interactions are those specified by x/z= 0.45. The three pointa plotted as in Figure 1 were computed using a lattice of dimensions 603, and the remainingtwenty points are from a lattice of dimensions 448. No dependence on the size of the lattice is apparent in Figure 1. The simulation in Figure 1 is extendedto a higher volume fraction in Figure 2. As the volume fraction rises well Withacontinuousincrease above -0.06, V~hdecr-. in Vm,the system must eventually pass through C*, thereby reaching the semidilute condition where no free chains remain and V u h 0. The mean square radius of gyration for A&o, calculated for a single chain on a 443 lattice when all pairwise interaction energies are zero, is 5.75 f 0.06. Thus, C* EJ 0.17. Hence, for this system, the concentration of free chains is approximatelyconstant when cmc < Vm < C*/3. T w o variables will be used to describe the size of the micelles. One variable, N-, is the number of chains in the largest micelle in eachsnapshot,and the other variable is M,IMo, where M, is the weight-average molecular weight of the aggregates, and MOis the molecular weight of a single chain. Thus,MW/Mo< N-, because the former variable is averaged over all aggregates in the system. Figures 3-5 compare the equilibration times of Vuh,
"+"
-
Wang et al.
68 Langmuit, Vol. 9, No. 1, 1993 O.OIO
25 I
20
15
4% >
0.004-0
0.002
r'
9
'
0.oOoJ.
.
'
.
'
.
. '
'
10
5
.
0
2
4
6 8 10 iteration (millions)
12
14
Figure 4. Behavior of MwlMo during a simulation on a U3lattice that starta with 200 free chains of A&lo with x l z = 0.45.
0.6
-
0.5
-
45 40
&
c
Q 0
:2or i
1
25
-
15
2
4
6
8
12
10
E
I
A
I
14
iteration (millions)
Figure 3. Fraction of free chains during a simulation on a U 3
lattice that starts with 200 free chains of A&o
MW/Mo,and N-.
0
with xlz = 0.45.
The volume fraction of free chains reaches ita equilibriumvalue very fast, within a few million iterations, but MW/Mocontinues to change in a systematic manner through the first 10million iterations. The value of N- converges at an intermediate rate. The micellization in the simulation occurs in two processes, with different time scales. The first faster step is the equilibration of the free chains with aggregates of all sizes, and the second slowerstep is the equilibrationof the aggregates with one another. The second step becomes very slow as one moves to conditions of strong segregation. This behavior is readily apparent if one arbitrarily selects one chain as a "tracer*,and then monitorsthe number of chains in the aggregate that contain the tracer, N*. At weak segregation Nt,experiences rapid fluctuations, and frequently can assumethe value of 1,meaning that the tracer has temporarily become a free chain. In contrast, at strongersegregationNechangesvery slowly after the tracer enters a micelle. This phenomenon is not an artifact of the simulation, but instead reflecta the dynamics of the exchange of chains between micelles and the population
I
I
I
I
I
I
I
2
4
6
8
10
12
14
iteration (millions)
Figure 5. History of N,, during a simulation on a 443lattice
that starts with 200 free chains of AloBro with xlz = 0.45.
of free chains, which can be monitored in an experiment that measures the efficiency of F6rster transfer in pop ulations of diblock copolymers labeled appropriately at the junction pointa.l6 Figure 6 depicts the dependence of the final values of M,IMo on the totalvolume fractionof the copolymer. From Figure 4 we conclude that the uncertainty in these values from the fluctuations is 10% , and that the very slow equilibration of MW/Momay cause some of the estimates to be a bit low. The values of M,IMo continue to increase as V- rises abovethe cmc, which was -0.008, and appear to be approachinga limitingvalue of -26. It is of interest to comparethisaggregationnumber with the one computed from a very simplemodel for the micelle. The mean square end-bend distance of the A block, ( r ~ is ~ 1)6.,6 when all
-
(16)Wang, Y.;Bahji, R.;Quirk, R. P.;Mattice, W.L. Polym. Bull. 1992,28,333.
Formation of Micelles by Diblock Copolymers 32
Langmuir, Vol. 9, No. 1, 1993 69
I
I
+ 24
0
-
+
-
0
10 15 20 25 30 35 45
0 0 0
0 16
00 0
OO 00 0
+
OO
"
"
0.02
.
'
I
.
0.04
0.06
0.08
0.10
VAS
Figure6. Dependenceof MJMoon Vm when x/x = 0.46. Points plotted aa + are from a W lattice, and the other points are from
a 4 4 3 lattice. NB 5 10 15 20
Table I. Cmc for Eight AIOBN- When x / e = 0.45 Vm-C V~cmc NB VmC'QC VA-C 0.0052 0.008 0.010 0.014
0.023 0.027 0.023 0.028 0.023 0.023 0.023
22 20 19 19 20 16 17
1.05 1.08
1.07 1.11 1.14 1.21 1.19
=Thenumber of chains in the micelle for which (rQ)miwb waa evaluated. Ratio of the mean square end-bend distance of the soluble blocka in the micelle to the mean square end-bend distance for the soluble block in the free chains with Em = E u = 0.
8-
O ' U.00
Table 11. Stretching of the Soluble Block in the Micelle8 of A ~ N When . x/z = 0.46
0.0034 O.OO40 0.0041 0.0046
25 30 35 45
0.014 0.018 0.023 0.029
0.0041 0.0045 0.0051 0.0052
pairwise interactiom are zero. The volume computed as (4/3)r(rA2)3/2 is 256, which is equivalent to the volume occupied by the A block in 25.6 chains. Thus, the number of chains in the micelle is well approximated by this simple estimate. Figure 1 used the results from 23 independent simulations to determine the cmc. An excellent approximation to that result could have been obtained as Vmk in a single simulation for which VU was slightly larger than the cmc, but well below C*. The lower limit of the cmc that can be studied by this approach is determined by the ratio of the sizes of the diblock copolymer and the lattice. If we insist that at least one chain of the diblock copolymer must exist as a free chain, the smallest accessible cmc is a volume fraction of (NA+ NB)L-~.With N A + NB = 20 and L3 = 443, the smallest accessible cmc is a volume fraction of order 2 X lo4. Accessto smaller cmc's requires use of a larger lattice. Influence of NBon the Cmc and Micelle Size. The cmc was evaluated for Ai& with NB = 5,10,15,20,25, 30,35,and 45. In each case, E f u = 0.45. The results are presented in Table I. The results for NB= 35 and 45were obtained on a 603 lattice, and all other results are from the 443lattice. If the cmc is expressed as the volume fraction of denoted Vmcmc, it increases as NB increases. However,the cmc is less dependent of NBif it is expressed as the volume fraction of A, V A ~especially ~ , from the data with 10 INB I30. The absence of a correlation between NBand V Ais dependent ~ ~ upon the assignment of EM = Em, with all other pairwise interactions being zero. If the pairwise interactions were reassigned so that EM # Em, the cmc would become more dependent on NB.'~Even with EM = Em, the cmc becomes a function of NB when the size of the soluble block is varied over a much larger range. As NB =, the large soluble blocks of B will shield the much smaller insoluble block of A from
-
(lS)Wang, Y.; Mntticr, W. L.; Napper, D. H. ACS Symp. Ser., in press.
participation in intermolecular interactions. The onset of this effect may begin at NB= 35,on the basis of the last two entries in Table I. The behavior of the simulations in Table I can be compared with the behavior of block copolymers of polybutadiene (A) and polystyrene (B),examined in a matrix of polystyrene using transmission electron microscopy and small-angle X-ray scattering.B When dispersed in a polystyrene of specified molecular weight, the copolymers denoted by SB 10/10,SB 23/10,and SB 40/10 by Thomas and co-workers have cmc's that are nearly constant, provided those cmc's are expressed in terms of the concentration of the polybutadiene block. (The compositionof SB X/Y is a polystyrene block of molecular weight approximately X kg mol-' and a polybutadiene block of molecular weight approximately Y kg mol-'.) For example, the cmc for all three of these copolymers in a polystyrene of molecular weight 7400 is in the range 0.42 0.07,when expressedin weight percent of polybutadiene. This constancy of the cmc does not extend to the sample denotedby SBSo/ 10,which has a larger cmc. Neverthelees, the experiments reported by Thomas and co-workersare consistent with the finding in the simulations that the variation in the size of the soluble block is not important in the specificationof the cmc, so long as it does not become greatly larger than the insoluble block. If N A and the number of chains in a micelle remain relatively constant, then the soluble block should become stretched as NB increases. The mean square end-bend distance was evaluated for the soluble block in micelles of that contained -20 chains. The results in Table I1 show that stretching of the soluble block is observed as NB increases, but the maximum stretching seen in the simulations is -lo%, on the basis of ((rB2)micelle/ (Q2)free)1/2.The values of @A2) are 12.5in all of these micelles. Therefore, the insoluble blocks in the micelles are compressed slightly from the dimensions they would have ((rA2) = 15.5)if all pairwise interactions were zero. Figure 7depicta the distributionfunctionsfor the volume fractions of A, B, and solvent, as a function of the radial distance from the center of the micelle formed by 16chains The volume fraction of A exceeds 0.9 in the of AI&. core of the micelle, and B is strongly excluded from the core. If all of the beads of A in the micelle were arranged in a compact sphere, that sphere would have a radius of 3.4. The volume fractions of all three species are changing rapidly in the vicinity of r = 3.4. Form of the Dependenceof the Cmc on XNA.Figure 8 depicts the dependence of the cmc on (x/z)NAusing NA = 8,10,and 12,and different values of x / z . The upper limit of X N Ain the figure is determined by the range where the cmc can be studied with lattices of sizes 443 and M3. The cmc is expressed as the natural logarithm of VA-~.
*
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IO hngmuir, Vol. 9, No. 1, 1993
Wang et al.
3
0
2
4
6
8
10
12
14
16
18
5
6
7
(W N A
1
0
4
20
r Figure 7. Volume fraction distribution aa a function of radial distance from the center of a micelle of 16 chains of A,,& when x/z = 0.46.
Figure 8. Cmc (expressedae In VA-~)for AN B N ~aa, a function of (x/z)NA.The values of N Aare (X) 8, ( 0 )la, and (A)12. The straight line with slope of -1.2 is the fit to the eight points for NA 10.
The value of NBremains in the range in which ita variation has little effect on the cmc, as shown in Table I. Figure 8contains results for systemswith different values of both x and NA. The approximation of the data by a single curve, drawn as a straight line with slope -1.2,shows that the important parameter is the product of the two, XNA. However, there are systematic deviations of data from this approximate description. When the value of N A decreases at constant XNA,the cmc increases, thereby causing departures from the simple description. Leibler has predicted an exponential dependence of the cmc on XNAfor symmetric diblock copolymers, under conditions of strong segregation,in a homopolymer matrix where the degree of polymerization of the homopolymer is much less than Ng.10 Thus, the prediction from his work is approximated in Figure 8,although very few of the pointa in that figure are for chains with N A= NB.His treatment predicts a slope of -1 when In Vmmcis plotted vs xN, and we find a slope of --1.2 when In VA- is plotted vs ( X / Z ) N A , z being about 4.
Conclusion By using reptation and Verdiedtockmayer moves on a cubic lattice, we can observe the cmc for diblock copolymers in a selectivesolvent under conditionsof weak segregation. The size of the micelle formed by is close to that estimated from a simple model of the core shelltype. In the micelle,the insoluble block has a slightly smaller mean square end-to-end distance than in an isolated chain. The soluble block becomes stretched as NBincreases. Variations in the size of the soluble block have a small effect on the cmc when Em = EM, but the cmc is strong$ a f f d by variation in the size of the insoluble block. The cmc can be approximated as an exponential dependence on XNA,but detailed examination detecta a small influence of N A on the cmc, at constant XNA. Acknowledgment. This research was supported by National Science Foundation Granta DMB 87-22238and INT 90-14836,and by the Australian Department of Industry, Technology and Commerce.
