Molecular simulation of ordered amphiphilic phases

Pergamon

Chemical Engineering

MOLECULAR

SIMULATION OF ORDERED PHASES AT&T

(First received

Vol. 49. No. 17, pp. 2833-2850, 1994 Copy&In 0 1994 Elscvicr Sctcna Ltd Printed in Great Britain. All ri&s ~pcrvcd ooo%2.w9/94 97.00 + o.ca

S&me,

AMPHIPHILIC

R. G. LARSON Bell Laboratories, Murray Hill, NJ 07974, U.S.A. 23 August

1993: accepted

in revised form 10 January

1994)

Abstract-Self-assembled equilibrium amphiphilic phases with one-, two-, and three-dimensional order are simulated by a Monte Carlo technique using a simple Flory lattice model to describe the configurations and interactions of idealized single-site “oil”, and “water”, molecules and multi-site amphiphilic molecules. Selfassembled patterns with at least one zero wavevector, namely lamellar or hexagonal, select orientations on the three-dimensional lattice so that the finite wavevectors of the pattern are roughly independent of lattice size. Patterns with three-dimensional order, including cubic and noncubic packings of spheres, and a tetragonal intermediate phase, also self-assemble when the lattices are of sizes nearly commensurate with the wavevectors of the bulk equilibrium patterns. The relative stability of sphere packings can be assessed by allowing patterns induced on small lattices to relax on large lattices that are periodic replications of the small ones; for a surfactant dissolved in “water” with a head group much larger than its tail group, the stability appears to increase in the following order: simple cubic, face-centered cubic, body-centered cubic, hexagonally close packed. The average confonnational states of the surfactant chain and its head and tail units are computed for various ordered phases; these results indicate that crowding of surfactant head groups, which can be swollen by solvent, is a significant factor influencing phase behavior. In lamellar phases with holes in the layers that contain the surfactant tails, defects in the patterns form easily; these facilitate changes in layer spacings when the temperature is changed.

1. INTRODUCTION

Surfactants, when mixed with water and oil, selfassemble into a variety of different ordered phases, including lamellar stacks, hexagonal arrays of cylinders, rectangular arrays of ribbons, cubic packings of spheres, and three-dimensionally ordered intermediate phases, including several different bicontinuous cubic phases, a tetragonal phase, and a rhombohedral phase (Kbkicheff and Cabane, 1987; Kekicheff and Tiddy, 1989; Luzzati and Spegt, 1967; Tardieu, 1972; Helfrich, 1987; Gruner, 1989; Turner, 1990, Charvolin and Sadoc, 1987; Hyde et al., 1984; Hagslitt et al., 1992; Blackburn and Kilpatrick, 1992). Block copolymers, which are long amphiphilic molecules, have also been found to form some of these phases (Thomas et al., 1986, 1990; Hasegawa et al., 1987; Almdal et al., 1992). Entropic and energetic factors govern the selection of the equilibrium pattern; for a surfactant these include the balance of hydrophilic and hydrophobic tendencies of the surfactant, the range of conformational states available to it, the volume fractions of oil, water, and surfactant, and, of course, the temperature. Incorporation of these factors into statistical-mechanical analyses sometimes yield predictions of the conditions under which lamellar, cylindrical, or spherical microstructures occur (Wang and Safran, 1990; Golubovii: and Lubensky, 1990; Huse and Leibler, 1988; Safran et al., 1984; Gompper and Schink, 1989; Helfand, 1975; Helfand and Wasserman, 1976; Fredrickson and Helfand, 1987); but these methods are not readily applicable to the prediction of more complex patterns, such as those of the intermediate phases. 2833

Recent increases in computer speed are improving the chances that molecular simulation could be used to predict pattern selection in these systems, but two major obstacles must first be overcome. The first obstacle is that self-assembly of an ordered pattern from a disordered starting state requires that the pattern’s phase information be transmitted across the entire simulation volume or box by collective molecular diffusion, and this process is exceedingly slow even for modern computers. The second obstacle is that conditions imposed on the boundaries of the box must be kept from artificially influencing the pattern selected. This second obstacle can be addressed by using periodic boundary conditions, but since the pattern that forms must be commensurate with the box dimensions, the size of the box will in general influence the pattern selected, unless the box is large. Enlarging the box, however, increases the distance over which order must be coherent, and hence further magnifies the first obstacle. Assuming that the time t, for diffusion of phase information scales as t, a L2, where L is the linear dimension of the three-dimensional system, and that the computer time T required for the simulation goes as te times the number of molecules in the simulation volume, we find that T - L5, and so these obstacles are serious ones. Nevertheless, we show here that they can be overcome on inexpensive workstation computers, if one uses a simplified lattice model of the surfactant-oil-water system. In this lattice model, described in detail elsewhere (Larson er al., 1985; Larson, 1992), idealized “oil” and “water” molecules occupy single sites of a simple cubic

2834

R.G. LARSON

lattice. The expected directional bias of the cubic lattice is suppressed by allowing a site to interact pairwise additively with all of its z = 26 nearest, facediagonal, and body-diagonal nearest neighbors, all with equal strength. The surfactant molecule occupies a sequence of lattice sites connected along any of these same 26 directions. With the inclusion of interactions between diagonally adjacent units, the rotational symmetry of continuous space is more closely approximated than would otherwise be the case. The amphiphile sites are occupied by either head (waterloving) or tail (oil-loving) units. For simplicity, we let the head and tail units interact with their neighbors just as do water and oil units, respectively. With these choices, the system can be characterized by a single dimensionless interaction energy parameter w, which is the interaction energy per oil-water contact, divided by k,T. The nomenclature H,T, defines a surfactant that consists of a string of i head units attached to j tail units. The usual periodic boundary conditions are used. Rearrangements of the molecules take place by oil-water interchanges, and by kink and reptation motions of the amphiphile, as described elsewhere (Larson et al., 1985; Larson, 1988). To prepare equilibrium structures, an initially random arrangement of molecules at infinite temperature (w = 0) is cooled slowly by increasing w in small increments, until an ordered pattern forms. To show that the pattern is that of an equilibrium bulk state, we repeat these runs on lattices of different sizes and at different cooling rates, and in some cases use as a starting state on a large box a pattern formed by combining eight replicas of a pattern generated on a box whose edges ate only half the length of those on the large box; see below. As described in an earlier publication (Larson, 1989), the number of attempted Monte Carlo (MC) moves-which we designate by T-that are needed to achieve an equilibrium pattern varies with the size of the box, L, the inverse dimensionless temperature, w, and the particular surfactant and its composition in oil and/or water. During simulations of cooling or heating, the system energy is monitored, and increments of temperature are taken only after the energy, averaged over many attempted MC moves, stabilizes at an approximately constant value; see Larson (1992). From our molecular simulations, we can extract detailed information about the molecular configurations in the various ordered and disordered states. Of particular interest is the degree to which the amphiphiles depart from their high-temperature random configurations when the solutions containing them are cooled into ordered states, and the degree to which the amphiphile configurations depend on the type of ordered phase in which they find themselves. To characterize the distribution of configurations, we compute R’, the mean-square separation of one end of the amphiphile from the other, R&, the meansquare separation of one end of the head group from the other, and R:, the mean-square separation of one

end of the tail group from the other. These quantities define the degree of stretching of the whole chain, its head group, and its tail group, respectively.

2. RESULTS

2.1. One-dimensional order: lamellar Let us first consider equilibrium self-assembled patterns with two infinite wavelengths, i.e. lamellat patterns. Figure 1 shows two mutually perpendicular slices through 9 x 9 x 9 and 15 x 15 x 15 lattices containing 60% (by volume) H,T, surfactant, 20% oil, and 20% water, cooled from an initially disordered state at w = 0 to w = 0.1385. In Fig. 1 and the following figures, the simulation areas are enclosed by dashed lines; the rest of the images shown in these figures are obtained by periodic replication of the image within dashed lines. Lamellar patterns ate evident in Fig 1. On the 9 x 9 x 9 lattice, the lamellae are oriented parallel to a face of the box, and only one period of the lamellar pattern fits in the box. When the edge of the box is increased to 15, which is incommensurate with the lamellar spacing found on the 9 x 9 x 9 box, the spontaneously self-assembled lamellae cannot orient parallel to a face of the box; instead the normal to the lamellae orient parallel to the body diagonal of the lattice; the repeat period, d, of the onedimensional pattern is then 15/d = 8.66 lattice sites. For a 20 x 20 x 20 lattice, shown in Fig. 2, the lamellae take on yet a different orientation, and the repeat period is 2013 = 8.94. For each lattice size, periodic boundary cond,itions make possible only a discrete set of orientations, and we find that the lamellar orientation and spacing vary with the size of the lattice, shown in Table 1. We find the elements of the vector m = (m,, m,, m3) to be the numbers of lamellae that intersect each of three mutually orthogonal edges of the periodic cubic box. The m,‘s ate therefore the “quantum numbers” that define the lamellar spacing and orientational state in a given finite box. Note from Table 1 that as the lattice size increases, new quantum states are selected so that the lamellar spacing L/m remains within certain bounds. For small lattices, the density of “quantum states” is sparse, and the lamellar spacing must change significantly, for example, from 7 to 9, as the lattice size increases. For some small lattices, such as the 10 x 10 x 10 lattice, no ordered phase forms, probably because on these small lattices, the only permissible “quantum numbers” require too much distortion from the optimal spacing for the lamellar phase to be stable. For bigger lattices, the density of “quantum states” is greater, and the lamellar spacing varies less with lattice size than it does on smaller lattices, and a single “classical” value, around 8.5, seems to be approached in the limit of a large lattice. The high variability of the ratio m, : tn2 : m3 in Table 1 shows that the orientation of the lamellae is controlled by the preferred lamellar spacing and not by lattice orientational bias. The lamellar ordering transition is expected to occur at roughly a fixed value of

Simulation of ordered amphiphilic phases

2835

0

9X9X9 Fig. 1. Two mutually perpendicular cross-sectional slice-s of 9 x 9 x 9 and 15 x 15 x 15 lattices containing 60% H,T, and equal parts of oil and water at w = 0.1385. In this and the following figures, the images were obtained by periodic replication of the actual simulated volume, which in this case is enclosed in dashed lines. (0 0 0) Tail units; (***) oil units.

zwN, where N is the length of the sutfactant chain. In the present case, where N = 6 and z = 26, the transition occurs when w = 0.1385. On a conventional cubic lattice with only nearest-neighbor interactions, so that z = 6, we expect the lamellar transition for H,T, to occur at w z 0.60, which is of order unity; and a significant lattice bias should thus be expected. For much longer “polymeric” amphiphiles, with N > 20, lattice bias should be small even on the conventional cubic lattice, and for them it is probably not necessary that z be large. For off-lattice simulations in continuous space, finite periodic boxes will, of course, produce effects comple’tely analogous to those discussed here; the lattice model, however, allows us to explore these box-size effects more completely than would be possible with the much smaller systems to which one is restricted in slower, off-lattice, simulations. We note here that for longer amphiphiles, representing diblockcopolytners, Fried and Binder (1991) have presented computer simulations of the structure of the disordered state near the transition to a lamellar phase.

2.2. Two-dimensional order: hexagonal arrays of cylinders The lattice model also allows self-assembly of patterns with one zero wavevector, namely hexagonal, or distorted hexagonal, packings of cylinders. For symmetric surfactants such as H,T, or H,T, in water, cylinders form at amphiphile volume fractions, C,, of 0.45-0.65. Because of the limited number of possible orientations on a small lattice, true hexagonal symmetry is often not attained in the cylinder-forming systems. In a true hexagonal pattern, passing through a single cylinder there are three infinite rows of cylinders, each oriented at an angle of 120” relative to each of the other two rows, and the spacing between cylinders is the same in all three rows. In the simulations, the constraints that the pattern fit the prescribed box, that periodic boundary conditions be satisfied, and that the preferred spacing be achieved, overdetermine the system. The pattern on a given finite box therefore can suffer a distortion that breaks the symmetry, so that the spacing between cylinders varies somewhat among the three rows. The intercylinder spacings for several quasi-hexagonal patterns

R. G. LARSON

2836 40

H3T3 CA=

0.6

CW/C~==l W

=

0.1385

0

20

0 0

20

40

Fig. 2. Two mutually perpendicular cross-sectional slices of a 20 x 20 x 20 lattice containing equal parts of oil and water at w = 0.1385.

Table 1. Spacings and orientations Lattice size L x L x L

Orientation

of lamellar H,T,

phases

m

Spacing N/m

morn

7X1X7 8X8X8 9X9X9

l,O,O l,O,O kO,O

1 1 1

7.00 8.00 9.00

12x 12x12 13x 13x13

lJ,O WO

2 2

8.49 9.19

14x 14x14 15x 15x1.5 16X16X16

l,l,l l,l,f 1,1,1

3 3

8.08 8.66 9.24

17X17X17 18X18X18

2,090 2,0,0

4 4

8.50 9.00

19x 19x19 20X20X20

2,f,O Zl,O

5 5

8.50 8.94

38x38x38

3,3,1

19

8.72

39X39X39 40X40X40

333.2 3,332

22 22

8.31 8.53

60% H,T,

and

Simulation

in Table 2; images of some of these systems can be found in Larson (1992). Note that in the first three rows of Table 2-those for the 60% I-I,T, systemthe distortion from hexagonal symmetry decreases with increasing box size. For lattices at least as big as 30 x 30 x 30, the distortions from the symmetry are no more than 5% for all compositions that form cylindrical microstructures; see the last group of four entries in Table 2. appear

2.3. Three-dimensional arrays of spheres

order;

cubic

and

2837

of ordered amphiphilic phases

noncubic

Patterns without an infinite wavelength, such as cubic, rhombohedral, or tetragonal phases, cannot readily be fit to a given lattice by reorientation of the pattern. A cubic phase will only form when the length L of a side of the cubic box is a multiple of the length of the cubic phase’s unit cell. Thus, for simulation of cubic phases, or other three-dimensionally ordered phases, the box size can strongly influence pattern selection.

At concentrations of I&T4 below that at which cylinders form, namely C, 6 0.4, spherical micelles form. Table 3 shows the patterns formed by a 35% solution of H,T4 in water as a function of box size. For small boxes, L < 20, the three simplest cubic symmetries, namely simple cubic (SC), body-centered cubic (BCC), and face-centered cubic (FCC), appear, each within its own range of box sizes. For the smallest boxes, L. = 11-13, usually a single spherical micelle forms, although sometimes runs on 11 x 11 x 11 and 12 x 12 x 12 boxes produce a single “cylindrical” micelle-i.e. one that spans these small boxes. Note in Table 3 that the number of micelles increases with L roughly as L3, so that the average number of surfactants in a micelle, given as the “aggregation number” in the last column of Table 3, generally remains within the range 60-l 10. The number M of micclles in the box controls the cubic symmetry selected. If M = 1, the symmetry is trivially simple cubic, while for M = 2, a BCC symmetry is obtained, and an FCC symmetry for M = 4. Since the

Table 2. Spacings of hexagonal and quasi-hexagonal H,T,, phases Surfactant concentration

Lattice size

LXLXL

Pattern symmetry

Inter-cylinder spacings

60% 60% 60%

20X20X20 30X30X30 40X40X40

Quasi-hexagonal Quasi-hexagonal Quasi-hexagonal

10.0~11.2~11.2 11.2~11.2~11.8 11.1~11.4~11.4

45% 50% 55% 65%

30X30X30 30X30X30 30X30X30 30X30X30

Hexagonal Hexagonal Quasi-hexagonal Quasi-hexagonal

12.3112.3112.3 12.3 I 12.3 112.3 10.8/10.8~11.1 10.8~10.8~11.1

Table 3. Symmetries and spacings for 35%

H,T, in water, w = 0.1385

No. of

LXLXL 10x10x10 llxllxll 12X12X12 13X13X13

micelles M

Symmetry

Spacing

L/M”’

Average aggregation no.

: 1 1

SC SC SC

10.0 11.0 12.0 13.0

43 58 75 96

14X14X14 16X16X16 17X17X17

2 2”

BCC BCC BCC

11.1 12.7 13.5

60 89 107

18x18~18 20X20X20

4 4

FCC FCC

11.3 12.6

64 87

22X22X22 23x23~23

4 6

-

13.9 12.7

116 89

24x24~24

8

Q-HCP

12.0

76

25x25~25 26x26~26 27x27~27 30X30X30 32x32~32 33 X 33 X 33

8 8 9

-

:: 15

12.5 13.0 13.0 13.1 13.0 13.4

85 96 96 98 96 105

40X40X40

32

12.6

87

SC

-

R. G. LARSON

2838

unit cells of the SC, BCC, and FCC symmetries contain 1, 2, and 4 spheres, respectively, the ordered cubic phases obtained on boxes with L g 20 contain only a single unit cell of the pattern, and the symmetry of that unit cell is controlled by the box size. Thus, these simulations on small lattices, unlike those producing lamellar or hexagonal symmetries, do not show which ordered patterns might be characteristic of a bulk system. On the 24 x 24 x 24 lattices, eight micelles are formed; these do not order into eight unit cells of a simple cubic pattern, but “quasi-ordered” patterns appear instead, which differ from run to run. One such quasi-ordered state is shown in Fig. 3. In Fig. 3, the eight spheres in this 24 x 24 x 24 box have arranged themselves into two layers containing four spheres each. Each layer is a distorted hexagonal arrangement with spacings along the three rows of spheres of 12: 13.4: 13.4, which may be as close to hexagonal as is allowed in the small box. The spheres in the second layer are staggered with respect to those in the first, and since there are only two layers of spheres, the pattern is repeated after two layers, which corresponds to a hexagonally close-packed (HCP) symmetry. However, the distance separating a layer from its repeat layer is 24 lattice units, so that the ratio of this spacing to the average spacing between spheres within a layer is 1.85, which differs from the corresponding ratio, 1.633, for a true HCP packing. This particular “quasi-HCP” pattern does not always occur on repeat runs, however. Sometimes a pattern similar to that shown in Fig. 5 occurs (discussed below), while in other runs a more disordered arrangement appear in which one or more pairs of spheres are fused into dimers. For boxes as large or larger than 22 x 22 x 22, the micelles do not develop any regular spatial order in the time period of the simulation, except for the 24 x 24 x 24 box, discussed above. The lack of a spatially ordered state is noted in column 3 of Table 3 by “-“.

H4T4,

cA=0.35,

co =0,

Micelles of H,T, that are not spatially ordered typically show large variations in shape and size with some of the micelles being cigar-shaped with aspect ratios of two or more. The diameters of these cigarshaped micelles are similar to those of the spheroidal micelles, but the cigars contain roughly twice as many surfactant molecules as do the spheroidal micelles, and so their lengths are twice their diameters. If the concentration C, of H,T, is increased from 35% to 40%, then longer, worm-like, micelles appear (Larson 1992); long micelles are also favored at lower dimensionless temperature, l/w. Since ordered patterns self-assemble on small boxes, but not on large ones, either the true bulk equilibrium state for 35% H,T, at w = 0.1385 is disordered and the order on the small lattices is a small-box artifact, or the simulations on the larger boxes were not run long enough for the bulk equilibrium ordered pattern to appear. To distinguish between these two possibilities, ordered starting states were prepared on large boxes by combining eight copies of the ordered states that had self-assembled on smaller boxes. Thus, Fig. 4(a) shows two superimposed slices of the ordered FCC state that self-assembled at w = 0.1385 on the 20 x 20 x 20 lattice; and Fig. 4(b) shows a slice of the 40 x 40 x 40 state prepared by putting together eight copies of the 20 x 20 x 20 system of Fig. 4(a). This larger FCC-ordered 40 x 40 x 40 starting state was then “annealed” by running a simulation holding the inverse dimensionless temperature$xed at w = 0.1385. Figure 4(c) and (d) shows the result after T = 3.2 x lo9 and 6.4 x lo9 attempted MC moves. Examining all slices of this system at T = 6.4 x 10’ confirms the impression drawn from a single slice, that the FCC symmetry has melted, leaving a disordered arrangement of spheroidal micelles. Since the original FCC pattern on the 20 x 20 x 20 systems formed spontaneously at w = 0.1385 from a random arrangement of molecules, and that same pattern spontaneously decayed at w

w-O.1385

Fig, 3. Two parallel slices of a 24 x 24 x 24 lattice containing 35%

H,T,

24X24x24

in water at w = 0.1385.

Simulation of ordered amphiphilic phases

H4T4,

CA=0.35,

Co=l,

.s%

d) T =

12.8 x IO9

2839

w =

“a:

^”

c) T =

0.1385

,

D

-900

I

3.2 x IO’

Fig. 4. (a) Two superimposedslices of the FCC pattern formed on a 20 x 20 x 20 lattice by 35% H,T, at w

= 0.1385. The asterisks here depict the tail units on a second slice of the lattice. (b) The FCC pattern in (a) has been periodically replicated to produce an FCC pattern on a 40 x 40 x 40 box, used as a starting state for a Monte Carlo “annealing” run at w = 0.1385. (c) and (d) The configuration after 3.2 x IO9 and 6.4 x lo9 Monte Carlo attempted moves.

= 0.1385 on the larger 40 x 40 x 40 lattice, we conclude that at this temperature the FCC symmetry is an artifact of the smallness of the 20 x 20 x 20 lattice. Even at a lower temperature, for which w = 0.1538, the FCC pattern melts into disorder, indicating that the FCC pattern is not stable even at lower temperatures. Likewise, by replicating the SC pattern formed on a 12 x 12 x 12 box, and annealing this pattern on a 24 x 24 x 24 box, we find that the SC pattern, like the FCC pattern, is unstable, and melts into the quasiCES49:17-I

ordered states typically of initially disordered runs on the 24 x 24 x 24 box. However, when the BCC pattern formed on the 16 x 16 x 16 lattice is replicated to produce a BCC pattern on a 32 x 32 x 32 lattice, annealing for a time as long as that for the 40 x 40 x40 lattice, namely T = 6.4 x 10q, does not lead to melting of the BCC pattern. Thus, for H,T, the BCC pattern is more stable than the FCC and SC patterns. Note, however, in Table 3, that the BCC pattern does not spontan-

R. G. LARSON

2840

eously form from a disordered starting state on the 32 x 32 x 32 lattice. Thus, the BCC pattern for H,T, appears to be only metastable at w = 0.1385. We therefore believe that the equilibrium bulk pattern for 35% H4T, consists of disordered micelles. It should not be too surprising that these H,T, micelles are spatially disordered in the bulk, since at a volume fraction of only 35%, significant interpenetration of any two micelles can be avoided easily, even when the micelles are spatially disordered. When we increase the concentration of H,T, to 40% to try to produce stronger excluded-volume interactions between adjacent micelles, the micelles begin to coalesce into oblong and worm-like micelles, rather than forming an ordered cubic phase. Even at 35% amphiphile, on boxes in which no ordered phase appears, a fraction of the micelles are “dimers”, i.e. oblong micelles with about twice the aggregation number as

are contained in spheroidal micelles. As a ‘result, the average aggregation number in boxes for which the micelles are disordered tends to be higher than that for ordered micelles; see Table 3 (and Tables 4 and 5). At a concentration of 45%, the H4T4 surfactants organize themselves into hexagonally packed cylinders. This apparently occurs because the junctions between the head and tail groups of the symmetric surfactant H,T, prefer to reside on an uncurved surface, and are content to form cylindrical micelles once the concentration becomes large enough that overlap of spheroidal micelles becomes probable. As a result, H,T, is unlikely to form an ordered packing of spherical micelles at any concentration or temperature. However, for surfactants with head groups much longer than their tail groups, the head-tail junctions prefer to reside on a surface that is curved towards the

Table 4. Symmetries and spacings for 55% HsT4 in water, w = 0.1308 No. of

LXLXL

micelles M

Symmetry

Spacing L/M’/’

Average aggregation no.

12x 12x12 13X13X13

1 1

SC SC

12.0 13.0

19 100

14 x 14 x 14 15xl5xlS 16X16X16

z 2

BCC BCC BCC

11.1 11.9 12.1

13.5

63 77 94

112

18 x 18 x 18 19 x 19 x 19 20X20X20 21X21X21 22X22X22 24 x 24 x 24

4 4 4 4 5 8

FCC FCC FCC FCC Q-HCP

11.3 12.0 12.6 13.2 12.9 12.0

61 79 92 106 93 19

16 31 47

BCC -

11.9 12.1 12.5

77 95 89

17X17X17

30X30X30 40X40X40 45X45X45

2

BCC

Table 5. Symmetries and spacings for 70% H,,T, LXLXL

No. of micelles M

12X12X12 14 x 14 x 14 15X15X15 16x16~16

1 1 2 2

18X18X18 19x 19x19 20X20X20 21 x21 x21

in water, w = 0.1077

Symmetry

Spacing L/M’13

SC

12.0

Average aggregation no.

76

BCC BCC

14.0 11.9 12.7

120 74 90

4 4 4 4

FCC FCC FCC FCC

11.3 12.0 12.6 13.2

64 75 88 101

24x24~24

8

Q-HCP

12.0

16

30X30X30 32x32~32 40X40X40 45X45X45 48X48X48

16 17 27 48 47

BCC -

11.9 12.4 13.2 12.4 13.3

14 84 104 83 103

Q-HCP -

2841

Simulation of ordered amphiphilic phases tail groups, and the surfactant concentration can be increased to a higher value without inducing a shape transition away from spherical micelles. For example, aqueous solutions of I-18T4 form spherical micelles at amphiphile concentrations as high as 55%, and for HIZT4, spherical micelles form even at a concentration of 70%. On small boxes, these produce the same ordered arrays of spherical micelles that are formed by 35% H,T,; compare Tables 3-5. The lattice sizes for which a given packing, SC, BCC, or FCC, is obtained are also similar for all three surfactants. This shows that H,T,, H,T,, and H,,T, have similar average micelle sixes. On the 24 x 24 x 24 lattice, micelles of 55% HsT, and 70% H,,T,, like those of 35% H,T,, form a “quasi-HCP” structure shown in Fig. 5. The pattern in Fig. 5 can be obtained from that of Fig. 3 by moving one sphere from each of the two layers in Fig. 3 into the region between the two layers, to create layers each with three spheres alternating with layers having just one isolated sphere each. Note that the isolated sphere is larger than the spheres in the more crowded layer containing three spheres.

H8Tq,

t&=0.55,

Co=O,

Because of the higher volume fraction of spherical micelles in the systems 55% HsT, and 70% Hr2T4, as compared to 35% H,T+, the two former systems have a greater tendency to form ordered arrays of spheres, even on larger boxes. On 30x 30 x 30 boxes, both 55% H8T4 and 70% H,,T, spontaneously self-assemble BCC-ordered arrays containing eight unit cells and 16 spheres. Two parallel slices of this BCC array formed by H8T4 are superimposed on each other and shown in Fig. 6. Recall that 35% H4T, forms only a mixture of spheroidal and cigar-shaped micelles from random starting states on a 30 x 30 x 30 box. Furthermore, when the FCC symmetry spontaneously formed by 55% H,T, on a 20 x 20 x 20 lattice is replicated to form an ordered starting state on a 40 x 40 x 40 box, this pattern remains stable after annealing at w = 0.1385, whereas for 35% H,T,, the FCC pattern spontaneously disordered on the 40 x 40 x 40 box; see Fig. 4. Thus, both the BCC and the FCC

symmetries are more stable for 55% HsT., than for 35% H4T4. These results show that an asymmetric surfactant that prefers to reside at an interface curved

w-0.1231

24x 24x 24

Fig. 5. Four parallel slices of a 24 x 24 x 24 lattice containing 55% HsT, in water at w = 0.1231. These slices are arranged in a clockwise sequence with respect to the order in which they appear in the threedimensional lattice.

2842

R.G.

LARSON

HsT4 cl*

=

0.55

co=0 w = 30x

0.1308 30x

30

Fig. 6. Two parallel slices showing the locations of the spheres of a BCC pattern spontaneously selfassembled on a 30 x 30 x 30 lattice containing 55% H,T, in water at w = 0.1308. (0 0 0) Tail units in spheres located in one slice, (*a*) those from the second slice.

towards the tails can be concentrated enough that the tendency to form three-dimensional arrays of micelles is enhanced. However, just as for 35% H,T, systems, for both 55% H,T, and 70% H,,TI systems the BCC systems seems to be more stable than the FCC symmetry. For both HsT4 and H,zT4, the FCC symmetry does not spontaneously self-assemble on 40 x 40 x 40 lattices, even though BCC symmetry self-assembled on 30 x 30 x 30 lattices for both of these amphiphiles. Thus, the BCC pattern appears to be more stable than the FCC pattern. On 45 x 45 x 45 lattices, however, neither 55% HBT4 nor 70% HrZT4 spontaneously self-assemble into BCC arrays. Instead, H,T, remains disordered at w = 0.1308, while H,*T, spontaneously self-assembles into a Q-HCP phase; see Fig. 7 and Table 5. In Fig. 7, the three in-plane spacings are 13.5: 13.5: 15, and the ratio of the spacing between repeat layers to the average in-plane spacing is 1.62, not far from the exact value of 1.633 for true HCP symmetry. Also, the nearest-neighbor spacing between a sphere in one plane and that in the next plane is 13.5, similar to the in-plane spacings. Note, however, that the bottom drawing of Fig. 7 shows that each sphere in the second layer is centered between two spheres of the first layer, rather than between three spheres as would be the case for true HCP symmetry. We believe, nevertheless, that the true bulk symmetry for an infinite box is likely to be HCP symmetry; distortions from this symmetry probably

occur because the cubic box is not compatible with HCP symmetry. If this conclusion is true, then the stability of ordered phases of 70% HiaT, appears to increase in the order SC < FCC < BCC < HCP. The relatively high stability of the HCP phase for 70% HI,T, may be caused by the high packing density of the micelles in this system, which would tend to favor the ordered phase with the greatest packing efficiency. For hard spheres, the HCP packing can accommodate a hard-sphere volume fraction of 0.74, higher than any other packing, cubic or noncubic. Table 6 gives the values of R2, the mean-square separation of one end of the amphiphile from the other, as well as the corresponding quantities Ri and R$ for the head and tail groups, for 35% H,T, and 70% H,,T, for SC, BCC, FCC, and Q-HCP packings. Note that the average conformations of 35% H,T, are essentially independent of the micellar packing, except that RZ for the SC packing is somewhat smaller than for the other packings. For 70% H 12T4r however, there are significant differences in RZ and in R& among the different sphere packings. These differences between H,T, and H,,T, can be explained by the differences in the packing of head groups. For 35% H,T,, the heads are surrounded by plenty of solvent that can rearrange spatially to accommodate differences in the spatial arrangement of the micelles, without requiring the chains to change their configurations. Hence, for 35% H,T4, the chains can relax to maximize their configurational entropy

2843

Simulation of ordered amphiphihc phases

J&T4

q4=0.70 c,=o w-0.1077 45X45X45

Fig. 7. (Top) Slice showing the locations of the spheres of a quasi-HCP pattern spontaneously selfassembled on a 45 x 45 x 45 lattice containing 70% H,,T, in water at w = 0.1077. (Bottom) The tail units in spheres located in one slice are denoted by circles; those from a second parallel slice are denoted by asterisks. The bottom image was not periodically replicated.

most part, to the state of For 70% H12T4, however, there is less solvent, and the head groups on one micelle must adjust their configurations to accommodate the head groups of another micelle. Hence, for 70% HlzTa the state of packing of micelles can influence the chain configurations. This influence on chain configurations suggests that the different packings have significantly different free energies; hence one is likely to be thermodynamically favored over the others. Significantly, the two most stable packings for 70% HIIT,, namely BCC and Q-HCP, have similar chain configurations statistics as shown in Table 6. Thus, the effect of micelle packing on chain

without packing

regard, for the of the micelles.

configuration helps explain why micelles assembled from 70% H,,T, form ordered phases on large Iattices. For 35% H,T,, all packings have almost the same chain statistics; hence bulk phases of 35% H,T, have little incentive to form any particular ordered phase. There are two rows in Table 6 for BCC packings of 35% H,T,; one for a 16 x 16 x 16 lattice, the other for a 32 x 32 x 32 packing formed by replicating the 16 x 16 x 16 packing and annealing, as discussed above. There is no statistically significant difference between results obtained on the larger and smaller lattices. A similar absence of lattice size effect was found for lamellar and cylindrical morphologies.

R. G. LARSON

2844

Table 6. Conformation properties of spherical micelles of H1T4 and H, zT, Surfactant

Concentration CA

LXLXL

W

Symmetry

R2

R:

R:

H&T, H;T;

0.35 0.35 0.35 0.35

12X12X12 20X20X20 16X16X16 32x32~32

0.1385 0.1385 0.1385 0.1385

SC FCC BCC BCC

22.9 23.8 23.8 23.7

7.2 7.2 7.3 7.2

6.8 6.9 6.9 6.9

I-LTz, H,zT, kT4 H,,T,

0.70 0.70 0.70 0.70

12x 12x 12 20X20X20 15x 13x 1.5 45X45X45

0.1077 0.1077 0.1077 0.1077

SC FCC BCC Q-HCP

37.0 39.0 40.2 40.2

26.7 26.4 27.1 27.7

6.6 7.2 6.6 6.7

HAT, _ _

H.tT4

all contain + j) x 0.18, where i and j are the head and tail lengths, respectively, and all three of these surfactants form densely packed spherical micelles with an average aggregation number of about 80. If the concentration of H,T, is increased to 40% or that of H,T, is increased to 60%, there is a strong tendency to form some oblong micelles on large lattices, and we do not find any selfassembled cubic phases on lattices large enough to contain more than one unit cell. At a 45% concentration of H,T,, a 70% concentration of H,T,, or a 90% concentration of H,,T,, quasi-hexagonal packings of cylinders form for all three surfactants. Thus, for both symmetric and asymmetric surfactants with heads longer than tails, densely packed spherical micelles form at a tail volume fraction of CT x 0.18, oblong micelles begin to form at C, x 0.20, and hexagonally packed cylinders appear at C, = 0.22. Winey et al. (1992) have found that in diblock copolymers that are mixed with a “solvent” consisting of a relatively low molecular weight homopolymer that is chemically identical to one of the blocks, a transition from spheres to hexagonal cylinders occurs when the volume fraction of the block that is dissimilar to the “solvent” reaches the same value that is found in our simulations, about 0.22! Thus, the volume fraction of the “tails” seems to control the spherical to cylindrical transition, both for small-molecule and polymeric amphiphiles, if the tail unit is no longer than the head group. On the other hand, in simulations in which the tail group is significantly longer than the head group, we observe that cylindrical or worm-like micelles are formed when the surfactant is mixed with water even at volume fractions of the tail group much lower than 0.22 (Larson, 1993). This is consistent with experiments showing that dilute, disordered, worm-like micelles are formed by long-tailed surfactants and diblock copolymers (Missel et al., 1980; Safran et al., 1984; Kinning et al., 19S8). Comparing Tables 3-5, one observes that the micelle aggregation numbers for H,T,, H8T4, and H,,T, are to be about the same, around 60-110. A computation of the distribution of micelle sizes for 20% H,T, at w = 0.1538 showed a mean size of 84.1 and a standard deviation about that mean of 20.8 35% H,T,,

55% HsT,,

and 70% H,,T,

a tail volume fraction of CT = CAj/(i

(Larson, 1992). The micelles of H4T4, HsT,, and H,,T, all have nearly the same sizes, probably because they all have the same tail lengths. The size of the largest possible spherical micelle with a sharp interface between head and tail units can be estimated from the length of the tail, by assuming that the tails fill a sphere whose radius R equals the tail lengthsince the center of a larger sphere could not be reached by the tails. The volume of this sphere, 45rR3/3, divided by the volume of a single tail, gives the maximum aggregation number for a spherical micelle with a sharp interface between head and tail units. This works out to be 67 for all three surfactants, H,T,, HsT,, I-I,,T,. The average micelle size for these surfactants turns out to bc somewhat larger than this, evidently because of micelle shape fluctuations, and penetrations of heads and water into the tail-containing core of the micelle. 2.4. Three-dimensional order: tetragonal intermediate phases At surfactant concentrations between those for which lamellae and hexagonally packed cylinders form, intermediate phases are typically observed in experimental surfactant and block copolymer systems. In an earlier publication, we found that for 75% H,T, in water, a phase consisting of lameflae with holes forms on cooling the originally disordered state. These holes are circular on average, and create gaps in the layers that contain the tails, so that there is a passage from one head-containing layer to the next through the hole in the tail-containing layer; examples of this structure are depicted shortly (in Figs 8-10). The concentration (75%) of H,T, at which these lamellae with holes form is intermediate between that required to form hexagonally ordered cylinders and that required for unbroken lamellae. We have found similar lamellar phases with holes for other surfactants at concentrations between that for hexagonal cylinders and that for lamellae without passages, including 75% H,T, in water, 75% H,TB in water, and the asymmetric surfactant H,T, in water at concentrations ranging from 45 to 52.5%. These passages are true equilibrium structures, and not quenched irregularities. This was shown for a 75% solution of H,T, in water which on most small lattices formed lamellae with passages, but on one small 17 x 17 x 17 lattice

Simulation of ordered amphiphilic phases formed lamellae without passages. This passage-free system was then replicated to create a large 34 x 34 x 34 lattice with initially passage-free lamellae; when this system was annealed, passages spontaneously appeared in the lamellae. This shows that the passages are an equilibrium features, and are not kinetically trapped defects produced during cooling from the disordered state. On large lattices, the hole-containing lamellae are sometimes irregular or distorted by defects in the pattern, apparently because the holes greatly reduce the energy cost of creating pattern defects, and so the defects can be difficult to eliminate. As an example,

consider the pattern with defects shown in Fig. 8, composed of 75% H,T, in water, which spontaneously self-assembled on a 34 x 34 x 34 lattice on slow cooling from w = 0 to w = 0.1192. When this pattern was reheated to w = 0.1154 and annealed, a wellordered lamellar pattern with holes appeared, two mutually perpendicular slices of which are shown in Fig. 9. The iamellar spacing of this pattern is 9.09. However, when this well-ordered pattern was cooled again, down to w = 0.1500, defects spontaneously nucleated, and a pattern similar to that of Fig. 8 reappeared. A long annealing period at w = 0.1500 resulted in the elimination of the defects, and pro-

J34T4 C*

=

0.75

co=0 w =

2845

0.1192

34X34X34

Fig. 8. The top two images (a’) and (a) are parallel slices taken from a 34 x 34 x 34 lattice containing 75% H,T, in water obtained by the usual slow cool from w = 0 to w = 0.1192. The bottom image (b) is a slice orthogonal to the other two slices.

2846

R. G. LARSON

(4

H4T4 c!*

=

0.75

c~/c~=l w =

0.1154

34X34X34

(b)

Fig. 9. Two mutually orthogonal slices of a 34 x 34 x 34 lattice containing 75% H,T, heating the pattern shown in Fig. 8 to w = 0.1154.

duced the pattern shown in Fig. IO. Comparing Fig. 9 with Fig. IO, one sees that both patterns consist of well-ordered lamellae with holes, but the patterns are oriented d@rently on the lattice, so that the hightemperature ordered state shows a lamellar spacing d = 9.09, while for the low-temperature state the spacing is larger, d = 9.43. The intermediate state, shown in Fig. 8, is a superposition of at least two different orientations. That is, the broken stripes in Fig. 8(a) have the same orientation as those in Figs 9(a) and IO(a), while Fig. 8(b) appears to be a superposition of the broken stripes in Figs 9(b) and IO(b). Because the intermediate state, Fig. 8, forms on cooling from a well-ordered state, that is from Fig. 9, we. conclude that the well-ordered state with d = 9.09 becomes unstable to the formation of defects, or is so weakly stable that thermal energy can easily produce the defects. At w = 0.1500, the defect-

in water formed by

ridden state is, in turn, unstable to the formation of a defect-free state with a larger spacing, d = 9.43. It is reasonable that thermodynamics should drive an increase in optimum spacing with increased w, because with increased w, a reduction in the number of hydrophilic-lyophobic contacts tends to occur; this can be accomplished by increasing the lamellar spacing d. The increase in spacing from d = 9.09 to d = 9.43 is the minimum increase possible for a box of this size, since the quantum numbers for the hightemperature state are m = (1,2,3), while for the lowtemperature state they are m = (0, 2, 3). Thus, at intermediate temperatures, a combination of the two states with an average spacing intermediate between the two pure states might be thermodynamically favored, if the energy of the defects is bw enough. The large number of lamellar holes evidently reduces the energetic cost of creating the pattern defects enough

Simulation of ordered amphiphilic phases

2847

H4

T4

C*

=

0.75

co=0 w =

0.1500

34X34X34

w

Fig. 10. Two mutually orthogonal slices of the pattern formed by slowly cooling that of Fig. 9 to w = 0.1500.

that the defect-ridden pattern shown in Fig. 8 is thermodynamically favored over defect-free states that have lamellar spacings that are not optimal. For well-ordered lamellar patterns without holes, the preferred spacing no doubt depends on w, just as it does for lamellar patterns with holes. But, so far, we have found that small changes in w do not cause a rotation, or a spontaneous generation of defects, in a lamellar pattern without holes, evidently because in lamellar patterns without holes, defects are so much more expensive energetically. On some lattices, we have found that a lamellar system with holes can be cooled without generation of defects. In particular, 75% H4T4 in water on a 40 x 40 x 40 lattice forms a lamellar pattern with holes with a lamellar spacing of d = 9.4 at w = 0.1347. This lamellar spacing is wide enough that the system can be cooled to w = 0.1614 without generation of lamellar

defects. When this is done, the holes within each layer order into a hexagonal pattern, as discussed in Larson (1992). The registry of the holes from one layer to the next on one layer are staggered, similar to that seen in the “R” phase of some experimental surfactant-water solutions (Luzzati et al., 1968; Ktkicheff and Cabane, 1987). Artificial inducement of tenagonally ordered holes also occurs on small lattices, and we are able to show that at least for one surfactant, H,T,, this tetragonal phase also occurs on large lattices, although at a temperature lower than that required on a small lattice. To show this, we plot in Fig. 11(a) the water and head units that fills the holes on two adjacent tail layers of a 22 x 22 x 22 lattice containing 75% H,T, in water. This system has ordered into a lamellar phase with holes in the tail layers, when cooled to w = 0.0923. In Fig. 11(a), the slices chosen are parallel

2848

KG.

LARSON

(4 W

04

=0.0923:22X22x22

w =0.0923:44x44X44

,

H6T6 C*

=

0.75

co=0

W

=0.1193:44x44X44 (c)

Fig. 11. (a) Pattern formed at w = 0.0923 by 75% H,T, on a 22 x 22 x 22 lattice. (0 0 0) Head and water units in the center of one tail layer; ( **a) those units in the adjacent tail layer. (b) Pattern formed after the pattern in (a) has been replicated on a 44 x 44 x 44 lattice and annealed at 0.0923. (c) Pattern formed after the pattern in (b) has been cooled slowly to w = 0.1193. Images in (b) and (c) were not periodically replicated. to,

and

pass

directly

through,

the

tail-containing

layers; the circles show the locations of the head and water units on one such slice, while the asterisks show these units in the neighboring tail layer. Thus, the circles and asterisks show the locations of the holes in

two adjacent tail-containing layers, in contrast to Figs 8-10, in which the tail units are plotted, so that the holes appear as gaps in the lamellae. On each slice of the 22 x 22 x 22 lattice, the holes or passages are ordered into a square array that is oriented diagonally with respect to the sides of the box. The passages on the second slice are positioned at the centers of the square formed by the passages on the first slice. Now, we create a 44 x 44 x 44 box with the same pattern as that of the 22 x 22 x 22 lattice by combining

eight copies of the 22 x 22 x 22 lattice. When this replicated tetragonal phase on the 44 x 44 x 44 lattice is annealed at w = 0.0923, the holes lose their spatial order, producing the disordered pattern shown in Fig. 1 l(b). Thus, like the FCC and SC sphere packings for HIT,, the tetragonal order for H,T, at w = 0.0923 is an artifact of the smallness of the lattice. However, if the disordered phase depicted in Fig. 11(b) is cooled slowly to w = 0.1193, a somewhat distorted tetragonal order reappears, shown in Fig. 11(c). Thus, on a large lattice the ordering temperature appears to be shifted with respect to that on a small lattice, but the symmetry of the bulk three-dimensionally ordered phase seems to be unaltered. In Fig. 1 l(c), the square array of passages is oriented parallel to the sides of the

Simulation of ordered amphiphilic phases Table 7. Conformations

Surfactant 2; HIT: J-J,Te

Concentration C” 0.35 0.60 0.75 0.80

2849

of H4T4 in water at +v = 0.1385 Symmetry

R2

R$

R:

Q-Hex BCC Lam(H) Lam

23.7 23.8 23.5 22.9

7.26 7.14 7.08

6.87 6.84 6.88 6.70

box, and there are nine passages on each slice of the lattice, rather than the eight that appear in Fig. 4. The average spacing between holes is 14.67 on the 44 x 44 x 44 lattice at w = 0.0923 and 15.56 on the 22 x 22 x 22 lattice at w = 0.1193. In experimental surfactant systems, phases consisting of lamellae with holes ordered on both hexagonal and square lattices have been observed as intermediate phases between hexagonally packed cylinders and lamellae (Luzzati et al., 1968; Ktkicheff and Tiddy, 1989). Cubic intermediate phases have also been seen for many surfactants and for block copolymers (Charvolin and Sadoc, 1987; Thomas et al., 1990; Hyde et al., 1984, Kikicheff and Cabane, 1987). Table 7 shows R2, R&, and Rs for H_+T, in water in four different ordered states at w = 0.1385: a BCC packing of spheres at C, = 0.35, a hexagonal packing of cylinders at CA = 0.60, a lamellar state with many holes at C, = 0.75, and a lamellar state with very few holes at C, = 0.8. The lamellar phase with many holes is designated in Table 7 by “Lam(H)“. Note that there is a discernible increase in R2 and R& as one moves from the flat surfactant layers in the lamellar morphology to the curved layers in the lamellae with holes and the cylindrical and spherical morphologies at w = 0.1385. This shows that the head groups are slightly more expanded when they are packed on the curved surfaces than they are in lamellae. Note that even in the lamellar state at C, = 0.8 the head group is more expanded than the tail, evidently because there is water available to swell the head group slightly, but no oi1 is present to swell the tails. The unequal swelling of the head and tail units may provide an incentive for holes to form in the tail layers, so that lateral expansion of the head layer, relative to the tail layer, can be accommodated. 3. SUMMARY Molecular simulations of self-assembled surfactant liquid crystalline phases, including those with one-, two-, or three-dimensional order, are now possible. The patterns with one- or two-dimensional order, namely lamellae and hexagonal cylinders, can be simulated even on relatively small simulation boxes without severe box-size artifacts. Patterns with threedimensional order require systematic studies of the effect of box size on the pattern selected, and larger boxes must be used to capture bulk behavior. However, in studying three-dimensional patterns on small

boxes, information learned about the relative stability of various possible competing phases cannot be obtained from experiments on real pattern forming systems. Thus, we find that at least for the surfactants H4T4. HsT,, and H,,T,, the most stable cubic phase is apparently BCC, followed by FCC and SC. At least for H,,T,, the noncubic HCP phase appears to be more stable than BCC. Ordered intermediate phases can also be simulated at compositions intermediate between that for lamellae and that for cylinders. So far, with our model, the only stable intermediate phases we have found are lamellae with holes. The formation of holes is evidently favored entropically, since they increase the volume available for the head groups, thereby allowing the head groups to swell with water. We find that when the temperature is changed so that a new lamellar spacing is thermodynamically favored over the original spacing, these holes so reduce the energy penalty for defects, that pattern defects spontaneously nucleate and allow the creation of the new, wider, lamellar spacing to appear, superposed onto the original one, but rotated with respect to it. This mixed state, containing the two superposed patterns, facilitates the transition from the original pattern to the new one. Hence, the holes in the lamellar pattern greatly reduce the kinetic barrier to thermodynamically favored changes in lamellar spacing. When the state containing lamellae with spatially disordered holes is cooled, the holes can order into hexagonal or square patterns within each layer, and the registry of the holes from layer to layer is staggered, forming a tetragonal phase for the holes ordered onto squares within layers. Acknowledgements-I gratefully acknowledge helpful comments and suggestions by Professor Helfand.

Bob Brown and Gene

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