Phase Behavior of an Amphiphilic Molecule in the Presence of Two Solvents by Dissipative Particle Dynamics

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Phase behavior of an amphiphilic molecule in the presence

of two solvents by dissipative particle dynamics

Ching-I Huang

*

, Yu-Jeng Chiou, Yi-Kang Lan

Institute of Polymer Science and Engineering, National Taiwan University, No. 1, Roosevelt Road, Sec. 4, Taipei 106, Taiwan Received 15 June 2006; received in revised form 20 October 2006; accepted 10 December 2006

Available online 8 January 2007

Abstract

We examine the phase behavior of AmBnamphiphilic molecules in the presence of two solvents X2and Y2, which are strongly selective for A

and B, respectively, by dissipative particle dynamics (DPD). We find that increasing the immiscibility parameter between the two solvents not only drives a macrophase separation into two phases X2-rich and Y2-rich for systems at less concentrated regimes, but also expands the ordered

microphase region at more concentrated regimes. It even induces a sequential transition of various ordered structures. This is not surprising since increasing the solvent immiscibility parameter enhances the preferentiality of X2for A and Y2for B, and thus qualitatively varies the degree of

molecular asymmetry in the amphiphilic molecules. In general, our current results reveal that the DPD simulation method has successfully cap-tured the phase separation behavior of an amphiphilic molecule in the presence of two solvents. However, we find that the packing order of the spherical micelles is greatly affected by the finite size of the simulation box. As such, it becomes difficult to examine the most stable packing array of spheres via the DPD method. Still, DPD reveals a possible spherical order of A15, which has been observed in some amphiphilic molecule systems.

Keywords: Dissipative particle dynamics; Amphiphilic; Solvent immiscibility

1. Introduction

Amphiphilic molecules, such as surfactants, lipids, and di-block copolymers, continue to attract a lot of attention due to the fact that they can self-assemble into a rich variety of mor-phologies [1e7]. Many of the amphiphilic molecule systems with valuable technological applications contain amphiphilic molecules, oil, and water[7]. Due to the fact that water and oil are strongly immiscible, the swelling ability of both hydro-philic and hydrophobic groups in the amphihydro-philic molecules varies with the solvent ratio and solvent amount. As a result, various microstructures with different interfacial curvatures, such as lamellae, normal (oil-in-water) and reverse (water-in-oil) bicontinuous phase, normal and reverse cylinders, and normal and reverse spheres, form in the ternary mixtures of

amphiphilic molecules, water, and oil[2,7]. Indeed, phase be-havior of an amphiphilic molecule in two solvents becomes very complicated as it involves the effects of the immiscibility between two solvents, solvent amount, solvent ratio, and the selectivity of two solvents for each amphiphilic group. Most of the theoretical research has focused on the examination of phase behavior and the interfacial properties of an amphiphilic molecule in the presence of two very immiscible solvents[8e 13]. To our knowledge, however, the effects of solvent misci-bility degree on the resulting microstructure formation of amphiphilic molecules have not been fully examined. In this paper, we employ dissipative particle dynamics (DPD) to sim-ulate the phase behavior of an amphiphilic molecule in the presence of two solvents. In particular, the ternary phase dia-grams at various degrees of the solvent immiscibility parame-ter are constructed.

DPD is a coarse-grained mesoscale simulation technique

[14], which has recently been successfully applied to study the * Corresponding author. Tel.:þ886 2 33665886; fax: þ886 2 33665237.

E-mail address:chingih@ntu.edu.tw(C.-I. Huang).

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mesophase behavior of a variety of amphiphilic molecule sys-tems [12,13,15e22]. Groot and Madden first applied DPD to examine the microphase separation behavior of linear AmBn

diblock copolymer melts[16]. By varying the A composition and the effective A/B segregation parameter, various ordered structures, such as lamellae (L), gyroid (G), perforated lamel-lae (PL), hexagonally packed cylinders (C), and spheres (S), have been obtained via DPD. In further, the phase diagram constructed by DPD is in a near quantitative agreement with that predicted by self-consistent mean-field theory [23]. In the amphiphilic molecule solutions, the related current DPD studies were mainly focused on the less concentrated regimes. For example, Cao et al. employed DPD to simulate the aggregation behavior of poly(ethylene oxide)epoly(propylene oxide) block copolymers in aqueous solutions[22]. In partic-ular, the effects of the copolymer architecture and concentra-tion on the formed micelle type and size were examined. Their results are in a qualitatively good agreement with exper-iments [24]. Rekvig et al. have demonstrated that the DPD simulations can successfully describe the partitioning behavior of the surfactant along the interface between water and oil

[12]. Schulz et al. applied DPD to simulate the self-assembling behavior of surfactant C10E4in the presence of water and oil

[13]. Their simulated structure results by varying the water/ oil ratio are in good agreement with the experimental phase diagram [25]. Though these solution studies demonstrated that the DPD simulation method is an appropriate technique to examine the phase behavior of amphiphilic molecules in the presence of two solvents, their results neither extend to the concentrated regimes nor consider the effects of solvent miscibility degree.

In this paper, we thus aim to simulate the phase behavior of an amphiphilic molecule in the presence of two solvents by DPD simulation method. In particular, the effects of the im-miscibility parameter between two solvents, the solvent ratio, and the volume fraction of amphiphilic molecules, are ana-lyzed. For simplicity, we consider a symmetric molecule and a asymmetric amphiphilic molecule, which are represented by two beads (A1B1) and four beads (A1B3), respectively.

Two solvents X2and Y2 are strongly selective for A and B,

respectively. Later, we will demonstrate that the effects of add-ing two strongly selective solvents on the resultadd-ing phase behavior of A1B1 and A1B3 are qualitatively consistent.

Hence, similar results should also hold true qualitatively in other amphiphilic molecules (AmBn) which initially form

dif-ferent ordered phases, such as G, PL, and S. Our DPD results provide a more complete understanding of the rich and com-plex phase behavior that amphiphilic molecules exhibit when two solvents are added.

2. DPD simulation method

In the DPD simulation, the time evolution of motion for a set of interacting particles is solved by Newton’s equation. For simplicity, we assume that the masses of all particles are equal to 1. The force acting on the ith particle fi

!

contains

three parts: a conservative force!FCij, a dissipative force F !D

ij, and a random force!FR_ij, i.e.,

fi ! ¼X isj F !C ijþ F !D ij þ F !R ij ð1Þ where the sum is over all other particles within a certain cut-off radiusrc. As this short-range cut-off counts only local

in-teractions,rcis usually set to 1 so that all lengths are measured

relative to the particle radius.

The conservative force !FC_ij is a soft repulsive force and given by F !C ij¼ aij 1rij rc nij ! rij< rc 0 rij rc ( ð2Þ where aij is the repulsive interaction parameter between

particles i and j, r! ¼ rij ! ri !; rj ij ¼ jr!j, and nij ! ¼ rij !=rij ij. The repulsion parameter aij is often related to the Florye

Huggins interaction parameter cij by the following equation

[15]

aijðTÞ ¼ aiiþ 3:497kBTcijðTÞ for r ¼ 3

aijðTÞ ¼ aiiþ 1:451kBTcijðTÞ for r ¼ 5

ð3Þ where r is the particle density of the system. The term aii,

which corresponds to the repulsion parameter between parti-cles of the same typei, is determined by matching the water compressibility as[15]

aii¼ 75kBT=r ð4Þ

The dissipative force!FDij is a hydrodynamic drag force and given by F !D ij ¼ guD_r ij nij ! ,v!nij ! rij ij< rc 0 rij rc ð5Þ where g is a friction parameter, uD is a r-dependent weight function vanishing forr rc, and! ¼ vvij ! vi !.j

The random force!FR_ij corresponds to the thermal noise and has the form of

F !R ij¼ suR_r ij qijn! rij ij< rc 0 rij rc ð6Þ where s is a parameter, uRis also a weight function, qij(t) is a

randomly fluctuating variable. Note that these two forces F

!D ij and F

!R

ij also act along the line of centers and conserve linear and angular momentums. There is an independent ran-dom function for each pair of particles. Also there is a relation between both constants g and s, which is as follows[15]

s2¼ 2gkBT ð7Þ

In our simulations, g¼ 4.5 and the temperature kBT¼ 1. As

such, s¼ 3.0 according to Eq.(7).

In order for the steady-state solution to the equation of motion to be the Gibbs ensemble and for the fluctuatione dissipation theorem to be satisfied, it has been shown [26]

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that only one of the two weight functions uDand uR can be chosen arbitrarily,

uDðrÞ ¼ uRðrÞ2 ð8Þ

which, in further, is usually taken as uD_{ðrÞ ¼} _uR_ðrÞ2_¼ rc rij 2 rij< rc 0 rij rc ð9Þ Finally, the spring force!f Si, which acts between the con-nected beads in a molecule, has the form of

f !S i ¼ X j Cr!ij ð10Þ

whereC is a harmonic type spring constant for the connecting pairs of beads in a molecule, and is chosen to be equal to 4 (in terms ofkBT )[15].

Note that a modified version of the velocityeVerlet algo-rithm is used here to solve the Newtonian equation of motion

[27] riðt þ DtÞ ¼ riðtÞ þ viðtÞ,Dt þ 1 2fiðtÞ,Dt 2 ~viðt þ DtÞ ¼ viðtÞ þ l fiðtÞ,Dt fiðt þ DtÞ ¼ fi½riðt þ DtÞ þ ~viðt þ DtÞ viðt þ DtÞ ¼ viðtÞ þ 1 2Dt,½ fiðtÞ þ fiðt þ DtÞ ð11Þ In particular, herein, we choose l¼ 0.65 and Dt ¼ 0.05.

3. System

In simulating the phase separation behavior of AmBn

amphiphilic molecules in the presence of two solvents X2

and Y2 by DPD, we assume that each component has the

same volume per segment (bead). We choose two types of molecules, one is symmetric A1B1and the other is asymmetric

A1B3. The solvents X2and Y2are strongly selective for A and

B, respectively, i.e., X2(Y2) likes A(B) instead of B(A). In

par-ticular, we set aAY¼ aBX¼ 100.54 > aAX¼ aBY¼ 25. When

the particle density r¼ 3, the dimensionless repulsion param-eter (i.e., in terms of kBT ) between equal particles aII in

Eq. (4) is set equal to 25 to resemble the Flory interaction parameter cII¼ 0, I ¼ A, B, X2, Y2. Parameter aAB is also

set at 100.54, which corresponds to cAB¼ 21.6 according to

Eq. (3). Therefore, the effective A/B interaction parameter (cABN )effcalculated by the following equation[16,19]

ðcABNÞeff¼

c_ABN

1þ 3:9N0:51 ð12Þ

for A1B1and A1B3is equal to 11.55 and 29.56, respectively.

According to the prediction by self-consistent mean-field

theory [23], these two amphiphilic molecules form the stable L phase and the hexagonally packed cylinders of AðCHEX

A Þ, re-spectively. In order to simulate the effects of the immiscibility parameter between the two solvents, we chooseaXY¼ 25, 30,

and 40, which corresponds to cXY¼ 0, 1.43 and 4.29,

respec-tively. As such, the systems with lower values of the volume fraction of amphiphilic molecules fCmay remain in the disor-dered solution state at aXY¼ 0, and undergo a macrophase

separation into X2-rich and Y2-rich phases whenaXYincreases

to 30 and 40. In addition, we adopt a 3D lattice with at least 10 10 10 grids to assure that the side length of our simu-lation box is significantly larger than the radius of gyration of these amphiphilic molecules. In each pattern, the red, green, blue, and purple colors correspond to component A, B, X, and Y, respectively.

4. Results and discussion

Fig. 1illustrates the morphology variation of A1B1with the

addition of solvent X2. When existing alone, A1B1forms the

stable L phase, as expected. As solvent X2is added, because

of the attraction of X2toward A, solvent X2 moves into the

A-rich layers. As a result, the systems with the surfactant vol-ume fraction fC 0:7 form a lamellar A/X2-rich and B-rich

segregated structures. With increasing amount of X2, because

of the increasing degree of swelling by X2, we observe that the

A-rich layers become thicker, and these amphiphilic mole-cules form a mutually parallel bilayer structure in the presence of X2 ð0:6 fC 0:2Þ. With continuously adding X2such

that fC 0:1, relatively small amounts of A1B1evolve from

the bilayer structure into a rod-like micelle with A facing out-ward and B facing inout-ward. Note that due to the significant im-miscibility between X2and B, the amount of X2which can be

dissolved in the A-rich layers is expected to have a maximum, and therefore the extra addition of X2may form a phase of its

own. That is, the bilayer structure in the solvent X2formed at

0:2 fC 0:6 may reveal the possibility of two-macrophase separation into 1 L phase and 1 disordered (D) X2-rich phase.

In order to further identify this issue, we expand our DPD sim-ulation box from 10 10 10 to 15 15 15, and present the resulting pattern for fC ¼ 0:5 in Fig. 1. It is clear that the thickness of the X2-rich layers simulated in a box of

15 15 15 is around 6.15 grids, which is significantly larger than 4.0 in 10 10 10. Similar results have also been observed in other systems at 0:2 fC 0:6, indicating that these systems actually form a two-phase coexistence of 1 L and 1 D. As solvents X2and Y2have opposite selectivity for

A and B, the morphology variation of A1B1with the addition

of solvent Y2is similar to that of A1B1þ X2. In general, our

current DPD results for a symmetric amphiphilic molecule in the presence of a strongly selective solvent are consistent with those for a symmetric AB block copolymer in the presence of a homopolymer A based on the self-consistent mean-field theory[28]. Note that when the solvent selectivity is slight, no macrophase separation occurs as fC decreases. The addition of an A-selective solvent X2thus acts in a

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compositionf. In this case, one may expect that a sequence of L/CHEX_B /B-formed spheres/D occurs, as frequently ob-served in the diblock copolymers in the presence of a selective solvent[29].

Fig. 2presents a series of phase diagrams for A1B1in the

presence of solvents X2and Y2, which are obtained by DPD

simulations and shown in ABeX2eY2composition triangles,

at various values of the immiscibility parameter between two solventsaXY. The Ith corner in the triangle represents a system

composed of 100% component I. When these two solvents are miscible, i.e.,aXY¼ 25, as shown inFig. 2(a), we observe that

for the solutions with f_C 0:7, no matter what the volume fraction ratio of X2 and Y2 ðfX2=fY2Þ is, they will evolve into a lamellar structure. The typical pattern is presented in

Fig. 3(a), where fC ¼ 0:7; fX2=fY2 ¼ 4=6, and aXY¼ 25. This is due to the fact that solvents X2and Y2have a very

strong affinity to A and B, respectively. Therefore, even if X2and Y2are miscible, a small amount of solvents X2 and

Y2 still accumulates into the A-rich and B-rich layers,

respectively. However, when the solutions are in the interme-diate concentration ðfC ¼ 0:2e0:6Þ, as the amount of these two solvents X2 and Y2 exceeds the maximum value which

can be dissolved, respectively, in the A-rich and B-rich do-mains, they undergo a macrophase separation into a lamellar A/X2-rich and B/Y2-rich phase and a disordered X2/Y2phase,

as shown inFig. 3(b), where fC ¼ 0:4; fX2=fY2 ¼ 4=6, and aXY¼ 25. Once the volume fraction of A1B1 becomes very

low, such as f_C 0:1, the molecules no longer form the la-mellar structure and instead a micelle-like structure with B facing outward and A facing inward (when Y2is more than

X2) or with B facing inward and A facing outward (when

X2is more than Y2) is formed.

AsaXYincreases to 30, the significant immiscibility

param-eter between two solvents not only drives a macrophase sepa-ration into X2-rich and Y2-rich phases for systems at lower

concentration values of fC, but also enlarges the 1 ordered L (A/X2-rich and B/Y2-rich) microphase region till the middle

of the phase triangle, as seen in the corresponding phase Fig. 1. Morphology variation of A1B1in the presence of A-selective solvent X2 with surfactant volume fraction fC. The interaction parameters are set as aAB¼ aBX¼ 100.54, and aAX¼ 25. The red, green, and blue colors represent A, B, and X, respectively. (For interpretation of the references to color in this figure

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diagram inFig. 2(b). This is not surprising since increasing the solvent immiscibility also enhances the preferentiality of X2

for A and Y2for B. Accordingly, asaXYincreases, more X2

and Y2can be dissolved into the A-rich and B-rich domains,

respectively. By varying the ratio of solvent amount f_X

2=fY2, we obtain various phase transition behaviors with f_C. For example, the systems in which the amount of X2is

much more than that of Y2, undergo a sequence of 1 L

(A/X2-rich and B/Y2-rich) / 1 L (A/X2-rich and B/Y2-rich)þ

1 D(solvent X2) / micelles in excess solvent X2 as fC de-creases. Similar phase transition behavior is also observed for the systems with Y2much more than X2except that the

disor-dered phase becomes rich in solvent Y2. When the amounts of

these two solvents X2and Y2 are comparable, the solutions

transform from 1 L (A/X2-rich and B/Y2-rich) /

microemul-sion / 2 disordered X2-rich and Y2-rich phases with

decreas-ing fC. Note that there exists a microemulsion region near the periphery of the 2 D (X2þ Y2) coexistence curve. As seen in

Fig. 3(c) where we present the morphological pattern for the sys-tem with fC¼ 0:2; fX2= fY2¼ 3=7, and aXY¼ 30, though A1B1 mainly assembles along the interface between the X2-rich and

Y2-rich segregated domains, but in fact the interface is not

very obvious. If we further increase the immiscibility parameter between solvents X2and Y2to 40 (the corresponding phase

di-agram is presented in Fig. 2(c)), we find that the two-macro-phase X2-rich and Y2-rich regimes significantly expands and

even eats up the microemulsion phase region, as expected. For example, inFig. 3(d) we present the corresponding morpholog-ical pattern for the same system as inFig. 3(c) butaXYincreases

to 40. It is clear that both X2-rich and Y2-rich domains have

sharp interfaces. Note that although the phase separation behav-ior of ternary mixtures of A1B1, X2, and Y2at higher

immiscibil-ity parameteraXYis not examined here, it is reasonable to expect

three-phase regions rich in each component, respectively, in the phase diagrams.

To examine the effects of adding two solvents X2and Y2on

the phase behavior of asymmetric amphiphilic molecules, we choose A1B3as a representative. In this case, pure A1B3forms

a stable CHEX

A .Fig. 4 presents a series of phase diagrams for A1B3in the presence of two solvents X2 and Y2 at various

values of the immiscibility parameter aXY. When only one

solvent is added, the resulting phase transitions are straightfor-ward, as have been discussed in the case of symmetric A1B1.

In the A-selective solvent X2 which is strongly immiscible

with the majority B blocks, we observe a stable CHEX_A phase at fC 0:8 and a macrophase separation into two phases rich in A1B3and X2, respectively, at 0:1 fC 0:7. In the B-selective solvent Y2, because of the fact that the strongly

immiscible A block is a minority component in the amphi-philic molecules, the addition of Y2can steadily partition into

the B-rich domains. As a result, a sequence of microphase transition of CHEX

A ðfC 0:8Þ / A-formed long or short mi-celles ðfC¼ 0:7; 0:6Þ / A-formed spheres (SA)ðfC 0:5Þ occurs. Note that this sequential microphase transition behavior induced by the addition of a selective solvent has been in a qual-itatively good agreement with our previous SCMF results except the spherical ordering phase[30]. Recall that both experimental

[29] and theoretical [30] studies have confirmed that the ‘‘normal’’ spheres (i.e., formed by the minority blocks) adopt a body-centered cubic (bcc) lattice while the ‘‘inverted’’ spheres (i.e., formed by the majority blocks) tend to pack from bcc to face-centered cubic (fcc) upon increasing the solvent selectivity and/or solvent amount. However, our DPD results demonstrate that the packing array of the spheres is strongly dependent on the size of the simulation box, which will be discussed later. Fig. 2. Phase diagrams of symmetric A1B1 in the presence of A-selective

solvent X2 and B-selective solvent Y2 for (a)aXY¼ 25, (b) aXY¼ 30, and

(c)aXY¼ 40. The other interaction parameters are set as aAB¼ aAY¼ aBX¼

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When both solvents are added simultaneously, the effects of the immiscibility parameter between two solventsaXYon the

phase behavior of asymmetric A1B3 (Fig. 4) are similar to

that of symmetric A1B1(Fig. 2). For example, when the

solu-tions are concentrated ðfC 0:8Þ so that the amounts of X2

and Y2which partition into A-rich and B-rich domains are

rel-atively small, the original microstructure formed by pure A1B3, i.e., CHEXA , can be preserved regardless of the value of aXYand fX2=fY2. The typical C

HEX

A structural pattern is pre-sented in Fig. 5(a), where f_C ¼ 0:9; fX2=fY2 ¼ 5=5, and aXY¼ 25. For the solutions with fC in the range between 0.1 and 0.7, the morphology type is greatly influenced by aXYand fX2=fY2. WhenaXY¼ 25, due to the fact that these two solvents are completely miscible, the extra amount of sol-vents forms a miscible X2/Y2disordered phase, as expected.

That is, the systems undergo a macrophase separation into an A1B3-rich phase and a disordered X2/Y2phase, as in the

A1B1systems. Although A1B3forms a phase itself, the

micro-structure type varies with the ratio of solvent amount. When X2is the major solvent, A1B3only aggregates as a macroscopic

domain, as presented inFig. 5(b), where fC¼ 0:6; fX2=fY2¼

7=3, andaXY¼ 25. Nevertheless, when Y2becomes the major

solvent, A1B3forms micelles with A and X2inside the cores

and coexists with a disordered X2/Y2 miscible phase. In

more detail, when the solutions are more concentrated, these A-formed micelles are cylindrical, and the typical pattern is shown in Fig. 5(c), where fC ¼ 0:6; fX2=fY2 ¼ 3=7, and aXY¼ 25. As fCdecreases and/or the proportion of Y2amount

in the solvents increases, these A-formed cylinders become shorter and even transform into spheres, as inFig. 5(d) where f_C ¼ 0:2; fX2=fY2 ¼ 2=8. This transition of cylinders into spheres is not surprising since increasing the ratio of Y2

amount in the two miscible solvents resembles a consequence of enhancing the solvent selectivity for B. As the solvent im-miscibility parameteraXYincreases, similar to that observed in

A1B1, the miscible X2/Y2disordered phase, which exists in the

regime of 0:1 fC 0:7 at aXY¼ 25, is no longer stable. It is

interesting to find that the solutions with Y2as the major

sol-vent, which originally separate into A1B3-rich and X2/Y2-rich

phases ataXY¼ 25, form spheres with minority A and X2in

the cores and majority B and Y2in the matrix with increasing

aXY. This is reasonable since increasing the solvent

Fig. 3. Morphological patterns of A1B1in the presence of A-selective solvent X2and B-selective solvent Y2with (a) fC¼ 0:7; fX2=fY2¼ 4=6, aXY¼ 25, (b) fC¼

0:4; f_X2=fY2¼ 4=6, aXY¼ 25, (c) fC¼ 0:2; fX2=fY2 ¼ 3=7, aXY¼ 30, and (d) fC¼ 0:2; fX2=fY2¼ 3=7, aXY¼ 40, simulated in a box of 10 10 10. The

other interaction parameters are set asaAB¼ aAY¼ aBX¼ 100.54, and aAX¼ aBY¼ 25. The red, green, blue, and purple colors represent A, B, X, and Y,

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immiscibility enhances the preferentiality of X2for A and Y2

for B and thus qualitatively increases the degree of molecular asymmetry for A1B3when Y2is the major solvent. As a result,

we observe a significantly enlarged microstructure region in-duced by increasing aXY in the region when Y2 amount is

more than X2.

Based on the fact that the effects of adding two strongly selective solvents on the resulting phase behavior of A1B1

and A1B3 are qualitatively consistent, we expect that similar

results also hold true qualitatively in other amphiphilic mole-cules (AmBn) which initially form different ordered phases,

such as G, PL, and S. As in the amphiphilic molecule solutions when only one solvent is added, in addition to the macrophase separation, a sequential transition of microstructures including L, G, PL, C, and S is possible by varying the immiscibility pa-rameter between two solventsaXY, the ratio of solvent amount

fX2=fY2, and the volume fraction of amphiphilic molecules fC. Though the addition of a second solvent cannot induce new morphology types, the presence of two solvents enriches the phase behavior further. When the added two solvents are miscible, a typical transition of 1 ordered phase / coexis-tence of AmBn-rich and X2/Y2-rich phases occurs with

de-creasing fC. Increasing the immiscibility between two solventsaXYsignificantly enlarges the 1 ordered phase regime,

in which a series of microphase transition can even be in-duced. This transition behavior is analogous to qualitatively varying the degree of molecular asymmetry for AmBn.

Further-more, we infer that these results associated with varying the solvent immiscibility parameter are quite general for the am-phiphilic molecules in the presence of two solvents, as they have also been observed in our previous phase behavior study of AB diblock copolymers in the presence of one neutral solvent and one slightly selective solvent[31].

The last issue to consider is whether the above mesophase behavior results are dependent on the finite size of the simula-tion box. In order to manifest this, we examine each micro-structure in different sizes of the simulation box L3 with L 10. When the systems form the lamellar and hexagonally packed cylindrical phases, as long as the simulation box size is significantly larger than the radius of gyration of amphiphilic molecules, we observe that these ordered structures are no lon-ger affected by the simulation box. However, when the amphi-philic molecule systems tend to form the spherical micelles, we find that the packing array of these spheres is strongly de-pendent on the size of the simulation box even though the box size is much larger than the radius of gyration of amphiphilic molecules. For example, in Fig. 6(a)e(c) we present the pat-terns for A1B3 in the solvents X2 and Y2 with fC ¼ 0:6; fX2=fY2 ¼ 2=8, and aXY¼ 30, simulated in a box of 103, 133, and 163, respectively. It is clear that the spherical mi-celles with the radius approximately equal to 2.6 grids are formed in the box of 103, 133, and 163, respectively, but they pack into an fcc, A15, and bcc lattice, respectively. With a further inspection ofFig. 6(a) the number of the effec-tive spheres formed in the simulation box of 10 10 10 is equal to 4, which simply corresponds to the value of effective spheres in an fcc lattice. Due to the fact that the radius of the formed spheres is independent of the simulation box, the num-ber of the effective spheres allowed to form in the box of 13 13 13 and 16 16 16 is thus expected to be equal to 4 (1.3)3

y 8 and 4 (1.6)3y 16, respectively. Recall that the number of effective spheres in the A15 lattice and bcc lattice is 8 and 2, respectively. Therefore, it is not surpris-ing that the spheres in the box of 13 13 13 form an A15 lattice (Fig. 6(b)), and they pack into a bcc array in the box of 16 16 16, which includes 8 unit cells (Fig. 6(c)). Fig. 4. Phase diagrams of asymmetric A1B3in the presence of A-selective

solvent X2 and B-selective solvent Y2 for (a)aXY¼ 25, (b) aXY¼ 30, and

(c)aXY¼ 40. The other interaction parameters are set as aAB¼ aAY¼ aBX¼

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The reason that these bcc, fcc, and A15 packing arrays are possible to occur for the same systems but in different sizes of the simulation box via DPD may be due to the fact that the free energy of these spherical packing lattices is quite close. Generally speaking, though DPD may not identify the most stable packing array of spheres due to the significant finite size effects, it reveals the possibility for the spheres packing into an A15 lattice. Indeed, in addition to fcc and bcc, A15 has been proposed as a quite possible state in the amphi-philic molecule systems[32].

In general, our current results reveal that the DPD simula-tion method has successfully captured the microphase separa-tion behavior associated with lamellae and hexagonally packed cylinders and the macrophase separation behavior of an amphiphilic molecule in the presence of two solvents. Var-ious transitions with decreasing fCoccur by varying the inter-action parameteraXYand solvent ratio fX2=fY2. However, we find that the packing order of the spherical micelles is greatly affected by the finite size of the simulation box. As such, it may become difficult to examine the stability of the spherical ordered phases via the DPD method.

5. Summary

We employ dissipative particle dynamics (DPD) to study the phase behavior of AmBnamphiphilic molecules in the

pres-ence of two solvents X2and Y2, which are strongly selective

for A and B, respectively. As in the amphiphilic molecule so-lutions when only one solvent is added, in addition to the mac-rophase separation, the possible formed phases include L, G, PL, C, and S. Though the addition of a second solvent cannot induce new morphology types, the phase behavior is strongly affected by the immiscibility parameter between two solvents aXY, the ratio of solvent amount fX2=fY2, and the volume frac-tion of amphiphilic molecules f_C. We observe that increasing aXYnot only drives a macrophase separation into X2-rich and

Y2-rich phases for systems at lower values of fC, but also enlarges the 1 ordered microphase region at higher fC. It even induces a sequential microphase transition, which is anal-ogous to qualitatively varying the degree of molecular asym-metry in the amphiphilic molecules. This is reasonable due to the fact that increasing the solvent immiscibility enhances the preferentiality of X2 for A and Y2 for B. Though the

Fig. 5. Morphological patterns of A1B3in the presence of A-selective solvent X2and B-selective solvent Y2whenaXY¼ 25, and (a) fC¼ 0:9; fX2=fY2¼ 5=5, (b)

fC¼ 0:6; fX2=fY2¼ 7=3, (c) fC¼ 0:6; fX2=fY2¼ 3=7, and (d) fC¼ 0:2; fX2=fY2¼ 2=8, simulated in a box of 10 10 10. The other interaction parameters

are set asaAB¼ aAY¼ aBX¼ 100.54, and aAX¼ aBY¼ 25. The red, green, blue, and purple colors represent A, B, X, and Y, respectively. The red surface

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packing order of the spherical micelles is greatly affected by the finite size of the simulation box, DPD reveals a possible order of A15. Indeed, in addition to fcc and bcc, A15 has been proposed as a quite possible state in the amphiphilic molecule systems.

Acknowledgements

This work was supported by the National Science Council of the Republic of China through grant NSC 94-2216-E-002-027.

Fig. 6. Morphological patterns of A1B3in the presence of X2and Y2with fC¼ 0:6; fX2=fY2¼ 2=8, and aXY¼ 30, simulated in a box of (a) 10 10 10, (b)

13 13 13, and (c) 16 16 16. The other interaction parameters are set as aAB¼ aAY¼ aBX¼ 100.54, and aAX¼ aBY¼ 25. The red, green, blue, and purple

colors represent A, B, X, and Y, respectively. The red surface corresponds to the isosurface of component A. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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