Dynamic density functional theory based on equation of state

Chemical Engineering Science 62 (2007) 3494 – 3501 www.elsevier.com/locate/ces

Dynamic density functional theory based on equation of state Hui Xu, Honglai Liu ∗ , Ying Hu State Key Laboratory of Chemical Engineering and Department of Chemistry, East China University of Science and Technology, Shanghai 200237, China Received 21 June 2006; received in revised form 2 February 2007; accepted 7 February 2007 Available online 28 March 2007

Abstract Dynamic density functional theory based on equation of state (EOS-based DDFT) for the calculation of microphase separation of block copolymer is presented. The free-energy functional is derived from the bonding potential and an equation of state for chain fluid; the latter can accurately describe the thermodynamic properties of real substances. Contrast to the traditional DDFT, the highlight aspect of this method is that the hypothetical external field is not used; the additional iterations of self-consistent nonlinear equations are therefore avoided. Although the free-energy functional form is a little complicated, the computing time is notably reduced. The applications to diblock copolymer melts and blends of copolymer and homopolymer are presented as illustrations. 䉷 2007 Elsevier Ltd. All rights reserved. Keywords: Dynamic density functional theory; Equation of state; Bonding potential; Block copolymer

1. Introduction Copolymers, surfactants and proteins are complex fluids; the most prominent character of them is that they form mesoscale structures in certain conditions. The macroscopic behavior of these systems is determined not only by their microscopic properties such as bonding, chain connectivity and flexibility, intersegment interactions and excluded volume effect, but also, more directly and decisively, by their mesoscale structures. At least three scales, micro, meso and macro exist simultaneously in these systems. Multiscale approach, especially building bridges between those scales, is essential in understanding the behavior of these systems. In the latest decades, mesoscale dynamic models have received increasing attention because of their connection to fast molecular kinetics with slow thermodynamic relaxations of macroscopic properties. Many simulation methods, such as dissipative particle dynamics (DPD) (Groot and Warren, 1997), cell dynamics system (CDS) method (Oono and Puri, 1988, 1990), and dynamic density functional theory (DDFT) (Fraaije, 1993; Fraaije et al., 1997; Horvat et al., 2004; Lyakhova et al., 2004; van Vlimmeren et al., 1999; Ren et al., 2002; ∗ Corresponding author. Tel./fax: +86 21 642 529 21.

E-mail address: [email protected] (H. Liu). 0009-2509/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2007.02.055

Maurits et al., 1998; Zvelindovsky et al., 1998; Xia et al., 2005; Morita et al., 2001), have been developed correspondingly to study and understand the properties of industrial copolymers, surfactants, and proteins. The DDFT originally proposed by Fraaije (Fraaije, 1993; Fraaije et al., 1997) has been successfully applied to study many dynamic phenomena of polymer systems, such as the effect of dissimilar interfaces on the behavior of cylinder forming block copolymers in thin films (Horvat et al., 2004; Lyakhova et al., 2004), microphase separation dynamics of aqueous solutions of the triblock polymer surfactants (van Vlimmeren et al., 1999), the effect of shear on polymer melts and polymer solutions (Ren et al., 2002; Maurits et al., 1998; Zvelindovsky et al., 1998), microphase ordering mechanisms in linear ABC triblock copolymer (Xia et al., 2005), structures of thin polymer blend films with a free surface (Morita et al., 2001), and so on. Fraaije’s DDFT is virtually a coupling of the dynamic formalism with the self-consistent field theory (SCFT). Introducing a hypothetical external field inherent in SCFT is therefore leading to a set of self-consistent nonlinear equations in DDFT that should be solved by various time-consuming iteration schemes. The later developed external potential dynamics (EPD) (Maurits and Fraaije, 1997) has made remarkable improvement by using nonlocal kinetic coupling to DDFT; the equation of motion for the density field is replaced by the

H. Xu et al. / Chemical Engineering Science 62 (2007) 3494 – 3501

corresponding equation for the hypothetical external field. However, the EPD did not bring improvement on computation efficiency. On the other hand, it is difficult to use the equation of motion for many systems having true external field, such as shear flow field, electronic field, and so on. It is worth pointing out that, Marconi and Tarazona (1999, 2000) also applied DDFT to simulate the microscopic properties of fluids in 1D; the results are consistent with that of Brownian dynamics simulations (Dzubiella and Likos, 2003; Penna and Tarazona, 2003). Differing from the traditional DDFT, the calculation of hypothetical external field is avoided in their work by employing an exact form of free-energy functional. This gives us enlightenment that looking for an accurate molecular model could be a good way to improve the calculating efficiency. In this work, we introduce an accurate form of free-energy functional in two steps. First, similar to Yu and Wu (2002), we employ the bonding potential to describe the contribution of ideal chain. By introducing the bonding potential, the much time-consuming simulation of single chain for obtaining the contribution of chain connectivity in the traditional DFT, is avoided, the computation efficiency is therefore improved obviously. Second, we express the nonideal free-energy by employing an equation of state (EOS). The EOS employed here is the square-well chain fluid EOS proposed by Liu and Hu (1997), which is a hard-sphere-chain model (Hu et al., 1996) modified by a square-well perturbation term. This molecular thermodynamic model has been satisfactorily tested by vapor pressures and volumetric properties for various pure fluids and polymers (Liu and Hu, 1996). It has also been used to describe accurately the experimental pVT data of binary liquid mixture including polymer blends, and the binary vapor–liquid equilibria data of volatile systems and polymer solutions (Liu and Hu, 1997). For the molecules with specific oriented interactions such as hydrogen bonding, the contribution of chemical association is added (Liu and Hu, 1998). A new multi-scale method, DDFT based on equation of state (EOS-based DDFT) is therefore presented in this work, which forms a more effective bridge between the macroscopic properties and the mesoscale structures. The numerical calculation of microphase separation in 3D of block copolymers, as well as blends of block copolymer and homopolymer, are shown as illustrations. Contrast to the traditional DDFT, the highlight aspect of EOS-based DDFT is that the iteration calculation of self-consistent nonlinear equations is avoided by employing the bonding potential and EOS. The free-energy functional, including ideal chain contribution, local interaction and excluded volume effect, is derived from EOS directly. The dynamics formalism satisfies the time-dependent Ginzburg–Landau theory. The mesoscale fluctuation is introduced by the white noise based on the fluctuation–dissipation theorem. 2. Theory and model

well chains with bead diameter  and chain length N = NA + NB . Beads A and B are indexed by I. I (r) and I (r) are the density field, chemical potential field of the bead I at position r, respectively. For a given density field, the free-energy functional F {} can be separated as contributions of an ideal free-energy functional and a nonideal free-energy functional: F {} = F id {} + F nid {}.

(1)

The ideal free-energy functional is expressed as follows (Yu and Wu, 2002):  I (r)[ln I (r) − 1] dr F id {} = −1 +

I N−1 

s (r)vb (|rs+1 − rs |) dr,

(2)

s=1

where the first term is the contribution of ideal gas. The second term takes into account chain connectivity where vb (|rs+1 −rs |) is the bonding potential between the beads of s at r and s + 1 at r (Yu and Wu, 2002). The nonideal free-energy functional F nid {} contains the excluded volume interactions and the cohesive interactions. The chemical potential fields I (r) are defined by the functional derivatives of the free-energy functional I (r) ≡

N−1  jF  K = −1 ln I (r) + I s vb (|rs+1 − rs |) jI (r) s=1

+ nid I (r),

(3)

where K I s is the Kronecker  function with value 1 when bead s is of type I and 0 otherwise. For an arbitrary chain configuration, the total bonding energy satisfies  N−1  N−1    exp − vb (|rs+1 − rs |) ∝ (|rs+1 − rs | − ), (4) s=1

s=1

where (r) is the Dirac  function. The proportionality constant in Eq. (4) can be determined from the normalization condition   N−1   1  vb (|rs+1 − rs |) = 1. (5) exp − V s=1

Substitution of Eq. (5) into Eq. (4) yields  N−1  N−1   (|rs+1 − rs | − ) exp − vb (|rs+1 − rs |) = . (6) 42 s=1

s=1

Eq. (3) can therefore be rewritten as  I (r) = −1 ln I (r) − −1 ln exp[−nid I (r)]

2.1. Thermodynamics Diblock copolymer is modeled as a compressible system with volume V containing n tangentially connected square-

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×

(|rs+1 − rs | − ) 42

N−1 

K Is

s=1

 .

(7)

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The key thermodynamic assumption of EOS-based DDFT is that the distribution of density field can be depicted by an EOS for fluid; therefore, the nonideal free-energy functional can be obtained from EOS. In this work, the EOS for square-well chain fluids proposed by Liu and Hu is adopted. For homogeneous chain fluids, the corresponding nonideal chemical potential is  8 − 92 + 33 (am + cm ) am 2 nid I = + + 1− (1 − )3 2(1 − )2  bm  + − cm ln(1 − ) (1 − )3  √ p 9  4  3 2 + (1+p+2q · LI )Apq p T˜ −q , (8)  p q

where the reduced bead density  = (/6) I I 3 , Apq s are numerical coefficients of the square-well interaction, I J is the interaction parameter between the beads of type I and J, am = a20 (N − 1)/N + a30 (N − 1)(N − 2)/N 2 , a20 = −0.61819, a30 = −10.25181,

(9)

bm = b20 (N − 1)/N + b30 (N − 1)(N − 2)/N 2 , b20 = 0.19421, b30 = 2.08257,

(10)

cm = c20 (N − 1)/N + c30 (N − 1)(N − 2)/N 2 , c20 = 2.75503, c30 = 4.83207, (11)

I J ( I J /kT )3 T˜ −1 = I J , (12) 3 I J I J    3 3 J J ( I J /kT ) J J  LI = , − 3 3 J K J K ( J K /kT ) J K J K 

(13)

where I = I / J J . The details can be referred to Liu and Hu (1997) and a summary is given in Appendix A. It has been known that the local density approximation is not accurate in the density functional theory for the inhomogeneous equilibrium systems. To improve the accuracy, an alternative approach is using the weighted density approximation (WDA). In this work, a simplest WDA, the Heaviside step function approximation, is used  (r) ˜ = (r )w(|r − r |) dr , (14) w(r) = 3( − r)/(43 ),

(15)

2.2. Dynamics Similar to the traditional DDFT, the time evolution of the density field is given by a time-dependent Ginzburg–Landautype equation jI (r, t) = ∇ · MI ∇I + I , jt

(17)

where M is a mobility coefficient, I is a stochastic noise which is distributed according to the fluctuation–dissipation theorem  I (r, t) = 0,  I (r, t) J (r , t  ) = 2−1 MI (r − r )(t − t  ).

(18)

2.3. Numerics The primary equations of EOS-based DDFT are the following: Eqs. (7) and (8) represent the chemical potential as a functional of density field, Eq. (17) relates the time evolution of density field jI /jt and the chemical potential, Eq. (18) poses a constraint to the exchange kinetic coefficient by the noise. These equations form a closed set. For the convenience of numerical calculation, the following dimensionless parameters are introduced:

≡ −1 Mh−2 t,

d ≡ h−1 ,

Tr = T /T0 ,

(19) −1

where is the dimensionless time, h is the mesh size,  M is the diffusion coefficient, d is the grid scaling, Tr is the reduced temperature, T0 is the reference temperature. The partial differential equation set can be integrated efficiently on a cubic mesh by a Crank–Nicolson scheme (Fraaije et al., 1997; van Vlimmeren and Fraaije, 1996). The corresponding Crank–Nicolson equations are k+1 k k k k+1 I r −  zI r = I r + (1 − )zI r + I r ,

k  k d [D I D ]rq I q , zI r =



(20) (21)

q

 kI r = [ 0 2 I wr ]k ,

(22)

where I ≡ I 3 /60 , 0 is the initial reduced bead density,  is the time step, D is the half-point finite-difference first derivative operator in lattice direction , d is the scalar weight factor, 0 is the noise parameter, wr is a random number satisfied normal distribution at r,  is the Crank–Nicolson parameter (in this work,  = 0.5), the superscript k means that all the enclosing terms are evaluated at time step k. The chemical potential is calculated by using once-integrated Green propagators (Yu and Wu, 2002; Doi and Edwards, 1986).

where w is a weighted function, and  is the Heaviside function. For inhomogeneous fluids, to calculate nid I (r), I in Eq. I (r) = −1 ln I (r) − −1 ln (8) should be replaced by the corresponding weighted densities N  (r); ˜ therefore, the chemical potential is a functional of density  K nid i × I s exp[−s (r)]Gs (r)Gs (r) , field. During the calculations, the weighted density functional is s=1 discretized as

 4 1 (x + 1, y, z) + (x, y + 1, z) + (x, y, z + 1) (x, ˜ y, z) = (x, y, z) + . (16) +(x − 1, y, z) + (x, y − 1, z) + (x, y, z − 1) 7 14

(23)

H. Xu et al. / Chemical Engineering Science 62 (2007) 3494 – 3501

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where G1 (r) = GiN (r) = 1,

(24)

Gs (r) = exp[−nid s (r)][Gs−1 ](r),

(25)

i Gis (r) = exp[−nid s (r)][Gs+1 ](r).

(26)

The connectivity operator is defined by  ( − |r − r |) [X](r) ≡ X(r ) dr . 2 4 V

(27)

For simplicity, the connectivity operator is calculated by using the following finite difference:smal

 1 X(x−1, y, z)+X(x, y−1, z)+X(x, y, z−1) [X](r)= . 6 +X(x+1, y, z)+X(x, y+1, z)+X(x, y, z+1) (28) 2.4. Algorithm

Fig. 1. Phase diagram for the diblock copolymer melts. Symbols “”, “•”, “” and “” refer to the disordered, spheres, cylinders and bicontinuous meso-structures, respectively. The morphologies of the marks a, b, c, d and e are shown correspondingly in Fig. 2.

The algorithm of EOS-based DDFT in an integration step mainly includes: Step 1: Input density fields {I }. Step 2: Calculate chemical potential fields {I } using Eqs. (7) and (8). Step 3: Generate the noise source according to Eq. (18). Step 4: Iterate the Crank–Nicolson equations (a) Check for convergence and exit or go to step 4b. (b) Select new {I } using conjugate gradient method. (c) Calculate new {I } using Eqs. (7) and (8) and go to step 4a. 3. Results and discussion

and Wasserman, 1975). However, what should be pointed out is that when the copolymer is symmetric or near symmetric, the morphology observed is not lamellae but bicontinuous. This is because that the driving force in EOS-based DDFT is chemical potential gradient, the same as in traditional DDFT and other TDGL methods, which is not large enough to make the morphology more ordered. To generate more ordered alignments, a simple steady shear field with a constant shear rate ˙ = dux /dy is added to the system, we then have ux = ˙ y, uy = 0, uz = 0 (Maurits et al., 1998; Zvelindovsky et al., 1998). The dynamics expression Eq. (17) is revised as follows:

3.1. Diblock copolymer melts

jI (r, t) j = −˙y I + ∇ · MI ∇I + I . jt jx

For diblock copolymer melts with different volume fractions, the following parameters are set for the numerical calculations: the copolymer chain length N = 10; the initial reduced bead density 0 = 0.4; the noise parameter 0 = 0.1; the grid scaling d = 1; the reference interaction parameter AA /kT 0 = BB /kT 0 = 0.10; AB /kT 0 = BA /kT 0 = 0.04; the dimensionless time step  = 0.02. All calculations are performed in a box of size 20 × 20 × 20, and the total time is 1000 dimensionless time steps. The start configuration is a homogeneous profile, I (r) = NI /N. Fig. 1 shows the calculated phase diagram of diblock copolymer melts. The corresponding morphologies are shown in Fig. 2. It can be seen from the figures that the EOS-based DDFT can generate the spherical (Fig. 2a and b), rodlike (Fig. 2c) and bicontinuous (Fig. 2d and e) morphologies for the cases with different fractions and temperatures. The results agree well with the strong segregation limit theory (SSLT) (Bates, 1991). In this work, the beads are modeled as hard spheres with square-well interaction, which leads to the micro-domains of pure A or B. This is in consistent with the main character of SSLT (Helfand

Fig. 3 shows the morphologies of diblock copolymer melts A5 B5 and A3 B7 under the steady shear flow. The direction of the shear is shown in Fig. 3c. The initial morphologies of both cases, before the shear added, correspond to Fig. 2e and c, respectively. It can be clearly seen that the initial messy phases, full of defects, change to the more ordered patterns with global orientation and fewer defects under shear. For the case of A5 B5 in Fig. 3a, lamella are exhibited. While for the case of A3 B7 in Fig. 3b, cylinders are formed. The application of shear in diblock polymer melts speeds up the formation of ordered structure enormously. One may note from Fig. 1 that the ordered–disordered transition (ODT) temperature Tr−1 ODT ≈ 2.5, when AA /kT = BB /kT ≈ 0.25 and AB /kT = BA /kT ≈ 0.10. It seems that the ODT temperature in EOS-based DDFT is different from that of the previous reports, NODT ≈ 10– 12, =( AA + BB − AB − BA )/2kT , N is the chain length. This is because that the square-well interaction in this work is different from the Flory–Huggins interaction used by other mesoscale simulation methods, such as DPD, CDS and MesoDyn.

(29)

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H. Xu et al. / Chemical Engineering Science 62 (2007) 3494 – 3501

Fig. 2. Morphologies of diblock copolymer melts with different fraction and temperature (A = 0.5). (a) A1 B9 , Tr−1 = 9, (b) A2 B8 , Tr−1 = 5, (c) A3 B7 , Tr−1 = 5, (d) A4 B6 , Tr−1 = 5, (e) A5 B5 , Tr−1 = 5. The color sheets are the boundaries between A and B.

Fig. 3. Morphologies of diblock copolymer melts A5 B5 (a) and A3 B7 (b) with the shear rate ˙ = 0.0002 (A = 0.5). (c) The spatial direction of steady shear flow. The color sheets are the boundaries between A and B.

Fig. 4. Morphologies of blend (fA5 B5 = 0.7 and fC5 = 0.3) at different times. (a) 100 steps, (b) 500 steps, (c) 1000 steps. Red, A; blue, B; green, C, their concentrations are greater than certain threshold.

3.2. Blend of diblock copolymer A5 B5 and homopolymer C5 It is well-known that after temperature decrease, a blend of copolymer and homopolymer exhibits simultaneously two kinds of phase separations, namely macrophase separation of diblock copolymer and homopolymer, and microphase separation in diblock copolymer. Generally, the microphase separa-

tion will happen prior to the macrophase separation if the fraction of copolymers is larger, while it is smaller, the macrophase separation will occur prior to the microphase separation on the contrary (Hashimoto et al., 1991; Ohta and Ito, 1995). The following parameters are set for the blend of a diblock copolymer and a homopolymer: the initial reduced bead density 0 = 0.4; the noise parameter 0 = 0.2; the grid scaling d = 1;

H. Xu et al. / Chemical Engineering Science 62 (2007) 3494 – 3501

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the interaction parameters AA /kT = BB /kT = CC /kT =0.50; AB /kT = BC /kT = 0.10; AC /kT = 0.45; the dimensionless time step  = 0.02. All calculations are performed in a box of size 30 × 30 × 30, and the total time is 1000 dimensionless time steps. The starting configuration is a homogeneous profile, I (r) = NI /N. The order parameter is defined as follows:

 1/2 1 (I (r) − I )2 , (30) pI = V V  1 I = I (r). (31) V V The order parameter of bead I is the root mean square deviation of volume density of bead I. Fig. 4 shows the morphologies of a blend with fA5 B5 = 0.7 and fC5 =0.3 at different times. Fig. 5 shows the time evolution of order parameter of different beads. Because A–C is more attractive than A–B and B–C, the order parameter of B-type bead increases faster than that of other beads and the B-type bead enriches at first. This leads to the microphase separation that happens in the block copolymer A5 B5 firstly as shown in Fig. 4a. However, because A–C is still a little bit repulsive

Fig. 5. Evolution of order parameter p of different beads. fA5 B5 = 0.7, fC5 = 0.3.

Fig. 7. Evolution of order parameters p of different beads. fA5 B5 = 0.3, fC5 = 0.7.

than A–A and C–C, after 500 steps, the homopolymer starts to enrich. Finally, a morphology is reached that micro-domains of C are dispersed in A matrix. When we increase the fraction of homopolymer to 0.7, i.e., fA5 B5 = 0.3 and fC5 = 0.7, as shown in Fig. 6, the macrophase transition occurs in the beginning period. The microphase transition then follows when the copolymer starts to enrich. Finally, morphology of shell–core type rod-like AB micro-domains with core B and shell A dispersed in C matrix is obtained. Fig. 7 is the corresponding evolution of order parameters. It can be found that the enrichment degree of A-type bead is much lower than that of B-type bead, the latter forms the core of micro-domains. The simulation results shown above reveal a strong dependence of the morphology of block copolymer on the fraction of homopolymer. Due to the compatibility between A and C, there are two typical phase separation processes which obeyed different mechanisms, which extend the types of the morphologies. In the present work, the mesoporous structure and the “core–shell” type sphere of AB copolymer, which cannot be observed in the pure symmetric diblock copolymer melts, are obtained with the different volume fractions of C. Because there is no chemical bond between copolymer and homopolymer,

Fig. 6. Morphologies of blend (fA5 B5 = 0.3 and fC5 = 0.7) at different times. (a) 100 steps, (b) 500 steps, (c) 1000 steps. Red, A; blue, B; green, C, their concentrations are greater than certain threshold.

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H. Xu et al. / Chemical Engineering Science 62 (2007) 3494 – 3501 Table 1 Computation CPU times for different approaches (A5 B5 , 1000 steps) Approach

a

b

c

CPU time

∼ 3h

∼ 15 min

∼ 4 min

It is clearly seen from Table 1 that the computation CPU time of EOS-based DDFT is much shorter than that of traditional DDFT. Because of the simpler form of the nonideal Flory–Huggins free-energy functional, the approach (c) spends further shorter time than the approach (b). 4. Conclusions

Fig. 8. CS as a function of the reduced beads density .

these morphologies can be separated from the blend through removing homopolymer by a selective solvent. 3.3. Selection of the time step In this work, the excluded volume effect is described by the Carnahan–Starling equation, the EOS-based DDFT is therefore sensitive to the dimensionless time step  . In Fig. 8, we plot the excluded volume potential calculated by Carnahan–Starling equation, CS , against the reduced beads density . It is seen from the figure that in the range of typical values of  in liquid region ( ≈ 0.4– 0.7), the excluded volume potential increases rapidly. For this highly compressible system, if the  is too large ( 0.1), the chemical potential gradient, the driving force of morphology evolution, will become too large, which leads to an unstable evolution process. Therefore, selecting a suitable  is crucial to the computing efficiency of EOS-based DDFT. In this work, an optimal value of  = 0.02 is selected for the systems studied.

We introduce a new multi-scale approach, an EOS-based DDFT for the calculation of microphase separation of copolymers. The model presents a further coarse-graining of DFT, and because EOS is directly related to the macroscale behavior of the material, the model is, therefore, a hybrid of mesoand macro-scale dynamics. By adopting an EOS for chain fluid to construct the nonideal free-energy functional and employing the bonding potential, the computation efficiency increases obviously than the traditional DDFT. The results are consistent with those by theories and experiments. The method can be extended to other systems including multicomponent copolymer mixture, surfactant solution, copolymer film, and so on. The method provides a new effective mean to study the dynamical properties of the free energy in the systems. This work improves the connection between microscopic properties and the meso-structures, which is a crucial contribution to the multiscale approach for complex fluids. The real advantage of this method highly relies on the accuracy of the EOS, also relies on the reliability of the WDA. Therefore, there are still rooms for improvements by adopting more accurate EOS and more reliable WDA. Besides, building the bridge between meso-structures and the macroscopic properties is worth paying more attention. Acknowledgments

3.4. Computation efficiency As a remarkable aspect of EOS-based DDFT, the computation efficiency is improved obviously. To compare the computation efficiency of different approaches, the following three models has been considered: (a) traditional DDFT with Flory–Huggins interaction and Helfand excluded volume interaction; (b) EOS-based DDFT with square-well interaction and hard sphere excluded volume interaction; (c) EOS-based DDFT with Flory–Huggins interaction and Helfand excluded volume interaction. The microphase separations of diblock copolymer A5 B5 were calculated on a PC with Celeron CPU 2.60 GHz by using three approaches, respectively. The corresponding CPU times are shown in Table 1. All of the three approaches were simulated in a box of size 30 × 30 × 30 by 1000 steps, which is enough to ensure that the microphase separation has reached equilibrium.

This work is supported by the National Natural Science Foundation of China (Projects no. 20490200), E-institute of Shanghai High Institution Grid (no. 200303) and Shanghai Municipal Education Commission of China. Appendix A The EOS for square-well chain fluids can be written as follows: A Aid AH S ASW = + + Ns Ns Ns Ns

(A.1)

where



Aid A0 Ns , = + ln Ns Ns V

(A.2)

H. Xu et al. / Chemical Engineering Science 62 (2007) 3494 – 3501

 1 3 − 1 + 1− (1 − )2   am  − bm bm − + − c ln(1 − ) , (A.3) m 2(1 − ) 2(1 − )2  √ p 9  4  3 2 = Apq p T˜ −q . (A.4)  p q

AH S = Ns

ASW Ns



A0 is the standard Helmholtz function, Ns is total bead number of the whole system. The chemical potential of I -type bead is 

jA I = (A.5) jNI T ,V ,NJ =I and nid I =



jAH S jNI



+ T ,V ,NJ =I

jASW jNI

 ,

(A.6)

T ,V ,NJ =I

where

  jAH S 8−92 +33 (am +cm ) = + 3 jNI 1− (1−) T ,V ,NJ =I +

am 2 2(1−)2

+

bm  (1−)

 −cm ln(1−) , 3 (A.7)



 9  4  jASW = (1 + p + 2q · LI ) jNI T ,V ,NJ =I p q  √ p 3 2 p T˜ −q . × Apq 

(A.8)

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