CHINESE JOURNAL OF CHEMICAL PHYSICS VOLUME 19, NUMBER 2 APRIL 27, 2006

LETTER

Thermal Induced Periodic Phase Separation in Polymer Blends

Zhe Sun, Hai-hua Song∗

School of Chemical Engineering & Technology, Tianjin University, Tianjin 300072, China

(Dated: Received on December 20, 2005; Accepted on March 13, 2005)

Simulations were carried out for studying the periodic phase separation of a symmetric binary polymer blendon the basis of Cahn-Hilliard-Cook theory. The time dependent interaction parameter χ(τ) was assumed toundergo a step-wise oscillation. The hierarchic structures composed of both large and small domains wereobtained. The mechanism of the periodic formation of hierarchic structures was also demonstrated.

Key words: Periodic spinodal decomposition, Simulations, Hierarchic structures

The ability to control the morphology of polymericmixtures is highly desirable for researchers becausethe mesoscale structure determines the mechanical andother properties of the materials [1-2]. Technically, ma-terials with a specific morphology are usually fabricatedunder a nonlinear and nonisothermal process. Thus,considerable studies have been devoted to investigatingthe phase separation driven by temperature modulation(e.g. double quench and continuous quench) [3-5].

Periodic quench is another controllable mode for driv-ing the nonisothermal phase separation. A typical peri-odic quench refers to quenching a binary mixture aboveand blow the critical temperature alternatively. Tanakaet al. performed the periodic quench experiments with asinusoidal temperature modulation on binary polymericmixtures [6]. They found that the hierarchic structures,i.e. structure-within-structure, emerge periodically solong as the average of temperature is lower than a spe-cial value.

To the best of our knowledge, no simulation on peri-odic phase separation has been reported so far. In thiswork, we model the kinetics of this process using theCahn-Hilliard-Cook theory. Moreover the mechanismof the formation and destruction of hierarchic struc-tures in each period is demonstrated, which has neverbeen involved in previous work [6-7]. We consider an in-compressible polymer blend with critical composition.Both components are assumed to have the same degreeof polymerization, N . And hydrodynamic interactionsare neglected for simplicity. The local fraction of com-ponent A, φ(r, t) obeys the CHC equation [8]:

∂φ(r, t)∂t

= D∇2 δFmix[φ(r, t)]δφ(r, t)

+ η(r, t) (1)

where D is the self-diffusion coefficient which we as-sume to be a constant, and η(r, t) is thermal noise. The

∗Author to whom correspondence should be addressed. E-mail:haihuasong@eyou.com

free energy of mixing Fmix[φ(r, t)] is taken as the Flory-Huggins-de Gennes form [9]:

Fmix[φ(r, t)]kBT

=∫

dr{

1N

[φ lnφ + (1− φ) ln(1− φ)]

+ χ(t)φ(1− φ) + κ(φ)(∇φ)2}

(2)

where χ(t) is the time dependent interaction parame-ter. And the coefficient of the gradient term is givenby κ=σ2/36φ(1 − φ) where σ is the Kuhn length forboth species A and B. Substituting Eq.(2) into Eq.(1)and rescaling into a dimensionless form, we arrive atthe following equation:

∂φ(x, τ)∂τ

=12∇2

[χc

2(χmax − χc)ln

φ

1− φ− 2χ(τ)φ

χmax − χc

− 118φ(1− φ)

∇2φ +1− 2φ

36φ2(1− φ)2(∇φ)2

]

+√

ες(x, τ) (3)

where x=√|χmax − χc|r/σ, τ=2D(χmax − χc)2t/σ2.

The critical point χc=1/N and χmax is the maximumof quench depth in our simulation. The noise intensityε =

√|χmax − χc| and ς is the white noise with Gaus-

sian components [10]. We assume that the time depen-dent interaction parameter undergoes a step-wise oscil-lation, χ(τ)=r+mW (τ/τp), where r is the average, mis the magnitude of the oscillation and τp is the quenchperiod. W (x) is defined as W (x)=1 for n<x≤n+1/2and W (x)=−1 for n+1/2<x≤n+1 where n is the ordi-nal number of period.

Using the finite difference method, we numericallysolve Eq.(3) in 2562 square lattices under periodicboundary conditions [11]. The temporal and spatialdiscretization were set as ∆τ=0.0005 and ∆x=0.5, re-spectively. In order to study the domain growth ofphases A and B, we calculated the circularly aver-aged structure factor S(k, τ), which is derived fromS(k, τ)=

∑k S(k, τ)/

∑′1, where the structure factor isdefined as S(k, τ)=〈φkφ−k〉 [8]. As is shown in Fig.1

ISSN 1003-7713/DOI:10.1360/cjcp2006.19(2).99.3 99 c©2006 Chinese Physical Society

100 Chin. J. Chem. Phys., Vol. 19, No. 2 Zhe Sun et al.

FIG. 1 (a)-(d). Snapshot pictures of periodic phase separa-tion with r=0.011, m=0.004 and τp=10. τin stands for theinner time within one period. (e)-(f). Temporal evolutionsof structure factor S(k, τ) with the same parameters.

(a)-(d), the white and black color regions represent A-rich and B-rich domains, respectively. We can clearlysee the hierarchic structures constituted by two length-scale domains. In Fig.1 (e)-(f), the evolution of the cir-cularly averaged structure factor S(k, τ) demonstratesthat small domains corresponding to large wave num-bers are created and destroyed in each period, and largedomains corresponding to small wave numbers growsuccessively [13]. Moreover, (a) and (b) depict thatsmall domains are developed from the large domainsdue to the deep quench in the first half period. Fig.1(b), (c) and (d) are obtained in the middle of the quenchperiod. At that moment, the size of small domains ap-proaches to its maximum value. In Fig.1 (b), smalldomains could be either lamellar phases or droplets,whereas only droplets can be discerned in Fig.1 (d).These results indicate that, along with the growth oflarge domains, the concentration of species A (B) in A

(B)-rich domains increases and the interfaces becomesharp. Therefore the coarsening of small domains tolamellar phases is restrained.

Figure 2 shows the temporal changes of S(k, τ) withinone period. It is seen that from Fig.2 (a) small domainsare developed during the first half period. As the in-ner time changes from 0.625 to 3.125, the intensity ofthe peaks with high wave numbers increases, and theposition of those peaks is invariable. This time regimeis equivalent to the early stages of the ordinary quenchduring which the linear theory is applicable. Further-more, as the inner time varies from 3.125 to 5.0, theintensity of the peaks with large wave numbers doesn’tchange greatly and the position of those peaks begins todecrease. These results indicate not only the coarseningof small domains but also the equilibrium of diffusionbetween the small domains and the large domains sur-round them. Tanaka et al. obtained the same results inthe two-step quench experiments [12]. Also the positionof the peaks with small wave numbers keeps increasingin Fig.2 (a). This doesn’t means the size of large do-mains decreases but means domains in small size grow

FIG. 2 Temporal changes of structure factor S(k, τ) withinthe 16th period. The parameters are as the same as thosein Fig.1. (a). The first half period, (b). The second halfperiod. The numerals in the curves mark the inner time τin

Chin. J. Chem. Phys., Vol. 19, No. 2 Thermal Induced Phase Separation in Polymer Blends 101

faster than that in large size. As shown in Fig.2 (b),both intensity and position of two characteristic peaksdecrease during the second half period. And the peakscorresponding to small domains merge into the peaksfor large domains gradually. It suggests that diffusion inthe process of dissolution prefer to eliminate the modeswith high wave numbers.

FIG. 3 The plots of two characteristic peak wave numbersvs. time for r=0.011 and τp=10. Filled symbols are forsmall domains, while open symbols are for large domains.

We fitted S(k, τ) with two Gaussian functions [6] atthe middlemost time of quench period and obtained thetwo peak wave numbers. The curves with open symbolsin Fig.3 show the coarsening of large domains. Andthe curves with filled symbols indicate that the magni-tude of the oscillation m determines the characteristic

size of small domains to some extent. This result couldbe understood qualitatively with the aid of the lineartheory. The characteristic wave numbers for small do-mains at the middlemost time of each period can beestimated as kmax∝(m + r − χc)1/2 so long as the timefor the coarsening of small domains is short. Hence kmax

for small domains increases with the enlargement of m,provided that both the average of interaction parameterand quench period are fixed.

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