Computational simulation of polymerization-induced phase separation under a temperature gradient

Computational and Theoretical Polymer Science 11 (2001) 205–217 www.elsevier.nl/locate/ctps

Computational simulation of polymerization-induced phase separation under a temperature gradient J. Oh, A.D. Rey* Department of Chemical Engineering, McGill University, 3610 University Street, Montreal, Quebec, Canada H3A 2B2 Received 2 December 1999; revised 24 April 2000; accepted 27 April 2000

Abstract Polymerization-induced phase separation (PIPS) via spinodal decomposition (SD) under a temperature gradient for the case of a monomer polymerizing in the presence of a non-reactive polymer is studied using high performance computational methods. An initial polymer (A)/ monomer (B) one-phase mixture, which has an upper critical solution temperature (UCST) and is maintained under a temperature gradient, phase-separates and evolves to form spatially inhomogeneous microstructures. The space-dependence of the phase-separated structures under the temperature gradient field is determined and characterized using quantitative visualization methods. It is found that a droplet-type phase-separated structure is formed in the high-temperature region, corresponding to the intermediate stage of SD. On the other hand, lamella or interconnected cylinder type of phase-separated structure is observed in the low-temperature region, corresponding to the early stage of SD structure, in the large or small temperature gradient field, respectively. The kinetics of the morphological evolution depends on the magnitude of the temperature gradient field. The non-uniform morphology induced by the temperature gradient is characterized using novel morphological techniques, such as the intensity and scale of segregation. It is found that significant non-uniform structures are formed in a temperature gradient in contrast to the uniform morphology formed under constant temperature. 䉷 2000 Elsevier Science Ltd. All rights reserved. Keywords: Polymerization-induced phase separation; Temperature gradient; Non-uniform structure

1. Introduction Polymerization-induce phase separation (PIPS) is an important and practical manufacturing route for multiphase material production [1–3]. With increasing demands and interests on multi-phase materials that have better properties and functionality, phase separation, particularly spinodal decomposition (SD), has been an active area of material science research. The PIPS process is a more complicated process than the classical thermal-induced phase separation (TIPS) method because phase separation and polymerization occur simultaneously in the PIPS method. For this reason, compared to the TIPS method, relatively small number of theoretical [1–6] and experimental [7–11] studies have been performed on the PIPS method despite many advantages over other phase separation techniques. In the PIPS process, an initial mixture is prepared in the one-phase (stable) region and the mixture is homogeneous. When the molecular weight of the components increases due to polymerization, the phase diagram, which has an upper critical solution temperature (UCST) in this study, * Corresponding author. Tel.: ⫹1-514-398-4196; fax: ⫹1-514-398-6678. E-mail address: [email protected] (A.D. Rey).

constantly shifts toward higher temperature and concentration. As a result, the single-phase mixture is thrust into the unstable or metastable region, and phase separation occurs. As a result of phase separation, various types of phase-separated structures can be formed depending on the initial composition and molecular weight. Typical examples of phase-separated morphologies in polymer systems are the droplet-type morphology and the interconnected cylindertype morphology. The type and characteristics of the phaseseparated morphology are critical in determining the mechanical and optical properties of multi-component composite materials. For example, the fracture toughness increases as dispersed particle sizes decrease. However, if the average droplet size decreases, scattering of multi-component materials increases because the number of scattering sites within the film increases [12]. Therefore, it is necessary to develop a characterization of phase-separated morphologies, and in this study, a novel and simple method has been suggested to eventually determine optimal properties more easily. The majority of works on phase separation has been performed in the absence of external fields, such as temperature gradient. However, temperature gradients are common during material processing and characterization experiments. Recently a few studies concerning phase separation

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behavior under external fields such as shear flows [13–15], electric fields [16,17], and temperature gradients [18–20], have been presented. In addition, composite materials, exhibiting inhomogeneous microstructures and space-dependent properties, known as functionally graded materials (FGM) [21–25], have received attentions because many current applications of materials require specific bulk/surface properties. For example, a body of turbine blade must be strong, tough and creep-resistant, whereas its outer surface must be refractory and oxidation-resistant [21]. FGMs are designed to take advantage of certain desirable features of each of the constituent phases and reduce the local concentration of stress induced by the abrupt changes in composition and microstructure [21,22]. The spatially graded structures are produced by several add-on processes. However, as shown here, transport-based processes, such as the polymerizedinduced phase separation method, can also form locally graded microstructures. Furthermore, it is expected that in the future similar space-dependent functionality may be applied to polymeric materials. Phase separation under a temperature gradient was studied experimentally by several authors in the last few years [18–20,26–28]. Platten and Chavepeyer [26] studied phase separation under a temperature gradient for a lowmolar mass binary solution. Kumaki et al. [28] also performed a study of phase separation under a small temperature gradient, and they found that phase separation can be induced by the temperature gradient even in the onephase region because macromolecules tend to move toward the colder surface. Tran-Cong and Okinaka [19,20] investigated the TIPS process of poly(2-chlorostyrene)/poly(vinyl methyl ether) (P2CS/PVME) blend under a temperature gradient. Since P2CS/PVME blend has a lower critical solution temperature (LCST), they found that the interconnected structure is formed slowly in the high-temperature side, and the droplet-type structure is formed as a result of the late stage of spinodal decomposition in the low-temperature side of the gradient. Xie et al. [18] observed the inhomogeneous particle size distribution of the dispersed phase along the direction from the center to the surface in PP (polypropylene)/EVAc (ethylene–vinyl acetate) blends. To our knowledge, however, no numerical study has yet been performed on polymerization-induced phase separation (PIPS) under a temperature gradient to obtain spatially inhomogeneous microstructure. The objectives of this study are: (1) to develop and solve a computational model of the PIPS process for binary composite materials under a temperature gradient; and (2) to develop guidelines for the formation of spatially inhomogeneous microstructures applied to produce FGMs, using the PIPS process.

is assumed that a linear temperature gradient exists along, say, the y-axis. The computational domain is a square of size: 0 ⬍ x ⬍ L and 0 ⬍ y ⬍ L: Therefore, the temperature field is given by T…y† ˆ

T2 ⫺ T1 y ⫹ T1 ; L

dT ˆ0 dx

T1 ⬍ T2

…1a†

…1b†

Under the temperature gradient field given in Eq. (1), one of the components (reactive monomer), say B, undergoes polymerization while the other component (non-reactive polymer A) does not participate in this polymerization reaction. In the PIPS method, the degree of polymerization of component B, NB, constantly increases due to polymerization. The growth rate of NB can be determined by solving the following kinetic rate equation [29]: dp ˆ k1 …1 ⫺ p†2 dt

…2†

where p is the extent of reaction, t is time, and k1 is the reaction rate constant. The solution to Eq. (2) is pˆ

k1 t 1 ⫹ k1 t

…3†

The reaction rate constant k1 depends on reaction temperature, following the Arrhenius equation, and it is given by k1 ˆ A0 exp‰⫺E0 =RTŠ

…4†

where A0 is the collision frequency factor, E0 is the Arrhenius activation energy, and R is the gas constant. The expressions for the weight average molecular size X w in terms of functionality of the monomer g is given by X w ˆ

1⫹a 1 ⫺ …g ⫺ 1†a

…5†

where a is the branching coefficient, defined as the probability that a given functional group of a branch unit leads to another branch unit. By assuming the degree of polymerization of component B, NB, can be represented by X w and applying Eq. (4) for k1, the growth rate of NB can be written as NB ˆ ˆ

1 ⫹ 2k1 t 1 ⫹ 2k1 t ⫺ gk1 t 1 ⫹ 2…A0 exp…⫺E0 =RT††t 1 ⫹ 2…A0 exp…⫺E0 =RT††t ⫺ g…A0 exp…⫺E0 =RT††t …6†

2. Theory of the PIPS process under a temperature gradient In this two-dimensional (2D) study of the PIPS process, it

Since k1 depends on the reaction temperature, following Eq. (4), NB is a function of both t and y. The phase separation phenomena via spinodal decomposition can be properly described by the non-linear

J. Oh, A.D. Rey / Computational and Theoretical Polymer Science 11 (2001) 205–217

Cahn–Hilliard equation as [30–32]    2F A 2f 2 ˆ 7· M7 ⫺ 2k7 F A 2t 2F A

…7†

where F A is the volume fraction of component A (for a binary mixture, F A ⫹ F B ˆ 1†; 7 denotes 2=2x~i ⫹ 2=2y~j where ~i and ~j are unit normal vectors for the x and y directions, respectively, M is mobility, f is free energy density of the system, and k is a positive interfacial constant. The bulk free energy f in Eq. (7), according to the Flory– Huggins theory, is written as [33,34]   k T FA F ln F A ⫹ B ln F B ⫹ xF A F B …8† f ˆ B NA NB v where kB is the Boltzmann constant, v is the volume of the reference unit, NA is the degree of polymerization of polymer A, and x is the temperature-dependent interaction parameter. The expression for x in terms of temperature can be written as [35]   1 Q …9† xˆ ⫺C 1⫺ T 2 where C is the dimensionless entropy and Q is the theta temperature. Note that the interaction parameter x also depends on temperature, thus it is also a function of y, in this study. The mobility M in Eq. (7) depends on the molecular weight and local concentration of the components, and, for the low molecular weight regime, it is defined as Mˆ 

D D N N F F  A B A B A B kB T …DA ⫹ DB †…NA F A ⫹ NB F B † v

2F Aⴱ …x; y; t† 2tⴱ Dⴱ T ⴱ ˆ …NA ⫹ NB †

…10†

207

where DA and DB are the self-diffusion coefficients of polymer segments A and B, respectively. Using the Rouse theory [36], and representing diffusion phenomena in the low molecular weight regime, Eq. (10) can be expressed as Mˆ

vNA NB F A F B z…NA ⫹ NB †…NA F A ⫹ NB F B †

…11†

where z is the friction coefficient. Furthermore, the molecular dependent interfacial parameter k in Eq. (7) is written as [1,2]

k ˆ k0 …NA ⫹ NB †

…12†

where k 0 is the interfacial parameter for a linear polymer. The dimensionless governing equation describing the polymerization process can be obtained from Eq. (6). When we introduce the dimensionless collision frequency factor as Aⴱ ˆ A0 L4 z=2k0 v and the dimensionless activation energy as Eⴱ ˆ E0 =RQ; Eq. (6) can be expressed in the dimensionless form as N B …yⴱ ; tⴱ † ˆ

1 ⫹ 2Aⴱ exp…⫺Eⴱ =T ⴱ †tⴱ 1 ⫹ 2Aⴱ exp…⫺Eⴱ =T ⴱ †tⴱ ⫺ gAⴱ exp…⫺Eⴱ =T ⴱ †tⴱ …13†

where xⴱ ˆ x=L; yⴱ ˆ y=L; tⴱ ˆ 2k0 vt=L4 z; and T ⴱ ˆ T=Q: The superscripted asterisks denote dimensionless variables. The governing equation describing spinodal decomposition is obtained by inserting Eqs. (8), (11) and (12) into Eq. (7), and by calculating the functional derivative. Thus the governing equation for phase separation is given by

"

⫺NB …NA F Aⴱ ⫹ NB …1 ⫺ F Aⴱ †† ⫺ NB …NA ⫺ NB †…1 ⫺ F Aⴱ † NA …NA F Aⴱ ⫹ NB …1 ⫺ F Aⴱ †† ⫺ NA …NA ⫺ NB †F Aⴱ ⫹ …NA F Aⴱ ⫹ NB …1 ⫺ F Aⴱ ††2 …NA F Aⴱ ⫹ NB …1 ⫺ F Aⴱ ††2 # ! NA NB …NA F Aⴱ ⫹ NB …1 ⫺ F Aⴱ ††…1 ⫺ 2F Aⴱ † ⫺ NA NB …NA ⫺ NB †F Aⴱ …1 ⫺ F Aⴱ † 2F Aⴱ 2F Aⴱ 2F Aⴱ 2F Aⴱ ⫺2x ⫹ 2xⴱ 2xⴱ 2yⴱ 2yⴱ …NA F Aⴱ ⫹ NB …1 ⫺ F Aⴱ ††2 " # 2 ⴱ ! Dⴱ T ⴱ NB …1 ⫺ F Aⴱ † NA F Aⴱ NA NB F Aⴱ …1 ⫺ F Aⴱ † 2 FA 22 F Aⴱ ⫹ ⫺ 2x ⫹ ⫹ …NA ⫹ NB † NA F Aⴱ ⫹ NB …1 ⫺ F Aⴱ † NA F Aⴱ ⫹ NB …1 ⫺ F Aⴱ † NA F Aⴱ ⫹ NB …1 ⫺ F Aⴱ † 2xⴱ2 2yⴱ2 NA NB …NA F Aⴱ ⫹ NB …1 ⫺ F Aⴱ ††…1 ⫺ 2F Aⴱ † ⫺ NA NB …NA ⫺ NB †F Aⴱ …1 ⫺ F Aⴱ † …NA F Aⴱ ⫹ NB …1 ⫺ F Aⴱ ††2 ! 23 F Aⴱ 2F Aⴱ 23 F Aⴱ 2F Aⴱ 23 F Aⴱ 2F Aⴱ 23 F Aⴱ 2F Aⴱ  ⫹ ⴱ ⴱ2 ⫹ ⴱ2 ⴱ ⫹ 2xⴱ3 2xⴱ 2x 2y 2xⴱ 2x 2y 2yⴱ 2yⴱ3 2yⴱ ! NA NB F Aⴱ …1 ⫺ F Aⴱ † 24 F Aⴱ 24 F Aⴱ 24 F Aⴱ ⫹ 2 ⴱ2 ⴱ2 ⫹ ⫺ …NA F Aⴱ ⫹ NB …1 ⫺ F Aⴱ †† 2xⴱ4 2x 2y 2yⴱ4

⫺

(14)

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Table 1 Parameter values used in this study. D ⴱ is the dimensionless diffusion coefficient, A ⴱ is the dimensionless collision frequency factor, E ⴱ is the ⴱ dimensionless activation energy, Q is the theta temperature, F A;0 is the dimensionless initial average concentration, C is dimensionless entropy, and T1ⴱ and T2ⴱ are the lower and higher dimensionless temperature, respectively. Three different temperature gradient fields are examined: Case A (high temperature gradient), Case B (low temperature gradient), and Case C (constant temperature) Parameter

Value

Dⴱ Aⴱ Eⴱ ⴱ F A;0 C Q NA T1ⴱ T2ⴱ Case A Case B Case C

2.0 × 10 6 2.0 × 10 7 10.0 0.2 1.0 273.0 100 1.500 1.510 1.502 1.500

where the dimensionless diffusion coefficient is defined as D ⴱ ˆ kB QL 2 =2vk0 : To solve Eq. (14), the following sets of natural and zero-mass boundary conditions are used: 2F Aⴱ ˆ 0; 2xⴱ

at tⴱ ⬎ 0; and xⴱ ˆ 0 and xⴱ ˆ 1

2F Aⴱ ˆ 0; 2yⴱ

at tⴱ ⬎ 0; and yⴱ ˆ 0 and yⴱ ˆ 1 …15b†

…15a†

and 2 F Aⴱ 2xⴱ3 3

ⴱ

2 F Aⴱ 2xⴱ 2yⴱ2 3

⫹

ˆ 0;

ⴱ

…16a† ⴱ

at t ⬎ 0; and x ˆ 0 and x ˆ 1 23 F Aⴱ 23 F Aⴱ ⫹ ˆ 0; 2yⴱ3 2yⴱ 2xⴱ2

…16b†

at tⴱ ⬎ 0; and yⴱ ˆ 0 and yⴱ ˆ 1 In addition, randomly generated initial conditions are used to reflect infinitesimal thermal concentration fluctuations present initially in the homogeneous phase mixture, as follows: ⴱ F Aⴱ …tⴱ ˆ 0† ˆ F A;0 ⫹ dF Aⴱ …tⴱ ˆ 0†

dence of T ⴱ, x and NB is given in Eqs. (1), (9) and (13), respectively. The governing equation describing the polymerization process given in Eq. (13) is the dimensionless algebraic equation and it can be solved analytically. However, the governing equation for the phase separation process, Eq. (14), has to be solved numerically for F Aⴱ …xⴱ ; yⴱ ; tⴱ † with the boundary condition given in Eqs. (15) and (16), and the initial condition given in Eq. (17). The numerical computation is carried in the 2D geometry …0 ⬍ xⴱ ⬍ 1; 0 ⬍ yⴱ ⬍ 1† with 21 × 21 nodes. The Galerkin finite element method with Hermitian bicubic basis functions is used for spatial discretization, and the Euler predictor–corrector method with the time step controller is used for time integration [37,38]. The dependent variables are F Aⴱ and NB, and the independent variables are x ⴱ, y ⴱ and t ⴱ. The parameters are the dimensionless diffusion coefficient D ⴱ, the dimensionless collision frequency factor A ⴱ, the dimensionless activation energy E ⴱ, the dimenⴱ ; the dimensionless initial average concentration F A;0 ⴱ ⴱ ⴱ ⴱ sionless temperature T1 …y ˆ 0† and T2 …y ˆ 1†; and the Flory–Huggins interaction parameter, x . The parameter values used in this study are listed in Table 1. Three representative cases corresponding to different magnitudes of the temperature gradient are studied, i.e. the strong and weak temperature gradient, and constant temperature.

3. Morphological characterization method Tadmor and Gogos [39] found that the textures would be fully characterized by measuring two values: the scale of segregation (s), and the intensity of segregation (I) [40]. A schematic representation of the scale of segregation and the intensity of segregation is shown in Fig. 1. It should be noted that a perfect compositional uniformity would be obtained by either reducing the scale of segregation to the scale of the ultimate particle or by reducing the intensity of segregation to zero. The scale of segregation is calculated by a process known as ‘dipole (needle) throwing’, dropping a dipole of length r on the textures and observing the frequency with which both ends of the dipole fall on the same phase. However, to avoid doing this tedious process, the coefficient of correlation is defined as [39]

…17†

N2 X

The linearization approximation and the equipartition theorem are employed for the infinitesimal concentration derivations dF Aⴱ ; and details are given in Refs. [1,2]. In Eq. (14), the dimensionless temperature, T ⴱ, the Flory–Huggins interaction parameter, x , and the degree of polymerization of component B, NB, are variables along the spatial direction y ⴱ under the temperature gradient. The temperature depen-

iˆ1

R…r† ˆ

ⴱ ⴱ …F 0A;i ⫺ F A;0 †…F 00A;i ⫺ F A;0 †

N 2 S2

…18†

where F 0A;i and F 00A;i are concentrations at two points at a distance of r from each other, N2 is the total number of couples of concentrations taken, and S is the variance which is calculated from the concentrations at all points,

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209

Fig. 1. Schematic representation of the intensity segregation and the scale of segregation. A perfect compositional uniformity is obtained by reducing either the scale of segregation or the intensity of segregation to zero. (Note that this figure has been reproduced from Ref. [36]).

where N1 is the total number of concentration taken. For a completely segregated state, I ˆ 1 and uniformly distributed state, I ˆ 0 since S2 ˆ 0:

and is given by 2N X2

S2 ˆ

ⴱ ⴱ 2 …F A;i ⫺ F A;0 †

iˆ1

2N2 ⫺ 1

…19†

Then, the scale of segregation s is defined as the integral of the coefficient of correlation R…r† over values of r from zero to 6 as Z6 …20† s ˆ R…r† dr 0

where 6 represents the dipole length at which R…6† ˆ 0: Note here that for a perfect correlation between the two phases, R…r† ˆ 1; and for no correlation between the two phases, R…r† ˆ 0: The intensity of segregation I is defined as the ratio of the measured variance to the variance of a completely segregated system [39]. Thus it is written as Iˆ

S2 s 02

…21†

where S 2 is the measured variance of the various points in the sample defined as S2 ˆ

N1 X 1 ⴱ 2 …F ⴱ ⫺ F A;0 † N1 ⫺ 1 iˆ1 A;i

…22†

and s 02 is the variance for completely unmixed state defined as ⴱ ⴱ s 02 ˆ F A;0 …1 ⫺ F A;0 †

…23†

4. Results and discussion 4.1. Typical phase-separated structures and patterns of PIPS under a temperature gradient Fig. 2 shows plots of 1/x versus F Aⴱ for Case A at the lowest temperature side …T 1ⴱ ˆ 1:50† and the highest temperature side …T2ⴱ ˆ 1:51†; at the dimensionless times of tⴱ ˆ 2:039 × 10⫺5 and tⴱ ˆ 2:091 × 10⫺5 : The long and short dashed lines represent the binodal and spinodal lines, respectively, at T1ⴱ, and the solid and dotted lines denote the binodal and spinodal lines, respectively, at T2ⴱ. The blank and filled circles represent two curing temperatures in the extreme, T1ⴱ and T2ⴱ, respectively. As polymerization proceeds, the molecular weight of component B, NB, increases, and the phase diagram moves up gradually, as indicated by the upward pointing arrow. Since NA ˆ 100 and NB ˆ 1 at t ⴱ ˆ 0, independently on the position in the system, the lower temperature side is located closer to the initial binodal line. However, as polymerization proceeds, NB increases faster in the high-temperature region than the low-temperature region because the dimensionless reaction rate constant K ⴱ is proportional to temperature following the Arrhenius equation shown in Eq. (4). As a result, for Case A, the curing point at the higher temperature side is thrust into

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Fig. 2. Phase diagrams of 1/x versus F Aⴱ , based on the Flory–Huggins equation, Eq. (8), for Case A, where x is the Flory–Huggins interaction parameter and F Aⴱ is dimensionless concentration of polymer A. The long and short dashed lines represent the binodal and spinodal lines, respectively, at the lowest temperature (T1ⴱ). The solid and dotted lines denote the binodal and spinodal lines, respectively, at the highest temperature (T2ⴱ). The phase diagram shifts toward higher temperature and concentration as time proceeds, and the representative times are t ⴱ ˆ 2:039 × 10⫺5 and tⴱ ˆ 2:091 × 10⫺5 ; in this figure. The blank and filled circles represent the two representative curing temperatures, T1ⴱ and T2ⴱ, respectively.

the unstable region earlier and phase separation occurs first in the high-temperature region. Fig. 3 shows the dimensionless concentration spatial profile, F Aⴱ …xⴱ ; yⴱ †; (first column) and patterns (second column) formed during phase separation for Case A in Table 1 at the following representative dimensionless times: (a) tⴱ ˆ 2:531 × 10⫺5 ; (b) tⴱ ˆ 2:570 × 10⫺5 ; and (c) tⴱ ˆ 2:598 × 10⫺5 : Darker regions represent polymer A-rich regions while brighter regions denote polymer Brich regions (see color scale on lower right). As we assumed previously, the temperature gradient exists along the y ⴱ-axis and the temperature at y ⴱ ˆ 0 …T1ⴱ ˆ 1:50† is lower than the temperature at y ⴱ ˆ 1 …T2ⴱ ˆ 1:51†; in this case. Fig. 3 shows that phase separation occurs earlier and more drastically in the high-temperature region. The location of the front, yfⴱ, indicated by the arrows and defined as the boundary between the phase-separated region and the region in which no phase separation occurs, constantly propagates toward the low-temperature region, leaving the high-temperature region behind, as phase separation proceeds. The front lines also indicate the interface between the unstable region (behind yfⴱ) and the stable region (ahead of yfⴱ). Due to the different values of the dimensionless reaction rate constants, K ⴱ, the quench depth, defined as the absolute difference between the critical and curing temperature, 兩Tc ⫺ T兩; increases more rapidly in the high-temperature region. Since the amount of phase separation is proportional to the quench depth, the amount of phase separation is large, and the droplet-type morphology forms and evolves, in the high temperature region. Pattern (c) indicates that phase separation already has reaches the intermediate stage of phase separation in the high-temperature

region. On the other hand, in the low-temperature region, no phase separation takes place until the front line passes, indicating the curing point is yet in the stable region. For the strong temperature gradient field, the propagation of the front line is slow. As a result, a lamella-type of phase-separated structure parallel to the x ⴱ-axis forms in the lowtemperature region. The droplet-type of phase-separated structure in Fig. 3 indicates that the curing point at the higher temperature side remains in the off-critical region during the polymerization and phase separation process. As phase separation and polymerization proceed, the lamella-type structures slowly break up during the phase separation process to form the droplet-type structure (see Fig. 5 for constant temperature). Fig. 4 represents the dimensionless concentration spatial profiles F Aⴱ …xⴱ ; yⴱ † (first column) and patterns (second column) formed during the phase separation phenomena for Case B, at the following dimensionless times: (a) tⴱ ˆ 2:590 × 10⫺5 ; (b) tⴱ ˆ 2:610 × 10⫺5 ; and (c) tⴱ ˆ 2:615 × 10⫺5 : In this case, the temperature at y ⴱ ˆ 0 …T1ⴱ ˆ 1:500† is lower than the temperature at y ⴱ ˆ 1 …T2ⴱ ˆ 1:502†: Since the temperature gradient for this case is weaker than that for Case A, the front line shown in Fig. 3 propagates toward the low-temperature region much faster than Case A. As a result, the initial one-phase mixture phase-separates and the initial concentration evolves almost simultaneously everywhere. However, the weak temperature gradient also induces the spatially different morphologies, that is the droplet-type phase-separated structures in the high-temperature region and the highly interconnected cylinder-type morphology in the lower-temperature region. This indicates that the initial curing point is located in the critical region at

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211

Fig. 3. Dimensionless concentration spatial profiles F Aⴱ …xⴱ ; yⴱ † (first column) and patterns (second column) formed during the phase separation phenomena under a temperature gradient for Case A.

given concentration and temperature. As polymerization proceeds, as mentioned earlier, the phase diagram shown in Fig. 2 shifts toward higher temperature and concentration, and the curing point moves off the critical region into the off-critical region. In Fig. 4, the droplet-type structure in the high-temperature region indicates the phase-separated structure, corresponding to the intermediate stage of phase

separation, while the interconnected cylinder-type structure represents the structure of the early stage of SD. Consequently, different temporal ranges of phase separation induce a space-dependent structure, in this case. The interconnected cylinder-type structure also breaks up (see pattern (c)) to form the droplet-type morphology. The dimensionless concentration spatial profiles and

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Fig. 4. Dimensionless concentration spatial profiles F Aⴱ …xⴱ ; yⴱ † (first column) and patterns (second column) formed during the phase separation phenomena under a temperature gradient for Case B.

patterns for constant temperature (Case C in Table 1) are presented in Fig. 5, at the following dimensionless times: (a) tⴱ ˆ 2:610 × 10⫺5 ; (b) tⴱ ˆ 2:617 × 10⫺5 ; and (c) tⴱ ˆ 2:640 × 10⫺5 : Compared to Case A (Fig. 3) and B (Fig. 4), the initial mixture phase-separates and evolves independently on the position during the polymerization and phase

separation process. At the early stage of phase separation, the random initial condition develops into an interconnected cylinder-type structure. However, the early phase-separated structure similar to the interconnected cylinder-type develops into the droplet-type structure as phase separation proceeds. In general, both for Cases A and B, the spatially

J. Oh, A.D. Rey / Computational and Theoretical Polymer Science 11 (2001) 205–217

213

Fig. 5. Dimensionless concentration spatial profiles F Aⴱ …xⴱ ; yⴱ † (first column) and patterns (second column) formed during the phase separation phenomena for Case C (constant temperature).

graded composite structures of the lamella- (interconnected cylinder) and droplet-type morphology can be obtained by the PIPS method under a temperature gradient. It is known that a dimensionless induction time tiⴱ exists in the PIPS process and that significant phase separation only occurs at tⴱ ⬎ tiⴱ [1,2]. The dimensionless induction time tiⴱ

can be represented by the sum of the dimensionless polymerization lag time tlpⴱ and the dimensionless phase separaⴱ ⫹ tlsⴱ : The dimensionless tion lag time tlsⴱ as tiⴱ ˆ tlp ⴱ polymerization lag time tlp represents the time it takes for the spinodal line to cross the initial curing point and the dimensionless phase separation lag time tlsⴱ is the time

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Fig. 6. Plots of the intensity of segregation I as a function of the position along the temperature gradient y ⴱ for Case A (X), Case B (O), and Case C (B) at the following characteristic times: (a) tⴱ ⫺ tiⴱ ˆ 0:019; (b) tⴱ ⫺ tiⴱ ˆ 0:026; and (c) tⴱ ⫺ tiⴱ ˆ 0:040:

required for the system to begin phase separation once it has been placed in the unstable region [1,2]. The dimensionless polymerization lag time tlpⴱ , the dimensionless phase separation lag time tlsⴱ and the dimensionless induction time tiⴱ for Cases A–C are obtained from the numerical results and are listed in Table 2. In this study, simulations are performed only for the early stage of phase separation because of numerical limitations. When the interconnected-cylinder type of structure breaks up in pattern (c) in Figs. 3 and 4, rapid variation of the

Table 2 Results for dynamical study of polymerization-induced phase separation where tlpⴱ , tlsⴱ and tiⴱ denote the dimensionless polymerization lag time, the dimensionless phase separation lag time, and the dimensionless induction time, respectively Case

ⴱ tlp × 105

tlsⴱ × 105

tiⴱ × 105

A B C

2.039 2.080 2.090

0.489 0.505 0.516

2.528 2.585 2.606

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integrand occurs at the pinching points. The time step Dti for stable integration must be less than the smallest time scale of the problem. The smallest time scale becomes infinitesimally small with rapid variation of the relaxation velocities 2F Aⴱ =2tⴱ at the pinching points, thus limiting the ability of the computational scheme to capture fast and small temporal scale procedures. However, the dominant morphological inhomogeneities depend on the location along the temperature gradient can be found in the very early stage of phase separation. Thus, these computational limitations do not preclude visualization of the main morphological transformations in the PIPS process. Table 2 shows that all the dimensionless kinetic values, tlpⴱ , tlsⴱ and tiⴱ, decrease as the magnitude of the temperature gradient increases. The dimensionless polymerization lag time tlpⴱ decreases with increasing K ⴱ, and the dimensionless phase separation lag time tlsⴱ is inversely proportional to both the dimensionless diffusion coefficient D ⴱ and the dimensionless rate constant K ⴱ. These results provide guidelines for the dependence of the time scales of the process on polymerization and diffusion rates. 4.2. Characterizations of the non-uniform phase-separated structures Fig. 6 shows the intensity of segregation I as a function of y ⴱ for Case A (X), Case B (O) and Case C (B) at the following representative characteristic times: (a) tⴱ ⫺ tiⴱ ˆ 0:019; (b) tⴱ ⫺ tiⴱ ˆ 0:026; and (c) tⴱ ⫺ tiⴱ ˆ 0:040: The intensity of segregation I has been defined in Eq. (21). In Fig. 6, the intensity of segregation I is calculated individually for the particular region along the temperature gradient. In addition, to eliminate the differences of the dimensionless induction time tiⴱ due to the temperature gradient for each case, tⴱ ⫺ tiⴱ is used for the characteristic time. Note that Cases A and B are under the strong and weak temperature gradient fields, respectively, while Case C is maintained under constant temperature. For all three cases, the intensity of segregation I increases with time. However, the time variations of I are significantly different for each case. For Case A, near the low-temperature side (between y ⴱ ˆ 0 and the front yfⴱ), I ˆ 0, since no phase-separation occurs. On the other hand, near the high-temperature side (y ⴱ ˆ 1), the value of I is considerably high because phase separation occurs significantly in the high-temperature region during the early stage of phase separation. It is also observed that the location of the front yfⴱ propagates slowly toward the lowtemperature side y ⴱ ˆ 0. In Fig. 6c, phase separation slows down in the high-temperature region, indicating the intermediate stage while phase separation initiates in the vicinity of the front. Consequently, the highly position-dependent microstructure forms in the strong temperature gradient field. Plots of the intensity of segregation for Case B clearly shows that the non-uniform structure forms during the phase separation process in the weak temperature gradient field. As expected, the value of I in the high-temperature side is

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much higher than that in the low-temperature side. The slopes of plots for Case B are considerably steep compared to those for Case C. Fig. 6 also shows that the phaseseparated morphology evolves uniformly for Case C, and almost same values of I are observed everywhere. Plots of the scale of segregation s as a function of y ⴱ are shown in Fig. 7 for Case A (X), Case B (O), and Case C (B) at the following characteristic times: (a) tⴱ ⫺ tiⴱ ˆ 0:019; (b) tⴱ ⫺ tiⴱ ˆ 0:026; and (c) tⴱ ⫺ tiⴱ ˆ 0:040: Recall that s is proportional to the average dispersed particle size, but it is inversely proportional to the number density of the particles as shown in Fig. 1. The scale of segregation s decreases as phase separation proceeds for all three cases. Since the random initial conditions develop into a phase-separated structure with a large particle size, the scale of segregation s is large. As phase separation proceeds, the average dispersed particle size decreases, thus s decreases significantly during the early phase separation stage. Fig. 7 also shows that the highly inhomogeneous microstructure forms during the phase separation process for Case A. In the hightemperature region, the value of s is much lower than that in the low-temperature region. This indicates that the small droplet-type morphology forms as a result of the intermediate stage of phase separation in the high-temperature region, while the large clustered lamella-type structure is formed in the low-temperature region. In the weak temperature gradient field, as shown in plots for Case B, the phase-separated particle size is different along the gradient. Since the value of s in the high-temperature region is slightly lower than that in the low-temperature region, it is expected that the small droplet-type morphology forms in the high-temperature region and the interconnected phase-separated structure develops in the low-temperature region. However, for Case C, the phase-separated morphology evolves uniformly everywhere, and the space-independent droplet-type morphology forms. In conclusion, the space-dependent morphologies, showing different particle size and shape, are obtained by the PIPS process under the temperature gradient. Since the mechanical and optical properties of multi-component composite material depend on the phase-separated morphology, it is expected that the presented numerical results can be applied to fabricate multi-component polymers with graded properties, and, in particular, FGMs with different skin/core properties. As general guidelines, it can be stated that the strong temperature gradient field should be applied for broad distributions of particle size and the highly inhomogeneous microstructures. However, the positiondependent structures with considerably inhomogeneous particle size and shape can be also produced in the weak temperature gradient field.

5. Conclusions This paper presents the computational modeling of

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Fig. 7. Plots of the scale of segregation s versus the position along the temperature gradient y ⴱ for Case A (X), Case B (O), and Case C (B) at the following characteristic times: (a) tⴱ ⫺ tiⴱ ˆ 0:019; (b) tⴱ ⫺ tiⴱ ˆ 0:026; and (c) tⴱ ⫺ tiⴱ ˆ 0:040:

polymerization-induced phase separation (PIPS) process under a temperature gradient. The time evolution of morphology under a temperature gradient describes how the initial homogeneous mixture evolves and phase-separates during the phase separation and polymerization process to eventually form a spatially inhomogeneous microstructure. The droplet type morphology (in the hightemperature region) and the lamella type morphology (in the low-temperature region) are obtained in a strong temperature gradient field. The evolution of phase separation follows a propagation front process. The location of the front propagates toward the low-temperature side, leaving the high-temperature region behind. In the region far behind

the front (high-temperature side) phase separation occurs much earlier and more significantly while no phase separation takes place ahead of the front line (low-temperature region). On the other hand, the droplet and interconnected cylinder type structure is found in the weak temperature gradient field. The kinetic measures of the PIPS process under a temperature gradient, such as the dimensionless polymerization lag time, the dimensionless phase separation lag time, and the dimensionless induction time, decrease as the highest temperature in the system increases, due to the increasing reaction rate constant. Lastly, the time variations of the intensity and scale of

J. Oh, A.D. Rey / Computational and Theoretical Polymer Science 11 (2001) 205–217

segregation as a function of the location along the temperature gradient, also indicate that the phase-separated structure evolves non-uniformly along the gradient. Under a temperature gradient field, the intensity and scale of segregation change more significantly in the high-temperature side than in the low-temperature side. However, the variations of the intensity and scale of segregation are uniformed at constant temperature. In conclusion, to design functionally graded polymer materials (FGPMs) with composite lamella-droplet morphologies, these simulations show that the temperature gradient should be greater than a critical value, so that front propagation behavior sets in. In this case, a droplet morphology forms in the hotter region while a lamella morphology with unit normal parallel to the temperature gradient forms in the cooler region. These results provide useful guidelines in the manufacturing of FGPMs by the PIPS method.

[13] [14] [15] [16] [17] [18] [19] [20] [21]

[22] [23] [24] [25] [26] [27] [28] [29]

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

Chan PK, Rey AD. Macromolecules 1996;29:8934. Chan PK, Rey AD. Macromolecules 1997;30:2135. Doane JW, Vaz NA, Wu B-G, Zumer S. Appl Phys Lett 1986;48:269. Lin B, Taylor PL. Polymer 1995;37(22):5099. Lin JC, Taylor PL. Phys Rev E 1994;49(3):2476. Glotzer SC, Di Marzio EA, Muthukumar M. Phys Rev Lett 1995;74(11):2034. Chen W, Tao Dong XL, Jiang M. Macromol Chem Phys 1998;199:327. Oyanguren PA, Riccardi CC, Williams RJJ, Mondragon I. J Polym Sci Part B 1998;36:1349. Inoue T. Prog Polym Sci 1995;20:119. Kim BS, Chiba T, Inoue T. Polymer 1995;36(1):43. Chen W, Kobayashi S, Inoue T, Ohnaga T, Ougizawa T. Polymer 1994;35(18):4015. Sperling LH. Polymeric multicomponent materials: an introduction. New York: Wiley, 1997.

[30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40]

217

Corberi F. Phys Rev Lett 1998;81(18):3852. Wagner AJ, Yeomans JM. Phys Rev E 1999;59(4):4366. Tirrell M. Fluid Phase Equilib 1986;30:367. Wirtz D, Fuller GG. Phys Rev Lett 1993;71:2236. Venugopal S, Krause S. Macromolecules 1992;25:4626. Xie X, Chen Y, Zhang Z, Tanioka A, Matsuoka M, Takemura K. Macromolecules 1999;32:4424. Okinaka J, Tran-Cong Q. Physica D 1995;84:23. Tran-Cong Q, Okinaka J. Polym Engng Sci 1999;39(2):365. Suresh S, Mortensen A. Fundamentals of functionally graded materials: processing and thermomechanical behaviour of graded metals and metal–ceramic composites. Cambridge, UK: Cambridge University Press, 1998. Markworth AJ, Ramesh KS, Parks Jr, W P. J Mater Sci 1995;30:2183. Tsai F, Torkelson JM. Macromolecules 1990;23:775. Caneba GT, Soong DS. Macromolecules 1985;18:2538. Caneba GT, Soong DS. Macromolecules 1985;18:2545. Platten JK, Chavepeyer G. Physica A 1995;213:110. Chavepeyer G, Platten JK, Salajan M. J Non-Equilib Thermodynam 1996;21(2):122. Kumaki J, Hashimoto T, Granick S. Phys Rev Lett 1996;77(10):1990. Barton JM. In: Dusek K, editor. Advances in polymer science, vol. 2. New York: Springer, 1985. Cahn JW, Hilliard JE. J Chem Phys 1958;28:258. Novick-Cohen A, Segel LA. Physica D 1984;10:277. Falk F. J Non-Equilib Thermodynam 1992;17:53. Cowie JMG. Polymers: chemistry and physics of modern materials. 2nd ed. New York: Chapman and Hall, 1991. Flory PJ. Principles of polymer chemistry. Ithaca, NY: Cornell University Press, 1953. Kurata M. Thermodynamics of polymer solutions. New York: Harwood Academic, 1982. Rouse PE. J Chem Phys 1953;21:1272. Fletcher CAJ. Computational Galerkin methods. New York: Springer, 1984. Finlayson BA. Nonlinear analysis in chemical engineering. New York: McGraw-Hill, 1980. Tadmor Z, Gogos CG. Principles of polymer processing. New York: Wiley-Interscience, 1979 (chap. 7). Danckwerts PV. Appl Sci Res 1952;A3:279.