numerical simulation of phase separation of immiscible polymer

8/7/2019 Numerical simulation of phase separation of immiscible polymer

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Numerical simulation of phase separation of immiscible polymer

blends on a heterogeneously functionalized substrate

Yingrui Shang, David Kazmer, Ming Wei, Joey Mead, and Carol Barry

Department of Plastics Engineering, University of Massachusetts at Lowell

Abstract

The spinodal phase decomposition of an immiscible binary polymer blends system is investigated

with numerical models in 2D and 3D. The effect of the elastic energy is included. The mechanism of

the evolution of the phase separation is studied and the characteristic length, R(t ) is shown to be

proportional to t 1/3 . In the case when the phase separation is directed by a heterogeneously

functionalized substrate, the increase of the characteristic length is divided into two stages by a

critical time. The R(t )~ t 1/3 diagram can be fitted with a straight line in both the first and second

stages. The slope of the fitting line significantly decreases after the critical time. The compatibility of

the resulting pattern to the substrate pattern is also measured by a factor C S. It is observed that there

is also a critical time in the evolution of the compatibility for the cases with and without elastic

energy. The critical time of C S is identical with the respective critical time of R(t ). The lateral and

vertical composition profiles functionalized substrate is observed with the 3D model. The difference

mechanism of the cases with and without elastic energy is discussed.


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Introduction

Polymer based structures with nano-sized features have various applications including lithographic

processes, biosensors, and semiconductor devices [1, 2]. The self-assembly of polymeric materials

with the method of heterogeneously patterned substrates are widely used for the manufacture of

nano-sized features [3, 4].

When a binary polymer blend is quenched into the immiscible gap on the phase diagram by

reducing the temperature, the spinodal decomposition can be self-induced from a system with a small

composition fluctuation. Generally, the immiscible polymers will separate into A- and B-rich

domains and coarsen with time. In our study, the system is diffusion controlled and there is no

predominant dynamic mass flow. Rather, the composition profile is determined by the free energy

minimization in the domain. This process can be described by the model first proposed by Cahn

[5-7].

According to the thermodynamic theory, the free energy of the system consists of two

components when no surface attraction exists. In our research, the local free energy is a function of

the composition and the temperature. The free energy penalty for the composition gradient is a

function of the composition, the temperature, and the gradient of the gradient. The elastic energy of

the domain can be added to the total free energy in a numerical simulation. To locally control the

pattern after phase separation, the substrate surface has functionalized patterns with different

attraction forces to the two types of polymers. The surface energy is specified as a function of the

coordinates and added to the total free energy to represent the contribution by the functionalized

substrate.


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In this article 2D and 3D numerical models are introduced to investigate template-directed

polymer self-assembly with a binary polymer blend.

Fundamentals

According to Cahn[5, 6], the mass flux, J , in the mixture can be denoted as

where M is mobility and is the chemical potential of the polymer blends. To find the chemical

potential, the expression of the total free energy has to be determined. According to Cahn and

Hilliard [8], the free energy, F, of a binary system can be written as:

where C is the mole fraction of one polymer component, f is the local free-energy density of

homogeneous material, f e is the elastic energy density, and is the gradient energy coefficient. Thus

the item (C )2 is the additional free energy density if the material is in a composition gradient.

Consecutively, the chemical potential is,

The mobility of the diffusion model is non-negative and evaluated as a positive constant in this

study. Substituting the chemical potential expression to the equation for the mass flux, the evolution

of the composition can be written as the function of local composition:

The free energy variation on the heterogeneously functionalized substrate is simulated by a free

energy term, f s, which is a function of the composition and the coordinates, r , as well. The surface

free energy is added to the total free energy on the surface of the substrate.

J M = 1

2( ) { ( ) }eV

F C f f C dV = + + 2

22ef f

C C C

= +

3

2 2ef dC f M C dt C C

= +

4


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The term represents the thermal noise in the phase separation may be added to the free energy, as

in indicated in the work of Shou and Chakrabarti [9]. But in this study the thermal fluctuations are

neglected since they are negligible for quenches far away from the critical points and/or spoinodal

lines [10, 11].

The Flory-Huggins type of free energy is used as the bulk free energy density [12]

where C i is the volume fraction of component i, mi is the degree of polymerization of component i, T

is the temperature in K, R is the ideal gas constant, AB is the Flory-Huggins interaction parameters

between two components, which is dependent on temperature, and vsite is the molar volume of the

reference site in the Flory-Huggins lattice model.

The elasticity is assumed isotropic in the domain. According to the Vegards law [13], the

stress-free strain is isotropic and depends linearly on the composition:

where eij0 is the stress-free strain, c0 is the average composition of the domain, ij is the Kronecker

delta function, and is the compositional expansion coefficient which is expected to be independent

of the composition and the composition gradient [14].

According to the linear elasticity, the stress, ij, is linear with the change of the strain by Hooks

law:

where cijkl represents the isothermal elastic tensor, which is independent of position and composition.

The elastic energy then can be expressed as follows, with no external anisotropic elastic applied,

( )2( ) { ( ) } ,e SSV

F C f f C dV f C dS = + + + r 5

( ln ln )A BA B AB A Bsite A B

C C RT f C C C C

v m m = + + 6

( )0 0ij ije C C = 7

( )0ij ijkl kl klc e e = 8


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The total strain can be evaluated by the local displacement, u [15].

The displacement of the reference lattice is then solved by the elastic equilibrium. Given the fast

relaxation time compared to the rate of morphology evolution, it can be assumed that the system is in

elastic equilibrium [16].

Numerical Method

The Cahn-Hilliard equation is known for its difficulty to solve due to its non-linearity and the

bihamonic term. The cosine transform method is applied to the spatial discretization:

where C and Sf f C C

+ represent the Fourier transform of the composition and the summation of

the first derivation of bulk free energy density and surface free energy both with the respect of the

composition. is the approximation of discrete Laplacian operator in the transform space [17].

The vector k denotes the discretisized spatial element position in all dimensions. Numerically,

k i=n i/N i, where ni is the element in the ith dimension and N i represents the number of elements in the

ith dimension. x is the spatial step in the numerical modeling.

By this means, the partial differential equation is transferred into an ordinary differential equation

( )( )0 012e ijkl ij ij kl kl

f c e e e e= 9

12

jiij

j i

uuer r

= +

10

0ij

jr

= 11

ef dC f M C

dt C C = +

12

2)(

2)2cos(2)(

x

k ii

i

= k 13


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in the discrete cosine space. A semi-implicit method is used to trade off the stability, computing time

and accuracy [18, 19]. To remove the shortcoming with the small time-step size associated with the

explicit Euler scheme, the linear fourth-order operators can be treated implicitly and the nonlinear

terms can be treated explicitly. The resulting first-order semi-implicit Fourier scheme is:

A second-order Adams-Bashforth method [4] was also used for the explicit treatment of the

nonlinear term (the sum of the bulk free energy density and template surface free energy

differentials). The equation can be discretisized into the form,

When the initial condition and the time length are selected, the composition profile of the domain

for the second time step can be calculated with Equation 14. After that, the third and following time

steps can be calculated with Equation 15.

For simplicity and generality, the equations are dimensionalized. The scaling dimensions and

time are chosen as follows,

The other parameters are treated accordingly. The factors for nondimensionalization are listed in

Table 1 .

Table 1 Characteristic Terms for Nondimentionalization

+

=+ +

C

C f

C C f

t M C C t M n

en

nn )()()1( 1 14

+

+

+=+

+

C

C f

C C f

C

C f

C C f

t M C C C t M n

enn

en

nnn )()()()(224)23(11

112 15

2 sitevLRT

=

2C site

C

L vt M RT

=

16


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where s0(r ) and s1(r ) are surface attraction parameters that represent the functionalization of the

substrate, and C ref is the reference composition. For non-dimensionalization, C ref and s0(r ) are chosen

to be 0. For example, as can be seen in Figure 1, the functionalized substrate energy factor s1(r ) is

alternating across the x direction of the substrate in an effort to develop a corresponding alternating

strip patterns in the self-assembling polymer near the substrate.

Figure 1 The spatial variation of surface free energy factor

Results and Discussion

To ensure the phase separation is induced spontaneously, the second derivative of the local free

energy with respect to the composition should be negative [24].

To meet this requirement, the shape of the free energy has to be tuned by temperature to quench

the mixture into a miscibility gap. A process of phase separation initiated from a random composition

distribution is shown in Figure 2. The effect of the functionalized substrate attraction is excluded. It

can be seen that in the case without the consideration of the elastic energy, the patterns evolve rapidly

in the early stage of the phase separation. The gradient in the interface of two phases increases very

quickly.

Figure 2 Evolution of phase separation in a 128128128 3D domain without patterned substrate

2

2 0f

C

numerical simulation of phase separation of immiscible polymer

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