article a modi ed phase-field model for polymer crystal growth

CHINESE JOURNAL OF CHEMICAL PHYSICS VOLUME 30, NUMBER 5 OCTOBER 27, 2017

ARTICLE

A Modified Phase-Field Model for Polymer Crystal Growth

Bin-xin Yang∗, Chen-hui Zhang, Fang Wang

Department of Mathematics, Taiyuan University of Science and Technology, Taiyuan 030024, China

(Dated: Received on March 23, 2017; Accepted on August 17, 2017)

The irrationality of existing phase field model is analyzed and a modified phase-field modelis proposed for polymer crystal growth, in which the parameters are obtained from realmaterials and very simple to use, and most importantly, no paradoxical parameters appearedin the model. Moreover, it can simulate different microstructure patterns owing to the use ofa new different free energy function for the simulation of morphologies of polymer. The newfree energy function considers both the cases of T<Tm and T≥Tm, which is more reasonablethan that in published literatures that all ignored the T≥Tm case. In order to show thevalidity of the modified model, the finite difference method is used to solve the model anddifferent crystallization morphologies during the solidification process of isotactic polystyreneare obtained under different conditions. Numerical results show that the growth rate of theinitial secondary arms is obviously increased as the anisotropy strength increases. But theanisotropy strength seems to have no apparent effect on the global growth rate. The wholegrowth process of the dendrite depends mainly upon the latent heat and the latent heat hasa direct effect on the tip radius and tip velocity of side branches.

Key words: Phase-field method, Polymer crystallization, Numerical simulation, Dendritic,Lamellar

I. INTRODUCTION

Microstructure evolution during solidification andcrystallization has attracted great research interests be-cause it has a significant effect on macro-scale proper-ties of materials [1, 2]. Since the different physical andchemical properties of polymers depend strongly on thecrystallinity, imperfection, and crystal morphology [3],it is very important to establish a powerful numericalmodel which can simulate the growth of polymer grainand predict morphologies of semi-crystalline polymers.Under ideal conditions, the concept of an ideal crys-tal of polymers requires infinite molecular mass, com-pletely regular constitution of the macromolecules, com-pletely regular configuration of the units in the macro-molecules, and completely regular conformation of thechains [4]. But it is almost impossible to find such anideal crystal.

Being the most common morphology, the spherulitecan be formed from the polymeric supercooling meltor concentrated solution. The phase field method hasbeen widely used in the simulation of the crystallizationof metal materials. Recently, the phase field theory hasalready demonstrated its ability to describe complexcrystal morphologies [5]. And different phase field mod-els have been proposed and applied to polymer crystal-

∗Author to whom correspondence should be addressed. E-mail:[email protected]

lization by modifying the phase-field equations of metalsolidification, such as the work of Granasy [6] and Kyu[7]. The models of Granasy et al. used several artifi-cially specified parameters which are not evaluated fromthe real material parameters. Kyu et al. obtained themodel parameters from those of the real materials andis very simple to use, but it has a severe scaling prob-lem caused by the fact that the heat diffusion is muchfaster than the solute diffusion. Wang et al. [8] pro-posed a modified phase-field model based on the Kyu’smodel, which also obtained the model parameters fromthose of the real materials. However some paradoxicaldefinitions of the parameters in both Kyu’s and Wang’smodels lead to the inaccuracy of their models. In thiswork, a modified phase field model is proposed, whichcan be viewed as a variant of Wang’s model. The modelparameters are obtained from those of the real materialsand very simple to use, and most importantly, no para-doxical parameters appear in the model. Moreover, itcan simulate different microstructure patterns owing tothe use of a different free energy functional form.

II. THE MODIFIED PHASE FIELD MODEL

A. Irrationality of existing phase field model

The basic idea of the phase field model is to use an or-der parameter to describe the phase transition of poly-mer. The range of the order parameter varies betweentwo specified values, which represent the amorphousstate and stable crystal state, respectively [8].

DOI:10.1063/1674-0068/30/cjcp1703050 538 c⃝2017 Chinese Physical Society

Chin. J. Chem. Phys., Vol. 30, No. 5 Phase-Field Model for Polymer Crystal Growth 539

According to the Ginzburg-Laudau theory, the tem-poral evolution of the crystal order parameter canbe modeled by the following non-conserved phase-fieldequation as

∂ψ (r, t)

∂t= −Γ

δF (ψ, T )

δψ (r, t)(1)

where ψ(r, t) represents the crystal order parameter attime t and position r. Γ is the mobility which is in-versely proportional to the viscosity of the melt. T isthe temperature. F (ψ, T ) is the total free energy of thecrystal ordering, defined in terms of a combination of alocal free energy density and anonlocal gradient term,which can be expressed as

F (ψ, T ) =

∫fcryst (ψ, T ) dV

=

∫[flocal (ψ, T ) + fgrad (ψ)] dV (2)

where flocal (ψ, T ) and fgrad (ψ) are the local free energydensity and the nonlocal free energy density, respec-tively. Both Xu et al. [9] and Wang et al. [8] adoptedthe local free energy density of Harrowell-Oxtoby [10]to account for the metastable states in polymer solidi-fication and spatio-temporal development of imperfectsemi-crystalline morphologies, which is described as

flocal (ψ, T ) =W

∫ ψ

0

φ (φ− ξ) (φ− ξ0)dφ

=W

(ξξ02ψ2 − ζ + ξ0

3ψ3 +

ψ4

4

)(3)

where ξ=ξ (T ) is the unstable energy barrier. ξ0 is thestable solidification potential, and can be regarded asthe degree of crystal perfection. It may be simply eval-uated as the ratio between melting temperature Tm andequilibrium melting point T 0

m of a polymer crystal [8].W is a dimensionless constant describing the height ofthe energy barrier for surface nucleation and it repre-sents the strength of the free energy density. It is worthpointing out that the coefficient of the third power termon the right side in Eq.(3) is nonzero, and hence thisfree energy should be applicable to the first order phasetransition including solidification [11]. The domain of ξshould be [0, ξ0] and that of ψ should be [0, ξ0]. How-ever, in both Xu’s and Wang’s studies, ξ is taken as

ξ =4ξ0ψ − 3ψ2

6ξ0 − 4ψ(4)

ψ =T 0m − TmT 0m − T

ξ0 (5)

Contradiction rises when we make a simple analysis.First, the range of ξ in Eq.(4) is [0, ξ0/2], while that

in Eq.(3) should be [0, ξ0]. Secondly, the range of ψ

FIG. 1 The change of ξ versus T .

in Eq.(5) is

[T 0m − TmT 0m − Tc

ξ0, ξ0

], while according to Eq.(3)

it should be [0, ξ0]. Thirdly, according to the range of

ψ in Eq.(5), the range of ξ is only a subinterval of [0,ξ0/2]. FIG. 1 shows the ξ-curve versus temperature Taccording to Eq.(4) and Eq.(5), in which T 0

m=242 ◦C,Tm=229 ◦ C, Tc=195 ◦C. It can be seen that when tem-perature T is less than Tm, the value of ξ is proper.However, a serious oscillation about ξ occurs and somevalues of ξ are negative when T is greater than Tm,which is obvious unreasonable. Moreover, the value ofξ increases exponentially when T is greater than 234 ◦C,which is not acceptable.

B. The modified phase field model

Since the ranges of ξ and ψ will affect the value ofthe local free energy density and in order to delete thecontradiction occurred in Refs. [8] and [9], a modifiedlocal free energy density is used in this work, which isgiven as

flocal (ψ, T ) =W

∫ ψ

0

φ(φ− ξ) (φ− ξ0) dφ

=W

∫ ψ

0

φ

(φ− ξ0

2+ ξ

)(φ− ξ0) dφ

=W

[(ξ204

− ξ0ξ

2

)ψ2 −

(ξ06

−

ξ − ξ03

)ψ3 +

ψ4

4

](6)

Where ξ is a function of ξ0 and ψ. As the value of ξ (or

the value of ξ) will change with the value of T , ξ (or ξ)should be constructed according to T . That what wediscuss will be divided into two parts, i.e. T<Tm andT≥Tm.

DOI:10.1063/1674-0068/30/cjcp1703050 c⃝2017 Chinese Physical Society

540 Chin. J. Chem. Phys., Vol. 30, No. 5 Bin-xin Yang et al.

1. T<Tm

When T<Tm, the local free energy density fora polymer melt generally satisfies the inequalitiesflocal (ξ, T )>0 and flocal (ξ0, T )<0, hence according to

“the zero point theorem”, there must exist a ψ∈(ξ, ξ0)such that flocal(ψ, T )=0. We can obtain that

ξ =3ξ20 − 6ξ0ψ + 3ψ2

6ξ0 − 4ψ(7)

by solving

W

[(ξ204

− ξ0ξ

2

)ψ2 −

(ξ06

− ξ − ξ03

)ψ3 +

ψ4

4

]=0(8)

Where

ξ =1

2ξ0 − ξ (9)

So we can get the same form as Wang et al. [9], i.e.

flocal (ψ, T ) =W

(ξξ02ψ2 − ξ + ξ0

3ψ3 +

ψ4

4

)(10)

Then, taking the first order derivative of flocal (ψ, T )with respect to ψ we have

f ′local (ψ, T ) = φ (φ− ξ) (φ− ξ0) (11)

This function shows two minima at ϕ=0, ϕ=ξ0 and a

maximum occurring at ϕ=ξ=1/2ξ0−ξ, respectively.At a given crystallization temperature T , a crystal

with an lamellar thickness l, (not the optimum lamellarthickness) is formed from a change in the free energy

∆flocal = 2Aσ −Al∆H

(1− T

Tm

)(12)

Where σ is the surface free energy per unit area of thefolded surface A. ∆H is the latent heat.

When ∆flocal=0, we can get

2σ

l−∆H

(1− T

Tm

)= 0 (13)

According to the Hoffman and Weeks [12, 13] relation-ship, the melting temperature Tm of the crystal solid-ified at a given crystallization temperature T can berelated to the lamellar thickness lz [7],

2σ

lz−∆H

(1− Tm

T 0m

)= 0 (14)

So, the stability order parameter could be determinedfrom Eq.(13) and Eq.(14) as described below

ψ =l

lz=T 0m − TmT 0m − T

ξ0 (15)

Where ψ is defined as ψ=l/lz. Finally, we find that the

value of ψ=l/lz is keeping in step with Wang et al. [8].

Therefore, the value of ξ can be given as

ξ =3ξ20 − 6ξ0ψ + 3ψ2

6ξ0 − 4ψ(16)

2. T≥Tm

When T≥Tm we will construct the value of ξ, which ismodeled on the structure (proposed by Ryo Kobayshi)of the phase field model of the metal [2]. In Ref.[2],

m(T ) =α

πarctan [γ(Tm − T )]

when T1>Tm and T2<Tm are symmetrical with Tm, i.e.T1+T2=Tm, one obtains

m(T1) =α

πarctan [γ(Tm − T1)]

=α

πarctan [γ(Tm − 2Tm + T2)]

= −απarctan [γ(Tm − T2)]

= −m(T2)

So we can construct the value of ξ according to thesymmetry. When T≥Tm, an alternative evaluation isgiven, that is

ψ =T 0m − Tm

T 0m − (2Tm − T )

ξ0 (17)

ξ = −3ξ20 − 6ξ0ψ + 3ψ2

6ξ0 − 4ψ(18)

Finally, we conclude that

ξ =1

2ξ0 −

3ξ20 − 6ξ0ψ + 3ψ2

6ξ0 − 4ψ

where ψ =T 0m − TmT 0m − T

ξ0, T < Tm

(19)

ξ =1

2ξ0 +

3ξ20 − 6ξ0ψ + 3ψ2

6ξ0 − 4ψ

where ψ =T 0m − Tm

T 0m − (2Tm − T )

ξ0, T ≥ Tm

(20)

The graph of Eq.(19) and Eq.(20) is shown in FIG. 2,in which T 0

m=242 ◦C, Tm=229 ◦C, Tc=195 ◦C.But in Ref.[8], when T approaches Tm, W will be

infinite. So we modified ξ as

ξ = ξ

{1 +

1

πarctan

[10

(T − TmTm − Tc

)]}The graph of ξ versus T is shown in FIG. 3.



FIG. 2 The change of ξ versus T with the modified model.

FIG. 3 The change of ξ versus T .

FIG. 4 shows the sketch map of local free energydensity with the melting temperature Tm=229 ◦C.The temperatures corresponding to the five curves areT1=245 ◦C, T2=233 ◦C, T3=229 ◦C, T4=225 ◦C, andT5=210 ◦C. It can be seen clearly that the formula for ξproposed in this work can compute the local free energydensity efficiently under the condition that T>Tm. Inthe cases of T1=245 ◦C and T2=233 ◦C, the curves oflocal free energy density meet the extreme condition atϕ=0 and ϕ=ξ0 (the black dashed line in FIG. 4).

The nonlocal gradient term can be written as

fgrad (ψ) =1

2κ20β

2 (θ) (∇ψ)2 (21)

Where κ0 is the coefficient of the interface gradient. θdefines the angle between the direction normal to theinterface and a reference axis [14]. The function β(θ)modulates the anisotropy of the interface width and in-terface kinetics time [14]. A convenient form that isoften used in the literature for square symmetry is

β (θ) = 1 + ε cos (jθ) (22)

where ε describes the degree of anisotropy of the surfacetension (or surface energy) and j is the anisotropy mode

FIG. 4 Sketch map of local free energy density.

[14]. And

θ = arctan

(∂ψ/∂y

∂ψ/∂x

)= arctan

(ψyψx

)(23)

Substituting Eqs. (2), (6) and (21) into Eq.(1), we havethe phase field equation as

∂ψ (x, t)

∂t= −Γ

{Wψ

(ψ − ξ0

2+ ξ

)(ψ − ξ0)−

κ20∇ · [β2(θ)∇ψ] + κ20∂

∂x

[β(θ)β′(θ)

∂ψ

∂y

]−

κ20∂

∂y

[β(θ)β′(θ)

∂ψ

∂x

]}(24)

C. Temperature equation

The two-dimensional transient differential equationgoverning the heat transfer within the calculation do-main is given by

∂T

∂t= α∇2T +K

∂ψ

∂t(25)

Where α=KT /(ρCp) is the thermal diffusivity, KT isthe thermal conductivity, ρ is the density, Cp is thespecific heat, K=∆H/Cp, and ∆H is the latent heat ofsolidification.

D. Dimensionless form

For computational convenience, we use non-dimensional groups identified by “overtildes”. Non-dimensional groups enhance versatility in accounting fordifferent polymer systems. Non-dimensional time andspace dimensions are: τ=Dt/d2, x=x/d, y=y/d. For aspecific polymer, τ and d can be related to the diffusioncoefficient and radius of gyration of a polymer chain[15]. Scaled temperature is U=(T−Tc)/(Tm−Tc). Tc is



the experimental temperature of crystallization. AndΓ=D/d2, with d being the characteristic length for sin-gle crystals, D being the diffusion coefficient. The finalgoverning equations can be represented in dimensionlessform

∂ψ

∂τ= −

[Wψ

(ψ − ξ0

2+ ξ

)(ψ − ξ0)− k20∇ ·(

β2(θ)∇ψ)+ k20

∂

∂x

(β(θ)β′(θ)

∂ψ

∂y

)−

k20∂

∂y

(β(θ)β′(θ)

∂ψ

∂x

)](26)

∂U

∂τ= α∇2U + K

∂ψ

∂τ(27)

Where

∇ =

(∂

∂x,∂

∂y

), k20 =

k20d2, α =

α

D, K =

K

Tm − Tc

The model parameters W and k0 may be expressed as

W = 6∆H

nRTξ30

(Tm − T

T 0m

)(ξ02

− ξ

)−1

(28)

k0 = 6σ

nRT

(2

W

)1/2

(29)

In Ref.[7], Xu et al. considered that the value of Kdepends on supercooling andK may be taken as Tm−Tcin the process of the simulation. In this work, α canalso change with respect to supercooling, crystallinityand imperfection.

E. DISCRETIZATION

Firstly, some expressions for derivative calculation inthe discretization process are given below.

β (θ) = 1 + ε cos (jθ) (30)

β′ (θ) = −jε sin (jθ) (31)

β′′ (θ) = −j2ε cos (jθ) (32)

θ = arctan

(ψyψx

)(33)

∂θ

∂ψx=

−ψyψ2x + ψ2

y

(34)

∂θ

∂ψy=

ψxψ2x + ψ2

y

(35)

∂θ

∂x=

∂θ

∂ψx

∂ψx∂x

+∂θ

∂ψy

∂ψy∂x

=ψxψxy − ψyψxx

ψ2x + ψ2

y

(36)

∂θ

∂y=

∂θ

∂ψx

∂ψx∂y

+∂θ

∂ψy

∂ψy∂y

=ψxψyy − ψyψxy

ψ2x + ψ2

y

(37)

∂β(θ)

∂x

∂ψ

∂x= β′(θ)

∂θ

∂x

∂ψ

∂x

= β′(θ) · ψxψxy − ψyψxxψ2x + ψ2

y

· ψx (38)

∂β(θ)

∂y

∂ψ

∂y= β′(θ)

∂θ

∂y

∂ψ

∂y

= β′(θ) · ψxψyy − ψyψxyψ2x + ψ2

y

· ψy (39)

∂

∂x[β(θ)β′(θ)] = (β′(θ))

2 ∂θ

∂x+ β(θ)β′′(θ)

∂θ

∂x

=[(β′(θ))

2+ β(θ)β′′(θ)

] ∂θ∂x

=[(β′(θ))

2+ β(θ)β′′(θ)

]·

ψxψxy − ψyψxxψ2x + ψ2

y

(40)

∂

∂y(β(θ)β′(θ)) = (β′(θ))

2 ∂θ

∂y+ β(θ)β′′(θ)

∂θ

∂y

=[(β′(θ))

2+ β(θ)β′′(θ)

] ∂θ∂y

=[(β′(θ))

2+ β(θ)β′′(θ)

]·

ψxψyy − ψyψxyψ2x + ψ2

y

(41)

For the phase field Eq.(26),

∂ψ

∂τ= −Wψ

(ψ − ξ0

2+ ξ

)(ψ − ξ0) + k20∇ ·

[β2(θ)∇ψ

]− k20

∂

∂x

[β(θ)β′(θ)

∂ψ

∂y

]+

k20∂

∂y

[β(θ)β′(θ)

∂ψ

∂x

](42)

the discretization forms for the second, third and fourthterm on the right side are symbolized as A, B, and Crespectively.

A = ∇ ·(β2(θ)∇ψ

)= β2(θ)∇2ψ +∇β2(θ) · ∇ψ= β2(θ)∇2ψ + 2β(θ)β′(θ)(∇θ · ∇ψ)

= β2(θ)∇2ψ + 2β(θ)β′(θ)

(∂θ

∂x

∂ψ

∂x+∂θ

∂y

∂ψ

∂y

)= β2(θ)∇2ψ + 2β(θ)β′(θ)

[(ψxψxy − ψyψxx

ψ2x + ψ2

y

· ψx)+(


y

· ψy)]

(43)

B =∂

∂x

(β(θ)β′(θ)

∂ψ

∂y

)= β(θ)β′(θ)ψxy + ψy

∂

∂x[β(θ)β′(θ)]

= β(θ)β′(θ)ψxy + ψy

[(β′(θ))

2+ β(θ)β′′(θ)

]·



ψxψxy − ψyψxxψ2x + ψ2

y

(44)

C =∂

∂y

(β(θ)β′(θ)

∂ψ

∂x

)= β(θ)β′(θ)ψxy + ψx

∂

∂y[β(θ)β′(θ)]

= β(θ)β′(θ)ψxy + ψx

[(β′(θ))

2+ β(θ)β′′(θ)

]·


y

(45)

Then the half-discretization form of the phase filedEq.(26) can be written as

∂ψ

∂τ= −Wψ

(ψ − ξ0

2+ ξ

)(ψ − ξ0) +

k20A− k20B + k20C (46)

or

∂ψ

∂τ= −Wψ

(ψ − ξ0

2+ ξ

)(ψ − ξ0) +

k20(A−B + C) (47)

The first-order and second-order finite differenceschemes are used for the derivatives in the half-discretization form of the phase field Eq.(25) and thetemperature Eq.(26), then the complete discretizationform of the phase field equation can be obtained.

III. RESULTS AND DISCUSSION

In order to show the validity of our modified phasefield model, the crystallization of isotactic polystyrene(iPS) is simulated.

A. Parameters setting and results

Eqs.(26) and (27) are solved over square domainsV with homogeneous Neumann boundary conditions∂ψ/∂τ=0 and ∂U/∂τ=0 imposed on the domain bound-aries ∂V . It is well established that nucleationcan be triggered through generation of strong ther-mal noise or seeded with a foreign object. In thepresent case, a nucleation event is triggered with asingle nucleus at the center of the lattice having aGaussian profile such that ψ(r)=exp(−r2/R2). Ris the radius of the initial nucleus [16]. A set ofmaterial parameters cited from Refs.[7] and [8] areused for our simulations of iPS solidification. Modelparameters are computed and specified as follows:∆H=9.4×104 kJ/mol, Cp=1.8 kJ/(kg·K), kT=0.128J/(m·s·K), σ=7.65×10−4 kJ/m2, D=1.0×10−9 m2/s,ρ=1.08×103 kg/m3, and d=1.0×10−7 m.

To solve the model, a number of numerical methods[17] can be used. In our simulations, the finite difference

TABLE I Set of model data at a different given experimen-tal temperature T .

T/◦C T 0m/◦C Tm/◦C Tc/

◦C W k20/10

−14m2 ξ0

200 242 230 200 3.13 0.81 0.952

210 242 233 210 2.20 1.10 0.963

195 242 229 195 3.59 1.43 0.946

180 242 225 180 5.26 0.53 0.929

method is employed on a grid size of 512×512. Thedimensionless temporal step size and spatial step sizeare fixed to be ∆τ=0.1, ∆x=∆y=3.0. Thermal noiseis imparted at the interface such that the melt-solidinterface retains some roughness by virtue of interfaceinstability, i.e. η=rψ(ξ0−ψ) [7]. r is a random numberbetween (−1/2, 1/2).

In recent years, it has been found that supercooling∆T=T 0

m−Tc has a strong effect on the crystallizationmorphology according to the theory of the crystalliza-tion kinetics of polymers [14]. The crystallization mor-phologies under different supercooling rates are givenbelow.

The set of the model data when the experimentaltemperature Tc=200 ◦C is given (Table I), and the nu-merical results is shown in FIG. 5(a1−c1).

The experimental results [18] is shown in FIG. 5(d1),which is in accordance with our numerical results.As pointed out in Ref.[18], when the supercooling∆T=42 ◦C, the iPS grain grows fast along the six spin-dle directions. Under influence of the interfacial noise,the solid-liquid interface is not stable and presents azigzag shape.

The set of the model data when the experimentaltemperature Tc=210 ◦C is given (Table I), and the nu-merical results are shown in FIG. 5(a2−c2). It can befound that the grain grows into a hexagon with a rela-tive smooth interface despite the existence of interfacenoise (ε=0.02). The numerical result is in accordancewith that of the experiments that is given in FIG. 5(d2).

The set of the model data when the experimentaltemperature Tc=195 ◦C is given (Table I), and the nu-merical results are shown in FIG. 5 (a3−c3). It can befound that the grain grows into anything like a flakeof snow with a stronger interface noise (ε=0.08). Thenumerical result is in accordance with that of the ex-periments that is given in FIG. 5(d3).

The set of the model data when the experimentaltemperature Tc=180 ◦C is given (Table I), and the nu-merical results are shown in FIG. 5(a4−c4). It can befound that the grain grows into a spherulite. The nu-merical result is in accordance with that of the experi-ments [18] that is given in FIG. 5(d4).



FIG. 5 The simulation results of (a) τ=1000∆τ , (b) τ=2000∆τ , (c) τ=3000∆τ and (d) experimental results [18] at Tc of(1) 200 ◦C, (2) 210 ◦C, (3) 195 ◦C, and (4) 180 ◦C.

B. The influence of the anisotropy strength oncrystallization morphologies

FIG. 6 shows the crystallization morphologies un-der different anisotropy strength parameters withanisotropy mode j=6 and the experimental tempera-ture Tc=195 ◦C. It is quite common to obtain crystalswith different shapes for the same polymer under thesame supercooling [18–20]. That is to say, there existother factors influencing the crystallization morphol-ogy. FIG. 6 shows the emergence of the dendrite-likesnowflakes crystal with different anisotropy strength(ε=0.02, 0.04, 0.06, 0.08). It is evident that theanisotropy strength strongly affects the interface struc-ture. As anisotropy strength increases, the branches of

the area range from 150◦ to 210◦ and the area rangesfrom −30◦ to 30◦ are very smooth at ε=0.02. With theanisotropy strength increasing, the edges of the mainbranches become rough and not smooth, appearing alot of secondary arms. And these secondary arms willgrow competitively. As can be seen, the growth rate ofthe initial secondary arms is obviously increased as theanisotropy strength increases. In addition, the growthrate of the vertical direction is obviously increased asthe anisotropy strength increases, which inhibits thegrowth rate of the horizontal direction. Above all, theanisotropy strength seems to have no apparent effect onthe global growth rate, which is in accordance with theresult of Wang and co-workers [8] who also investigatedthe crystallization shapes of iPS.



FIG. 6 Crystallization morphologies under different anisotropy strength parameters with anisotropy mode and the experi-mental temperature Tc=195 ◦C: (a) ε=0.02, (b) ε=0.04, (c) ε=0.06, (d) ε=0.08.

FIG. 7 Crystallization morphologies under different latent heat values with anisotropy mode and the experimental temper-ature Tc: (a) K=0.8, (b) K=1.0, (c) K=1.2, (d) K=1.4, and (e) K=1.6.

FIG. 8 The simulation results with different interface thickness: (a) k20=0.6, (b) k2

0=0.8, (c) k20=1.0, (d) k2

0=1.2.

C. The influence of the latent heat on crystallizationmorphologies

FIG. 7 shows the crystallization morphologies un-der different latent heat values (K=0.8, 1.0, 1.2, 1.4,1.6) obtained by the current phase-field model withanisotropy mode j=6 and the experimental tempera-ture Tc=195 ◦C. Since the value of K changes only whenthe value of K changes, so the influence of K on thecrystal shape can replace the role of the latent heat.In FIG. 7, the first morphology of the crystal resem-bles a hexagon shape with faceted fronts, and then thedents appear in the middle of each side of the hexagon.Finally, the dents turn into a narrow crack and the den-drite shape is formed. So we conclude that the wholegrowth process of the dendrite depends mainly uponthe latent heat and the latent heat has a direct effecton the tip radius and tip velocity of side branches bycomparing the following four pictures in FIG. 7. As K

increases from 0.8 to 1.6, the tip radius and tip velocityof side branches also increase.

D. The influence of the interface thickness oncrystallization morphologies

Taking the solid-liquid interface as the diffusion in-terface is the most particular character in the phasefield model. The diffusion interface has a finite thick-ness although it is very thin and is expressed as a steepinternal layer of a phase indication function [2]. How-ever, what the optimal value of the interface thicknessshould be still unknown. In the phase field model, theinterface thickness relates to k20. FIG. 8 shows four pic-tures which are the crystal morphologies obtained atk20=0.6, 0.8, 1.0, 1.2 respectively. The secondary armsare few and the major branches are a little smooth whenk20=0.6. As can be seen, the number of the secondary

arms changes more and more as k20 increases. In addi-



tion, the growth rate of the vertical direction is obvi-ously increased as k20 increases, which also inhibits thegrowth rate of the horizontal direction.

IV. CONCLUSION

In this work, a modified phase-field model is proposedfor polymer crystal growth, in which the parametersare obtained from those of the real materials and verysimple to use, and the most importantly, no paradoxi-cal parameters appeared in the model. A new differentfree energy functional form is proposed for the simula-tion of morphologies of polymer. The finite differencemethod is used to solve the model and different crystal-lization morphologies during the solidification processof isotactic polystyrene is obtained under different con-ditions. The influence of different parameters, such asthe anisotropy strength, the latent heat, and the inter-face thickness, on the crystallization morphologies arediscussed in detail. These results are in accordance withthe experimental results in Refs.[18, 20].

V. ACKNOWLEDGEMENTS

This work is supported by the National Nat-ural Science Foundation of China (No.11402210),the Natural Science Foundation of Shanxi Province(No.2012011019-2), and the Doctoral Fund of TaiyuanUniversity of Science and Technology (No.20152024).

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article a modi ed phase-field model for polymer crystal growth

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