고분자 가공에 따른 결정성고분자의 사슬배향에 관한 현상학적 해석

우종표, J.L. White*

명지대학교 화학공학과, *애크론대학교 고분자공학과

Phenomenological Analysis of Polymer Chain Orientations in Semicrystalline Polymers During Fabrication Process

Jong-Pyo Wu, James L. White*

Dept. of Chem. Eng., Myong Ji University

*Dept. of Polymer Eng., The University of Akron

Introduction

Molecular orientation has long been known to have a major effect on the mechanical, optical, and electrical properties of polymers. Structural characterization and structure development as function of processing conditions have received significant attention for systems with uniaxial and biaxial symmetry such as melt spun fiber and tubular blown film processes.

It has been generally accepted that for flowing polymer solutions and melts of flexible polymers the chain orientation is quantitatively related to the stress field through the rheo-optical law.[1] Furthermore the chain orientation in glassy polymers vitrified under stress can be related to stress field at vitrification through the rheo-optical law. This has been verified for various vitrifying polymers. For semicrystalline polymers, however, it does not follow that chain orientation of crystalline part may be related in terms of the rheo-optical law. The reason is that there exists a chain rearrangement during the process of crystallization. The experimental observations of melt spun high density polyethylene fiber and tubular blown film indicated a significant increase of orientation level compared with the predicted values from the rheo-optical law.[2,3] There has been no attempt to correlate the orientation states of the melt with those of the crystalline part in semicrystalline polymers.

Though the chain orientation of a crystalline polymer does not obey the rheo-optical law directly, it will, however, primarily depend upon the orientation of the melt prior to the onset of crystallization. The purpose of this study is to propose one approach for establishing the relationship between the chain orientation of crystalline part and the orientation of the polymer melt prior to the onset of crystallization.

Phenomenological Theory

Specification of Chain Orientation by a Tensor

We can define a set of orientation tensors by defining dyadic products of the vector p and integrating the products of these tensors with the distribution function as proposed by Advani and Tucker [4]. The second order tensor may be written as

aij = ∫pipj □(p)dp	(1)

The tensorial nature of aij is evident from their definition.

Generalized Linear Relationship In Semicrystalline Polymers

In semicrystalline polymers one can represent the amorphous orientation state of polymer melt and the crystalline orientation state of subsequently crystallized polymer in terms of aAij and aCij. Here the superscript A and C mean the amorphous melt and the crystalline state, respectively. We may relate the second order tensors aAij for amorphous polymer melt prior to the onset of crystallization to aCij for crystallized polymer by a generalized linear relationship such as[5],

aCij = W*ijkl aAkl	(2)

where fourth rank tensor W*ijkl defines the change in orientation associated with crystallization. Due to the inherent nature of symmetric aij, the fourth rank tensor W*ijkl satisfies

W*ijkl = W*jikl,  W*ijkl = W*ijlk,  W*ijkl = W*jilk	(3)

for the most general case. This means there exist 36 independent coefficients. It also suggests the representation of the tensor aij as a vector ai(i runs from 1 to 6) and the fourth rank tensor W*ijkl as a second order tensor Wij with 36 components, i.e.,

bCi = Wij bAj	(4)

where b1=a11, b2=a22, b3=a33, b4=a12, b5=a13, and b6=a23. This means

W11 W12 W13  W14 W15 W16

W21 W22 W23  W24 W25 W26

W31 W32 W33  W34 W35 W36

Wij =	W41 W42 W43  W44 W45 W46	(5)

W51 W52 W53  W54 W55 W56

W61 W62 W63  W64 W65 W66

The upper-left side 9 components in the Wij matrix imply the axial changes of orientation and the lower-right side 9 components indicate the crossaxial changes of orientation, and the remaining components imply the axial-crossaxial changes of orientation during crystallization.

One may apply Equation(4) to the uniaxial, the planar orientation states, and the biaxial orientation states. The orientation factors can be correlated with the following relationship[6]

f1B = a11 - a33	(6)

f2B = a22 - a33	(7)

Discussions

The results derived by the generalized linear relationships between two orientation tensors aAij and aCij for each different mode of orientation are shown in Table 1. These equations imply the crystalline orientation factor may have a linear relationship with stresses and also imply the possible change of crystalline orientation level compared with that of polymer melt prior to the onset of crystallization depending on the component W11. This can possibly explain the earlier experimental observations [2,3], shown in Fig.1. Their experimental data implies that W11 □ 10 for melt spun high density polyethylene fibers.

The components of Wij might be primarily dependent on the crystal structure and its morphology of polymers. We believe that crystallization concept such as nucleation and growth theory of polymer crystal introduced by Hoffman[7] can be used to understand Wij. The component W11 implies the contribution of melt orientation along the machine(1) direction to the crystalline orientation along the machine(1)-direction, while the component W12 implies the contribution of polymer melt orientation along the transverse(2)-direction to the crystalline orientation along the machine(1)-direction. We can define all the possible ranges of component W11 for the uniaxial case into the following three categories

CASE I	:	fC < fHA (W11<1 ,W12>0)

CASE Ⅱ:	fC = fHA	(W11=1, W12=0)

CASE Ⅲ:	fC > fHA	(W11>1, W12<0)

Previous experimental observation for polypropylene in melt spinning process shows that it belongs to the CASE Ⅲ(W11>1, W12<0), where any chain orientation of melt state greatly enhances the crystalline phase orientation.

Conclusions

A phenomenological approach for establishing the relationship between the chain orientation of a crystalline polymer and the orientation of polymer melt prior to the onset of crystallization was proposed.

A generalized linear relationship between second order orientation tensors aAij and aCij representing the polymer melt state and crystallized state, respectively, was developed by the fourth order tensor W*ijkl. The crystalline orientation factors were derived from the components of W*ijkl and the rheo-optical law for the polymer melt prior to the onset of crystallization.

The results obtained by this approach can explain the experimental observations such as the jump of crystalline orientaion level and the change of orientation mode compared with the predicted values from the rheo-optical law.

References

1) J.L.S.Wales and W.Philippoff, Rheol. Acta, 12, 25 (1973)

2) J.R.Dees and J.E.Spruiell, J. Appl. Polym. Sci., 18, 1053 (1974)

3) K.J.Choi, J.E.Spruiell, and J.L.White, Polym. Eng. Sci., 29, 463 (1989)

4) S.G.Advani and C.L.Tccker,Ⅲ, J. Rheol., 31, 751 (1987)

5) I.M.Ward and D.W.Hadley, "An Introduction to the Mechanical Properties of Solid     Polymers", Wiley (1993)

6) J.L.White and J.E.Spruiell, Polym. Eng. Sci., 21, 859 (1981)

7) J.D.Hoffman, Polymer, 24, 3 (1983)

Table 1.    Results of Generalized Linear Relationship Between Melt and

Crystalline Orientation Tensors

Orientation Mode	Orientation Factors of Melts	Crystalline Orientation Factors
Uniaxial Orientation	fHA = (C/□o)□11	fC = ½(3W11 - 1)(C/□o)□11
Planar Orientation	fPA = (C/□o)(□11-□22)	fPC = (2W11-1)(C/□o)(□11-□22)
Biaxial Orientation	fB1,A = (C/□o)(□11-□33) fB2,A = (C/□o)(□22-□33)	fB1,C = (2W11+W21-1)(C/□o)(□11-□33) + (W22+2W12-1)(C/□o)(□22-□33) fB2,C = (W11+2W21-1)(C/□o)(□11-□33) + (W12+2W22-1)(C/□o)(□22-□33)

Fig. 1.   Orientation Factors of Melt Spun High Density Polyethylene Fiber

As a Function of Spinline Stresses