Theories and Applications of Chem. Eng., 2002, Vol. 8, No. 1

화학공학의 이론과 응용 제8권 제1호 2002년

네트웍 모델에서의 LAOS 수치모사

Theories and Applications of Chem. Eng., 2002, Vol. 8, No. 1

김승하, 심훈구, 안경현, 이승종

서울대학교 응용화학부

Large amplitude oscillatory shear behavior investigated by a network model

Seung Ha Kim, Hoon Ku Sim, Kyung Hyun Ahn, Seung Jong Lee

School of Chemical Engineering, Seoul National University

Introduction

The zero-shear-rate viscosity increases much faster than linearly with concentration, indicating strong polymer-polymer interactions at larger concentration. There are two approaches in the large concentrations system. There is the approach in modeling the interactions of the polymer molecules which builds on the classical network theories of rubber elasticity to analyze the melt and concentrated system. In the network theories of macromolecular solids, chemical crosslinks are described as points, or junctions, at which connecting portions of molecules are forced to move together for all times. In adapting this theory to liquids we assume the junctions are not permanent, but are continually being created and destroyed. The melt and concentrated system need to analyze the nonlinear area because the systems involve large, rapid deformations which cannot be modeled using the theory of linear viscoelasticity, which predicts a superimposable, linear dependence of the amplitude of the stress response on the amplitude of the imposed strain. Among the nonlinear experiments, step shear rate test (measurement of shear viscosity as a function of shear rate) is often conducted. As it provide the information how readily the material can be processed or shaped into a useful product [Larson (1999)], it can be helpful in polymer processing as in injection molding. However, as deformation is too fast at high shear rate, the material doesn't have enough time to respond to the environment and hardly provides the information on its microstructure. Another nonlinear experiment is large amplitude oscillatory shear (LAOS). It is normally generated by strain sweep test, which is used to check the linear viscoelastic regime before the frequency sweep test. As storage modulus G' and loss modulus G''□ are defined only in the linear viscoelastic regime, the moduli at large strain lose mathematical background, and have normally been neglected. However, provided that enough care is taken, LAOS test can provide plentiful amount of additional information. In addition, it allows both strain amplitude and time scale to be varied independently, and it is easy to generate because it does not involve any sudden jump in speed or position. We calculated the LAOS test by network model of Marrucci paper.

Theory

Mathematic formation (Network model by Marrucci paper)

[eqn: {upsilon }_{a }+ {upsilon }_{p } =upsilon]	(1)

[eqn: {y }_{a }= { { upsilon }_{ a} } over { upsilon} , {y }_{p }= { { upsilon }_{ p} } over { upsilon} , { y}_{a }+ { y}_{p }=1]	(2)

The equation for the distribution of active chains

[eqn: { PARTIAL psi } over { PARTIAL t}=- { PARTIAL } over { PARTIAL R } CDOT ( psi k CDOTR)- beta (R) psi + alpha (R) phi]	(3)

The equation for the distribution of pendent chains

[eqn: { PARTIAL phi } over { PARTIAL t }=- { PARTIAL } over { PARTIAL R } CDOT (-D { PARTIAL phi } over { PARTIAL R }+ phi { D} over {kT }F(R)+ phi k CDOT R)- alpha (R) phi + beta (R) psi]	(4)

The stretching force

[eqn: F(R)=- { 3kT} over {N {b }^{2 } } { R} over {1- { R}^{2 } / { (Nb)}^{2 } }]	(5)

The rate coefficients

[eqn: alpha (R)= { alpha }_{0 }+ { alpha }_{1 } { R} over { { R}_{0 } }]	(6)

[eqn: beta (R)= { { beta }_{0 } } over {1- { R}^{2 } /{(Nb) }^{2 } }]	(7)

The imposed shear strain in an oscillatory shear

[eqn: gamma (t)= { gamma }_{0 }sin( omega t)]	(8)

[eqn: { .} atop { gamma } = { gamma }_{ 0} omega cos(omegat)]	(9)

A linear shear stress response is sinusoidal

[eqn: sigma (t)= { sigma}_{0 }sin( omega t+ delta )]	(10)

A nonlinear response is given by a Fourier series

[eqn: sigma (t)= SUM from { {j }=1,odd} to inf { sigma }_{j }sin(jt omega+ { delta }_{j })]	(11)

Results

We calculated the data for the parameters of T(relaxation time), A(creation rate). T is the term of loss rate and A is the term of creation rate. We have the test changed the parameter scale.

Fig 1. Simulated G' as A=5(A1=5, A0=0) and various scale T at crossover.

Fig 2. Simulated G'' as A=5(A1=5, A0=0) and various scale T at crossover.

Fig 3. Simulated G' as A=5(A1=0, A0=5) and various scale T at crossover.

Fig 4. Simulated G'' as A=5(A1=0, A0=5) and various scale T at crossover.

Above graphs(Fig 1, 2, 3, 4) we knew the correlation between T and A. Because T parameter means relaxation time, G' decreases and G'' increases as T increases.

We observed 4 types of LAOS behavior.

Fig 5. LAOS behavior at A=5(A0=5, A1=0) and T=0.01 in the crossover.

Fig 6. LAOS behavior at A=5(A0=0, A1=5) and T=0.01 in the crossover.

Fig 7. LAOS behavior at A=20(A0=0, A1=20) and T=0.01.

Theories and Applications of Chem. Eng., 2002, Vol. 8, No. 1

Fig 8. LAOS behavior at A=5(A0=0, A1=5) and T=0.1.

We compared the graphs(Fig 5, 6, 7, 8) with experiment data.

References

(1) R. Byron Bird: Dynamics of polymeric liquids, volume 2, 353-391

(2) A. Vaccaro, G. Marrucci: A model for the nonlinear rheology of associating polymers, J. Non-Newtonian Fluid Mech. 92(2000) 261-273

(3) Joseph A. Yosick, A. Jeffrey Giacomin, Paula Moldenaers: A kinetic network model for nonlinear flow behavior of molten plastics in both shear and extension, J. Non-Newtonian Fluid Mech. 70(1997) 103-123