Effect of Friction Reduction Due to Weak Segment Orientation
under Equibiaxial Elongational Flow
Keiko TAKEDA, Sathish Kumar SUKUMARAN, Masataka SUGIMOTO, Kiyohito KOYAMA,and Yuichi MASUBUCHI*
Graduate School of Science and Engineering, Yamagata University, Yonezawa 992-8510, Japan*National Composite Center, Nagoya University, Furocho, Chikusa, Nagoya 464-8603, Japan
kei@yz.yamagata-u.ac.jp
1. Introduction:
Dynamics of polymers under the elongational flow has not been fully elucidated yet. For
instance, in the high strain rates it has been known that the polystyrene (PS) melts shows the strain
rate thickening of the steady-state elongational viscosity, while PS solutions indicates the strain rate
thinning [1]. The steady-state viscosities at the high Weissenberg number defined by using Rousetime (WiR) possess a qualitative difference between polymer melts and solutions. Yaoita et al. [2]focused on the friction reduction for this issue. They analized the step-strain behavior under
uniaxial elongational flow after cessation of strain for PS melts [3] and solutions [4] using PCNmodel [5] and found that molecular friction changed by the chain stretch and orientation(stretch/orientation-induced reduction of monomeric friction: SORF). They concluded that thedifference of SORF effect between melts and solutions under the high strain rates made thenoticeable difference in the uniaxial elongational viscosity. According to this concept, themolecular friction of the polymer melts reduces due to SORF under the high rates, leading to thethinning of the viscosities. In the solutions a presence of solvent suppresses the molecularorientation, and the friction reduction is less than melts. This small friction reduction in solutions
results in the thickening of the viscosities as the rate increases. The SORF effect on molecularbehavior under fast elongational flow is expressed as,$" )! ( FSO*
11 '*"= fFENE#+{1%tanh&F%F}()
SOSO,+$ )! (0)(1+ # ) ( 2(1)where ! is the segmental friction, and !(0) is the equilibrium value. fFENE is the FENE factor, Fso is
the stretch/orientation order parameter, FSO = !! 2 S with !! is the average stretch ratio normalized
by the maximum stretch of entangled sub-chains, S is the average orientational anisotropy of thecomponents, F’SO=FSOfFENE, and "=20, ! = 5 !10 "9 , #=0.15 and F*so=0.14 [2].
In our previous paper [6], we simulated the transient equibiaxial viscosities of PS melts byTakahashi et al. [7] by using PCN simulation, our results reasonably agreed with the experimental
data in the limited strain rates. In this study we investigated the steady-state viscosity of equibiaxial
elongational flow in the wide strain rate region 10-3< WiR <10 by the PCN simulation in which theSORF effect was taken into account to reveal the strain rate dependency of the equibiaxial
elongational viscosity. We examined the stress under elongational flow using the decoupling
approximation [2] as follows
follows,#Z&! " % ( fFENE )! 2 S$ Z0 '(2)Here, Z is the average number of entanglement segments, fFENE is the spring bead constant, $ is thestretch normalized by the maximum stretch, and S is the orientation anisotropy. In this study, weused two types of definition of S to examine the effect of SORF under equibiaxial elongational flow
as below,
S1 = u y2 + uz2 ! 2 u x2
(3)
222
S1/2 = u y + uz / 2 ! u x(4)where u is the unit bond vector, and x is the squeezing direction, and y and z are the elongational directions.

(())

2. Model:
In the PCN model, the polymer melt is expected
as a set of primitive chains in 3-Dimentional spacesthat are entangled with each other through slip-links.
Consecutive strands (sub-chains) including two nodes
which connect chains forming a group with otherchains. The motion of nodes is described by aLangevin-type equation that considers the force
balance, sub-chain tension, osmotic force and random
Brownian force. The finite chain extensibility (FENE)
is accounted as an elastic force with FENE-P
approximation [8]. Chain hooking and unhooking at
the chain ends leads to creation and distraction ofslip-links (entanglements, Z) respectively. Therefore,
the number of Z fluctuates with time. Molecularmechanisms for entangled polymers, such asreputation, contour length fluctuation, and thermal
convective constraint release, are considered.
The relevant parameters used in this simulation
are listed in Table 1. We neglected the molecularweight distribution for simplicity. The molecularweight of Kuhn step for PS melt (Mk=700) [9], n0 is
M0/Mk=15. n0 determines the maximum stretch ratio(FENE). Shift factor G0=ρRT/M0 is 0.31 MPa (whereρ is the density). The unit time τ0 is 0.01 sec. The
Rouse relaxation time τR (that is 1/2 of the relaxation
time for chain contraction) is as Z 02τ 0 / 2π 2 = 0.27 sec.

Table 1 Parameters of PS used in the simulations
Mw

M0Z0G0[MPa]τ0[s]

T[K]

250k

11k

23

0.31

0.01

433

Fig.1 Normalized steady state viscosity,orientation, squared reduced stretch, reduced
segment number and reduced friction of PS
Melt from top to bottom. Solid and dottedcurves are for the results with and without
friction reduction.

3. Results and Discussion:
Fig. 1 shows the normalized steady-state viscositywith the zero rate value, orientation, normalized square stretch with the maximum stretch, FENE
factor and reduced friction with the equilibrium value from top to bottom. Taking account of SORF,
η/η(0) under uniaxial flow successfully showed strain rate thinning at high WiR [2]. We should notethat under biaxial flow η/η(0) showed thinning at high WiR, while η/η(0) showed thickening
without SORF. The calculated η/η(0) with SORF by using S1 showed relatively strong WiR
dependency, comparing with that by S1/2. Friction reduction of S1 decreased by saturated orientation.
S1/2 orientation was also saturated, but the stretch growth is higher than of S1 and suppressed thedecrease of friction reduction and η/η(0).
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