Relationship between Viscoelastic Relaxation Function and Segmental Displacement Function for Entangled Linear Chain in Early Stage of the Tube Dilation Process

Abstract

For highly entangled linear chains, the mean-square displacement of the primitive chain segments [d(n,t)] 2 (n ) segment index) not including a contribution from a nondiffusive part of the contour length fluctuation (CLF) was analyzed on the basis of the tube concept. The analysis was made by just considering the Gaussian nature of the chains (without detailed assumptions about the chain motion). A simple analytic relationship between this [d(n,t)] 2 and the normalized viscoelastic relaxation function µ(t) was derived in a range of µ(t) > 0.5 and at t longer than the Rouse time τRouse for CLF. This relationship was significantly affected by the dynamic tube dilation (DTD) mechanism. For example, the end segment exhibited d 2 ) 〈R 2 〉(1µ) in the absence of DTD and d 2 = 〈R 2 〉(1µ 1/2 ) in the presence of DTD, with 〈R 2 〉 being the mean-square end-to-end distance of the chain. This difference should enable a test of the DTD picture at short t (but still longer than τ Rouse) through comparison of the viscoelastic and segmental displacement data.

Test of Tube Dilation Picture at Short Times through Dynamic Scattering and Viscoelastic Experiments

“…15 This É is normalized to satisfy R Nð1À'Þ 0 dÉðÞ ¼ 1. Utilizing this É, we can express the mean-square displacement function defined in eq.…”

“…On substituting ¼ 0 we recover the expression of the self-displacement function derived in the earlier paper. 15 The approximation underlying eqs. 21-23 may look crude.…”

“…We explored various possibilities and found that the mean-square displacement of the primitive chain segments, ½dðn; tÞ 2 (with n = segment index) without the contribution of non-diffusive part of the CLF, can be easily analyzed. Focusing on this feature of ½dðn; tÞ 2 we formulated the DTD and non-DTD relationships between ðtÞ and ½dðn; tÞ 2 at short t. 15 The incoherent intermediate scattering function of actual chains, I s ðq; tÞ with q being the wave vector, can be expressed in terms of this ½dðn; tÞ 2 , enabling us to test the DTD picture at short t through comparison of the viscoelastic and dynamic scattering data ( and I s ).…”

“…15 The Ár ð1Þ and D are not correlated, and the conditional average of ðÁrÞ 2 for the given m, m 0 , n and n 0 values is obtained as…”

“…Since the distribution of the lateral displacement is determined by the dilated tube diameter a 0 and independent of the m, m 0 , n and n 0 values, the average hD 2 i can be calculated in the following way. 15 If the n-th segment is in the surviving portion of the dilated tube after step 1 (Figure 1c), the segment position after steps 1 and 2 should be uniformly distributed in the internal cross-section of this tube (shown with the dotted ellipse). Then, D is equivalent to a vector connecting arbitrarily chosen two points in a circular region of the diameter a 0 À a, the region available for the center of the n-th segment.…”