=> Return to Physics.html

Both the elastic and viscous properties of a chain are modified by entanglements as can be seen by the breadth of the spectrum of effects that are observed from dramatic, power-law changes in the viscosity, to completely different mechanical behavior i.e. wax versus polyethylene. Entanglements do not effect properties that depend on local motion of the chain such as the glass transition temperature or a-relaxations. All properties that can be related to the entire chain, normal mode relaxations, are effected by entanglements. Between entanglements the Rouse model works quite well and it can be used, for instance to model relaxations between crosslink points in an elastomer.

where q = 4p/l sin(q/2), FT is the Fourier transform, g(r, t) is the time dependent pair correlation function.

The quasi-elastic neutron scattering experiment allows one to observe the time correlation of dynamics for a single chain in an entangled mat as a function of both time and distance of correlation. It is expected that as one approaches the smallest scales of size, a Kuhn-unit, that the correlations will drop. By Bragg's law, d = 2p/d, high-q is related to small sizes (Kuhn size) and small-q is related to larger sizes (coil size).

For unrestricted chain motion, as time goes to ° the correlations approach <c_m> so we can write,

g(r, t => °) = <c_m> (Unrestricted chain motion)

and

S(q, t=> °) = 0 (Unrestricted chain motion)

For confined chains such as chains confined by the reptation tube

g(r, t => °) ­ <c_m> (Restricted chain motion)

and this will hold for all sizes (q values) where the chains are confined to a tube. Then measurement of the quasi-elastic neutron scattering pattern can be used to determine the extent of applicability of the Rouse model at small sizes and the effects on dymanics of reptation at larger distances. Strobl's figure 6.8 shows quarielastic neutron scattering data that demonstrates this behavior.

Understanding figure 6.8 requires a bit of background. The series of curves in each plot corresponds to different values of q, or different size scales of observation by Bragg's law, size scale = 2p/q. The highest q's reflect an approach to Rouse dynamics at size scales smaller than that of the entanglements or at size scales where the tube model is not appropriate because the chains have significant lateral motion. At low-q (top curves) significant deviation from the unconstrained motion occurs due to the presence of the reptation tube.

t_R = ((z_R/a_R²) N₀²)/(3kTp²)

and through the use of Bragg's law an association between q² and 1/a_R² can be made so q²a_R² is a dimensionless and reduced q, while t/t_R is a dimensionless and reduced time. The parameter "u" depends on both time and q in a natural way and is defined as,

u = q² (12kT a_R² t/z_R)^1/2

The use of the parameter "u" should lead to a universal curve for all of the different times and q-values in the top plot. The curves come close to collapsing in this approach. At small-u (short times, the curves approach Rouse-dynamics. At longer times (large-u) the curves deviate from Rouse behavior due to reptation based constraints.

By extrapolating the quasi-elastic neutron scattering measurements on the time domain to t => °, the point of deviation from 0 can be obtained in q, q*, and d* = 1/q* can be calculated as a measure of the size scale of confinement associated with the reptation tube. This is shown in figure 6.9 of Strobl.

The size scale of confinement has a linear temperature dependence form these measurements, and increases with increasing temperature. As d is the cutoff size for unconstrained motion, increasing d means that constraints are less effective at higher temperatures as might be expected.

This quasi-elastic neutron scattering data indicates that the dynamics of a polymer chain in the entangled melt should be broken down into at least two regimes of behavior, an unconstrained regime at small sizes and a constrained regime at larger sizes. At even smaller scales we expect a third regime associated with the a-processes associated with local chain chemical composition and bonding, DG_mic(t). The time dependent shear modulus becomes then,

G(t) = DG_mic(t) + r_pkT S exp(-2t/t_m) + r_pkT m* F(t/t_d)

where the summation is from m = m* to N_R - 1, i.e. the high order Rouse modes beyond the critical size scale for entanglements and where m* is associated with the lowest order (largest size) where Rouse dynamics are appropriate. The Rouse regime extends to sizes where the chains are still considered to be Gaussian coils, obeying rubber elasticity on the small end, i.e. to the Kuhn segment size. m* is given by (N_R - 1)/(N_R,c -1), where N_R,c is the contour length in Rouse-units corresponding to the critical molecular weight for entanglements, M_c. The relaxation time t_m* is given by t_R {(N_R,c -1)/(N_R - 1)}².

The third term in the equation for the time dependent modulus reflects dynamics in the confined regime. The expression F(t/t_d) means that this regime is assume to depend on a single relaxation time, t_d, that is associated with the time for "disentanglement". F(t/t_d) is a general normalized function meaning that F(0) = 1. t_d is the integral width of the arbitrary function, F(t/t_d), so,

Empirically (from experiment),

t_d = K M^3.4

where M is the molar mass of the polymer.

There is a gap in the relaxation behavior between t_m*, the longest relaxation time for Rouse behavior, and t_d, the relaxation time for tube renewal or release of entanglement constraints. This gap in the relaxation spectrum leads to the plateau modulus region. The length of this region in time is reflected by t_d/t_m* Å M^3.4/M_c^3.4.

The effect of entanglements on the zero shear rate viscosity, h₀, can be calculated using the previously derived expression,

and the expression given above for the three regimes of dynamics in an entangled melt. Ignoring DG_mic(t), and using a mean relaxation time, t_a, for the Rouse modes that extend up to M_c,

h₀ = r_pkT [(N_R - m*) t_a + M* t_d]

Since m* is associated with the plateau modulus, G_pl, and the N_R term is associated with the shortest time relaxations, G(0), the expression can be rewritten,

h₀ = G(0) [(G(0) - G_pl) t_a/G(0) + G_pl t_d/G(0)]

The first term in brackets is constant, b₁, so,

h₀ = b₁ + b₂ M^3.4

At high molecular weights,

h₀ Å t_d Å b₂ M^3.4

Strobl's figure 6.10 shows that the motion of a chain can be considered to consist of two discernable components, the rapid, Rouse components normal to the chain, within the reptation tube, and the slower and discrete (i.e. there is a gap in time) motions in the direction of the reptation tube or the time for the tube to refresh or renew itself (tube renewal time). The path of the reptation tube is termed the "primitive path" for the chain.

The primitive path represents a redefinition of the chain, just as the Rouse-model redefined the chain in terms of rubber elastic units. For the primitive path the redefinition is based on the longest time Rouse mode that reflects unconstrained dynamics. Since the primitive path is the chain, albeit renormalized to the Rouse unit, it displays Gaussian scaling. Then we can write,

R₀/a_R = N_R^1/2

where the power 1/2 reflects 1/d_f and d_f is the mass-fractal dimension for the chain.

The number of Rouse units in the chain, N_R, is related to the length of the primitive path along the contour of the tube, l_pr, by,

N_R Å a_prl_pr

where a_pr reflects the persistence length for the primitive path or the stiffness of the primitive path.

Within the reptation tube the chain thermally diffuses along the primitive path following the Einstein relationship,

D' = kT/z_p

where D' is the diffusion coefficient for the chain within the tube along the primitive path, and z_p is the friction factor for the chain. By definition there are no entanglements within the tube, so the Rouse result can be used,

z_p = N_R z_R

so,

D' = kT/N_R z_R

The time for the chain to completely diffuse out of the reptation tube is called the "reptation time", t_d, as shown in Strobl's figure 6.11 on pp. 284.

Such random diffusion is governed by laws for Brownian motion so,

t_d = l_pr²/D'

Using the definitions of l_pr, and D', we have,

t_d = z_R N_R³

Various modifications to the reptation model have been proposed, most prominent of these is the "constraint-release" approach where some dynamic behavior is introduced to describe the tube itself. These have met limited success.

Strobl's figure 6.12, p. 286, shows results for the measurement of the diffusion coefficient for polymer chains in an entangled melt.

The left graph shows that the diffusion coefficient scaled with M to a power -2. This has been verified in a wide range of entangled polymers. The Rouse model (using the Einstein relationship) indicates that the diffusion coefficient should scale with 1/M,

D = kT/z_p = kT/(N_Rz_R) Å 1/M

while the reptation theory correctly predicts the M^-2 dependence. This dependence can be obtained by considering motion of the center of mass of the polymer over a distance of the order of l_pr, along the primitive path. The mean-squared distance moved, <Dr_c²>, is given by,

<Dr_c²> Å R₀² = l_pr a_pr

In 3-dimensions the diffusion coefficient is given by,

D = <Dr_c²>/(6 Dt) = <Dr_c²>/t_d Å l_pr a_pr/t_d Å N_R/N_R³ Å M^-2

Combination of the reptation model and the Rouse model predict a change in the power-law behavior at the entanglement molecular weight of M^-1 to M^-2 dependence for the diffusion coefficient.

Strobl's figure 6.13 shows a classic experiment (that has been repeated a number of times in the literature) using fluorescently tagged DNA chains (a polyelectrolyte).

The data shows a time sequence of the dynamics of a DNA chain (a micron-scale biopolymer). The chain is pulled towards the bottom of the figure by the bright spot. These images support the reptation model for obvious reasons.

=> Return to Physics.html