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At low frequency the storage shear modulus, G’(w), follows w².

Polymers are generally power-law fluids that display a strong dependence of observed viscosity on rate of strain. At low rates of strain, the viscosity plateaus at the extrapolated, zero shear rate viscosity, h₀, that is associated with J_e⁰.

Discussion of the strain rate dependence of the viscosity, shown above, will require the development of theoretical models for polymer flow. Here we are interested in generic descriptions of dynamic phenomena so will focus on the zero shear rate viscosity, h₀, and the plateau shear compliance, J_e⁰, specifically, how do these values relate to the primary response function, m(t).

For a Newtonian fluid, s_xy = h₀ dg_xy/dt. If a Newtonian fluid is measured in a dynamic compliance experiment, where, s_xy = s_xy⁰ exp(iwt) and g = J*s_xy⁰ exp(iwt). Taking the derivative of the strain and inserting into Newton’s law of viscosity yields,

J* = -i/(h₀w)

J*(w=>0) = J_e⁰ - i/(h₀w)

G*(w=>0) = J_e⁰ (wh₀)² + i (wh₀)

Then we have well defined power-laws in frequency for G* and J* at the zero shear rate limit (low frequency regime) and the recoverable shear compliance and zero shear rate viscosity are the primary parameters of importance. The w² dependence is shown in the first plot of this chapter (G' versus w in figure 5.15).

and the viscosity, h, is the stress, y(t), divided by the strain rate, de/dt, so the zero shear rate viscosity, h₀, (at infinite time, t = °) in a shear experiment where there is no strain before time t=0, is given by,

m(t) = dG(t)/dt

J_e⁰ can also be related to the time dependent modulus and the primary response function by considering a dynamic mechanical experiment where e(t) = e₀ exp(iwt) and s(t) = G* e(t). Using the above equation for the Boltzman superposition principle:

taking the derivative of the strain, de/dt = iw exp(iwt) and substituting t" = t-t',

For w => 0, the exponential can be expanded, exp(-iA) = 1 + A + ...., so,

The first expression can be coupled with the previous expression for G', G' = J_e⁰ (wh₀)², to yield a description of J_e⁰,

The ratios of the two integrals that define J_e⁰ and h₀, yields a time,

a_T would appear to be an empirical parameter as presented. Consider the equation for the complex modulus in the "terminal zone" (flow zone) given above, G*(w=>0) = J_e⁰ (wh₀)² + i (wh₀); and the equation for the complex compliance in the terminal zone, J*(w=>0) = J_e⁰ - i/(h₀w). In both cases the frequency and viscosity appear always as a pair so that the effects of viscosity and frequency can not be experimentally distinguished using complex modulus or complex compliance. Temperature is not directly expressed, however, we know that viscosity is generally a function of temperature in Newtonian fluids, following the Arrhenius exponential law,

h₀(T) = B exp(T_A/T) (Newtonian Fluid)

h₀(T) = B exp(T_A/{T - T_V}) (Vogel-Fulcher Fluid, i.e. polymers)

where T_V is called the Vogel temperature, and is loosely related to the glass transition temperature (typically T_V is 50 C° below T_g). The Vogel temperature implies that there is a discontinuity at some finite temperature where flow can no longer occur. This behavior is associated with the chain-like nature of polymers, and loss of long range mobility lateral to the chain. Strobl's Figure 5.16 demonstrates the validity of the Vogel-Fulcher law both for viscosity and for the a-relaxation time (glass transition) in polyisobutylene. (The Arrhenius law would be flat in this temperature range, i.e. constant viscosity and relaxation time.)

The shift in frequency response at different temperatures is then, solely due to changes in the viscosity as described by the Vogel-Fulcher equation. a_T involves the mechanical or dielectric response at two temperatures, the temperature for which the master curve is to be constructed, T₀, and the temperature at which the measurement was made, T. The inverse relationship between frequency and viscosity, implied by the presence of h₀w as a pair in all terminal relaxation equations, leads directly to a definition of the frequency shift factor in terms of viscosity,

a_T = h₀(T)/h₀(T₀) = t(T)/t(T₀)

The complex modulus measured at temperature T, can be expressed at temperature T₀ using a_T as, G*(w, T) = G*(a_Tw, T₀) or, more typically on a log w scale as, G*(log w, T) = G*(log a_T+ log w, T₀). The time dependent shear modulus is shifted to T₀ by, G(t, T) = G(T₀, t/a_T), or on a log scale, G(log t, T) = G(log t - log a_T, T₀).

A fundamental principle of our understanding of polymers is that the mechanical properties are based on molecular structure and dynamics. This means that other analytic techniques that can be tied to molecular structure should mimic the response observed in mechanical measurements. We will later look at the stress optical law and the existence of a constant stress optical coefficient as verification of this principle. Suffice it to say at this point that there is extremely strong experimental evidence that there is a molecular basis for mechanical properties in polymers centered on the Kuhn segment (or dynamic mer-unit). For this reason we might expect to observed relaxations in dynamic dielectric measurements that strongly parallel relaxations observed in dynamic mechanical experiments. The a-transition, at the glass transition temperature, that serves as a boundary between solid and liquid behavior, can be used as a case study of parallel features in dynamic dielectric and dynamic mechanical measurements. Strobl's figure 5.17 shows the frequency and temperature dependence of the loss and storage dielectric constant for polyvinylacetate (PVA) near the a-transition. The behavior is completely analogous to the observation in the dynamic modulus or compliance.

From the maximum in the dielectric loss curve occurs at a frequency that reflects the inverse of the time constant associated with the a-relaxation. This frequency at maximum can be termed the "rate" of the a-relaxation at the indicated temperature. Such a rate of relaxation can also be determined from dynamic modulus (G*) and compliance (J*) measurements. Strobl's figure 5.18 shows that these rates parallel each other but are shifted by a constant, with the dielectric rate observed at an intermediate value between the compliance (higher temperature, lower rate) and modulus (lower temperature and higher rate).

In addition to the a-transition, other molecular based transitions are observed, generally at lower-frequencies compared to the a-transition. Cis-polyisoprene (PIP), for example, shows a prominent normal-mode transition shown as a function of temperature, rate and molecular weight in Strobl's figure 5.20. Most importantly, the normal-mode transition is a strong function of the molecular weight, while the a-transition shows no molecular weight dependence.

For the low molecular weight (oligomer) regime, the power law is t = M² and for the high molecular weight (polymer) regime, the power law is t = M^3.7. The crossover from oligomer to polymer behavior occurs at about 10,000 g/mole for cis-PIP. This corresponds to the entanglement molecular weight where topological constraints between chains (intertwining or entanglements) dominate the normal mode relaxation.

If the polymer chain's asymmetry is considered, the basis for and relationship between the a-transition and the normal mode transition can be easily understood. Given that the chain is topologically a one dimensional object, that is, it is connected linearly, we can consider dynamics associated with motion lateral to the chain axis, a-transition, and motions associated with the direction of the chain path. While lateral motion can be associated with basically a single relaxation time, or a simple combination of relaxation times, pathwise relaxations involve a number of different modes associated with different lengths of chain segments. The longer these Kuhn-segment groupings, the slower the relaxation. The fastest pathwise relaxation is associated with a single Kuhn-segment and would have a relaxation time similar to the lateral a-relaxation. The slowest relaxation occurs for the entire chain and is associated with the normal-mode relaxation and involves complete reorganization of the end-to-end vector for the chain. Chain motion along the path can be considered a superposition of these various modes. The analysis of chain pathwise relaxation in terms of modes is the basis of the Rouse and the Reptation models for polymer dynamics. The dielectric normal mode is associated with the lowest order Rouse-mode for oligomers and the lowest order reptation mode for polymers.

P(t) =e₀(e_u -1) E₀ +e₀ De(t) E₀

The orientational polarization, P_or is related to r_N, the number density of dipoles, p, in the electric field E at temperature T. The polymer chain is a chain composed of Kuhn dipoles and the transverse and longitudinal components of the resulting orientational dipole moment are related to the a-process and the normal mode relaxations respectively,

a-process

Normal Mode

where r_s is the number density of Kuhn segments.

Since neither the a-process or the normal mode relaxations are well described by a single relaxation time, the Havriliak-Nagami equation is used to describe the frequency dependence of the relaxation strength,

In a Cole-Cole plot, at the low frequency side (high e'), b₁ Å d(log e")/d(log w) and at the high frequency side (low e'), b₂ Å - d(log e")/d(log w). Typically, b₁ Å 0.5 and b₂ Å 0.7 for the a-process and b₁ Å 1 and b₂ Å 0.4 for the normal mode.

From a dynamics perspective, the glass transition is a kinetic phenomena. That is, if the view of the WLF equation is taken, the properties of a glass could be considered to represent an exceedingly viscous liquid. In fact this is not exactly the case since well below the glass transition temperature a-relaxations are physically prevented through steric interference. That is, at some point tin cooling, a-type relaxations will not occur no matter how long one waits. An alternative interpretation of the glass transition involves the view that it is a second order transition. A second order transition displays a discontinuity in the derivative of the specific volume (1/r) or the enthalpy. Figure 5.24 of Strobl shows the discontinuity in specific volume for polyvinylacetate that occurs at 25 to 30°C.