Mechanical retardation and relaxation in liquids

Summary

According to linear viscoelastic theory, the expressions for the complex compliance in cyclic shear at angular frequency,ω, and for the creep function are:

$$J*\sqrt 2 \left( {j\omega } \right) = J_\infty + 1/j\omega \eta + J_r \chi \left( {j\omega } \right)$$

([1])

$$J\left( t \right) = J_\infty + t/\eta + J_r \psi \left( t \right)$$

([2])

The normalised functionsχ(jω) andψ(t) are related through the spectrum of retardation times.

Recent work in the author's laboratory has shown that for non-polymeric liquidsχ(jω) can be closely represented by the empiricalDavidson-Cole expression used extensively to describe dielectric relaxation:

$$\chi \left( {j\omega } \right) = \frac{1}{{\left( {1 + j\omega \tau _r } \right)^\beta }}$$

([3])

τ _r is a characteristic retardation time parameter and 0 <β < 1. For supercooled liquids,β is typically 0.5, in which case eq. [1] may be rewritten:

$$J*\left( {j\omega } \right) = J_\infty \left[ {1 + \frac{1}{{j\omega \tau _m }}} \right] + \frac{{J_r }}{{\left( {1 + j\omega \tau _r } \right)^{\tfrac{1}{2}} }}$$

([4])

τ _m =ηJ _∞ is theMaxwell relaxation time.

Eq. [4] is fitted to experimental results for a number of liquids: the corresponding functionψ(t) in the time domain is found to fit the creep data ofPlazek for 1,3,5-tri-(α-naphthyl) benzene.

In practiceτ _r >τ _m below the “Arrhenius temperature” and forωτ _r ≫ 1 eq. [4] reduces to the form of theBarlow, Erginsav andLamb equation used previously:

$$J*\left( {j\omega } \right) = J_\infty \left[ {1 + \frac{1}{{j\omega \tau _m }}} \right] + \frac{{2J_r }}{{\left( {1 + j\omega \tau _r } \right)^{\tfrac{1}{2}} }}$$

([5])

withJ _r J _∞ =2(τ _r /τ _m )^1/2.

A review is given of available experimental results which confirm the applicability of eq. [4] or, in more approximate form, eq. [5]. It has also been shown that within experimental accuracy:

• a) J _∞ is a linear function of temperature, and

• b) G _∞(=1J _∞) is a linear function of pressure.

In the case of polymer melts, the behaviour at high frequencies where molecular movements are restricted to small elements of the polymer chain is found to be similar to that observed in non-polymeric liquids, eq. [5]. Additional contributions are found at lower frequencies due to entanglement effects. Results are given of preliminary measurements on a range of polystyrenes of narrow molecular weight distribution.