Relaxation patterns of long, linear, flexible, monodisperse polymers: BSW spectrum revisited

Abstract

Theoretical predictions for the dynamic moduli of long, linear, flexible, monodisperse polymers are summarized and compared with experimental observations. Surprisingly, the predicted 1/2 power scaling of the long-time modes of the relaxation spectrum is not found in the experiments. Instead, scaling with a power of about 1/4 extends all the way up to the longest relaxation times near τ/τ _max = 1. This is expressed in the empirical relaxation time spectrum of Baumgaertel-Schausberger-Winter, denoted as “BSW spectrum,” and justifies a closer look at the properties of the BSW spectrum. Working with the BSW spectrum, however, is made difficult by the fact that hypergeometric functions occur naturally in BSW-based rheological material functions. BSW provides no explicit solutions for the dynamic moduli, G ^′(ω), G ^″(ω), or the relaxation modulus G(t). To overcome this problem, close approximations of simple analytical form are shown for these moduli. With these approximations, analysis of linear viscoelastic data allows the direct determination of BSW parameters.

Appendices

Appendix 1: Asymptotes to the dynamic moduli

The approximate solution has to satisfy Eqs. 20 and 21 at low and at high frequencies, (ωτ _max < <1) and (ωτ _max > >1). The low frequency asymptotes of the dynamic moduli of BSW are

$$\begin{array}[b]{ll} &{\kern-5pt} G_{\omega \tau _{\max } <<1}^\prime \left( \omega \right)=\displaystyle\frac{n_{\rm e} }{2+n_{\rm e} }\;G_N^0 \left( {\omega \tau _{\max } } \right)^2;\\[17pt] &{\kern-5pt} G_{\omega \tau _{\max } <<1}^{\prime\prime} \left( \omega \right)=\displaystyle\frac{n_{\rm e} }{1+n_{\rm e} }G_N^0 \,\omega \tau _{\max } . \end{array}$$

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The low-frequency asymptotes intersect at (see Fig. 5)

$$\omega _x =\displaystyle\frac{1}{\tau _{\max } }\frac{2+n_{\rm e} }{1+n_{\rm e} }.$$

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The corresponding high frequency asymptotes are

$$\begin{array}{rll}G_{\omega \tau _{\max } >>1}^\prime \left( \omega \right)&=&\;G_N^0 ; \\ G_{\omega \tau _{\max } >>1}^{\prime\prime} \left( \omega \right)&=&\frac{{n_{\rm e} \pi } \mathord{\left/ {\vphantom {{n_{\rm e} \pi } 2}} \right. \kern-\nulldelimiterspace} 2}{\cos \left( {{n_{\rm e} \pi } \mathord{\left/ {\vphantom {{n_{\rm e} \pi } 2}} \right. \kern-\nulldelimiterspace} 2} \right)}G_N^0 \left( {\omega \tau _{\max } } \right)^{-n_{\rm e} }. \end{array}$$

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Appendix 2: Incomplete gamma function for G(t) and approximate solution

For completeness, we present the approximation of the relaxation function obtained for BSW spectrum. Again, we start out with the expression of Carri and Winter (1997). The BSW-spectrum solution for Eq. 2,

$$G\left( t \right)=n_{\rm e} G_N^0 \left( {\frac{t}{\tau _{\max }^{\rm BSW} }} \right)^{n_{\rm e} }\Gamma \left( {-n_{\rm e} ,\frac{t}{\tau _{\max }^{\rm BSW} }} \right),$$

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includes an incomplete Gamma function that can be replaced by a suitable approximation. The result is a polynomial expression multiplied by an exponential decay:

$$G\left( t \right)_{\rm app}\! =\!G_N^0 \!\left[\! {1\!+\!\left( {\frac{t}{n_{\rm e} \tau _{\max }^{\rm BSW} }}\!\! \right)^{\!\frac{\alpha n_{\rm e}^\beta }{3}}\!+\!\left( {\frac{t}{n_{\rm e} \tau _{\max }^{\rm BSW} }}\!\! \right)^{\!\alpha \,n_{\rm e}^\beta }}\! \right]^{-\frac{1}{\alpha n_{\rm e}^\beta }}\!\!e^{-\frac{t}{\tau _{\max }^{\rm BSW} }}\!.$$

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An optimized fit is achieved with α = 1.23 and β = 0.18. The approximation is chosen such that the equation is an exact solution for the short time and the long time asymptotes (while neglecting the glass transition as stated in the beginning). These two limits are

$$G_{\rm app} \left( {t \mathord{\left/ {\vphantom {t {\tau _{\max } }}} \right. \kern-\nulldelimiterspace} {\tau _{\max } }<<1} \right)=G\left( {t \mathord{\left/ {\vphantom {t {\tau _{\max } }}} \right. \kern-\nulldelimiterspace} {\tau _{\max } }<<1} \right)=G_N^0 {, {\rm and}} $$

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$$\begin{array}{rll}G_{\rm app} \left( {t \mathord{\left/ {\vphantom {t {\tau _{\max } }}} \right. \kern-\nulldelimiterspace} {\tau _{\max } }>>1} \right)&=&G\left( {t \mathord{\left/ {\vphantom {t {\tau _{\max } }}} \right. \kern-\nulldelimiterspace} {\tau _{\max } }>>1} \right) \\ &=&n_{\rm e} G_N^0 \,e^{-\;\frac{t}{\tau _{\max }^{{BSW}} }}\left( {\frac{t}{\tau _{\max }^{{BSW}} }} \right)^{-1}. \end{array}$$

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Appendix 3: Same other interesting properties of BSW related moduli

The maximum of the loss modulus is a distinct data point from dynamic mechanical experiments. ω _max is the particular frequency for which $G^{\prime\prime}=G_{\max }^{\prime\prime} =G^{\prime\prime}(\omega_{\rm max})$. When predicted with BSW, $G_{\max }^{\prime\prime}$ posses properties that resemble the Maxwell model:

$$\frac{G_{\max }^{\prime\prime} }{G_N^0 }=\frac{x_{\max } }{1+x_{\max }^2 }$$

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with x _max = ω _max τ _max. Although this equation is of Maxwellian structure, the Maxwell result is not recovered because, for the same material parameters, ω _max is different for the Maxwell and the BSW spectra. Equation 32 can be derived from the integral equation for $G_{\max }^{\prime\prime}$,

$$\frac{G_{\max }^{\prime\prime} }{G_N^0 }=n_{\rm e} x_{\max }^{-n{ }_{\rm e}} \;\int\limits_0^{x_{\max } } {dx\frac{x^{n_{\rm e} }}{1+x^2}} ,$$

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together with the defining equation for the abscissa. The condition ${{d}G^{\prime\prime}} \mathord{\!\left/\! {\vphantom {{{d}G^{\prime\prime}} {\left. {{d}\omega } \right|_{\omega =\omega \max } }}} \right. \kern-\nulldelimiterspace} {\left. {{d}\omega } \right|_{\omega =\omega \max } }=0$ results in

$$-n_{\rm e} x_{\max }^{-1-n_{\rm e} } \int\limits_0^{x_{\max } } {dx\frac{x^{n_{\rm e} }}{1+x^2}} +\frac{1}{1+x_{\max }^2 }=0.$$

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The solution of Eq. 34 results in the value for x _max. For this purpose, we substitute the integral by the corresponding hypergeometric function and then transform the equation to receive the following result:

$$F\left( {1,1;\frac{n_{\rm e} +3}{2};\frac{x_{\max }^2 }{1+x_{\max }^2 }} \right)=\frac{n_{\rm e} +1}{n_{\rm e} }.$$

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Again, we approximate the hypergeometric function by a simpler function and arrive at

$$\left( {1+\left( {1+n_{\rm e} } \right)x_{\max }^2 } \right)^{0.5}\cong \frac{n_{\rm e} +1}{n_{\rm e} },$$

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which leads to an explicit expression for x _max:

$$x_{\max } \cong \frac{1}{n_{\rm e} }\sqrt {\frac{2n_{\rm e} +1}{n_{\rm e} +1}} \approx \frac{1}{n_{\rm e} }.$$

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Together with Eq. 32, we obtain the final result:

$$\frac{G_{\max }^{\prime\prime} }{G_N^0 }=\frac{n_{\rm e} \left( {1+n_{\rm e} } \right)^{0.5}\left( {1+2n_{\rm e} } \right)^{0.5}}{1+2n_{\rm e} +n_{\rm e}^2 +n_{\rm e}^3 },$$

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or

$$\frac{G_{\max }^{\prime\prime} }{G_N^0 }\approx \frac{n_{\rm e} }{1+n_{\rm e}^2 }$$

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that relates the value of the plateau modulus to the maximal value of the G ^″ for a known relaxation exponent n _e.