Derivation of the “switch function” in the Mead–Larson–Doi theory

Abstract

The “switch function” is rigorously derived from first principles and is shown to be a fundamental and essential feature of any constraint release model for Doi–Edwards-type molecular models that invoke the concept of a discrete slip-link tube with separate descriptions of orientation and stretch. The switch function self-consistently apportions the fraction of Kuhn bond orientation relaxation attributed to tube stretch and orientation such that the same net fraction of Kuhn bond orientational relaxation per constraint release event occurs independent of the level of tube stretch, provided the chain obeys Gaussian statistics. The switch function is derived for the particular case of the Mead–Larson–Doi (MLD) model, with constraint release-driven tube shortening quantified by $\frac{1}{2}(\lambda -1)$. The MLD switch function is generalized to account for arbitrary non-Gaussian, levels of finite tube stretch. The MLD model tube-shortening process is also impacted by non-Gaussian, finite extensibility, effects and is generalized utilizing a second-moment Kuhn–Grün analysis. It is shown by performing a fourth-moment analysis of the Kuhn bond orientation that the same switch function derived by analyzing the second moment also applies to the fourth moment in the small stretch Gaussian limit.

Appendix: Asymptotic orientation in fast uniaxial flow

In this appendix, we calculate the steady-state asymptotic orientation in fast, steady uniaxial extensional flow of the monodisperse MLD toy model (Mead et al. 1998; Mead 2007). In this way, we demonstrate that using the non-Gaussian switch function derived above rather than the Gaussian switch function is absolutely necessary to achieve a physically sensible result for the segmental orientation in very fast flows. The principal point to this simple exercise is to demonstrate that even small amounts of non-Gaussian character can significantly alter the nature of the predicted material response. This is the first, but by no means the last, demonstration of the importance of including the effects of non-Gaussian chain statistics in molecular models of polymers.

We consider the case of a monodisperse polymer in steady fast uniaxial extension where $\dot{{\varepsilon }}\tau_{\rm s} \gg 1$. Here τ _s is the stretch relaxation time. We will assume that the flow is so fast that the finitely extensible chain is nearly fully stretched, λ ≈ λ _max. The steady-state monodisperse orientational relaxation time τ can be approximated as

$$ \frac{1}{\tau }\approx \frac{1}{\lambda_{\max } }\dot{{\varepsilon }}\left( {S_{zz}^\infty -S_{xx}^\infty } \right) $$

(23)

where λ _max is the maximum stretch of a finitely extensible chain. Note that we have used the Gaussian form of the switch function in Eq. 23. With this expression for the steady-state relaxation time, we can calculate the steady-state asymptotic orientation, $S_{zz}^\infty -S_{xx}^\infty$, in fast stretching uniaxial flow as

$$ S_{zz}^\infty -S_{xx}^\infty =\int\limits_o^\infty {\frac{1}{\dot{{\varepsilon }}\tau }\exp \left\{ {-\frac{x^\prime }{\dot{{\varepsilon }}\tau }} \right\}} \left( {Q_{zz} \left( {x^\prime } \right)-Q_{xx} \left( {x^\prime } \right)} \right)\mbox{d}x^\prime $$

(24)

where $x^\prime \equiv \dot{{\varepsilon }}\left( {t-t^\prime } \right)$.

If $\dot{{\varepsilon }}\tau \gg 1$, then Q _zz  − Q _xx →1 for all x ^′, and the above equation can be trivially integrated yielding $S_{zz}^\infty -S_{xx}^\infty \;\approx 1$. Of course, this result is physically satisfying since it is known that fast uniaxial flow will result in essentially full orientation of the chain segments. However, when the Gaussian switch function is used, $\dot{{\varepsilon }}\tau \approx \frac{\lambda_{\max } }{S_{zz}^\infty -S_{xx}^\infty }$. Although this parameter is greater than unity, it is certainly not much greater than unity. Hence, the criteria for the validity of the above analysis yielding $S_{zz}^\infty -S_{xx}^\infty \;\approx 1$ are not met. Therefore, whenever the Gaussian switch function is used, the asymptotic orientation will be significantly less than unity, $S_{zz}^\infty -S_{xx}^\infty \;<1$, despite the fact that significant stretch is occurring. This result is unphysical since from simple symmetry considerations, it is obvious that $S_{zz}^\infty -S_{xx}^\infty \;\to 1$ for$\,\dot{{\varepsilon }}\tau_s \gg 1$, that is, full orientation whenever stretch occurs. An asymptotic analysis can be performed to quantitatively determine the asymptotic orientation as a function of λ _max. This analysis will determine the precise magnitude of the error involved in calculating the orientation in fast flows when the Gaussian switch function is used. An example of how to perform this analysis can be found in the appendix of the original MLD article (Mead et al. 1998). Essentially, the analysis consists of iteratively solving for $S_{zz}^\infty -S_{xx}^\infty \;$ given the prescribed values of $\dot{{\varepsilon }}$ and λ _max in Eq. 3.

$${\begin{array}{rll} S_{zz}^\infty -S_{xx}^\infty &=&\int\limits_o^\infty {\frac{\left( {S_{zz}^\infty -S_{xx}^\infty } \right)}{\lambda_{\max } }\exp \left\{ {-x^\prime \frac{\left( {S_{zz}^\infty -S_{xx}^\infty } \right)}{\lambda _{\max } }} \right\}} \notag \\ &&\times \left( {Q_{zz} \left( {x^\prime } \right)-Q_{xx} \left( {x^\prime } \right)} \right)\mbox{d}x^\prime \end{array}} $$

(25)

In this manner, calculations of the steady-state asymptotic orientation, $S_{zz}^\infty -S_{xx}^\infty$, versus λ _max can be made by letting $\dot{{\varepsilon }}\to \infty$ and determining the limiting value of $S_{zz}^\infty -S_{xx}^\infty$. More generally, the orientation will be a function of both $\dot{{\varepsilon }}$ and λ. One point that should be noted is that, unlike in the original monodisperse Doi-Edwards-Marrucci-Grizzuti model, orientation and stretch are not widely separated events. Stretch commences prior to full orientation of the tube. Full orientation of the tube only occurs after significant stretch has already occurred. The cause of this change is Convective Constraint Release, which brings the orientation and stretch processes closer together.

The cause of the above difficulty in calculating the asymptotic orientation is that we have not used the non-Gaussian form of the switch function. When this is done, the expression for the dimensionless parameter $\dot{{\varepsilon }}\tau$ for fast flows is modified to $\dot{{\varepsilon }}\tau \approx \frac{\frac{\lambda_{\max } }{3}L^{-1}\left( {\frac{\lambda }{\lambda_{\max } }} \right)}{S_{zz}^\infty -S_{xx}^\infty } \gg 1$. This parameter is necessarily extremely large and much greater than unity in fast stretching flows due to the singular nature of the inverse Langevin function when the argument approaches unity. Since $\dot{{\varepsilon }}\tau_s \gg 1$, we know that the inverse Langevin function will be extremely large. Hence, the criteria for full orientation in fast flows are met when the non-Gaussian switch function is used but not when the Gaussian version is used. The differences between using the Gaussian or non-Gaussian switch functions in the calculated orientation can be quite significant.

The principal point to this simple calculation is to demonstrate that the non-Gaussian switch function is an essential physical feature of the model whose importance is manifested in many disparate ways even when relatively low levels of non-Gaussian character are present. The above calculation clearly demonstrates that without the non-Gaussian switch function, even the most basic model parameters, in this case, the segmental orientation, are corrupted in very fast flows. Since the orientation enters into calculation of the stretch, this model parameter will also be corrupted in fast flows. Although this calculation was performed for uniaxial flows, the same conclusions apply to any fast extensional flow field where significant stretch is anticipated.