Analytic slip-link expressions for universal dynamic modulus predictions of linear monodisperse polymer melts

Abstract

The discrete slip-link model (DSM) is a robust mesoscopic theory that has great success predicting the rheology of flexible entangled polymer liquids and gels. In the most coarse-grained version of the DSM, we exploit heavily the universality observed in the shape of the relaxation modulus of linear monodisperse melts. For this type of polymer, we present here analytic expressions for the relaxation modulus. The high-frequency dynamics which are typically coarse-grained out from the DSM are added back into these expressions by using a Rouse chain with fixed ends to represent the fast motion of Kuhn steps between entanglements. We find consistency in the friction used for both fast and slow modes. We test these expressions against experimental data for three chemistries and molecular weights with good agreement. Using these analytic expressions, the polymer density, the molecular weight of a Kuhn step, M _K, and the low-frequency cross-over between the storage and loss moduli, $G^{\prime }$ and $G^{\prime \prime }$, it is now straightforward to estimate model parameter values and obtain predictions over the experimentally accessible frequency range without performing expensive numerical calculations.

Appendices

Appendix 1: Derivation of rouse modes

We assume that the SD and CD processes are much slower than the motion of individual Kuhn steps so that we can treat the entanglements as effective cross-links for each strand of the chain. Thus, we have a Rouse strand (Rouse 1953) with fixed ends. In the continuous limit, the Langevin equation for r _s is given by

$$\begin{array}{@{}rcl@{}} \frac{\partial{\boldsymbol r}_{s}}{\partial t}=\frac{1}{\tau_{\mathrm{K}}} \frac{\partial^{2}{\boldsymbol r}_{s}}{\partial s^{2}}+{\boldsymbol f}_{s}(t)~, \end{array} $$

(A1)

where r _s is the position in space of a bead at position s (0≤s≤l) along the strand contour, $\tau _{\mathrm {K}}=\frac {\zeta _{\mathrm {K}}a_{\mathrm {K}}^{2}}{k_{\mathrm {B}}T}$, and f _s (t) is the fluctuating force representing the interactions of the beads with the mean field. According to the fluctuation dissipation theorem (FDT) (Kubo 1966; Van Kampen 1992), the moments of the random forces are given by

$$\begin{array}{@{}rcl@{}} \left<{\boldsymbol f}_{s}(t)\right>&=&{\boldsymbol 0}\\ \left<{\boldsymbol f}_{s}(t){\boldsymbol f}_{u}(t^{\prime})\right>&=&\frac{2a_{\mathrm{K}}^{2}} {\tau_{\mathrm{K}}}\delta(s-u)\delta(t-t^{\prime}){\boldsymbol \delta}. \end{array} $$

(A2)

The boundary conditions for the network strand are

$$\begin{array}{@{}rcl@{}} {\boldsymbol r}_{0}&=&{\boldsymbol 0} \end{array} $$

(A3)

$$\begin{array}{@{}rcl@{}} {\boldsymbol r}_{l}&=&{\boldsymbol Q}. \end{array} $$

(A4)

These boundary conditions are satisfied by the normal mode solution

$$\begin{array}{@{}rcl@{}} {\boldsymbol r}_{s}=\frac{s}{l}{\boldsymbol Q}+\sum\limits_{p=1}^{\infty}{\boldsymbol x}_{p} \sin\left(\frac{s\pi p}{l}\right), \end{array} $$

(A5)

in terms of the normal mode coordinates x _p defined by

$$\begin{array}{@{}rcl@{}} {\boldsymbol x}_{p}:={{\int}_{0}^{l}}{\boldsymbol r}_{s}\sin\left(\frac{s\pi p}{l}\right)\mathrm{d}s \,\,\text{with}\,\, p=0,1,2,... \end{array} $$

(A6)

We multiply each side of Eq. A1 by $\sin \left (\frac {s\pi m}{l}\right )$ and integrate s from 0 to l to find

$$\begin{array}{@{}rcl@{}} \frac{\mathrm{d}{\boldsymbol x}_{m}}{\mathrm{d}t}&=&-\frac{1}{\lambda_{m}}{\boldsymbol x}_{m} +\hat{{\boldsymbol f}}_{m} \end{array} $$

(A7)

where $\lambda _{m}:=\left (\frac {l}{\pi m}\right )^{2}\tau _{\mathrm {K}}$ and ${\hat {\boldsymbol f}_{m}}:=\frac {2\pi }{l} {{\int }_{0}^{l}}{\boldsymbol f}_{s}(t)\sin \left (\frac {s\pi m}{l}\right )\mathrm {d}s$.

Using the FDT (Eq. A2), the Langevin equation can be written as the stochastic differential equation

$$\begin{array}{@{}rcl@{}} \mathrm{d}{\boldsymbol x}_{m}(t)&=&-\frac{1}{\lambda_{m}}{\boldsymbol x}_{m}(t)\mathrm{d}t +\sqrt{\frac{4\pi a_{\mathrm{K}}^{2}}{l\tau_{\mathrm{K}}}}\mathrm{d}{\boldsymbol W}_{m}(t)~. \end{array} $$

(A8)

This is an Ornstein-Uhlenbeck process where dW _m(t) is a Wiener increment. Equation A8 can be solved using Itô calculus with solution (Gardiner 2009)

$$\begin{array}{@{}rcl@{}} {\boldsymbol x}_{m}(t)={\boldsymbol x}_{m}(0)e^{-t/\lambda_{m}}+ \sqrt{\frac{4\pi a_{\mathrm{K}}^{2}}{l\tau_{\mathrm{K}}}} {{\int}_{0}^{t}}e^{-(t-t^{\prime})/\lambda_{m}}\mathrm{d}{\boldsymbol W}_{m}(t^{\prime})~, \end{array} $$

(A9)

and cross-correlation functions given by

$$\begin{array}{@{}rcl@{}} \left<{\boldsymbol x}_{m}(t){\boldsymbol x}_{n}(0)\right>_{\text{eq}}=\left<{\boldsymbol x}_{m}(0){\boldsymbol x}_{n}(0)\right>_{\text{eq}}e^{-t/\lambda_{m}}\delta_{mn}~. \end{array} $$

(A10)

We need the second moment which can be found from the free energy of a strand given in Eq. 8. Using the same procedure as before, we can write the free energy in normal modes

$$\begin{array}{@{}rcl@{}} F_{s}&=&\frac{3k_{\mathrm{B}}T}{2a_{\mathrm{K}}^{2}}\left(\frac{Q^{2}}{l} +\sum\limits_{p=1}^{\infty}\left(\frac{\pi p}{l}\right)^{2}{\boldsymbol x}_{p}\cdot{\boldsymbol x}_{p} \frac{l}{2\pi}\right) \end{array} $$

(A11)

$$\begin{array}{@{}rcl@{}} &=&\frac{3k_{\mathrm{B}}T}{2a_{\mathrm{K}}^{2}}\left(\frac{Q^{2}}{l} +\frac{\pi}{2l} \sum\limits_{p=1}^{\infty} p^{2}{x_{p}^{2}}\right) \end{array} $$

(A12)

From here, we can see that the second and first moments for a Gaussian distributed function are given by

$$\begin{array}{@{}rcl@{}} \left<{\boldsymbol x}_{p}(0){\boldsymbol x}_{q}(0)\right>=\frac{2la_{\mathrm{K}}^{2}}{3p^{2}\pi}{\boldsymbol\delta}\delta_{pq};~~~~ \left<{\boldsymbol x}_{p}(0)\right>={\boldsymbol 0}~, \end{array} $$

(A13)

where δ is the identity tensor and δ _pq is the Kronecker delta function. The stress in a strand is calculated according to Eq. 9

$$\begin{array}{@{}rcl@{}} \hat{{\boldsymbol \tau}_{s}}&=&{-}\frac{3k_{\mathrm{B}}T}{a_{\mathrm{K}}^{2}} {{\int}_{0}^{l}}\frac{\partial{\boldsymbol r}_{s}}{\partial s}\frac{\partial{\boldsymbol r}_{s}}{\partial s}\mathrm{d}s \end{array} $$

(A14)

$$\begin{array}{@{}rcl@{}} &=&{-}\frac{3k_{\mathrm{B}}T}{a_{\mathrm{K}}^{2}}\left[\frac{{\boldsymbol{QQ}}} {l}+\frac{\pi}{2l} \sum\limits_{p=1}^{\infty} p^{2}{\boldsymbol x}_{p}{\boldsymbol x}_{p}\right] \end{array} $$

(A15)

The relaxation of a strand can be calculated from the Green-Kubo expression resulting in Eq. 10. For the entire chain, the relaxation modulus is given by

$$\begin{array}{@{}rcl@{}} G_{\mathrm{R}}(t)&=&n_{c} \sum\limits_{Z=1}^{N_{\mathrm{K}}}\sum\limits_{s=1}^{Z} \sum\limits_{l=1}^{N_{\mathrm{K}}}{\int}_{-\infty}^{\infty} p(l,Z,{\boldsymbol Q})G_{s}^{\mathrm{R}}(t|l,{\boldsymbol Q})\mathrm{d}{\boldsymbol Q}~. \end{array} $$

(A16)

By definition of the conditional probability,

$$\begin{array}{@{}rcl@{}} \sum\limits_{Z=1}^{N_{\mathrm{K}}}\sum\limits_{s=1}^{Z} \sum\limits_{l=1}^{N_{\mathrm{K}}}p(l,Z,\boldsymbol{Q})\!\!&=&\!\!\sum\limits_{Z=1}^{N_{\mathrm{K}}} \sum\limits_{s=1}^{Z}\sum\limits_{l=1}^{N_{\mathrm{K}}}p\!(Z)\!p(l,\!\boldsymbol{Q}|Z) \end{array} $$

(A17)

$$\begin{array}{@{}rcl@{}} &\cong&\sum\limits_{Z=1}^{N_{\mathrm{K}}}\sum\limits_{l=1}^{N_{\mathrm{K}}}p(Z)Zp(l,\boldsymbol{Q}) \end{array} $$

(A18)

$$\begin{array}{@{}rcl@{}} &=&\left<Z\right>_{\text{eq}}\sum\limits_{l=1}^{N_{\mathrm{K}}}p(l,\boldsymbol{Q}) \end{array} $$

(A19)

for chains where Z is large. Using these results in Eq. A16, we obtain the final expressions shown in “High-frequency dynamics.”

Appendix 2: Analysis of relaxation modulus predictions

The relaxation modulus, G(t), can be represented using a continuous spectrum of relaxation times, h(τ),

$$\begin{array}{@{}rcl@{}} G(t)=G_{0}{\int}_{0}^{\infty}\frac{h(\tau)} {\tau}\exp\left(-\frac{t}{\tau}\right)\mathrm{d}\tau \end{array} $$

(B1)

as suggested by Baumgaertel et al. (1990) (BSW). This function allows us to convert the G(t) of a DSM prediction from the time to the frequency domain. Here, we use a modified BSW spectrum with m modes, which has the form

$$\begin{array}{@{}rcl@{}} h(\tau)=\frac{\displaystyle\sum\limits_{i=1}^{m} \tau^{\alpha_{i}} H(\tau_{i}-\tau)H (\tau-\tau_{i-1}){\prod}_{j=1}^{i-1} \tau_{j}^{\alpha_{j}-\alpha_{j+1}}} {\displaystyle\sum\limits_{i=1}^{m}\frac{\tau_{i}^{\alpha_{i}} -\tau_{i-1}^{\alpha_{i}}}{\alpha_{i}} {\prod}_{j=1}^{i-1}\tau_{j}^{\alpha_{j}-\alpha_{j+1}}}~, \end{array} $$

(B2)

where H(x) is the Heaviside step function. The complex modulus, G ^∗, is calculated from the one-sided Fourier transform of the relaxation modulus multiplied by i ω, with {α _i ,τ _i } as fitting parameters (Khaliullin and Schieber 2009)

$$\begin{array}{@{}rcl@{}} G^{\ast}&=& i\omega{\int}_{0}^{\infty}G(t) e^{-i\omega t}\mathrm{d}t~, \end{array} $$

(B3)

where the storage modulus, $G^{\prime }$, is given by

$$\begin{array}{@{}rcl@{}} G^{\prime}(\omega)\!&=&\!G_{0}\omega^{2} \displaystyle\sum\limits_{i=1}^{m}\frac{\displaystyle {\prod}_{j=0}^{i-1}\tau_{j}^{\alpha_{j}-\alpha_{j+1}}} {\alpha_{i}+2}\left[{~}{2}F_{1}\left(1,\frac{\alpha_{i}+2}{2}; \frac{\alpha_{i}+4}{2};-\omega^{2}{\tau_{i}^{2}}\right) \tau_{i}^{\alpha_{i}+2}\right.\\ & &\left.-{~}_{2}F_{1}\left(1,\frac{\alpha_{i}+2}{2}; \frac{\alpha_{i}+4}{2};-\omega^{2}\tau_{i-1}^{2}\right) \tau_{i-1}^{\alpha_{i}+2}\right]\bigg/\displaystyle \sum\limits_{i=1}^{m}\frac{\displaystyle{\prod}_{j=0}^{i-1} \tau_{j}^{\alpha_{j}-\alpha_{j+1}}(\tau_{i}^{\alpha_{i}} -\tau_{i-1}^{\alpha_{i}})}{\alpha_{i}}~, \end{array} $$

(B4)

and the loss modulus, $G^{\prime \prime }$, is given by

$$\begin{array}{@{}rcl@{}} G^{\prime\prime}(\omega)&=&G_{0}\omega \displaystyle\sum\limits_{i=1}^{m}\frac{\displaystyle {\prod}_{j=0}^{i-1}\tau_{j}^{\alpha_{j}-\alpha_{j+1}}} {\alpha_{i}+1}\left[{~}_{2}F_{1}\left(1,\frac{\alpha_{i}+1} {2};\frac{\alpha_{i}+3}{2};-\omega^{2}{\tau_{i}^{2}}\right) \tau_{i}^{\alpha_{i}+1}\right.\\ & &\left.-{~}_{2}F_{1}\left(1,\frac{\alpha_{i}+1}{2}; \frac{\alpha_{i}+3}{2};-\omega^{2}\tau_{i-1}^{2}\right) \tau_{i-1}^{\alpha_{i}+1}\right]\bigg/\displaystyle \sum\limits_{i=1}^{m}\frac{\displaystyle{\prod}_{j=0}^{i-1} \tau_{j}^{\alpha_{j}-\alpha_{j+1}}(\tau_{i}^{\alpha_{i}} -\tau_{i-1}^{\alpha_{i}})}{\alpha_{i}}~, \end{array} $$

(B5)

where ₂ F ₁(a,b;c;z) is the hypergeometric function (Abramowitz et al. 1972).