Fractional modelling of functionalized CNT suspensions

Abstract

Experimental findings and rheological modelling of chemically treated single-wall carbon nanotubes suspended in an epoxy resin were addressed in a recent publication (Ma et al., J Rheol 53:547–573, 2009). The shear-thinning behaviour was successfully modelled by a Fokker-Planck-based orientation model. However, the proposed model failed to describe linear viscoelasticity using a single mode as well as the relaxation after applying a finite step strain. Both experiments revealed a power-law behaviour for the storage and relaxation moduli. In this paper, we show that a single-mode fractional diffusion model is able to predict these experimental observations.

Appendices

Appendix: A: On fractional derivatives

There are many books on fractional calculus and fractional differential equations (e.g. Kilbas et al. 2006; Podlubny 1999). We summarize here the main concepts needed to understand the developments carried out in this paper.

We start with the formula usually attributed to Cauchy for evaluating the nth integration, $n \in \mathbb {N}$, of a function f(t)

$$ J^{n}f(t):=\int {\cdots} {\int}_{\!\!\!\!0}^{t} f(\tau) \ d\tau = \frac{1}{(n-1)!} {\int}_{\!\!\!\!0}^{t} (t-\tau)^{n-1} f(\tau) \ d\tau. $$

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This can be rewritten as

$$ J^{n}f(t) = \frac{1}{\Gamma(n)} {{\int}_{\!\!\!\!0}^{t}} (t-\tau)^{n-1} f(\tau) \ d\tau $$

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where Γ(n) = (n−1)! is the gamma function. In the latter being in fact defined for any real value $\alpha \in \mathbb {R}$, we can define the fractional integral from

$$ J^{\alpha} f(t) := \frac{1}{\Gamma(\alpha)} {{\int}_{\!\!\!\!0}^{t}} (t-\tau)^{\alpha-1} f(\tau) \ d\tau. $$

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Now, if we consider the fractional derivative order (α), we select an integer $m \in \mathbb {N}$ such that m− 1 < α < m, and it suffices to consider an integer m-order derivative combined with a (m−α) fractional integral. Obviously, we could take the derivative of the integral or the integral of the derivative, resulting in the left- and right-hand definitions of the fractional derivative usually denoted by D ^α f(t) and $D^{\alpha }_{\ast } f(t)$, respectively.

Because these approaches to the fractional derivative began with an expression for the repeated integration of a function, one could consider a similar approach for the derivative. This was the route considered by Grunwald and Letnikov (GL) that defined the so-called ‘differintegral’ that leads to the fractional counterpart of the usual finite differences. In the present work, we use the GL definition of the fractional derivative.

It turns out that the composition of fractional derivatives follows a rule similar to that for standard derivatives. On the other hand, the Fourier transform of a fractional derivative of order α reads $\mathcal {F}(g(t);\omega ) = (i\omega )^{\alpha }\mathcal {G}(\omega )$. This property is particularly useful when addressing harmonic responses as in the case of LVE experiments.

Appendix: B: Derivation of the fractional derivative of the orientation tensor

We discuss the contribution of fractional diffusion to the rod rotary velocity (the flow-induced contribution remains unchanged)

$$ \left . \frac{d^{\alpha} \mathbf{p}}{dt^{\alpha}} \right |^{B} = - D_{r} \frac{\frac{\partial \psi}{\partial \mathbf {p}}}{\psi}. $$

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Now, we consider the second-order orientation tensor

$$ \mathbf{a} = {\int}_{\!\!\!\!\mathcal{S}} \mathbf{p} \otimes \mathbf{p} \ \psi \ d\mathbf{p} $$

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whose time derivative can be rewritten as

$$ \dot{\mathbf{a}} |^{B} = \frac{d^{1-\alpha}}{d^{1-\alpha}} \left\{ \frac{d^{\alpha}}{dt^{\alpha}} \left\{ {\int}_{\!\!\!\!\mathcal{S}} \left( \mathbf{p} \otimes \mathbf{p} + \mathbf{p} \otimes \mathbf{p} \right) \psi \ d\mathbf{p} \right\} \right\} $$

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or

$$ \dot{\mathbf{a}} |^{B} = \frac{d^{1-\alpha}}{d^{1-\alpha}} \left\{ {\int}_{\!\!\!\!\mathcal{S}} \frac{d^{\alpha}}{dt^{\alpha}} \left( \mathbf{p} \otimes \mathbf{p} + \mathbf{p} \otimes \mathbf{p} \right) \psi \ d\mathbf{p} \right\}. $$

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Considering the first term of Leibnitz’s rule related to the fractional derivative of a product of functions (it is easy to prove that the second one leads to the standard diffusion integer term, while the others can be neglected), we obtain

$$ \dot{\mathbf{a}} |^{B} \approx \frac{d^{1-\alpha}}{d^{1-\alpha}} \left\{ {\int}_{\!\!\!\!\mathcal{S}} \left( \frac{d^{\alpha} \mathbf{p}}{dt^{\alpha}} \otimes \mathbf{p} + \mathbf{p} \otimes \frac{d^{\alpha} \mathbf{p}}{dt^{\alpha}} \right) \psi \ d\mathbf p \right\} $$

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or

$$ \dot{\mathbf{a}} |^{B} \approx -D_{r}\frac{d^{1-\alpha}}{d^{1-\alpha}} \left\{ {\int}_{\!\!\!\!\mathcal{S}} \left(\frac{\frac{\partial \psi}{\partial \mathbf{p}}}{\psi} \otimes\mathbf{p} + \mathbf{p} \otimes \frac{\frac{\partial \psi}{\partial\mathbf{p}}}{\psi} \right) \psi \ d\mathbf{p} \right\} , $$

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which finally gives

$$ \dot{\mathbf{a}} |^{B} \approx -2 d D_{r}\frac{d^{1-\alpha}}{d^{1-\alpha}} \left( \mathbf{a} - \frac{\mathbf{I}}{d} \right) . $$

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