The rheological modelling of carbon nanotube (CNT) suspensions in steady shear flows

This paper is concerned with the rheological modelling of both chemically treated and untreated carbon nanotube (CNT) suspended in a Newtonian epoxy resin. CNT suspensions generally exhibited shear-thinning characteristic—the apparent viscosity decreases as shear rate increases—when subject to steady shear flows. Chemically treated CNT suspensions with little optical microstructure were found to exhibit a less significant shear-thinning effect compared with untreated CNT suspensions where clear optical aggregates were observed. In the case of treated CNT suspensions, the shear-thinning characteristic could be described using a Fokker–Planck based orientation model. The model assumed that the treated CNTs behaved as high aspect ratio rods and that shear flow was able to align the CNTs in the flow direction, thereby resulting in a decrease in the shear viscosity. Despite the success in describing the rheological response of treated CNT in steady shear flows, the orientation model failed to explain the more pronounced shear-thinning effect observed in untreated CNT suspensions having a hierarchy of aggregate structures. A new model called the aggregation/orientation (AO) model was formulated by modifying the Fokker–Planck equation. The AO model considered elements of aggregation as well as CNT orientation and it was capable of capturing the steady shear response of untreated CNT suspensions.


Introduction
The experimental rheology of carbon nanotube (CNT) suspensions has been studied by a number of authors (see for example [1][2][3]). In terms of steady shear measurements, addition of CNT to a Newtonian matrix has been found to increase the apparent viscosity at low shear and the viscosity decreases asymptotically to the matrix viscosity as shear rate increases [3][4][5]. This is known as steady shear-thinning and this type of behaviour is commonly observed in suspensions [6]. The extent of viscosity enhancement at low shear depends on a number of factors such as the state of dispersion as well as the type and aspect ratios of CNT used [7][8][9]. For untreated CNT suspensions where clear optical aggregates were present, an order of magnitude of increase in low shear viscosity could result by adding only 0.1% of CNTs [3,4]. In the case of chemically treated CNT suspensions, the viscosity enhancement effect tends to be less pronounced [5,7,9]. In this paper, the treated CNTs were supplied by Nanocomposites Inc. (USA) and they were chemically treated in a way such that aggregation was prevented by electrostatic repulsion between CNTs [10,11]. The untreated CNTs were produced by chemical vapour deposition (CVD) method [12] and were provided by the Department of Materials Science and Metallurgy, University of Cambridge.
This paper describes the modelling of steady shearthinning for both treated and untreated CNT suspensions.  Fig. 3 Fokker Planck based orientation model fitting to experimental data of untreated CNT suspensions at two selected CNT concentrations (concentra-tion=0.25% for a and c and concentration=0.1% for b and d). a and b showed the fittings with a fixed rotary diffusion coefficient (D r =0.0005 s −1 ) and a changing N p , whereas c and d showed the fittings with a fixed N p , but with an adjustable D r . Experimental data are represented by unfilled circles and the lines are the orientation model fits [4] where n 2 0; 1 ½ describes the state of aggregation (n=0 corresponds to CNTs that are free from entanglement and n=1 represents a CNT aggregate network); v c is the aggregation velocity and v d is the disaggregation velocity. The constitutive equation becomes: where the contribution of the rotary diffusion has been neglected and To simplify the analysis and minimize the number of rheological parameters, the following assumptions are made: 1. N p varies linearly with the population variable n. Moreover, as N min % 0 leading to Eq. 7a where N p ðnÞ % N max p n. 2. D r is mainly due to CNT interactions and it vanishes in the limit case of n=1 where the only orientation mechanism is affine deformation, as typically assumed in the case of associative polymers [21,22]. A linear dependence on the parameter n is assumed as described by Eq. 7b. 3. v d increases linearly with the shear rate and it takes the form as given in Eq. 7c where Á g max is a characteristic shear rate above which the suspension viscosity coincides with the matrix viscosity. At high shear rates, v d remains constant and v d ¼ v max d (Eq. 7d) 4. v c decreases linearly with the shear rate and it takes the form as given in Eq. 7e. It can be noted from Eq. 7f that as the shear rate becomes higher than Á g max , the aggregation velocity becomes zero.
The rheological model contains three independent fitting parameters: the concentration and aspect-ratio parameter N max p , the rotational diffusion coefficient D max r À Á and the ratio between the aggregation and disaggregation velocities b ¼ v c = v d . Figure 4 shows the best fits to experimental data with different CNT concentrations. In these fittings, it was assumed that only N max p depended on the CNT concentration and the other two rheological parameters were not a function of the concentration (i.e. D max As shown in Fig. 4, reasonable agreement between experimental results and the AO model was obtained with D max r =0.001 s −1 and β=0.004, further supporting the belief that the more pronounced shear-thinning in untreated CNT suspensions was due to the orientation as well as the aggregation of CNTs.

Conclusions
The steady shear-thinning characteristics of both chemically treated and untreated carbon nanotube (CNT) suspensions have been modelled. Experimentally, untreated CNT suspensions were found to exhibit a more pronounced shear-thinning effect compared with treated CNT suspensions. The comparatively moderate shear-thinning behaviour of treated CNT suspensions was successfully modelled using a Fokker-Planck based orientation model with two fitting parameters (N p and D r ). The model essentially assumed treated CNTs as short and rigid fibres that can align in shear flow and shear thinning was believed to be a result of progressive alignment of CNTs in the flow direction.
In the case of untreated CNT suspensions, the Fokker-Planck orientation model failed to describe the experimental shear thinning. In-situ optical observations clearly showed the presence of a hierarchy of CNT aggregate structures which evolved as the magnitude of shear rate changed. A new model named aggregation/orientation (AO) model was formulated. The model considered the effect of CNT orientation and CNT aggregation and the Fokker-Planck equation was modified accordingly. An additional fitting parameter representing the kinetics of CNT aggregation/disaggregation was introduced in the modelling and the resulting AO model with three fitting parameters was capable of describing experimental data of untreated CNT suspensions. The model was supported by experimental optical observations and offered a plausible explanation to the significant shear-thinning characteristic of aggregating untreated CNT suspensions.