Shear Thickening of Dense Suspensions: The Role of Friction

TL;DR

This paper investigates how inter-particle friction affects shear thickening in dense suspensions, revealing that the effective friction coefficient remains largely unaffected by microscopic friction variations, which aligns with recent theories and experiments.

Contribution

It introduces a model linking effective friction to jamming distance and shear stress, showing insensitivity to microscopic friction in dense suspensions.

Findings

Effective friction coefficient is insensitive to interparticle friction.

Proposed expressions relate effective friction to jamming and shear stress.

Results align with recent theoretical and experimental findings.

Abstract

Shear thickening of particle suspensions is characterized by a transition between lubricated and frictional contacts between the particles. Using 3D numerical simulations, we study how the inter-particle friction coefficient influences the effective macroscopic friction coefficient and hence the microstructure and rheology of dense shear thickening suspensions. We propose expressions for effective friction coefficient in terms of distance to jamming for varying shear stresses and particle friction coefficient values. We find effective friction coefficient to be rather insensitive to interparticle friction, which is perhaps surprising but agrees with recent theory and experiments.

Equations

$$\displaystyle\mathbf{F}_{\mathrm{rep}}=\left\{\begin{array}[]{ll}-c_{0}\dfrac{(d-d_{\mathrm{c}})^{2}}{dd_{\mathrm{c}}^{2}}\mathbf{e}_{h},&d\leq d_{\mathrm{c}}\\ 0&\mathrm{otherwise}\end{array}\right.$$

$$F_{\mathrm{norm,fric}}=\left\{\begin{array}[]{ll}|\mathbf{F}_{\mathrm{rep}}|-F_{\mathrm{CL}}&\quad\text{if }|\mathbf{F}_{\mathrm{rep}}|\geq F_{\mathrm{CL}},\\ 0&\quad\mathrm{otherwise}.\end{array}\right.$$

$$\mu (I_v)=\underset{{\mu }_c}{\underbrace{{\mu }_1+\frac{{\mu }_2-{\mu }_1}{1+I_0/I_v}}}+\underset{{\mu }_h}{\underbrace{I_v+\frac{5}{2}{\varphi }_m{I}_v^{\frac{1}{2}}}}$$

$${\sigma }_0$$

$$\tilde{\sigma }$$

$$f$$

$${\varphi }_m(\tilde{\sigma },{\mu }_m)$$

$${\varphi }_J({\mu }_m)$$

$$\varphi (I_v)=\frac{{\varphi }_m}{1+I_v^{0.5}}$$

$$\mu (\varphi ,{\varphi }_m)=\underset{{\mu }_c}{\underbrace{{\mu }_1+\frac{{\mu }_2-{\mu }_1}{1+I_0{\varphi }^2/({\varphi }_m-\varphi {)}^2}}}+\underset{{\mu }_h}{\underbrace{{\left(\frac{{\varphi }_m-\varphi }{\varphi }\right)}^2+\frac{5}{2}\frac{{\varphi }_m}{\varphi }({\varphi }_m-\varphi )}}$$

$$\frac{Z}{Z_J}=1-(1+I_v^{{\alpha }_1}{)}^{-{\beta }_1}$$

$$Z_J=6-2(1+{\mu }_m^{{\alpha }_2}{)}^{-{\beta }_2}$$

$$\frac{\mu -\mu 1}{\mu }=\frac{\sqrt{I_v}}{\sqrt{I_v}+\sqrt{I_v^{2{\mu }_1}}}$$

$$\mu ={\mu }_1\left(1+\sqrt{\frac{I_v}{I_v^{2{\mu }_1}}}\right)$$

Full text

††preprint: APS/123-QED

I Introduction

Understanding the rheological properties of shear thickening suspensions is scientifically challenging and highly relevant from the viewpoint of several applications Lee et al. (2003); Li et al. (2015); Decker et al. (2007); Cwalina et al. (2016). The phenomenon of shear thickening Denn et al. (2018); Fall et al. (2012, 2015); Brown and Jaeger (2014); Denn and Morris (2014); Cwalina and Wagner (2014); Boyer et al. (2011) in which the viscosity increases with increasing shear rate and shear stress, is attributed to the formation of frictional contacts between the particles as suggested by computational results Mari et al. (2014); Seto et al. (2013); Ness and Sun (2015) and confirmed by experiments Comtet et al. (2017); Lin et al. (2015); Pan et al. (2015); Royer et al. (2016); Huang et al. (2005). Shear thickening suspensions can be characterized by their macroscopic friction coefficient $\mu$, given by $\mu=\sigma_{\mathrm{shear}}/P$, with $\sigma_{\mathrm{shear}}$ the shear stress and $P$ the confining pressure. Using suspensions under constant confining pressure, Boyer et al. Boyer et al. (2011) demonstrated that $\mu$ is a unique function of a viscous parameter $I_{v}$ defined as $I_{v}=\eta_{f}\dot{\gamma}/P$, where $\eta_{f}$ and $\dot{\gamma}$ are the fluid viscosity and the shear rate respectively. They observe similar $\mu(I_{v})$ behavior for different materials (polystyrene, PMMA) and particle sizes. Gallier et al.Gallier et al. (2014) studied $\mu(I_{v})$ rheology in simulations for $\phi<0.45$ ($\phi$ being the particle volume fraction) and their simulations agree quantitatively with the experimental results. However, a more detailed analysis of $\mu$ and associated changes in the microstructure of the suspension is needed to shed further light on the behavior of the macroscopic friction coefficient $\mu$ and notably its relation with the microscopic inter-particle friction coefficient $\mu_{m}$. Here, we perform 3D numerical simulations of dense shear thickening suspensions with varying inter-particle friction coefficients to study associated changes on $\mu$. Based on recent results on constitutive relationships for shear thickening systems Singh et al. (2018); Wyart and Cates (2014), we propose analytic expressions for $\mu$ in terms of distance to jamming ($\phi_{m}-\phi$, where $\phi_{m}$ is the jamming volume fraction) for constant volume systems with varying pressure, shear stress and $\mu_{m}$ values. Using the average coordination number as a parameter, the microstructure of the particles in the system is analyzed to assess its influence on $\mu$. Finally, simulations of non-spherical particles are performed to study the effect of non-sphericity on the behavior of the macroscopic friction coefficient.

II Methods

The numerical simulations were performed using the simulation framework SuSi Lorenz et al. (2018). We use the Lattice Boltzmann Method (LBM) based fluid to simulate the fluid field and Lagrangian particles as the solid phase. The fluid-particle interactions are modelled with the Noble Torczynski Method Noble and Torczynski (1998). Lubrication forces are calculated explicitly at particle gaps smaller than the LBM lattice spacing. Adaptive refinement of timesteps is performed in order to ensure numerical stability and accuracy, as the inter-particle forces diverge at small particle gaps. The contact normal force $\mathbf{F}_{\mathrm{rep}}$ between particles is calculated from the overlap of a contact repulsion layer Lorenz et al. (2018) of specified thickness $d_{c}\approx 0.001R$ Mari et al. (2014), where $R$ is the mean radius of particles.

[TABLE]

where $c_{0}$ is the repulsion coefficient, $d$ is the gap between the particles, $d_{c}$ is the repulsion layer thickness and $\mathbf{e}_{h}$ is the connecting unit vector between the particles. The static and kinetic friction between particles is modeled as proposed by Luding Luding (2008). Upon initiation of frictional contact between particle pairs, a linear spring of length $\xi$ is initialized between the closest surface points to model static friction and is updated using the relative tangential velocity between the two contacting surface points. The maximum static friction is $F_{\mathrm{s}}\leq\mu_{\mathrm{s}}|\mathbf{F}_{\mathrm{norm,fric}}|$, as given by Coulomb’s Law. The spring force $\mathbf{F}_{\mathrm{spr}}$ is applied if the amplitude of $\mathbf{F}_{\mathrm{spr}}=-k\mathbf{\xi}$ is smaller than the maximum possible static friction force $F_{\mathrm{s}}$. Kinetic friction $F_{\mathrm{k}}=\mu_{\mathrm{k}}|\mathbf{F}_{\mathrm{norm,fric}}|$ is applied as a tangential force at the surface points if $F_{\mathrm{spr}}$ exceeds $F_{\mathrm{s}}$. For kinetic friction, the static friction spring length is rescaled so that $F_{\mathrm{spr}}=F_{\mathrm{k}}$. In our simulations, we keep $\mu_{s}=\mu_{k}=\mu_{\mathrm{m}}$, where $\mu_{m}$ is referred to as the microscopic friction coefficient.

The interacting particles are deemed frictional based on a Critical Load Model Mari et al. (2014), where two particles are considered to be in friction if the normal force ($F_{\mathrm{rep}}$) between the contacting particles exceeds a threshold value ($F_{\mathrm{CL}}$). The static and kinetic friction is based on the normal force for friction ($F_{\mathrm{norm,fric}}$), calculated as Mari et al. (2014):

[TABLE]

For the simulations discussed in the subsequent sections, a $96\mu m\times 64\mu m\times 96\mu m$ system is used, which contains $\approx 650$ particles for $\phi=0.56$. The particles are have a mean diameter of $8\mu m$ with a standard deviation of $0.4\mu m$ to avoid crystallization. The particles are neutrally buoyant in the suspending fluid, which mimics water (fluid viscosity $\eta_{f}=1.002\times 10^{-3}Pa.s$, density $\rho_{f}=1000kg/m^{3}$). The simulated systems have a characteristic stress for frictional contacts, given by $\sigma_{0}=F_{\mathrm{CL}}/(6\pi R^{2})$, where $R$ is the average particle radius. For the performed analysis, we choose instances of the system with average shear stress greater than $\sigma_{0}$, so that frictional interactions are significant.

III Results and Discussion

The macroscopic friction coefficient ($\mu$) of suspensions is characterized by the viscous number ($I_{v}$) of the suspension flow. $I_{v}$ is defined as $I_{v}=\eta_{f}\dot{\gamma}/P$, where $\eta_{f}$ is the fluid viscosity, $\dot{\gamma}$ is the shear rate and $P$ is the pressure in the system. The viscous number can be seen as the ratio of the internal timescale of microscopic particle rearrangements in a viscous system ($\eta_{f}/P$), to the macroscopic flow timescale ($1/\dot{\gamma}$). Boyer et al. Boyer et al. (2011) used pressure imposed flows to study variation in $\mu$ with $I_{v}$, where systems of hard spheres were sheared at constant pressure ($P$) and shear rate ($\dot{\gamma}$) while the system was allowed to dilate (changing $\phi$) in order to keep $P$ constant. They demonstrated that $\mu$ of suspensions is the sum of contact ($\mu_{c}$) and hydrodynamic ($\mu_{h}$) stress contributions, as shown in Eq.5.

[TABLE]

Where, $\mu_{1}$ is the limit of of the particle contact contribution to macroscopic friction ($\mu_{c}$) at vanishing viscous numbers, and $\mu_{2}$ is the maximum $\mu_{c}$ at $I_{v}\rightarrow\infty$ as observed in granular flows Cassar et al. (2005); Jop (2015). $I_{0}$ represents the scale over which $\mu_{c}(I_{v})$ changes and is observed to be constant for a given particle shape. $\phi_{m}$ is the jamming volume fraction.

$\mu_{h}(I_{v})$ is designed to reproduce the Einstein viscosity at low $\phi$ and be non-saturating at high $I_{v}$. Here, simulations of constant $\phi$ and $\dot{\gamma}$ with varying $P$ are performed to study $\mu(I_{v})$. In this study, we define $P$ as the average of the diagonal elements of the stress tensor in the system i.e. $P=\sum_{i=1}^{3}\sigma_{ii}/3$. We systematically vary the microscopic friction coefficient $\mu_{m}$ and compare to the predictions of $\mu(I_{v})$ rheology (Eq. 5), to see if the constant $\phi$ and $\dot{\gamma}$ simulations conform to the predictions of $\mu(I_{v})$ rheology.

Fig.1(a) compares the the results from our simulations to the $\mu(I_{v})$ rheology predicted by Eq.5, and the experimental results from Boyer et al. Boyer et al. (2011). Suspensions of different $\phi$ values were simulated to obtain the range of $I_{v}$ values. It can be observed that $\mu\approx 0.34$ at vanishing $I_{v}$, which is similar to the values obtained in experiments Boyer et al. (2011); Cassar et al. (2005). Using $\mu_{2}=0.7$ and $I_{0}\approx 0.009$ provides a good fit to the simulation data. The value for $\mu_{2}$ is the same as that observed previously in experiments and simulations of spherical particles Boyer et al. (2011); Gallier et al. (2014).

At vanishing $I_{v}$, we find high corresponding $\phi$ values similar to that in experiments Boyer et al. (2011). Under constant $\phi$ settings, the range of $I_{v}$ values accessible for each $\phi$ value is limited (as seen in Fig. 1(b)), and multiple simulations of varying $\phi$ values are required to capture $I_{v}$ values varying in orders of magnitude. This issue can be overcome by allowing the system to dilate in order to change $\phi$, as done in experiments. The variation in $\phi$ with $I_{v}$ is shown in Fig. 1(b), along with the experimental observation from Boyer et al.Boyer et al. (2011). The simulations show good agreement with the experimental results.

III.1 Effect of varying microscopic friction coefficient

Earlier simulation studies of the role of the microscopic friction coefficient ($\mu_{m}$) were performed at large viscous numbers ($I_{v}>0.1$) with limited overlap between $I_{v}$ ranges studied in experiments Gallier et al. (2014). Here, a larger range of $I_{v}$ values is accessed, allowing comparisons with experimental results at lower $I_{v}$ values. In order to study the effect of changing $\mu_{m}$ on $\mu$, simulations of $0.01\leq\mu_{m}\leq 10$ are performed, while keeping all other system parameters the same. This amounts to over 500 individual simulations, the results of which are presented in Fig. 2.

In Fig. 2(a) the simulation results of $\mu(I_{v})$ for various $\mu_{m}$ values are shown. At large $I_{v}$ values ($I_{v}>0.1$), $\mu(I_{v})$ is similar for all $\mu_{m}$ values. At vanishing $I_{v}$ values ($I_{v}<10^{-4}$), the minimum $\mu(I_{v})$ (i.e. $\mu_{1}$) reduces with decreasing $\mu_{m}$, as shown in Fig. 2(c). This observation is in agreement to that made in past simulations of 2D granular and suspension flows Da Cruz et al. (2005); Trulsson et al. (2017). Interestingly, the relationship between $\mu-\mu_{1}$ and $I_{v}$ collapses to the same curve for all $\mu_{m}$ values in this system (see Fig. 2(b)). Such a collapse was not observed when spherical particle suspensions studied in this section are compared against non-spherical particle suspensions (see Section III.4), suggesting that particle shape is a factor here. The change in $\mu_{1}$ with $\mu_{m}$ follows a sigmoidal relationship, as observed in Fig. 2(c).

The collapse of $\mu-\mu_{1}$ for $I_{v}<10^{-3}$ with the viscous number is obviously due to $\mu$ being constant and equal to $\mu_{1}$ in this range. Within the intermediate viscous number range ($10^{-3}\leq I_{v}\leq 10^{-1}$) where the particle contact contribution ($\mu_{c}$ in Eq. 5) to $\mu$ is dominant, the variation in $\mu$ with the microscopic friction coefficient $\mu_{m}$ is dictated by the variation in $\mu_{2}-\mu_{1}$ with $\mu_{m}$. Seeing that $\mu_{2}$ is rather insensitive to microscopic inter-particle friction coefficients ($\mu_{2}$ varies between 0.7 and 0.8 for completely frictionless and frictional particles respectively Gallier et al. (2014)), we estimate that the largest difference in $\mu-\mu_{1}$ between systems of $\mu_{m}=0.01$ and $\mu_{m}=10.0$ should be $\approx 0.2$, which agrees with the observed variations in $\mu-\mu_{1}$ with $\mu_{m}$ at $I_{v}\approx 10^{-1}$. For large viscous number range ($I_{v}>10^{-1}$), the variations in $\mu$ are dominated by the hydrodynamic component ($\mu_{h}$ in Eq. 5), and does not depend on the friction.

The main contribution to $\mu-\mu_{1}$ is therefore given by the distance to jamming. The collapse of the data for $\mu-\mu_{1}$ as a function of $I_{v}$ for $0.01\leq\mu_{m}\leq 10$ implies that at the same microscopic to macroscopic particle rearrangement timescale ratios (i.e. $I_{v}$), all systems will have the same distance to jamming, regardless of their microscopic friction coefficient. This also entails that if $\mu-\mu_{1}$ indeed is a measure of the distance of a system from jamming, it should have a mapping to some other measure of distance to jamming, such as $\phi_{m}-\phi$. We shall explore this in the following section.

III.2 Macroscopic friction coefficient and distance to jamming

In the simulations, a range of shear stresses ($\sigma_{\mathrm{shear}}$), volume fractions ($\phi$) and microscopic friction coefficients ($\mu_{m}$) are studied. From previous experiments and simulations Singh et al. (2018); Wyart and Cates (2014), we understand the effect of changing each of these parameters on the rheology, especially on the jamming volume fraction ($\phi_{m}$). Shear thickening is due to the formation of system spanning frictional networks, and the best way to describe this is to look at the fraction of frictional particles in the system. Beyond a characteristic shear stress $\sigma_{0}$, the fraction of particles in the system that have frictional contacts ($f$) increases until all particles become frictional Mari et al. (2014). This increase in $f$ with shear stress $\sigma_{\mathrm{shear}}$ can be described Singh et al. (2017) as

[TABLE]

where R is the average radius of the particles, $F_{\mathrm{CL}}$ is the onset normal force between particles to initiate friction, and $\sigma_{0}=F_{\mathrm{CL}}/(6\pi R^{2})$ is the characteristic stress for the onset of friction. Increasing the fraction of frictional particles leads to a lower jamming volume fraction $\phi_{m}$, as $\phi_{m}$ for frictional particles is lower than non frictional particles Wyart and Cates (2014); Singh et al. (2018). This is a result of the frictional particles requiring a smaller number of inter-particle contacts to be arrested in comparison with frictionless particles Song et al. (2008). The average coordination number for jamming ($Z_{J}$) in suspensions varies continuously between $Z_{J}(\mu_{m}=\infty)=4$ and $Z_{J}(\mu_{m}=0)=6$ in suspensions. Increasing the fraction of frictional particles in the system reduces the jamming volume fraction $\phi_{m}$ from that of a lubricated, non-frictional suspension ($\phi^{0}_{J}$) to that of a frictional suspension ($\phi_{J}$). $\phi_{J}(\mu_{m})$ is the jamming volume fraction in a suspension with all particles in frictional contact and is a decreasing function of the microscopic friction coefficient $\mu_{m}$. Hence, the volume fraction associated with jamming varies with $\mu_{m}$ and the fraction of frictional particles $f$ in the system, and can be described Singh et al. (2018) by

[TABLE]

where $\phi_{J}(\mu_{m})$ represents the jamming volume fraction when $f=1$ for a given microscopic friction coefficient $\mu_{m}$. $\phi^{0}_{J}$ is the jamming volume fraction when $f=0$, which is equivalent to a $\mu_{m}=0$ (frictionless) state. Changing the microscopic friction coefficient $\mu_{m}$ influences $\phi_{m}$, as lowering $\mu_{m}$ increases $\phi_{J}$, according to Eq. 10 Singh et al. (2018)

[TABLE]

Here, $\phi^{\infty}_{J}$ is the jamming volume fraction at large $\mu_{m}$ values, and $\mu_{\phi}$ is a constant. Boyer et al. Boyer et al. (2011) proposed a model for $I_{v}$ in terms of $\phi_{m}$ and $\phi$ as:

[TABLE]

when substituted in Eq. 5, this gives $\mu$ as a function of $\phi_{m}$ and $\phi$:

[TABLE]

Under constant volume settings, the fraction of the frictional contacts varies with shear stress (or shear rate) in the system, which in turn varies $\phi_{m}$. We can account for this variation in $\phi_{m}$ by employing Eqs. 6, 7, 8, 9 and 10. This helps to predict $\phi_{m}$ in our constant volume system in terms of $\sigma_{\mathrm{shear}}$ and $\mu_{m}$ which in turn enables an analysis of $\mu$ as a function of $\phi_{m}$-$\phi$ (i.e. a distance to jamming metric) and compare against the predictions from Eq. 12.

Fig. 3(a) shows the $\mu-\mu_{1}$ as a function of $\phi_{m}$- $\phi$ compiled over a range of $\sigma_{\mathrm{shear}}$, $\phi$ and $\mu$ values. The simulation results show agreement with the predictions from theory outlined in Eqs. 12, 6, 7, 8, 9, 11 and 10. The changes in $\phi_{J}$ with $\mu_{m}$ are taken into account by using their relationship outlined in Eq. 10, as shown in Fig. 3(b). The simulation results agree with the theoretical assumption that, by accounting for changes in $\phi_{m}$ with $\sigma_{\mathrm{shear}}$ and $\mu_{m}$, the values of $\mu$ across different $\sigma_{\mathrm{shear}}$ and $\mu_{m}$ values collapse to the regime outlined in Fig. 3(a). The change in the frictional jamming volume fraction $\phi_{J}$ with $\mu_{m}$ is shown in Fig. 3(b), along with the model presented in Eq. 10. The results also show that $\mu-\mu_{1}$ is indeed a measure for the distance to jamming, as suggested in the previous section.

III.3 Microstructure changes

The microscopic friction coefficient plays an important role in the nature of contact networks formed at jamming. The mean coordination number at which the suspension jams ($Z_{J}$), is inversely dependent on $\mu_{m}$, as $Z_{J}(\mu_{m}=0)=6$ and $Z_{J}(\mu_{m}=\infty)=4$ Song et al. (2008). The evolution of $\mu$ with average coordination number ($Z$) under varying $\mu_{m}$ values thus, is of interest. It is also compelling to view $\mu(I_{v})$ rheology in terms of the evolution of $Z$.

Fig. 4(a) shows average coordination number $Z(I_{v})$ under various $\mu_{m}$ values. $Z$ is calculated per particle by counting the number contacts it makes, i.e. cases where $r_{ij}-R_{i}-R_{j}\leq d_{c}$ where $r_{i,j}$ are the distance between the particles and and $R_{i,j}$ are their radii. Even though the data is compiled from various $\phi$ and $\sigma_{\mathrm{shear}}$ values, $Z(I_{v},\mu_{m})$ collapses to unique curves depending on $\mu_{m}$. The maximum coordination number is $Z\approx 4$ at $\mu_{m}=10.0$ and saturates at higher maximum values ($Z_{J}$) with reducing $\mu_{m}$ as expected from $Z_{J}(\mu_{m})$ relationship described before. The low $Z$ values at large $I_{v}$ sheds light on the the insensitivity of $\mu(I_{v})$ rheology to changes in $\mu_{m}$ in these $I_{v}$ ranges. $\mu(I_{v})$ rheology hence is essentially the process of varying coordination numbers between zero and $Z_{J}(\mu_{m})$. Upon normalizing $Z$ by $Z_{J}(\mu_{m})$, the different $Z(\mu_{m})$ curves collapse to a single curve, which can be modeled as:

[TABLE]

where $\alpha_{1}=0.77$ and $\beta_{1}=0.3$. The variation in $Z_{J}$ between 6 and 4 depending on $\mu_{m}$ can also be modeled using the expression:

[TABLE]

where $\alpha_{2}=-1.72$ and $\beta_{2}=0.27$. Fig. 4(b) shows $Z/Z_{J}$ as a function of $I_{v}$, and it can be observed that the data collapses to a single curve, modeled by Eq. 13. The variation in $Z_{J}$ with $\mu_{m}$, modeled by Eq. 14 is shown in Fig. 4(c). It is relevant to note that the variation in $Z_{J}$ with $\mu_{m}$ is found to be quite similar to the change in the coordination numbers associated with minimum random loose packing (RLP) limit observed in dry granular systems Song et al. (2008). The minimum RLP coordination number corresponds to the minimum coordination number required to obtain a disordered, mechanically stable jammed system. As the limits of jamming are prescribed entirely by the properties of the particles, it is conceivable that the characteristics related to jamming in granular systems devoid of fluid is to be expected in suspensions as well.

The effect of changing $Z$ on $\mu$, under various $\mu_{m}$ values is shown in Fig. 5(a). $\mu(Z)$ values reasonably collapses into a single curve for all values of $\mu_{m}$ studied. This demonstrates that the the minimum $\mu$ achieved at low $I_{v}$ values (i.e. $\mu_{1}$) is determined by $Z_{J}$. As $Z_{J}$ is inversely related to $\mu_{m}$, the relationship between $\mu_{1}$ and $\mu_{m}$ depicted in Fig. 2(b) can be rationalized. Assuming a range of $I_{v}$ values, one can calculate and compare $\mu$ against $Z$ for a given $\mu_{m}$ value using the relationships outlined in Eqs. 14, 13, 10 and 5. As shown in Fig. 5(a), the theoretical predictions of $\mu(Z,\mu_{m}=0.5)$ is in agreement with the simulation results. Consequently, the variation in $\mu$ with $\phi$ also collapses reasonably onto a simple curve across the various $\mu_{m}$ values studied, as seen in Fig. 5(b). This behavior is observed in 2D simulations of sheared suspensions and dense granular systems Thomas et al. (2018); Da Cruz et al. (2005) and experimentally by Boyer et al.Boyer et al. (2011). With increasing volume fraction, under a given shear rate, the shear stress and normal stresses become larger, but their ratio ($\mu$) reduces till $\mu=\mu_{1}$ at jamming (see Fig. 5(d-f)).

This implies that the jamming volume fraction determines $\mu_{1}$, the minimum macroscopic friction coefficient. The lower the jamming volume fraction, the higher the observed $\mu_{1}$; see Fig. 5(c). Our simulations of non-spherical particle suspensions (see next section) that jam at a lower volume fraction compared to spherical particles also agree with this observation, as shown in Fig. 5(c).

III.4 Non spherical particles

Particle shapes have significant effects on the shear thickening behavior of the suspensions. Cornstarch particles are observed to shear thicken at much lower $\phi_{m}$ values ($\phi_{m}$ $\approx 0.44$) Fall et al. (2012) in comparison to suspensions of spherical particles which shear thicken around $\phi_{m}=0.56$. Simulation results Lorenz et al. (2018) show that frictional jamming volume fraction $\phi_{J}^{\infty}$ is lowered when particles shapes become ’cornstarch-like’. In the interest of comparing the macroscopic friction coefficient variation in spherical particles to that of non-spherical particles, simulations of ’cornstarch-like’ non-spherical particle suspensions were performed. The ’cornstarch-like’ particles were created using overlapping spheres of varying sizes, as outlined in Lorenz et al. (2018). A representation of the non-spherical particles used is provided in Fig. 6(a)(inset).

Fig. 6(a) compares $\mu(I_{v})$ for spherical particle suspensions and non-spherical particle suspensions. At high viscous numbers, $\mu(I_{v})$ for spherical and non-spherical particle suspensions tends to be the same. This is understandable, as at high $I_{v}$ values the coordination numbers of the particles (spherical or non-spherical) in the suspensions reduces and particle shapes become increasingly less relevant. However, at small $I_{v}$ values, $\mu(I_{v})$ behavior of non-spherical particle suspensions deviates from that of spherical particle suspensions, for any constant $\mu_{m}$ value. Naturally, these deviations become apparent at $I_{v}$ values where particle interactions become relevant, i.e. $I_{v}<10^{-1}$. Results suggests that the macroscopic friction coefficient of non-spherical particle suspensions plateaus to $\mu_{1}$ at higher viscous numbers in comparison to the spherical particle suspensions. Also, at vanishing viscous numbers, the macroscopic friction coefficient of the non-spherical particle suspensions saturates to a higher $\mu_{1}$ in comparison with spherical particle suspensions, for a given $\mu_{m}$ value. This agrees with measurements of the macroscopic friction coefficient for cornstarch suspensions close to jamming Fall et al. (2012), where $\mu_{1}\approx 0.62$ in the experimental systems and $\mu_{1}\approx 0.6$ in the simulations. In the previous section, it was concluded that the jamming volume fraction determines the minimum value of the macroscopic friction coefficient. Considering that the non-spherical suspension simulated here jams around $\phi^{non-spherical}_{J}=0.53$, which is lower than the jamming volume fraction for spherical particles ($\phi^{spherical}_{J}=0.576$) at the same $\mu_{m}$ value ($\mu_{m}=1$), the larger $\mu_{1}$ observed here can be rationalized.

It is intriguing to see whether one can generalize these variations in $\mu$ with particle shapes and microscopic friction coefficients to arrive at a common curve for all available data. By (a) normalizing $I_{v}$ with $I_{v}^{2\mu_{1}}$ (where $I_{v}^{2\mu_{1}}=I_{v}(\mu=2\mu_{1})$) to account for the shift in $I_{v}$ values at which $\mu$ plateaus to $\mu_{1}$, and (b) setting upper and lower bounds to the variation in $\mu$ by using $(\mu-\mu_{1})/\mu$ as the measure of the variation of $\mu$ with $I_{v}$, the results collapses nicely to a single curve, for both spherical and non-spherical particle suspensions, across varying $\mu_{m}$ values (see Fig. 6(b)). The results of Boyer et al. Boyer et al. (2011) are shown for comparison, and also agrees with the curve. This common relationship can be fitted using the curve given by:

[TABLE]

which in turn gives:

[TABLE]

Even though the simulation results conform to the expression given by Eq. 15, it should be mentioned that the validity of the expression at high viscous numbers ($I_{v}>0.5$) is suspect, as we have no experimental data in this regime. Experimental data for non-spherical particles at viscous numbers high enough to obtain $I^{2\mu_{1}}_{v}$ is also absent, which prevents us from further validation.

IV Conclusion

We analyze the behavior of the macroscopic friction coefficient ($\mu$) under different microscopic friction coefficients ($\mu_{m}$) using 3D numerical simulations. The predictions of $\mu$ from simulations agree with earlier predictions of viscous number granular suspension rheology. We find that when $\mu_{m}>0.3$, that viscous number rheology is largely insensitive to the value of $\mu_{m}$. By changing the jamming volume fraction $\phi_{m}$ with the changes in shear stresses and $\mu_{m}$, we analyze $\mu$ in terms of distance to jamming ($\phi_{m}-\phi$) and provide phenomenological but analytic formulae that match the observations. Our results also suggest the behavior of $\mu$ across various $\mu_{m}$ and viscous numbers ($I_{v}$) can be reduced to effects of distance to jamming. The study of changes in the average coordination number ($Z$) with viscous number ($I_{v}$) shows that $Z$ smoothly decreases from $Z_{J}$ ($Z$ at jamming) to zero with increasing viscous number, where $Z_{J}$ is again determined by $\mu_{m}$. Our results suggest that the minimum $\mu$ achieved is inversely related to the jamming volume fraction and $Z_{J}$. Finally, we show that with appropriate scaling, a common curve for the variation of $\mu$ with $I_{v}$ emerges for both spherical and non-spherical particles under varying $\mu_{m}$ values.

Conflicts of Interest

There are no conflicts of interests.

ACKNOWLEDGEMENTS

We would like to thank SURFsara for using their HPC infrastructure and for providing support (project number 00231267). Author VS acknowledges funding by NWO, Netherlands under the CSER program (project number 14CSER026).