#### 3.1. Steady State Shear Rheology

The relative viscosity (

${\eta}_{r}$) values of the three types of suspensions i.e., GS2, PS52, and PS226 in glycerol is illustrated in

Figure 2 as a function of shear rate (

$\dot{\gamma}$) for various particle volume fractions (

$\phi $). The results illustrate a non-monotonic influence of the particle specific surface area on the behaviour of the suspensions. The PS52 particles are characterized by a pronounced shear thinning behaviour as the volume fraction increases (

Figure 2b), with the onset observed at

$\phi \ge 0.25$, whereas glass spheres and PS226 particle suspensions exhibit almost negligible or weak shear thinning (

Figure 2a,c) at

$\phi \ge 0.30$ and

$\phi \ge 0.10$, respectively. The shear thinning response of the GS2 and PS226 is also followed by shear thickening at higher particle volume fractions (

$\phi \ge 0.50$ and

$\phi \ge 0.15$), while no evidence of shear thickening is observed for the PS52 suspensions, even at the highest particle volume fractions studied. The PS226 suspensions (

Figure 2c) also exhibit the highest relative viscosities among the three particle suspensions. The shear thinning response of the silicas can be described by the Carreau equation (Equation (3)) sufficiently well and the corresponding fittings are shown as continuous lines in

Figure 2.

The Carreau fittings are used to estimate the zero- and infinite-shear rate relative viscosities,

${\eta}_{r,0}$ and

${\eta}_{r,\infty}$, respectively. Their values are summarised in

Figure 3a as a function of

$\phi $; the Quemada (Equation (4)) fittings are also included indicated by continuous and dashed lines. The

${\eta}_{r,0}$ values for the glass sphere suspensions were determined at

$\dot{\gamma}={10}^{-3}{s}^{-1}$. Similarly, the

${\eta}_{r,\infty}$ values for suspensions exhibiting both shear thinning and shear thickening, were taken in the shear rate region, just before the onset of the shear thickening regime. It can be seen that the relative viscosity values at zero and infinite shear (

${\eta}_{r,0}$ and

${\eta}_{r,\infty}$) almost overlap for the glass sphere suspensions, since they exhibit negligible shear thinning. In contrast, they differ substantially for the PS52 particle suspensions illustrating their pronounced shear thinning at

$\phi \ge 0.25$ and only slightly for the PS226 ones due to their mild shear thinning behaviour. The PS52 and PS226 particles both increase the viscosity of glycerol to a greater extent than glass spheres due to their surface asperities, which offer a higher area available for particle contacts [

3,

9], and porosity with the PS226 inducing a much steeper increase at lower particle volume fractions as compared with the other two particle types.

Table 2 summarises the maximum packing fraction (

${\phi}_{m}$) obtained from the Quemada fittings to the relative viscosities in the infinite shear rate regimes

${\eta}_{r,\infty}$. Increasing the specific surface area of the particles results in a decrease in the

${\phi}_{m}$ values. The glass sphere (GS2) suspensions exhibit the highest values of

${\phi}_{m}$ out of the three suspensions and these are similar to those reported in the literature for randomly closed packed monodisperse and polydisperse spheres [

33,

34,

35]. The

${\phi}_{m}$ values for both the PS52 and PS226 particles are lower as compared with the GS2. This can be attributed to the increased particle contacts enabled by their irregular surfaces which can limit the particle loading capacity. The origins of particle contacts and their relevance to the observed suspension rheology are discussed in more detail in

Section 4.

Figure 3b replots the

${\eta}_{r,\infty}$ values as a function of the particle volume fraction, this time normalised with the maximum packing fraction (

${\phi}_{m}$) at infinite shear rate. Although normalising

$\phi $ nearly collapses the relative viscosities of the three types of suspensions, small variations in the

${\eta}_{r,\infty}$ values between the glass spheres (GS2) and the irregular commercial silicas (PS52 and PS226) can still be discerned, highlighting the strong effect of particle surface characteristics on suspension rheology.

In our previous work [

12], we showed that the strong shear thinning behaviour of the PS52 suspensions, as opposed to that of GS2, can be attributed to the surface asperities being elastically deformed due to the frictional contacts between the former particles. According to a theory introduced by Chatté et al. [

36] and Lobry et al. [

28], this leads to a decrease in the microscopic friction coefficient and shear thinning. The shear thinning of the glass spheres is, however, suppressed due to the presence of a solvation layer as the hydroxyl groups (-OH) of the glycerol molecules bind onto the inherently hydrophilic silicas through hydrogen bonding. The PS226 particles are also irregularly shaped, similar to the PS52, as shown in

Figure 1, and exhibit much higher specific surface area as compared with the latter; thus, one would expect strong shear thinning behaviour for these particles at even lower particle concentrations as compared with PS52 ones. However, it seems that the increased specific surface area of the PS226 competes with the frictional contacts induced by the particle irregular surface and leads to the PS226 suspensions exhibiting weak shear thinning behaviour, followed by shear thickening at the highest

$\phi $ values. This might be associated with the limited loading capacity of the PS226 particles as compared with the other two particle types, as only volume fractions up to

$\phi =0.20$ could be achieved. It should be noted that the solvation layer, which acts as a lubricant to particle contacts, exists for all types of particle suspensions, but its effectiveness appears to depend on the particle surface area as indicated by the non-monotonic results reported herein.

To further probe the frictional contacts between the different types of particles, the friction coefficients (

$\mu $) were estimated using the bulk rheology data. The friction coefficient was estimated by the ratio of the shear stress (τ) to the particle pressure (P), assumed equal to the normal stress as measured with the rheometer [

9,

37,

38].

Figure 4 plots the estimated

$\mu $ values for selected and concentrated suspensions of the three particle types as a function of a normalised shear rate, namely the viscous number (

${I}_{v}={\eta}_{f}\dot{\gamma}/P$, where

${\eta}_{f}$ is the viscosity of the suspending medium, glycerol,

${\eta}_{f}=1.3\text{}\mathrm{P}\mathrm{a}\xb7\mathrm{s}$). Two distinct regimes can be observed in the friction coefficient (

$\mu $) values in the low

${I}_{v}$ region, which seem to be related to the extent of shear thinning in each case; while the coefficient estimates of the three types of particles seem to match and vary linearly with

${I}_{v}$ for

${I}_{v}\ge 0.01$, a deviation is observed for lower viscous numbers. An asymptotic behaviour characterized by higher

$\mu $ values is exhibited by the PS52 suspensions reaching a value of

$\mu \approx 0.02$ as

${I}_{v}$ tends to 0; this behaviour indicates the presence of frictional contacts and explains the strong shear thinning response of these suspensions, despite the relatively low particle volume fraction as compared with the GS2 suspensions. The asymptotic value estimated for PS52 is lower than that reported by Boyer et al. (2011), i.e.,

$\mu =0.32$, for PMMA (polymethylmethacrylate,

$d=1.1\pm 0.05\text{}\mathrm{m}\mathrm{m}$) and PS (polystyrene,

$d=0.58\pm 0.01\text{}\mathrm{m}\mathrm{m}$) spheres suspended in index-matched Newtonian fluids at room temperature. This may be attributed to the smaller size of the particles used in the present study (

$d=17.5\pm 15.5\text{}\mathsf{\mu}\mathrm{m}$) and the assumptions of the particle pressure being equal to the measured normal stress; local particle pressure can also be influenced by particle surface asperities and porosity. It should be recalled that, in the study of Boyer et al., the particle pressure was directly measured by a pressure-imposed shear cell [

37].

On the contrary, no asymptotic

$\mu $ value was observed for the GS2 and PS226 suspensions, which is consistent with their low, almost negligible degree of shear thinning. However, as PS226 particles exhibit similar surface morphology to the PS52 particles (

Figure 1a,b), some frictional contacts are expected. The absence of such contacts inducing a shear thinning response indicates a more complex rheology for the highly porous PS226 particles and highlights the need for more parameters to be explored to fully explain it; these are discussed in

Section 4. It can, thus, be concluded from

Figure 4 that although the PS52 suspensions are in the frictional regime, a lubrication effect seems to govern the rheology of the PS226 suspensions similar to that of the highly concentrated glass sphere suspensions. The deviations of

$\mu $ values from a single master curve, as shown in

Figure 4, have generally been observed in numerical studies with varying dynamic friction coefficients, highlighting the strong effects of friction on suspension rheology [

38].

Porous particles further complicate suspension rheology as part of the solvent becoming absorbed into the pores and leading to an increase in the effective particle volume fraction as compared with the apparent one. The effective volume fraction of the suspensions,

${\phi}_{eff}$, is given by [

15]:

where,

${w}_{p}$ is the weight fraction of the particles in suspension,

$\epsilon $ represents the particle porosity,

${\rho}_{p}$ is the density of the solid particles,

${\rho}_{f}$ the density of the suspending medium, and

${\rho}_{susp}$ is the suspension density defined as [

39]:

with

$\phi $ the apparent particle volume fraction.

The estimated

${\phi}_{eff}$ values are presented in

Figure 5a as a function of the apparent volume fraction, alongside a plot of the maximum packing fraction,

${\phi}_{m}$, as a function of particle porosity, ε, and specific surface area,

$S$ (

Figure 5b). Two sets of data are shown in

Figure 5a for the PS226 particles. These refer to the

${\phi}_{eff}$ estimates using either the material’s density (filled green diamonds), i.e.,

${\rho}_{p}=2\text{}\mathrm{g}/\mathrm{m}\mathrm{L}$, or the experimentally derived particle density, i.e.,

${\rho}_{p}=1.24\text{}\mathrm{g}/\mathrm{m}\mathrm{L}$, through volumetric measurements of the weighted particle. A volumetric cylinder was filled with a known volume of water, and then 1 gr of weighted mass was added each time and the increase in the water volume was recorded. The experimentally derived density of the PS226 particles is considerably lower as compared with the pure silicon dioxide density probably due to the high porosity of these particles. Interestingly, the

${\phi}_{eff}$ values for the PS226 using

${\rho}_{p}=1.24\text{}\mathrm{g}/\mathrm{m}\mathrm{L}$ appear lower as compared with that of the PS52 independently of the higher porosity of the former. This is because not only the density values in Equations (5) and (6) but also the apparent volume fraction needs be recalculated to correspond to

${\rho}_{p}=1.24\text{}\mathrm{g}/\mathrm{m}\mathrm{L}$. Therefore, the increased

${\phi}_{eff}$ of the PS52 and PS226 suspensions as compared with the glass spheres due to particle porosity, can be responsible for the onset of non-Newtonian rheological phenomena at lower apparent

$\phi $ values for the commercial silicas, as shown in

Figure 2. The

${\phi}_{m}$ values also appear to decrease with particle porosity and specific surface area as expected due to an increase in the available particle-particle and particle-solvent areas for contact.

The extent of shear thinning and shear thickening response of the suspensions exhibiting non-Newtonian rheology, as shown in

Figure 2, is further described through the parameters

${\eta}_{r,e}$ (Equation (7)) and

${\eta}_{r,t}$ (Equation (8)), respectively. The

${\eta}_{r,e}$ is defined as:

${\eta}_{r,t}$ is defined in a similar manner:

with

${\eta}_{r,peak}$ representing the peak relative viscosity in the shear thickening area.

The dependence of these parameters on the particle volume fraction is presented in

Figure 6a,b. respectively. The GS2 suspensions exhibit almost negligible shear thinning, while the PS52 particle suspensions exhibit the most pronounced shear thinning as compared with the other two types of particles. Shear thickening (

Figure 6b) is only observed for the GS2 and PS226 suspensions, with the latter showing higher

${\eta}_{r,t}$ values.

As discussed above, particle porosity influences interparticle interactions, and thus suspension rheology through increasing the effective volume fraction of the particles in the suspension. By replotting the

${\eta}_{r,e}$ and

${\eta}_{r,t}$ values as a function of

${\phi}_{eff}$ for the two silicas, they appear to approach those of the glass sphere suspensions (

Figure 6c,d, respectively). However, considering only porosity as the main factor to estimate the effective volume fraction of the suspensions is not adequate to fully capture their non-Newtonian behaviour. The irregular silicas still show stronger shear thinning and shear thickening response as compared with the non-porous glass spheres despite scaling the data by

${\phi}_{eff}$.

#### 3.2. Viscoelasticity

Particle suspensions are likely to exhibit viscoelastic properties at sufficiently high particle volume fractions, in which interparticle interactions are strong giving rise to the elastic component of the material. The viscoelastic properties of selected highly concentrated suspensions of the three particle types were investigated under oscillatory frequency sweeps in the linear viscoelastic region (strain amplitude,

${\gamma}_{0}=0.1\%$). The measured viscoelastic moduli, i.e., storage,

${G}^{\prime}$, and loss modulus,

${G}^{\u2033}$, are presented in

Figure 7a as a function of the angular frequency (

$\omega $) for all three suspensions at a fixed particle volume fraction of

$\phi =0.30$. The suspension at

$\phi =0.20$ is used for the PS226 particles, as this is the highest volume fraction achieved to provide a homogeneous sample based on visual inspection. The suspensions exhibit viscous dominated behaviour, with the

${G}^{\u2033}$ values being higher than the

${G}^{\prime}$, at all experimental conditions and particle characteristics, indicating the hindrance of direct particle contacts due to the presence of the solvation layer [

23] in the whole range of angular frequencies investigated. Increasing the particle specific surface area leads to increasing both

${G}^{\prime}$ and

${G}^{\u2033}$ values as compared with the glass sphere suspension. However, the PS226 exhibit slightly lower values of the viscoelastic moduli as compared with the PS52, which can be attributed to the lower

${\phi}_{eff}$ of the former.

The suspension viscoelasticity can be further evaluated from the corresponding phase angles (

$\delta $) obtained through the frequency sweeps (

Figure 7b). In general, a value of

$\delta \approx 0\xb0$ indicates that the material behaves as an elastic solid, while

$\delta \approx 90\xb0$ represents a liquid-like behaviour. The phase angle values, varying from

$0\xb0$ to

$90\xb0$, indicate that the material shows properties between a solid and a liquid, i.e., it is viscoelastic. The GS2 suspension shows

$\delta $ values very close to

$90\xb0$ and which are almost independent of the applied angular frequency. In contrast, PS52 and PS226 exhibit lower phase angles (

$\delta \le 80\xb0$), indicating an increase in suspension viscoelasticity. The lower phase angles for the PS52 can originate from interparticle interactions also responsible for the shear thinning response of these suspensions at low shear rates under steady state (

Figure 2b). The low phase angles observed for the PS226 suspension, similar to those of the PS52, might also arise from the weak shear thinning of the former.