3.1 Weak hydrodynamic interactions, $\unicode[STIX]{x1D705}\rightarrow \infty$

Structural and rheological relaxation upon microrheological flow cessation in the freely draining limit, $\unicode[STIX]{x1D705}\rightarrow \infty$, was studied by Zia & Brady (Reference Zia and Brady2013). In this limit, repulsive forces are so long-ranged that two particles cannot get close enough to feel disturbance flows from one another. The governing Smoluchowski equations (2.18) take a simple form, which, when inserted into the viscosity expressions in equations (2.17), give the viscosity relaxation with time. They found that the viscosity decays over at least two characteristic time scales: an early-time fast decay corresponding to the relaxation of the particle-rich boundary layer near the probe surface, and a longer-time slower relaxation that arises as Brownian motion drives particles towards a uniform configuration. A new idea to emerge from this study focused on the long-time terminal relaxation and its connection to memory loss as the structure relaxes and approaches its equilibrium configuration; Zia & Brady (Reference Zia and Brady2013) found that the duration of microstructural memory decreases with increasing pre-cessation flow strength. They showed that the probe travels backwards after the external force is removed and the back-travel of the probe is proportional to the entropic energy stored in the distorted configuration, minus viscous solvent drag. To form a foundation to study the effect of hydrodynamic interactions, we provide a brief review of their results in appendix A, with additional structural analysis.

3.2 Strong ($\unicode[STIX]{x1D705}\rightarrow 0$) versus weak ($\unicode[STIX]{x1D705}\rightarrow \infty$) hydrodynamic interactions

Figure 2.

Contour plots showing relaxation of structural deformation around the probe after removal of the external force, with strong hydrodynamic interactions, $\unicode[STIX]{x1D705}\rightarrow 0$, for $Pe=$ 1, 5, 10 and 50. Each image is a snapshot in time, scaled diffusively as shown at the bottom of the figure. The probe is at the centre of the black circle, which indicates the volume excluded to bath particle centres. Colour varies from red to green to blue for particle accumulation, undisturbed structure and particle depletion. Colour scaled for maximum contrast.

In this section, we examine structural relaxation after the removal of the external force driving the probe, at short and long times, and its effect on probe motion in the limit of very strong hydrodynamic interactions, $\unicode[STIX]{x1D705}\rightarrow 0$. To understand how long structural deformation is remembered after shutoff of the external flow, we examine the contour plots of the relaxing structure at several instants in time after the force is removed. Contour plots of the relaxing microstructure, $f(\boldsymbol{r},t;\unicode[STIX]{x1D705}\rightarrow 0,Pe)$, obtained by numerical solutions of equations (2.18), are shown in figure 2, where the four rows correspond to pre-cessation forces $Pe=$ 1, 5, 10 and 50 (probe driven to the right in the plots). The black circle signifies the volume excluded to bath particle centres. Red shows bath particle accumulation ($f(\boldsymbol{r},t)>0$), blue represents depletion ($f(\boldsymbol{r},t)<0$) and green indicates an undisturbed structure ($f(\boldsymbol{r},t)=0$) around the probe. Since Brownian motion drives relaxation, time has been scaled diffusively as $t_{diff}=(a_{th}^{2}/2D_{0})$. The time corresponding to each of the snapshots is labelled at the bottom of the figure. Note that the colour scale is chosen for maximum contrast at each value of $Pe$, for best visual representation of structural gradients. Thus the density corresponding to dark red or dark blue differs from one value of $Pe$ to another.

Going down the left-most column ($\hat{t}_{diff}=0$) of figure 2, the steady-state structural deformation for weak probe forcing is weak but propagates over long distances from the probe. In contrast, when forcing is strong, particle accumulation is concentrated at the upstream face of the probe and a depletion wake trails it, with very steep radial gradients in front of the probe and very steep lateral gradients across the trailing wake. Once the flow is shut off, Brownian motion acts to distribute the bath particles uniformly around the probe via random displacements and Brownian drift. For $Pe=1$ and $Pe=5$, the upstream and downstream regions appear to relax at the same rate. But for the more strongly distorted structure, $Pe=50$, the boundary layer appears to dissipate well before the wake fills in. The contour plots also indicate that the relaxation in the radial direction is faster than that in the angular direction, because angular gradients are weaker. This also influences filling-in of the wake, which occurs over long times. This is consistent with the theory and simulation results of Zia & Brady (Reference Zia and Brady2013), and later simulations by Marenne et al. (Reference Marenne, Morris, Foss and Brady2017), who examined the relaxation of a concentration suspension upon cessation of an external shear flow. In all cases, by the time it takes a particle to diffuse its size, $\hat{t}_{diff}=1$, the wakes in all four rows have mostly healed. This makes sense because the wake is approximately one particle size wide and lateral diffusion fills it in. But upstream relaxation is strongly dependent on pre-cessation $Pe$ – more strongly distorted structure at contact relaxes faster. This is consistent with the freely draining relaxation results of Zia & Brady (Reference Zia and Brady2013) discussed in § 3.1. However, comparison of the contour plots suggests that boundary-layer relaxation is slower when hydrodynamic interactions are stronger. This arises from the lubrication force at close particle distances that resists the separation of an interacting pair, and is illustrated in figure 3.

Figure 3.

Stagnation-point density normalized by its value at $\hat{t}_{diff}=0$ is plotted in the limits of weak ($\unicode[STIX]{x1D705}\rightarrow \infty$) and strong ($\unicode[STIX]{x1D705}\rightarrow 0$) hydrodynamic interactions on the left axis, revealing the faster rate of relaxation in the absence of hydrodynamics. The two right axes show the absolute value of the deformation at the stagnation point, revealing that a stronger deformation drives the faster relaxation rate. Note that the right axis for weak hydrodynamics ($\unicode[STIX]{x1D705}\rightarrow \infty$) spans $0\geqslant f\gtrsim 60$ and the right axis for strong hydrodynamics ($\unicode[STIX]{x1D705}\rightarrow 0$) spans only $0\gtrsim f\geqslant 16$.

Figure 4.

Flow cessation, strong hydrodynamics and arbitrary $Pe$. (a) Microviscosity with diffusively scaled time for various pre-cessation $Pe$, where the external flow is shut off at $\hat{t}_{diff}=0$. (b) Decay of the entropic contributions to the viscosity after flow cessation.

Figure 3 shows the stagnation-point density, the structural distortion at the upstream face of the probe ($r=2$ and $\unicode[STIX]{x1D703}=0$), in the limit of weak ($\unicode[STIX]{x1D705}\rightarrow \infty$, right axis) and strong ($\unicode[STIX]{x1D705}\rightarrow 0$, second right axis) hydrodynamic interactions (both at $Pe=50$) as a function of time. The steady-state structure prior to flow cessation is less distorted in the regime of strong hydrodynamic interactions. This weak gradient in the structure, along with the slower relative particle motion between particles in the presence of stronger hydrodynamic interactions, leads to a slower structural relaxation upon flow cessation. The strong impact of hydrodynamic interactions on the rate of structural relaxation will affect the probe motion, and in turn the viscosity of the colloidal dispersion. After the removal of the external force, deterministic probe motion arises solely from relaxation of the non-equilibrium structure (Zia & Brady Reference Zia and Brady2013). Hydrodynamic interactions slow this motion directly because they reduce relative particle mobility, and indirectly because they suppress microstructural distortion, leading to weaker gradients at a given value of $Pe$, and consequently slower relaxation.

We now examine the effect of structural relaxation on the viscosity. The solution in figure 2 is inserted in the viscosity expressions in equations (2.17) to obtain the decay in the viscosity with time, plotted in figure 4(a) as a function of diffusively scaled time for pre-cessation Péclet numbers ranging from 0.5 to 100, along with the asymptotically weak forcing limit, $Pe\rightarrow 0$. As soon as the flow is shut off at $\hat{t}_{diff}=0$, there is an instantaneous and precipitous drop in the viscosity because the direct hydrodynamic contributions, $\unicode[STIX]{x1D702}^{H,0}$ and $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D702}^{H}$, vanish immediately. This intrinsic viscosity arising from the presence of the particles always remains, and pertains to relaxation of thermal fluctuations of $O(kT)$, but because it is intransient it exerts no effect on the cessation behaviour. We thus remove it from the plotted quantity. However, any viscous drag on the probe proportional to flow strength vanishes instantaneously upon flow shutoff, owing to the linearity of Stokes flow. Thus, any forces or velocities linear in $Pe$ vanish when flow is shut off.

This is not the case for interparticle forces and velocities, or for Brownian forces and velocities, because these care nothing about the flow, only relative particle positions. These forces produce random and deterministic relaxation after flow shutoff, the latter owing to structural gradients. Hydrodynamic interactions influence relative motion and slow this relaxation. In contrast, the non-equilibrium hydrodynamic contribution to the velocity and viscous drag (equation (2.17b)) is proportional to flow, and vanishes when flow ceases. This is no different from the corresponding contribution to the shear stress and shear viscosity, which vanishes instantaneously upon flow shutoff (Foss Reference Foss1999; Marenne et al. Reference Marenne, Morris, Foss and Brady2017). Subsequent evolution comes only from the entropic contributions to the steady-state viscosity, $\unicode[STIX]{x1D702}^{P}$ and $\unicode[STIX]{x1D702}^{B}$, which relax monotonically in time, following structural relaxation (cf. figure 2).

An arrow indicates that the entropic contribution remaining after shutoff varies with $Pe$, illustrated in the inset of figure 4(a), which plots the fraction of the total viscosity prior to shutoff that was entropic:

(3.1)$$\begin{eqnarray}\frac{\unicode[STIX]{x1D702}_{SS}^{entropic}}{\unicode[STIX]{x1D702}_{SS}^{micro}}=\frac{\unicode[STIX]{x1D702}_{SS}^{P}+\unicode[STIX]{x1D702}_{SS}^{B}}{\unicode[STIX]{x1D702}_{SS}^{micro}}.\end{eqnarray}$$

In the limit of weak external flow, $Pe\ll 1$, and strong hydrodynamic interactions, $\unicode[STIX]{x1D705}\rightarrow 0$, one has $\unicode[STIX]{x1D702}_{SS}^{entropic}/\unicode[STIX]{x1D702}_{SS}^{micro}\rightarrow 0.128$, and since $\unicode[STIX]{x1D702}_{SS}^{P}+\unicode[STIX]{x1D702}_{SS}^{B}=0.128\unicode[STIX]{x1D702}_{SS}^{micro}=0.128\times 2.1\unicode[STIX]{x1D702}_{s}\unicode[STIX]{x1D719}=0.27\unicode[STIX]{x1D702}_{s}\unicode[STIX]{x1D719}$, we recover the steady-state results of Khair & Brady (Reference Khair and Brady2006). As the strength of the external force on the probe increases, the effect of Brownian motion gets weaker and, as a result, the entropic contribution to the steady-state viscosity decreases. This is reflected in the monotonic decrease of the relative entropic contribution, $\unicode[STIX]{x1D702}_{SS}^{entropic}/\unicode[STIX]{x1D702}_{SS}^{micro}$, with increasing $Pe$, shown in the inset of figure 4(a). While this ratio is easily calculated theoretically from steady-state analysis (Bergenholtz et al. Reference Bergenholtz, Brady and Vicic2002; Khair & Brady Reference Khair and Brady2006), such decomposition of the viscosity in experiments would be quite challenging. This illustrates the power of flow cessation to obtain the individual microscopic contributions to the steady-state viscosity.

Figure 4(b) shows the relaxation of only the entropic contributions to the steady-state viscosity as a function of diffusively scaled time for various strengths of pre-cessation external flow, $Pe$. Flow-thickening behaviour that was present prior to flow cessation now vanishes as direct hydrodynamic contributions to the steady-state viscosity decay instantaneously upon flow cessation. However, hydrodynamic coupling between particles itself does not go away; it continues to influence the rate at which entropic contributions dissipate. We normalize the post-cessation viscosity with its value at $\hat{t}_{diff}=0$ and plot it as a function of time in figure 5. Curves for each $Pe$ exhibit at least two relaxation modes: initially viscosity relaxes at a fast rate, followed by a slower long-time decay. At very early times after shutoff, relaxation is linear in time,

(3.2)$$\begin{eqnarray}\frac{\unicode[STIX]{x1D702}^{micro}(t=0)-\unicode[STIX]{x1D702}^{micro}(t)}{\unicode[STIX]{x1D702}^{micro}(t=0)}\sim t\quad \text{as }t\rightarrow 0,\end{eqnarray}$$

suggesting that Brownian drift sets the early-time viscosity relaxation in the limit of strong hydrodynamic interactions, $\unicode[STIX]{x1D705}\rightarrow 0$ (inset in figure 5). This makes a stronger statement however: the early-time rate of viscosity decay is independent of the pre-cessation flow strength, $Pe$. The $Pe$ independence of the relaxation is also found in the limit of weak hydrodynamic interactions, $\unicode[STIX]{x1D705}\rightarrow \infty$ (see appendix A). Comparing figure 5 and figure 14 (in appendix A) shows that hydrodynamics slows the rate of relaxation. When $\unicode[STIX]{x1D705}\rightarrow 0$, $\unicode[STIX]{x1D702}^{micro}\sim t$ at $t\rightarrow 0$, but when $\unicode[STIX]{x1D705}\rightarrow \infty$, $\unicode[STIX]{x1D702}^{micro}\sim t^{1/2}$. This hydrodynamic coupling persists over the entire relaxation process, slowing the long-time relaxation – irrespective of the pre-cessation $Pe$. This is discussed in § 3.3.

Figure 5.

Normalization of post-cessation entropic viscosity on its initial value reveals multiple relaxation processes, regardless of pre-cessation $Pe$. Inset: early-time decay is advective. Strong hydrodynamics ($\unicode[STIX]{x1D705}\rightarrow 0$).

The relaxation rate also depends on pre-cessation flow strength, with faster relaxation after stronger flow. This behaviour is similar to the relaxation observed in shear rheology flow cessation for concentrated suspensions (Foss Reference Foss1999; Marenne et al. Reference Marenne, Morris, Foss and Brady2017) and microrheological flow cessation in the absence of hydrodynamic interactions (Zia & Brady Reference Zia and Brady2013). To examine the micromechanical origins of this behaviour, we recall that stronger flow produces more localized distortion and stronger asymmetry as long as $\unicode[STIX]{x1D705}\neq 0$ (Batchelor Reference Batchelor1976; Khair & Brady Reference Khair and Brady2006; Chu & Zia Reference Chu and Zia2016). Strong flow produces a boundary layer that gets compressed as flow strength increases, with a thickness that scales as $\unicode[STIX]{x1D6FF}\sim 1/Pe$ (Squires & Brady Reference Squires and Brady2005). The densest region of the boundary layer is at the contact stagnation point, $f(2,\unicode[STIX]{x03C0},t)$, which increases with increasing $Pe$. Together, these drive a strong particle gradient across the boundary layer. To relax it, the probe need only diffuse a distance $\unicode[STIX]{x1D6FF}\sim 1/Pe$, and a stronger gradient produces a faster push to do so (Zia & Brady Reference Zia and Brady2013). This is quantified in figure 6.

Figure 6.

Comparison of structural relaxation rates to viscosity relaxation rate, with and without hydrodynamic interactions. (a,b) Structural relaxation at the stagnation point for (a) weak and (b) strong hydrodynamics. (c,d) Viscosity (on left axes) and stagnation-point density (on right axes) for (c) weak and (d) strong hydrodynamics. Panel (c) shows that, in a freely draining system, structure and rheology relax at the same rate. However, when hydrodynamics are strong (d), stagnation-point density relaxation leads viscosity relaxation. This illustrates one role of long-range hydrodynamics in viscosity relaxation, where downstream coupling in the more slowly relaxing wake region slows probe motion.

The maximum boundary-layer density, which occurs at the stagnation point, $f(2,\unicode[STIX]{x03C0},t)$, normalized by its initial value, is plotted as it decays with time for various values of $Pe$ in the limits of weak ($\unicode[STIX]{x1D705}\rightarrow \infty$) and strong ($\unicode[STIX]{x1D705}\rightarrow 0$) hydrodynamic interactions in figure 6(a) and (b), respectively. These plots show the presence of multiple relaxation time scales with or without hydrodynamics: an early-time fast relaxation followed by a long-time slower decay (Zia & Brady Reference Zia and Brady2013). The relaxation of the stagnation-point density is faster with increasing $Pe$ – irrespective of $\unicode[STIX]{x1D705}$ – showing that a more deformed structure relaxes faster, which owes its origin to boundary-layer dynamics, discussed further below. We next compare the relaxation rate of viscosity versus the relaxation rate of structure, for two representative flow strengths, $Pe=1$ and $Pe=50$. In figure 6(c) and (d), we plot the viscosity (left axis) versus the stagnation-point density (right axis), for weak (c) and strong (d) hydrodynamics. In the freely draining system (figure 6c), structure and viscosity relax at the same rate, irrespective of $Pe$, confirming the idea that contact structure sets the viscosity relaxation in the limit of $\unicode[STIX]{x1D705}\rightarrow \infty$ (Zia & Brady Reference Zia and Brady2013). In contrast, when hydrodynamic interactions matter (figure 6d), the stagnation-point density relaxes faster than the viscosity; this arises from long-ranged hydrodynamic interactions, where a particle that has relaxed away from contact still interacts with the probe through solvent-mediated hydrodynamic interactions. This illustrates one role of long-range hydrodynamics in viscosity relaxation, where downstream coupling in the more slowly relaxing wake region slows probe motion.

Figure 7.

Flow cessation for strong hydrodynamics and arbitrary $Pe$. (a) Normalized microviscosity as a function of $\hat{t}_{BL}=t/(\unicode[STIX]{x1D6FF}^{2}/2D_{0})$. For comparison, the weak hydrodynamics behaviour ($\unicode[STIX]{x1D705}\rightarrow \infty$) is shown in the inset. Comparison of the two reveals that, when relaxation time is scaled on the diffusivity of a particle on the length scale $\unicode[STIX]{x1D6FF}\sim 1/Pe$ of the boundary layer, only in the absence of hydrodynamics (inset) do the curves collapse at high $Pe$. This suggests that the scaling of particle mobility, rather than just the distance travelled to relax the boundary layer, must be taken into account. (b) The diffusivity is rescaled with the $Pe$-dependent concentration in the boundary layer, where now the curves align for all $Pe$.

We now take a closer look at the relaxation time scale that evidently changes with structural deformation. In studies of freely draining suspensions, a simple rescaling of time – where the diffusive process setting relaxation occurs through the boundary layer – collapses viscosity relaxation curves together, for values of $Pe$ sufficiently high to produce a proper boundary layer. Similar rescalings have been applied in hydrodynamically interacting systems with varying degrees of success (Foss Reference Foss1999; Marenne et al. Reference Marenne, Morris, Foss and Brady2017). The idea is that the probe need execute only one diffusive step out of the wake to relax the majority of the suspension stress. In figure 7(a), the post-cessation viscosity is scaled on the boundary-layer thickness, $\unicode[STIX]{x1D6FF}\sim 1/Pe$ (Khair & Brady Reference Khair and Brady2006), but fails to unite any of the curves. However, the rescaling works quite well at high $Pe$ in the freely draining system (inset), a phenomenon that has also been observed in sheared suspensions (Foss Reference Foss1999; Marenne et al. Reference Marenne, Morris, Foss and Brady2017). The difference between the two is the density inside the boundary layer, which is $g\sim Pe$ for freely draining (Squires & Brady Reference Squires and Brady2005) and $g\sim Pe^{\unicode[STIX]{x1D708}}$, with $\unicode[STIX]{x1D708}=0.799$, for strong hydrodynamics (Khair & Brady Reference Khair and Brady2006). Thus it makes sense to rescale not only the distance the probe must travel as it diffusively relaxes, but also the diffusive hindrance. That is, the boundary layer is less crowded, although hydrodynamic coupling slows relative motion there. We rescale time on the hindered diffusion in figure 7(b), where the high-$Pe$ viscosity curves are plotted as a function of time made dimensionless on boundary-layer relaxation, $t/(\unicode[STIX]{x1D6FF}^{2}/2D_{0})\times (1/Pe^{0.799})^{2}$. This reflects the time required for a particle to diffuse the thickness $\unicode[STIX]{x1D6FF}$ of a boundary layer of density $g\sim Pe^{\unicode[STIX]{x1D708}}$, where $\unicode[STIX]{x1D708}=0.799$ for $\unicode[STIX]{x1D705}=0$ (Khair & Brady Reference Khair and Brady2006). This scaling effectively adjusts the diffusive relaxation process for both boundary-layer hindrance $g\sim Pe^{\unicode[STIX]{x1D708}}$ and the distance $\unicode[STIX]{x1D6FF}$ the probe must travel to relax it. This collapses the high-$Pe$ curves together quite well in figure 7(b). The rescaling argument depends on the formation of a thin boundary layer through which the probe must execute a diffusive step to relax both structure and stress, making this scaling most appropriate when $Pe$ is appropriately large.

This analysis provides a direct way to compare the effective boundary-layer thickness in the presence of strong hydrodynamic interactions to the regime of weak hydrodynamics, $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D705}\rightarrow 0)\approx Pe^{0.799}\times \unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D705}\rightarrow \infty )$. This further emphasizes the importance of the boundary layer in the relaxation behaviour in dilute suspensions. The probe need move only the size of the boundary layer to relax the majority of the stress. However, for concentrated suspensions, the particle gradient across the wake will also play a prominent role in the relaxation process in both microrheology (Zia & Brady Reference Zia and Brady2013) and shear macrorheology (Foss Reference Foss1999; Marenne et al. Reference Marenne, Morris, Foss and Brady2017). The important implication of the difference between the strong hydrodynamics (main plot in figure 7a) and freely draining limits (inset in figure 7a) is that, even though the hydrodynamic contribution arising directly from the imposed flow vanishes immediately upon flow cessation, its impact on particle mobility and structural deformation persists well after flow shutoff.

Figure 8.

Flow cessation, strong hydrodynamics and arbitrary $Pe$: viscosity has been multiplied by the pre-cessation $Pe$ and is plotted against the diffusively scaled time.

The relaxation behaviour upon flow cessation gives the extent to which microstructural distortions persist in suspensions and memory behaviour in colloidal dispersions. We further examine the long-time viscosity relaxation over diffusive time scales in figure 8, where the viscosity is multiplied by $Pe$ (giving a measure of the stress associated with the deformed microstructure) and is plotted as a function of the diffusively scaled time for various $Pe$ in the limit of strong hydrodynamic interactions, $\unicode[STIX]{x1D705}\rightarrow 0$. Since the steady-state viscosities at different $Pe$ are of the same order, the vertical axis in figure 8 gives a measure of how far the structure is driven from its equilibrium configuration prior to flow cessation (Foss Reference Foss1999; Zia & Brady Reference Zia and Brady2013; Marenne et al. Reference Marenne, Morris, Foss and Brady2017). Earlier we established that the early-time viscosity decay is set by the boundary-layer relaxation, whereas the long-time viscosity relaxation reflects the structural rearrangement as Brownian motion drives the particles towards a uniform distribution. At long times, all the curves in figure 8, corresponding to different pre-cessation $Pe$, align well with each other, signalling the loss of memory in the distorted suspension. This behaviour is similar to that found in the regime of weak hydrodynamic interactions (Zia & Brady Reference Zia and Brady2013) and suggests that, independent of the initial distortion, the microstructural relaxation follows the same path at long times even for colloidal particles interacting via strong hydrodynamic interactions. The early-time relaxation is faster with increasing $Pe$, and therefore the time at which the curves begin to follow the long-time relaxation behaviour decreases with increasing $Pe$, signalling a decrease in the duration of memory in the colloidal dispersion. This also indicates that the memory is stored entropically in the distorted microstructure, which will be further explored in § 3.4. The loss of memory at long times recovers previous results of microrheology (Zia & Brady Reference Zia and Brady2013) and shear rheology (Foss Reference Foss1999; Marenne et al. Reference Marenne, Morris, Foss and Brady2017).

In summary, hydrodynamic interactions slow all modes of structural relaxation because such coupling slows relative particle motion. But the same interactions preserve structural symmetry pre-cessation and the two effects combined produce qualitatively different relaxation than that found in a freely draining suspension. The most distinct of these effects is the long-range hydrodynamic coupling that wraps structure around the probe, providing a pronounced temporal lag of rheological relaxation behind structural relaxation in the boundary layer – in contrast to freely draining systems where the two relax together. Flow thickening is forgotten instantaneously after flow shutoff, reaffirming the hydrodynamic nature of that phenomenon.

3.3 Arbitrary forcing and strength of hydrodynamics

Figure 9.

Normalized viscosity as a function of time for a range of $\unicode[STIX]{x1D705}$ for (a) $Pe=1$, (b) $Pe=10$ and $Pe=100$. The insets show the absolute values of the viscosity in a log–log plot to focus on the early-time relaxation and long-time algebraic decay.

Real materials in industry and science are typified by intermediate flow strengths and particle surface conditions that give moderately strong hydrodynamic interactions. Here we study relaxation at these finite values of $Pe$ and $\unicode[STIX]{x1D705}$. In figure 9, we plot the normalized viscosity as a function of the diffusively scaled time after flow shutoff, for $\unicode[STIX]{x1D705}\in [10^{-5},10^{3}]$, with one value of $Pe$ in each plot: $Pe=1$, $Pe=10$ and $Pe=100$. All three plots show at least two relaxation modes: an early-time faster relaxation, which becomes more pronounced with increasing $Pe$, and a slower long-time decay of $\unicode[STIX]{x1D702}^{micro}$ towards zero. All relaxation modes slow down as hydrodynamic interactions become stronger, because they slow relative diffusion. However, the viscosity relaxes back to equilibrium faster with increasing pre-cessation $Pe$. We expect the latter based on the asymptotic results in the previous section, but here we analyse it quantitatively, with a focus on the time required for the microviscosity to completely relax, $\hat{t}_{diff}^{relax}$, in detail.

In figure 10(a), we plot $\hat{t}_{diff}^{relax}$ as a function of the pre-cessation probe forcing, $Pe$, for various strengths of hydrodynamic interactions between particles, $\unicode[STIX]{x1D705}$. Relaxation is faster following higher-$Pe$ flow (at fixed $\unicode[STIX]{x1D705}$), but this effect weakens as hydrodynamic interactions grow stronger and hinder diffusive relaxation.

Figure 10.

Time required for the viscosity to fully relax, $\hat{t}_{diff}^{relax}$, as a function of (a) $Pe$ for various $\unicode[STIX]{x1D705}$, and of (b) $\unicode[STIX]{x1D705}$ for various $Pe$.

The parameter space is ‘flipped’ in figure 10(b) to highlight the primary effects of hydrodynamic coupling on structural relaxation. First, hydrodynamics preserve symmetry, which suppresses the strong gradients that drive fast relaxation. Second, they impede relative particle motion. These two effects combined slow down structural and rheological relaxation. For higher pre-cessation flow, these effects become more pronounced, approaching the pure hydrodynamics limit where structure is symmetric and Brownian motion is absent, removing all mechanisms for relaxation. As a result $\unicode[STIX]{x1D705}$ has a very strong effect on $\hat{t}_{diff}^{relax}$ as pre-cessation $Pe$ increases.

Figure 11.

Flow cessation, arbitrary $Pe$ and $\unicode[STIX]{x1D705}$: (a) stretched exponent, $\unicode[STIX]{x1D6FD}$, as a function of $Pe$ for various values of $\unicode[STIX]{x1D705}$, and (b) characteristic relaxation time scale, $\unicode[STIX]{x1D70F}_{rel}$, as a function of $Pe$ for various strengths of hydrodynamic interactions.

Relaxation processes are sometimes fitted to a stretched exponential following the Kohlrausch–Williams–Watts function form (Bouchaud Reference Bouchaud2008):

(3.3)$$\begin{eqnarray}\frac{\unicode[STIX]{x1D702}^{micro}}{\unicode[STIX]{x1D702}^{micro}(t=0)}\sim \text{e}^{-(t/\unicode[STIX]{x1D70F}^{rel})^{\unicode[STIX]{x1D6FD}}}.\end{eqnarray}$$

This functional form can be solved for both the stretching exponent, $\unicode[STIX]{x1D6FD}$, and the characteristic relaxation time scale, $\unicode[STIX]{x1D70F}_{rel}$. The stretching exponent, $0\leqslant \unicode[STIX]{x1D6FD}\leqslant 1$, indicates a disparity in the different rates at which a system relaxes. For a system with one relaxation process (or one relaxation time scale), $\unicode[STIX]{x1D6FD}=1$, and $\unicode[STIX]{x1D6FD}$ decreases if a structure relaxes over multiple disparate time scales. Prior cessation relaxation studies of sheared suspensions mention that, in the limit of very strong hydrodynamic interactions, $\unicode[STIX]{x1D70F}_{rel}$ and $\unicode[STIX]{x1D6FD}$ both decrease monotonically with increasing $Pe$ (Marenne et al. Reference Marenne, Morris, Foss and Brady2017). Here we seek a mechanistic explanation for this behaviour. We fit the data in figure 9 to equation (3.3) and plot the stretching exponent, $\unicode[STIX]{x1D6FD}$, in figure 11(a), for several values of $\unicode[STIX]{x1D705}$, and the characteristic relaxation time scale, $\unicode[STIX]{x1D70F}_{rel}$, in figure 11(b). Our work recovers the $Pe$-dependent behaviour found in sheared suspensions, and additionally finds that both $\unicode[STIX]{x1D6FD}$ and $\unicode[STIX]{x1D70F}_{rel}$ increase monotonically with decreasing $\unicode[STIX]{x1D705}$ (strengthening hydrodynamic interactions), supporting our conclusion that hindered relative mobility in the boundary layer slows stress relaxation. The stretched exponent, $\unicode[STIX]{x1D6FD}$ in figure 11(a), decreases with increasing $Pe$, indicating the disparity between the early-time faster and long-time slower relaxation processes for high $Pe$ (Zia & Brady Reference Zia and Brady2013). With increasing strength of hydrodynamic interactions, the disparity between the faster and slower relaxation modes decreases, leading to a larger $\unicode[STIX]{x1D6FD}$.

3.4 Probe motion and non-equilibrium osmotic pressure

Figure 12.

Non-equilibrium osmotic pressure drives deterministic probe motion after the external force has been removed. (a) The probe displacement (left axis) and the steady-state osmotic pressure (right axis) from Chu & Zia (Reference Chu and Zia2017, with permission), as a function of $Pe$. (b) Ratio of the pre-cessation osmotic pressure to the probe displacement as a function of $Pe$.

In the previous sections, we examined the viscous behaviour of colloidal dispersions with a focus on the relaxation of the viscosity upon flow cessation. Here, we explore the elastic behaviour in suspensions by examining the stress and energy storage in a distorted microstructure, and their effect on particle motion. At equilibrium, thermal fluctuations produce temporary gradients in number density that relax via Brownian diffusion, giving rise to osmotic pressure. The reduction in entropy associated with local compression of structure temporarily stores energy entropically, but this chemical potential is quickly dissipated viscously back into the solvent. Zia & Brady (Reference Zia and Brady2013) showed that flow energy stored entropically in the steady-state deformed microstructure can be partially recovered to deterministically back-drive the probe after removal of the external force, for a freely draining suspension. When hydrodynamic interactions play a role, the compressed boundary layer, and the rate at which it relaxes, change (cf. §§ 3.2 and 3.3). Here we investigate whether hydrodynamics changes the useful work the non-equilibrium osmotic pressure can do post-cessation. We monitor probe position after removal of the external force, and plot the total probe displacement, $\langle x\rangle (Pe,\unicode[STIX]{x1D705})$, where the angle brackets signify an average over realizations, on the left axis in figure 12(a). When pre-cessation $Pe$ is small, very little deterministic probe motion occurs after cessation, because the relaxing structure was a weak diffusive dipole. As $Pe$ increases, the same trend as observed in freely draining systems (Zia & Brady Reference Zia and Brady2013) is observed: probe displacement grows. However, the stronger the hydrodynamic interactions, the less post-cessation total displacement at any fixed $Pe$. The preservation of structural symmetry by hydrodynamic interactions is responsible for this behaviour. But as $Pe$ continues to grow, total displacement actually decreases, because now the boundary layer has wrapped around the probe, accumulating particles in the downstream region of the probe. The probe thus gets a more symmetric distribution of kicks from behind as well as in front, and, in consequence, the osmotic force is more uniformly distributed on the probe. The probe displacement now decreases with increasing $Pe$. From an entropic perspective, a less distorted structure stores less energy.

The strength of hydrodynamic interactions between particles, $\unicode[STIX]{x1D705}$, has a strong influence on the steady-state structure around the probe, and in turn on the average probe displacement after flow cessation. Stronger hydrodynamic coupling preserves symmetry, leading to a weaker entropic force, and thence less average probe displacement. Overall, while the total mean probe displacement upon flow cessation decreases with increasing strength of hydrodynamic interactions, $\unicode[STIX]{x1D705}$, it shows a non-monotonic dependence on the pre-cessation probe forcing strength, $Pe$, owing to the wrapping of the boundary layer around the probe.

Since the probe motion after flow cessation arises solely from the deformed structure, one can connect the entropic force that drives the probe with the osmotic compressibility of the suspension (Zia & Brady Reference Zia and Brady2013). The steady-state osmotic pressure was calculated by Chu & Zia (Reference Chu and Zia2017) as a function of $Pe$ for various values of $\unicode[STIX]{x1D705}$ and we plot those values on the right axis in figure 12(a). A comparison between the post-cessation mean probe displacement, $\langle x\rangle (Pe,\unicode[STIX]{x1D705})$, and the non-equilibrium osmotic pressure, $\langle \unicode[STIX]{x1D6F1}^{neq}\rangle (Pe,\unicode[STIX]{x1D705})$, reveals that the two quantities are strongly correlated: both $\langle x\rangle (Pe,\unicode[STIX]{x1D705})$ and $\langle \unicode[STIX]{x1D6F1}^{neq}\rangle (Pe,\unicode[STIX]{x1D705})$ approach zero as $Pe\ll 1$, both increase with increasing $Pe$ for $Pe<5$ and decrease thereafter; for weak hydrodynamic interactions, both quantities approach an asymptotic value at very high $Pe$. Increasing strength of hydrodynamic coupling between particles suppresses both $\langle x\rangle (Pe,\unicode[STIX]{x1D705})$ and $\langle \unicode[STIX]{x1D6F1}^{neq}\rangle (Pe,\unicode[STIX]{x1D705})$ at all $Pe$. In the limit of strong hydrodynamic interactions, $\unicode[STIX]{x1D705}\rightarrow 0$, and strong probe forcing, $Pe$, both quantities approach zero. This behaviour in the dual limits of strong hydrodynamic interactions and strong probe forcing is expected, as these asymptotic limits correspond to the pure Stokes flow limit, where the structure becomes spherically symmetric around the probe particle. This symmetric structure leads to zero mean probe displacement upon removal of the external force.

Overall, both the probe back-travel, $\langle x\rangle (Pe,\unicode[STIX]{x1D705})$, and the steady-state osmotic pressure, $\langle \unicode[STIX]{x1D6F1}^{neq}\rangle (Pe,\unicode[STIX]{x1D705})$, are attenuated by hydrodynamic interactions, and exhibit a non-monotonic behaviour in $Pe$ associated with wrapping of the boundary layer. There is a strong one-to-one qualitative connection between the osmotic pressure and probe back-travel. In fact, the ratio is near unity when hydrodynamic interactions are strong, which is more clearly seen in figure 12(b) where we plot the ratio of the osmotic pressure to the probe back-travel as a function of $Pe$ for various $\unicode[STIX]{x1D705}$. This ratio, which signifies the entropic force density arising from the distorted structure, exhibits a strong dependence on the strength of hydrodynamic interactions, $\unicode[STIX]{x1D705}$. However, this ratio does not change much as $Pe$ is varied, suggesting that the probe back-travel gives a very clear experimental measure of the osmotic pressure in a suspension. This further illustrates that the stored energy can do useful work and the probe back-travel gives a direct measure of the flow energy density associated with the deformed structure.

In summary, probe back-travel upon flow cessation can provide a measure of the osmotic pressure in a colloidal dispersion, a link inferred from the connection of the osmotic pressure to the chemical potential. The osmotic pressure is energy density stored entropically in the distorted hard-sphere suspension. Hydrodynamic interactions hinder this entropic energy storage in colloidal dispersions by preserving structural symmetry and, in addition, they reduce the useful work available upon flow cessation by hindering relative particle motion.