Experimental Realization of the Green-Kubo Relation in Colloidal Suspensions Enabled by Image-based Stress Measurements

TL;DR

This study experimentally verifies the Green-Kubo relation in colloidal suspensions by using a novel image-based stress measurement technique, enabling precise stress fluctuation analysis and viscosity determination in 3D colloidal liquids.

Contribution

We developed a non-invasive method combining confocal microscopy and SALSA to measure stresses in 3D colloids, validating the Green-Kubo relation for nearly hard-sphere suspensions.

Findings

Green-Kubo relation accurately predicts viscosity in colloidal suspensions.

Our method measures forces as low as 50 fN with high spatial resolution.

Results agree with simulations and traditional rheometry.

Abstract

By combining confocal microscopy and Stress Assessment from Local Structural Anisotropy (SALSA), we directly measure stresses in 3D quiescent colloidal liquids. Our non-invasive and non-perturbative method allows us to measure forces $\lesssim$ 50 fN with a small and tunable probing volume, enabling us to resolve the stress fluctuations arising from particle thermal motions. We use the Green-Kubo relation to relate these measured stress fluctuations to the bulk Brownian viscosity at different volume fractions and comparing against simulations and conventional rheometry measurements. We demonstrate that the Green-Kubo analysis gives excellent agreement with these prior results. This agreement provides a strong demonstration of the applicability of the Green-Kubo relation in nearly hard-sphere suspensions and opens the door to investigations of local flow properties in many poorly understood far-from-equilibrium systems, including suspensions that are glassy, strongly-sheared, or highly-confined.

Equations

$${\eta }_B^{bulk}$$

$$\sim {\int }_0^{\infty }{C}_{\Pi }{e}^{-\Delta t/\tau }d\Delta t\sim ⟨{\Pi }^2⟩-⟨\Pi {⟩}^2$$

Full text

All quiescent thermal systems may seem static macroscopically, but microscopically they fluctuate strongly. By observing the system’s response to these thermal fluctuations, a material’s linear transport coefficients can be predicted using the Green-Kubo relation Kubo (1957); Green (1954); Morriss and Evans (2007); Hansen and McDonald (2013). This foundational relation – a central achievement of nonequilibrium statistical mechanics – has enabled numerous diverse theoretical calculations ranging from electrical and magnetic susceptibilities in quantum systems Baranger and Stone (1989); Alhassid (2000) to thermal conductivities in nanotubes Berber et al. (2000); Savin et al. (2009); Yao et al. (2005); Zhang et al. (2004). In particular, it has been widely used to theoretically determine the viscosities in bulk Hoover et al. (1980), confined Neek-Amal et al. (2016); Huang and Szlufarska (2014), supercooled Kushima et al. (2009); Sosso et al. (2012), and quantum Reichman and Rabani (2001) liquids, where external load is problematic or heterogeneities play a crucial role. Unfortunately, these applications have remained strictly theoretical due to the difficulties in experimentally observing fluctuations in atomic systems, which are too rapid ($\sim$ ps) and weak ($\sim$ $\mu$N) to mechanically resolve in experiments.

Here, by using high-speed confocal microscopy in conjunction with Stress Assessment from Local Structural Anisotropy (SALSA) Lin et al. (2016), we directly measure the stress fluctuations in nearly hard-sphere colloidal liquids. Colloidal suspensions are comprised of particles that are small enough to demonstrate Brownian motions, while large enough to be optically imaged, providing length- and time-scales that are associated with system relaxation Foss and Brady (2000). To measure a suspension’s stress fluctuations, we use a confocal microscope to image the 3D microstructure of the sample, then use SALSA to determine its Brownian stress arising from interparticle thermal collisions. Since SALSA is image-based, non-invasive, non-perturbative, and able to measure the suspension stress with a tunable probing volume, it can resolve the weak stress fluctuations that are usually averaged out in conventional bulk measurements due to the requisite large probing volume.

The suspension samples are comprised of silica spheres with a radius $a=$ 490 nm in a water-glycerine mixture that has a matched refractive index and viscosity $\eta_{0}=60$ mPa$\cdot$s. We add 1.25 mg/ml of fluorescein sodium salt to the solvent to shorten screening length ($\leq$10 nm) and obtain nearly hard sphere interactions. The added fluorescein also makes the solvent fluorescent, so the solvent appears bright and the particles appear dark. We then image the particle configuration using a high-speed confocal microscope with a hyper-fine scanner that maximizes the stability in the vertical ($z$-axis) scanning position (schematic in Fig. 1(a)). To ensure that the suspension structure remains homogenous throughout the experiment, we image the sample within a minute after the sample cell is made. We capture 216 frames per second and acquire stacks of 100 images within $0.5~{}\rm{s}\sim 0.02\tau_{B}$, where $\tau_{B}=6\pi a^{3}\eta_{0}/k_{B}T$ is the self-diffusion time of the sample.

By implementing the previously developed SALSA method Lin et al. (2016), we determine the stress in our 3D suspensions. SALSA uses the featured particle positions to calculate the local structural anisotropy or fabric tensor $\psi_{ij}^{\alpha}(\Delta)=\sum_{\beta\in nn}\hat{r}_{i}^{\alpha\beta}\hat{r}^{\alpha\beta}_{j}$ of particle $\alpha$, where $nn$ is the set of colliding neighbors that lie within a distance $2a+\Delta$ from particle $\alpha$ ($\Delta=106$ nm in the current work), $i$,$j$ are spatial indices, and $\hat{r}_{ij}$ is the unit vector between particles (see Fig. 1(b) and SI). Scaling the ensemble-averaged $\psi_{ij}^{\alpha}(\Delta)$ by $\Delta$ enables us to estimate the probability of thermal collisions between particles. Consequently, the instantaneous Brownian stress of the sample can be approximated: $\sigma_{ij}(V,\Delta)=\frac{k_{B}T}{V}\frac{a}{\Delta}\sum_{\alpha\in V}\psi_{ij}^{\alpha}(\Delta)+nk_{B}T\delta_{ij}$, where $V$ is the averaging window volume, $k_{B}T$ is thermal energy, $n$ is number density, and $\delta_{ij}$ is Kronecker delta function. Here, $nk_{B}T$ is simply the ideal gas term.

The typical volume of our probed region $V=61\mu$m$\times 15\mu$m$\times 12\mu$m $\sim 10~{}\rm{pL}$ contains approximately 6,000 particles at a volume fraction $\phi\sim 0.27$. This small volume ensures that the stress fluctuations are not suppressed by the volume averaging, $\propto 1/V$, while preserving bulk behavior. We plot the instantaneous stress $\sigma_{xz}$ and $\sigma_{xy}$ in Fig. 1(b), where $\hat{z}$ is the gravitational axis and $\hat{x}$ and $\hat{y}$ are horizontal. In contrast to a flat line at zero level anticipated in a macroscopic measurement, we find that both $\sigma_{xz}$ and $\sigma_{xy}$ fluctuate up to $\pm$ 0.5 mPa. We note that the force fluctuations corresponding to these stresses are less than 50 fN, difficult to resolve using mechanical methods.

We calculate the time-time autocorrelation function $\langle\sigma_{ij}(t+\Delta t)\sigma_{ij}(t)\rangle$ for the stress components $\sigma_{xz}$ and $\sigma_{xy}$, and show the correlation decay in a log-linear plot, see Fig. 2 (a). Despite the slight sedimentation due to the density mismatch between the particle and solvent, both autocorrelation functions decay in the same fashion indicating an isotropic viscosity of the sample (see SI). We further examine the cross-correlation $\langle\sigma_{xz}(t+\Delta t)\sigma_{xy}(t)\rangle$ and find it negligibly small, which is consistent with the system symmetry. While the exact function form of the autocorrelation decay cannot be determined from the current data due to the limited measurement time span, we use an exponential decay ($\sim e^{-\Delta t/\tau}$) to quantify the correlation time. In doing this, we find that the correlation time $\tau$ varies weakly with the suspension volume fraction $\phi$ (see Fig. 2 (b)). We compare our observed trend with previous simulations of short-time diffusivity $D_{ss}$ where $a^{2}/D_{ss}$ roughly sets the relaxation time-scale of the system [19,20]. In simulations, $D_{ss}$ decays approximately as $D_{ss}\sim D_{0}(1-b\phi)$ (red line, Fig. 2 (b)) with $b$ on the order of 1.5 at intermediate volume fractions. Here, we find a weaker trend $b\sim 0.60\pm 0.23$ (blue dashed line) indicating either our measurements are not sufficiently precise to determine $b$ accurately or that the functional form changes at volume fractions approaching close-packing.

With the measured stress fluctuations, we can directly calculate the shear viscosity of our sample via the Green-Kubo formula $\eta_{B}=\langle\frac{V}{k_{B}T}\int\langle\sigma_{ij}(t+\Delta t)\sigma_{ij}(t)\rangle d\Delta t\rangle_{i\neq j}$, where $\eta_{B}$ is the Brownian contribution to the total shear viscosity $\eta_{tot}$. Since our suspension systems are nearly hard-sphere, we anticipate that the stresses are weakly correlated in space, and thus the sample viscosity is roughly independent of probe window size. To verify this, we change our probing (averaging) volume $V$, and investigate how the stress fluctuations vary. In Fig. 2(c), we plot the mean variance of shear stress $C_{ij}=\langle\sigma_{ij}(t)\sigma_{ij}(t)\rangle_{t,i\neq j}$ as a function of $V/V_{p}$ where $V_{p}$ is the particle volume $(4/3)\pi a^{3}$. We find that $C_{ij}$ is inversely proportional to $V/V_{p}$ when $V/V_{p}\geq 200$ corresponding to a cubic volume that is approximately six particles across. This inverse proportionality and constant viscosity shown in the inset of Fig. 2(c) are consistent with the Green-Kubo formula. When $V/V_{p}\leq 200$, we find that the viscosity slightly deviates from its bulk value. The viscosity reduction is around $20\%$ of the mean for the smallest probing volume explored – a three-particle wide cube. While this reduction is reminiscent of the system size-dependent viscosity associated with long-ranged stress correlations in atomic simulations Levashov et al. (2011); Yeh and Hummer (2004); Erpenbeck (1995); Meier et al. (2004); Heyes (2007); Chattoraj and Lemaitre (2013); Maloney and Lemaître (2006), in our nearly hard-sphere liquid system we do not anticipate such long-ranged correlations that lead to nonlocal viscosities. Instead, at small volumes, the stress fluctuations are strongly influenced by changes in particle number as particles pass into and out of the constrained field of view.

To compare our results with macroscopic flow measurements and simulations, we use the measured stress autocorrelation in conjunction with the Green-Kubo relation to determine the Brownian viscosity $\eta_{B}$ of suspensions at eight different volume fractions $0.12\leq\phi\leq 0.45$ (see Fig. 3). The resulting viscosities (red circles) show excellent agreement with previous hydrodynamic Stokesian simulations (blue squares)  Banchio and Brady (2003). To further confirm the accuracy of our SALSA stress measurement, we also use Brownian Dynamics simulations to generate sets of particle configurations matching the experimental parameters (e.g. particle size, solvent viscosity, and temperature), and compare the stresses calculated from actual virials $F_{ij}X_{ij}$ (purple diamonds) with those calculated on the same data set with SALSA (green diamonds) Guazzelli and Morris (2011). Both results again show a quantitative agreement with the experimental measurements. Finally, the measured Brownian viscosities are compared with conventional mechanical measurements by subtracting the hydrodynamic contribution $\eta_{H}$ from the total viscosity $\eta_{tot}$ determined using rheometry (purple crosses) Cheng et al. (2002, 2011). The rheology data points (colloidal PMMA and silica systems) are obtained from previous experiments Cheng et al. (2002) and the hydrodynamic contribution is calculated from previous analytical approximation for the high frequency viscosity Cheng et al. (2002); Sierou and Brady (2001); Mewis and Wagner (2012); Lin et al. (2014). We find good agreement between the viscosities determined by our stress fluctuation measurements and conventional rheometry at all volume fractions explored. Collectively, the agreements between our results, simulations, and bulk measurements provide a clear demonstration of the Green-Kubo formula in hard-sphere systems.

In contrast to conventional mechanical measurements, which can only measure the flow-gradient stress and the difference between normal stresses, SALSA measures all stress components simultaneously. In Fig. 4(a) we report the pressure of the suspension at the eight volume fractions explored in Fig. 3. We find that the measured osmotic pressure arising from Brownian collisions (red disks) is well described by the Carnahan-Starling equation of state (gray line) 111We note that in previous confocal measurements where the pressure is determined by calculating the available volume to insert an additional sphere into a system, and its surface area Dullens et al. (2006), the volume and corresponding surface area become exceedingly small and difficult to measure when suspensions are dense. This uncertainty results in a mismatch between data and theory. In our experiment, SALSA accurately captures the particle collision probabilities and correctly reports the pressure at all tested volume fractions.. In addition, we find that both the shear ($\eta_{B}$, light red points) and bulk ($\eta_{B}^{bulk}$, dark red points) viscosities roughly exhibit $\Pi^{2}$ scaling (dashed black line), as shown in Fig. 4(b). While the underlying mechanism of such an empirical scaling remains an open question, we can qualitatively understand this scaling for the bulk viscosity using a dimensional analysis. Since the correlations in the Green-Kubo formula decay approximately exponentially in time, and the relaxation time $\tau$ does not increases significantly with increasing pressure over the range measured, we have

[TABLE]

where $C_{\Pi}$ is the variance of pressure Heyes (1992); Fernandez et al. (2004).

While many previous studies have made analogies between the transport phenomena of colloidal systems and simple liquids Pham et al. (2002); Ediger et al. (1996); Cheng et al. (2011); Lin et al. (2013), we find that the observed $\Pi^{2}$ scaling is actually absent in atomic systems. Specifically, the atomic viscosity (blue curve in Fig. 4(b)) exhibits a similar scaling behavior, but only at high pressures corresponding to large $\phi$. At low pressures, the viscosity trend deviates from the $\Pi^{2}$ scaling. We conjecture that this deviation is associated with the kinetic contribution to the viscosity  Chapman and Cowling (1970); Hansen and McDonald (2013), which is associated with atom velocity, insensitive to $\Pi$, and dominates in the dilute limit (see SI). Collectively, our findings, which are made possible by SALSA, suggest that even the Brownian contribution to the colloidal viscosity can have a distinct transport mechanism than that in simple liquids.

In conclusion, we measure the stress fluctuation in colloidal liquids with SALSA, and experimentally demonstrate the well-known Green-Kubo relation Kubo (1957); Green (1954). Our measurements essentially show that “as far as linear responses are concerned, the admittance is reduced to the calculation of time-fluctuations in equilibrium” Kubo (1957). Previous pioneering experiments were able to combine the Green-Kubo relation with numerical simulations to extract the viscosity of a 2D dusty plasma Feng et al. (2011). These measurements, however, relied on assumptions for the interparticle potentials and ignored power-law decays in the stress correlation characteristic of 2D systems, which are known to lead to diverging integrals Alder and Wainwright (1970); Evans (1980); Clercx and Schram (1992). The analysis presented here avoids many of these complications and opens the door to further investigations of stress distributions in liquids under shear, confinement, and at high densities where the suspension becomes glassy Kegel and van Blaaderen (2000); Weeks et al. (2000); Hunter and Weeks (2012). In such situations SALSA is still applicable since the solvent remains in equilibrium. More importantly, since the SALSA measurement is non-invasive, it also allows for probing the mechanical heterogeneity in a 3D colloidal glass Wyart et al. (2005); Schall et al. (2007); Falk and Langer (1998); Bowick and Chaikin (2016); Chen et al. (2011); Tan et al. (2012); Chen et al. (2010), in which we can perform a time-average for particle-scale stress calculation. Measuring the temporal and spatial stress fluctuations in such a system will shed light on the generalization of the Green-Kubo relation in far-from-equilibrium systems and elucidate the mechanisms that underly the flow behaviors of disordered systems.

The authors thank James Sethna, Brian Leahy, James Swan, John Brady, and Wilson Poon for helpful discussions. I.C. and N.Y.C.L. gratefully acknowledge the Poon Laboratory at School of Physics & Astronomy, University of Edinburgh for generous use of their PMMA suspensions. I.C. and N.Y.C.L. acknowledge funding from National Science Foundation (NSF) NSF CBET-PMP Award 1509308. M. B. was supported by Department of Energy DOE-DE-FG02-07ER46393 and continued support from NSF DMR-1507607.