In this section, we report the results of three rheological experiments: (i) flow sweeps of corn stover slurries before and after fermentation; (ii) flow sweeps of samples taken over the course of fermentation; and (iii) large amplitude oscillatory shear (LAOS). During a *flow sweep*, the rheometer turns at steady state while torque is measured, and the torque and rotation rate data are used to infer strain rate, stress, and viscosity. In an *oscillatory shear* test, the rheometer oscillates at a specified frequency while measuring torque and rotation speed. All rheological experiments were conducted for corn stover slurries at various *insoluble solids concentrations* measured as described in the Methods section.

### Experiment 1: Rheology before and after fermentation

In this experiment, the rheological properties of unpretreated corn stover (with 0.2 mm maximum particle size) were examined without and with fermentation by *C. thermocellum*, a thermophilic, cellulolytic anaerobic bacterium. Steady shear flow sweeps were performed on slurries before and after fermentation. Slurries were prepared at four different insoluble solid concentrations \( c \) = 10, 12.5, 15, and 17.5 wt.%. These concentrations span the capability range of the rheometer and are representative of conditions anticipated in an industrial process. Fermentation was carried out using *C. thermocellum*. After reaction, the solids remaining in the reactor were reconstituted to the same four concentrations as the unfermented slurries to allow for a direct comparison of viscosity before and after fermentation.

As illustrated in Fig. 1, the slurries ranged from being a pourable liquid to being a clay-like paste, depending on the solids concentration and state of fermentation. These extremes posed challenges to collection of repeatable shear sweep data: Low-solids-concentration slurries were liquid-like, so they had to be well mixed and then immediately loaded onto the rheometer to mitigate particle settling. High-solids-concentration slurries had no visible free water; so, care had to be taken to load the appropriate amount of material into the rheometer to maintain good contact between the top plate and the material yet prevent the top plate from squeezing water out of the slurry.

Figure 1 reports viscosity and shear stress versus strain rate for flow sweep experiments, with power law fits displayed as dashed lines overlaid on the data. The black dotted lines represent the rheometer’s low-torque limit, 10 μNm, and the red dotted lines take into account manufacturing errors with 20 times this low-torque limit [9]. Viscosity is observed to decrease after fermentation and with lower solids concentration. For the range of strain rates considered, these corn stover slurries are well described by a power law constitutive model,

$$ \tau = m\dot{\gamma }^{n} $$

where \( \tau \) is the shear stress (Pa), \( \dot{\gamma } \) is the strain rate (1/s), \( m \) is the plastic viscosity (Pa s), and \( n \) is the power law index. Interestingly, all slurries have similar \( n \) values of 0.10, indicative of highly shear-thinning behavior. Figure 2 reports the plastic viscosity of both the unfermented and fermented material. Also shown are literature data: Viamajala et al. [38] report data for corn stover slurries milled to − 80 mesh (0.177 mm) size and treated with dilute acid (1.5% H_{2}SO4) at 25 °C and 190 °C. Dunaway et al. [6] provide data for before and after hydrolysis of a corn stover slurry with pretreatment via dilute acid (1.6% H_{2}SO4) at 190 °C, hydrolysis for 170 h by fungal cellulase (*T. reesei*), and 0.03 mm particle size (Berson, personal communication).

It may be observed that the plastic viscosity of our fermented 10 wt.% corn stover slurry is 40-fold lower than that of its unfermented counterpart. At higher solids concentrations (for instance 17.5 wt.%), this viscosity reduction is less pronounced (twofold decrease). Plastic viscosity is observed to increase with increasing solids concentrations but plateau for very high concentrations. This “stacking” stems from frictional forces (between granular particles) dominating once the solids concentration is so high that the slurry contains no free water [38].

### Experiment 2: Rheology over the course of fermentation

In this experiment, flow sweeps and LAOS measurements were made on slurry samples drawn from the bioreactor during the course of fermentation to characterize the changes in rheology versus extent of fermentation. This section presents results of flow sweep tests, and the results of LAOS tests are presented in the next section. Duplicate fermentations for this experiment were carried out at an initial solids concentration of 8 wt.% (80 g/L), which is near the maximum allowable limit for batch cultures in light of initial mixing and sampling constraints in the laboratory bioreactor set-up employed. All slurry samples were concentrated twofold following fermentation and prior to rheological measurements to enable measuring viscosity above the low-torque limit of the rheometer. The solid concentrations referred to in this section are these twofold-concentrated values.

Figure 3 presents the changes in viscosity as fermentation proceeds for a corn stover slurry with an initial insoluble solids concentration of 16 wt.%. Data for two separate fermentation trials are overlaid along with their power law fits. Plastic viscosity is plotted on a linear scale (c) and a log scale (d) as a function of fractional conversion \( X \). Error bars represent 1 standard deviation. Data points for which error bars are not visible had standard deviations smaller than the symbols for the data. The solid concentrations at several fermentation time points are also indicated on Fig. 3a, b, with a lower solids concentration corresponding to a sample collected later in time. Consistent with the results presented in Fig. 1 (obtained at specified solids concentrations), results in Fig. 3a, b (at variable solids concentrations) exhibit several key features: shear-thinning behavior; viscosity well represented by a power law fit; and viscosity that decreases with decreasing solids concentration.

In Fig. 3c, d, plastic viscosity values are plotted versus fractional carbohydrate conversion on linear and log scales, respectively. The plastic viscosity falls from roughly 1920 to 1 Pa s over the course of the fermentation. Most of this nearly-2000-fold decrease takes place before 20% conversion, which corresponds to a fermentation time of approximately 2 days in these experiments. Interestingly, this 2000-fold decrease in viscosity is similar to that observed in Fig. 2 (therein considering a hypothetical fermentation run starting at 17.5 wt.% solids and reacting until a final concentration of 10 wt.% is reached). Here, the fermentation run started at 16 wt.% solids and ended at 10 wt.%; so, it is reasonable that the results of the two figures are in agreement.

The data were fit (via nonlinear least squares) with a double exponential curve, \( m\, = \, 1700{\text{e}}^{{ - 23{\text{X}}}} \, + \,220{\text{e}}^{{ - 9.5{\text{X}}}} \), which illustrates the initial rapid decay of viscosity in the first stages of conversion \( 0 \le X \le 0.2 \). The viscosity decreases eightfold (from 1920 to 250 Pa s) in the first 10% of conversion. A goodness-of-fit metric appropriate for a nonlinear regression such as this is the canonical angle \( \psi \), which is the angle between the vector of data \( \tilde{m}_{n} \) and the vector of model predictions, \( m_{n} \). This angle can be computed via \( \cos \psi \, = \,{{\left( { \mathop \sum \nolimits m_{n} \tilde{m}_{n} } \right)} \mathord{\left/ {\vphantom {{\left( { \mathop \sum \nolimits m_{n} \tilde{m}_{n} } \right)} {\left( {\sqrt {\mathop \sum \nolimits m_{n}^{2} } \sqrt {\mathop \sum \nolimits \tilde{m}_{n}^{2} } } \right)}}} \right. \kern-0pt} {\left( {\sqrt {\mathop \sum \nolimits m_{n}^{2} } \sqrt {\mathop \sum \nolimits \tilde{m}_{n}^{2} } } \right)}} \), where \( \psi = 0^{ \circ } \) would indicate a perfect fit and \( \psi = 90^{ \circ } \) would indicate that the model provides no explanation of the data. For these data and fit, the canonical angle is \( \psi = 12^{ \circ } \). This is a reasonable fit; by analogy with linear data *y*(*x*) = * x * + * N*(0,σ^{2}), the canonical angle depends linearly on the standard deviation σ (for \( \psi < 20^{ \circ } \)), and \( \psi = 12^{ \circ } \) corresponds to *σ* = 0.122 and a *coefficient of determination* of *R*^{2} = 0.85.

### LAOS experiments

Large amplitude oscillatory shear tests were conducted on the corn stover slurries sampled during *Experiment 2* to further probe their rheological properties. In each LAOS experiment, stress is measured while the material is harmonically strained, \( \gamma \left( t \right) \, = \, \gamma_{0} { \sin }\left( {\omega t} \right) \). The stress response is plotted as stress versus strain (elastic Lissajous–Bowditch (ELB) curve) or stress versus strain rate (viscous Lissajous–Bowditch (VLB) curve). A perfectly elastic body would have a straight-line ELB curve and a circular VLB curve, and vice versa for a perfectly viscous fluid. Each subfigure in Fig. 4 contains a family of L–B (Lissajous–Bowditch) curves, with each curve centered around its frequency \( \omega \) (rad/s) and amplitude \( \gamma_{0} \) (%) in a Pipkin map [31]. LAOS data are typically filtered using a three-odd-term Fourier series [8]:

$$ \sigma \left( t \right)\, = \,\sigma^{\prime}\left( t \right)\, + \,\sigma^{\prime\prime}\left( t \right)\, = \,\mathop \sum \limits_{n = 1,3,5} \gamma_{0} G^{\prime}_{n} \sin n\omega t\, + \, \gamma_{0} G^{\prime\prime}_{n} \cos n\omega t$$

where \( \sigma^{\prime}\left( t \right) \) and \( \sigma^{\prime\prime}\left( t \right) \) represent the elastic and viscous stress contributions, and the \( G_{n}^{'} \left( {\gamma_{0} ,\omega } \right) \) and \( G_{n}^{''} \left( {\gamma_{0} ,\omega } \right) \) are the viscoelastic moduli.

Figure 4 shows ELB and VLB curves for corn stover slurries at time = 0 h (16 wt.% solids, \( X = 0 \), *m* = 1920 Pa) (a, b) and time = 44 h (13 wt.% solids, \( X = 0.14 \), *m* = 112 Pa) (c, d) after the start of fermentation. Each curve’s raw data and filtered data are plotted in black and in color, respectively. The maximum stress \( \tau_{ \hbox{max} } \) (Pa) is shown above each curve. The elastic stress \( \sigma^{\prime}\left( t \right) \) or viscous stress \( \sigma^{\prime\prime}\left( t \right) \) is plotted in dotted lines. In (a) and (b), the dotted line partitions the linear viscoelastic region and the nonlinear viscoelastic region. Also, the solid black line indicates the crossover line that delineates the predominantly elastic region (below) and the predominantly viscous region (above). Figure 4a, b show ELB and VLB curves, respectively, for a 16 wt.% solids “hour 0” slurry, which is essentially unfermented. These Pipkin maps show three distinct regions: linear viscoelastic (LVE); nonlinear viscoelastic (NLVE) yet still elastically dominated; and viscous dominated. The LVE region (defined herein as that for which the third- and fifth-order Fourier coefficients are less than 5% of the first-order coefficients) occurs for small strain amplitudes, with the cutoff amplitude depending on the frequency (as shown in Fig. 4) but being roughly \( \gamma_{0} \,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{ < } \, 10\% \). In the LVE region, the narrowly elliptical shape of the ELB curves indicates that elastic behavior dominates at these low strain amplitudes.

In the NLVE region, the shapes of the L–B curves illustrate the complex rheology of the slurry. For example, the inward curvature of the viscous stress (dashed lines in the VLB curves) suggests that the deviations from linearity are due to shear-thinning, which reaffirms the results in *Experiment 1*. Also, in the NLVE region, both the ELB and VLB curves are tilted near the edges, as previously observed by [35] for pretreated, unfermented slurries at similar solids concentrations. For moderate strain amplitudes \( 10\% \,{ \lesssim }\,\gamma_{0} \,{ \lesssim }\,100{\text{\% }} \), the NLVE region is characterized by predominantly elastic behavior.

The transition from elastic to viscous behavior is given where the phase amplitude \( \tan \delta = G_{1}^{''} /G_{1}^{'} \) is unity. Figure 5b shows \( \tan \delta = 1 \) at roughly \( \gamma_{0} \approx 100\% \). It is interesting to note that the noisiest data occur on this elastic–viscous crossover boundary; this is not surprising given that for smaller strain amplitudes, the rotor is moving too little to disrupt the matrix of solids particles, whereas for much larger strain amplitudes, the matrix is disrupted so much that the displacements of individual particles are averaged out.

For high strain amplitudes and frequencies, the flow is dominated by viscous effects. In this region, some VLB curves are observed to self-intersect, which is indicative of a strong elastic nonlinearity caused by a viscoelastic stress overshoot, essentially meaning that the rate at which the slurry is unloading stress is faster than its rate of deformation [10]. Such nonlinearities are typically present due to reversible changes in the material’s disposition (with one possibility being that the slurry’s entrained water is being squeezed out upon deformation). At very high amplitudes and frequencies, the square-shaped elastic curves show that the slurry is plastically deforming.

Figure 4c, d shows the corresponding LAOS data for a 13 wt.% solids “hour 44” slurry. This 13 wt.% solids sample had visible free water, making data collection quite challenging. Most of the raw data were too noisy to be included in the Pipkin map, but with the available data, a few comparisons can be made. The LVE–NLVE transition occurs at a lower strain amplitude for the “hour 44” slurry, and this transition is due to its shear-thinning behavior, similar to the “hour 0” slurry. Overall, the shapes of the L–B curves are surprisingly similar for the “hour 0” and “hour 44” slurries, given that the steady shear viscosity has fallen by a factor of seventeen.

Having characterized the nonlinear response of slurries using Lissajous–Bowditch curves, parameters like the *perfect plastic dissipation ratio* \( \phi \) and *phase angle* \( \delta \) were computed to assess yielding behavior. Figure 5 displays a checkerboard plot of \( \phi \) and \( \tan \delta \), with each square in Fig. 5 corresponding to a pair of Lissajous–Bowditch curves in Fig. 4. Like Fig. 4, Fig. 5 displays data for corn stover slurries at time = 0 h (16 wt.% solids, \( X = 0 \), *m* = 1920 Pa) and time = 44 h (13 wt.% solids, \( X = 0.14 \), *m* = 112 Pa) (c, d) after the start of fermentation, plotted on a Pipkin space. White squares on the Pipkin space indicate that the data were too noisy to be reported. The perfect plastic dissipation ratio compares the actual energy dissipated by an L–B curve to its theoretical value [11],

$$ \phi \, = \, \frac{{\pi \gamma_{0}^{2} G^{\prime\prime}_{1} }}{{4\gamma_{0} \sigma_{\text {max}} }}\, = \, \frac{{\pi \gamma_{0} G^{\prime\prime}_{1} }}{{4\sigma_{\text {max}} }} $$

where \( \gamma_{0} \) is the strain amplitude (unitless), \( G_{1 }^{''} \) is the loss modulus (Pa), and \( \sigma_{ \text{max} } \) is the maximum stress (Pa). Graphically, \( \phi \) is the ratio of the area inside the L–B curve to the area of the smallest rectangle that can be drawn around the L–B curve, with \( \phi = 0 \) indicating a perfectly elastic response, \( \phi \, = \,\frac{\pi }{4} \,\sim \,0.785 \) corresponding to a Newtonian fluid, and \( \phi = 1 \) indicating a perfectly plastic response.

In Fig. 5a, the 16 wt.% “hour 0” slurry shows \( 0.30\, \le \,\phi \, \le \,0.83 \), with this upper limit of \( \phi = 0.83 \) being between Newtonian and perfectly plastic. The slurry transitions from exhibiting elastoplastic behavior (\( \phi \le \frac{\pi }{4} ) \) to exhibiting pseudoplasticity \( \left( {\phi \ge \frac{\pi }{4} } \right) \) at roughly \( \gamma_{0} \approx 100 \)%, which is consistent with the elastic–viscous crossover at \( \tan \delta = 1 \). In Fig. 5c, the 13 wt.% “hour 44” slurry’s perfect plastic dissipation ratios range from \( 0.32 \le \phi \le 0.87 \), which again is remarkably similar to the “hour 0” slurry. Like its unfermented counterpart, it did not exhibit perfect plastic yielding, but showed elastoviscoplastic deformation.