Scientific questions surrounding the shear-dependent microstructure of carbon black suspensions are motivated by a desire to predict and control complex rheological and electrical properties encountered under shear. In this work, direct structural measurements over a hierarchy of length scales spanning from nanometers to tens of micrometers are used to determine the microstructural origin of the suspension viscosity measured at high shear rates. These experiments were performed on a series of dense suspensions consisting of high-structured carbon blacks from two commercial sources suspended in two Newtonian fluids, propylene carbonate and light mineral oil. The shear-induced microstructure was measured at a range of applied shear rates using Rheo-VSANS (very small angle neutron scattering) and Rheo-USANS (ultra-small angle neutron scattering) techniques. A shear-thinning viscosity is found to arise due to the self-similar break up of micrometer-sized agglomerates with increasing shear intensity. This self-similarity yields a master curve for the shear-dependent agglomerate size when plotted against the Mason number, which compares the shear force acting to break particle-particle bonds to the cohesive force holding bonds together. It is found that the agglomerate size scales as R g , agg ∼ M n − 1. Inclusion of the particle stress contribution extends the relevance of the Mason number to concentrated suspensions such as those relevant to the processing of carbon black suspensions for various applications.

## I. INTRODUCTION

The ability to predict and control the behavior of colloidal suspensions under changing formulation and flow conditions is important for applications ranging from ceramics, inks, coatings, and personal care products to personal protective equipment and sports apparel [1–3]. While the rheological behavior of these suspensions is widely studied, a more foundational understanding of the shear-dependent behavior requires microstructural information [4,5]. Many suspensions of interest are unstable, resulting in the formation of flocculated or aggregated structures [6,7]. In these cases, the rheology becomes directly related to the size and number density of particle flocs [5]. The evolution of these floc characteristics in response to an applied deformation depends on factors such as particle size [8–10], the nature of the particle-particle interactions [8,10–12], and the shear history [5,13,14]. Therefore, a need exists to develop relationships between these factors and the shear-induced floc structure. To this end, many studies have focused on understanding floc formation under quiescent conditions [7,15], as well as the shear-dependent floc structure in dilute colloidal suspensions [12,16–20]. However, it has proven challenging to measure the shear-dependent floc structure in concentrated suspensions. As these higher particle loadings are relevant for a wide range of applications, understanding the effect of shear on the floc structure is necessary to improve their formulation and processing.

It is known that in flocculated colloidal suspensions, the forces experienced in the shear flow, or processing, drive structural rearrangements that, in turn, affect the rheological properties of the suspension [5,14]. These structural rearrangements take place over a system-dependent period of time [13,21] and consist of changes in the floc size [12,19,20], the internal floc structure [12,16,22,23], and the degree of floc anisotropy [24–32]. For dilute suspensions of polystyrene latex particles, Sonntag and Russel studied the effect of fluid stresses and floc cohesive strength on these rearrangements by performing *ex situ* small angle light scattering measurements [12]. By normalizing the fluid stresses to the cohesive strength of a floc, the measured trend of decreasing the floc size with increasing the shear intensity and decreasing the cohesive strength could be expressed by a single curve [12]. Additionally, measurements of the floc structure before and after shear showed a shear-induced densification from a fractal dimension of 2.2 to 2.48 [12]. Similar densification has been reported by Harshe *et al*. and by Sommer, where the fractal dimension was found to be both system and processing dependent [16,23]. Some studies of concentrated suspensions have focused on understanding the shear-induced formation of anisotropic floc structures under shear and the resulting effect on the rheological properties [14,26,32]. These anisotropic structures form in specific shear regimes [25,27–29] and are oriented in either the flow or the vorticity direction [24–32]. Fewer studies have addressed the effect of shear on the internal floc structure and the floc size at a high effective volume fraction with significant yield stresses [22,26,33]. However, it has been shown through experimental work as well as simulations that flocs densify with a high fractal dimension and eventually break up with increasing shear intensity [22,26]. While these studies point to a systematic behavior of shear-induced floc restructuring, a unified experimental understanding of the floc structure evolution during shear flow and its relationship to rheological properties across a broad range of suspensions has not yet been established.

Carbon black suspensions are one class of industrial, flocculated suspensions that find many applications, e.g., tire rubbers, inks and paints, plastics, battery electrode slurries, and fuel cell catalyst inks [34–36]. These suspensions exhibit many useful, or sometimes undesirable, shear-dependent behaviors such as thixotropy, tunability of conductivity and yield stress, apparent shear-thickening behavior, and shear-thinning behavior that are attributed to shear-induced changes in floc or agglomerate structure [27,28,37–40]. Under shear in a rotational rheometer, the structural motifs observed in carbon black suspensions can be summarized by three structural regimes as shown by Osuji and Weitz and supported by other studies [25,27,28,39,41]. With increasing shear intensity, these three regimes consist of (1) vorticity-aligned flocs, which disintegrate to form (2) large, dense agglomerates that eventually break up into (3) smaller, more open agglomerates [25,27,28,39,41]. This transformation from large, dense agglomerates to small, open agglomerates has been measured for a suspension of carbon black in propylene carbonate using Rheo-USANS measurements [41]. Analysis shows that this transition as well as flow stability can be identified by the inverse Bingham number, $B i \u2212 1$, which compares the measured bulk stress to the yield stress of the suspension [41]. At $B i \u2212 1<1$, in a low shear rate regime, large and dense agglomerates are formed, which sediment under shear, leading to rheological anomalies. The transition to higher shear rates, where $B i \u2212 1>1$, leads to the formation of stable, small, and open agglomerates and an apparent shear-thickening. In this high shear intensity regime, the viscosity exhibits a shear-thinning behavior, where the internal structure, or fractal dimension, of agglomerates is shear-independent and the size of agglomerates decreases with increasing shear rate as expected for reversibly thixotropic suspensions [41]. Questions remain, however, as to the generality of this detailed structure-property relationship across a broader range of carbon black types and suspending fluids, or in other words, across systems with large variations in the primary particle structure and colloidal interactions.

This work addresses structural evolution in the third, high shear intensity regime by quantifying the shear-induced agglomerate structure simultaneously with the rheology in a variety of high-structured carbon black suspensions. The factors that are considered to control the agglomerate break up in this work include primary particle and primary aggregate nano- to microstructures, interparticle interaction strength, suspending medium characteristics, and volume fraction. These factors were explored by studying eight suspensions consisting of two commonly used conductive carbon blacks, Vulcan XC-72 and KetjenBlack EC-600JD, with different nanoscale building block structures suspended in two Newtonian fluids with different dielectric properties, over a range of volume fractions. Rheo-VSANS and Rheo-USANS measurements were used to directly measure the shear-dependent primary particle, primary aggregate, and agglomerate structure at a range of steady shear rates. The observed self-similar agglomerate break up is found to depend on a modified Mason number that includes the bulk stress, allowing for the construction of a master curve for the microstructure and the derivation of a relation between the bulk viscosity, the shear intensity, and the microstructure.

## II. MATERIALS AND METHODS

### A. Suspensions studied

The eight suspensions studied are listed in Table I. These suspensions consist of two conductive carbon blacks, Vulcan XC-72 (Cabot Corporation) and KetjenBlack EC-600JD (AkzoNobel), suspended in two Newtonian fluids, propylene carbonate (Acros, $ \eta f=0.0025 Pa s$, $ \epsilon f=64$, $ n f=1.421$, $ \rho f=1.204 g c m \u2212 3$) and light mineral oil (Sigma, $ \eta f=0.026 Pa s$, $ \epsilon f=2.4$, $ n f=1.467$, $ \rho f=0.838 g c m \u2212 3$), at three volume fractions above the mechanical percolation threshold, $ \varphi eff=0.12, 0.20, and0.27$. These volume fractions were chosen because they correspond to weak gels with measurable yield stresses. The shear-independent effective volume fraction, $ \varphi eff$, is defined in Richards *et al*. [42] as $ \varphi eff= \varphi C B , dry/ \Phi C B , pagg$, which accounts for the porosity of the primary particle and primary aggregate structures by normalizing to the volume fraction of carbon black in a primary aggregate volume, $ \Phi C B , pagg$. These suspensions were selected to evaluate the combined effect of particle loading, interaction potential, suspending medium, and building block characteristics on the shear-dependent agglomerate structure. As documented in a previous paper, these two carbons have primary particle and primary aggregate structures that vary in porosity, fractal dimension, size, and polydispersity [42]. The two suspending fluids were not only chosen due to their vastly different dielectric properties, which provides an opportunity to develop structure-property relationships across a wide range of fluid types but also because these fluids represent the existing body of the literature regarding the shear-dependent behavior of carbon black suspensions. Propylene carbonate reflects the high dielectric constant fluids used for suspensions containing carbon black for electrochemical energy storage devices [35,36,38,42] and light mineral oil reflects the nonpolar fluids used to study carbon black suspensions as model rheological systems [25,28,37,39,40,43,44]. Stock suspensions were prepared at the highest volume fraction studied by high shear mixing using an IKA T 18 ULTRA-TURRAX mixer with a high shear screen dispersing tool for 10 min at 11 000 rpm (1150 rad s^{−1}). This preparation protocol effectively disperses the carbon black powder and leads to reversible thixotropic behavior as the shear rate applied in the mixing step is far above the shear rates applied during characterization [41,45]. To maintain a constant preparation history across suspensions of the same family, lower volume fractions were prepared by appropriate dilution of the stock sample.

Carbon black type . | Suspending fluid . | Weight % . | Volume fraction, ϕ_{eff}
. |
---|---|---|---|

Vulcan XC-72 | Propylene carbonate | 3.5, 5.75, 8.0 | 0.12, 0.20, 0.27 |

Vulcan XC-72 | Light mineral oil | 4.9, 8.0, 11.1 | 0.12, 0.20, 0.27 |

KetjenBlack EC-600JD | Light mineral oil | 2.5, 3.5 | 0.20, 0.27 |

Carbon black type . | Suspending fluid . | Weight % . | Volume fraction, ϕ_{eff}
. |
---|---|---|---|

Vulcan XC-72 | Propylene carbonate | 3.5, 5.75, 8.0 | 0.12, 0.20, 0.27 |

Vulcan XC-72 | Light mineral oil | 4.9, 8.0, 11.1 | 0.12, 0.20, 0.27 |

KetjenBlack EC-600JD | Light mineral oil | 2.5, 3.5 | 0.20, 0.27 |

### B. Rheological measurements

Rheological measurements were performed using an Anton-Paar MCR-301 stress-controlled rheometer at 25 °C equipped with a quartz cup and 60 mm long titanium bob (OD = 50 mm, ID = 49 mm, truncation gap = 0.05 mm) available as a sample environment at the NIST Center for Neutron Research in Gaithersburg, MD, USA [46]. Self-similar structure flow curves were constructed for these thixotropic suspensions following a protocol described previously by Hipp *et al*. [41]. Briefly after loading the sample, a conditioning protocol of twenty 30 s flow ramps from 500 to 0.1 s^{−1} was performed followed by shearing at a shear rate of 2500 s^{−1} until a steady stress was achieved. Then, a chosen preshear shear rate of $ \gamma \u02d9=2500 s \u2212 1$ was applied for 600 s followed by a 300 s waiting period before the transient stress was measured at a desired shear rate. During the waiting period, a small-amplitude oscillatory shear (SAOS) measurement was performed at a strain of $\gamma =0.1%$ over a frequency range of $\omega =1 rad s \u2212 1$ to 100 rad s^{−1} to monitor restructuring (see supplementary material) [81]. Following this waiting period, the shear rate of interest was applied for 300 s and the stress response was measured. The resulting transient stress at each shear rate was then fit to a double exponential and extrapolated to a stress at t = 0 s, which was used to construct a self-similar structure flow curve.

*et al*. [41] and detailed in the supplementary material [81],

### C. Rheo-VSANS measurements

The shear-induced building block and internal agglomerate structure were measured using the Rheo-VSANS sample environment on the NG3 VSANS diffractometer at the NIST Center for Neutron Research in Gaithersburg, MD, USA [48]. The measurements were performed over a *q* range of $1.5\xd7 10 \u2212 1>q>3.3\xd7 10 \u2212 4 \xc5 \u2212 1$, with $\lambda =6.7\xc5$ and $ \Delta \lambda /\lambda =12%$. Four converging neutron beams pass through a 45 mm tall by 12 mm wide aperture to enable simultaneous measurement of the entire *q* range in 5 min increments. *In situ* rheological measurements were performed using an Anton-Paar MCR-501 stress-controlled rheometer at 25 °C equipped with a quartz cup and 60 mm long bob (OD = 50 mm, ID = 49 mm, truncation gap = 0.05 mm) aligned in the flow-vorticity (1–3) plane. These measurements were performed by applying a preshear of $ \gamma \u02d9=2500 s \u2212 1$ for 600 s and immediately stepping to the shear rate of interest, which was held for 3000 s while scattering data were collected. The VSANS data were reduced to absolute scale using IGOR Pro reduction protocols [48].

### D. Rheo-USANS measurements

The shear-induced agglomerate structure was measured using the Rheo-USANS sample environment on the BT5 diffractometer at the NIST Center for Neutron Research in Gaithersburg, MD, USA [49]. Five buffers were used to probe a *q* range from 1 × 10^{−3} Å^{−1 }> *q *> 3 × 10^{−5} Å^{−1} where $q=4\pi /\lambda sin( \theta / 2)$ with $\lambda =2.4\xc5$ and $ \Delta \lambda /\lambda =6%$. The vertically collimated neutron beam used by this instrument results in a slit-smeared scattering curve where a scattered intensity reported at a single *q* value contains scattering contributions from a *q* range that is dictated by the vertical resolution for the specific instrument [49,50]. *In situ* rheological measurements were performed using an Anton-Paar MCR-301 stress-controlled rheometer at 25 °C equipped with a quartz cup and 60 mm long bob (OD = 50 mm, ID = 49 mm, truncation gap = 0.05 mm) aligned in the flow-vorticity (1–3) plane and positioned between the analyzer and monochromator crystals [41,46]. For Vulcan XC-72 suspensions, a titanium bob was used and for KetjenBlack EC-600JD suspensions a quartz bob was used to decrease background scattering. These measurements were performed by applying a preshear of $ \gamma \u02d9=2500 s \u2212 1$ for 600 s before stepping to the shear rate of interest. The structure evolved to a steady state for 600 s before the USANS measurement was initiated. The measurements required approximately 10 000 s for Vulcan XC-72 suspensions and, due to the higher primary structure porosity, 32 000 s for KetjenBlack EC-600JD suspensions for each shear condition. The USANS data were reduced to absolute scale using standard IGOR Pro reduction protocols [50].

### E. USANS data modeling: Hierarchical fractal model

*q*range for a suspension of Vulcan XC-72 in propylene carbonate at $ \varphi eff=0.22$ [42] is shown in Fig. 1. The model agrees closely with the data for nearly all

*q*, except at

*q*values close to the primary aggregate size (10

^{−3}Å

^{−1}). This is due to a soft potential between primary aggregates that is not included in the model in favor of a structure factor that instead describes the agglomerate structure [42,52,53].

The form factor of primary particles, $ P p p(q)$, is modeled using a Guinier–Porod model to account for surface roughness of the primary particles [54] as shown in Fig. 1 as a (red) dashed line. The scattered intensity from primary aggregates is constructed using this form factor as $ P pagg(q)= P p p(q) S intra(q)$, as shown in Fig. 1 as a (blue) dotted-dashed line. This form factor has previously been thoroughly characterized and modeled for both carbon blacks studied here [42]. The intra-aggregate structure factor, $ S intra(q)$, was modeled using a Teixeira fractal structure factor where the average primary particle radius was used as the lower cutoff size to the primary aggregate fractal dimension [42,55].

At high concentrations, primary aggregates interact to form agglomerates, the structure of which, $ S inter(q)$, is again represented by a Teixeira fractal structure factor, but where the lower cutoff size for the agglomerate fractal dimension is set as three times the correlation length of the primary aggregate, $ \xi pagg$ [55]. This length scale was chosen due to the origin of the correlation length, $\xi $, in the Teixeira fractal model, which sets an upper cutoff size to the fractality of an aggregate [55]. As described by Freltoft, Kjems, and Sinha, to account for the finite size of an aggregate, the particle pair correlation function describing its fractal structure is weighted by the probability that a fractal exists for length scales greater than or equal to the center of mass of the aggregate, *r*, by using an exponential decay function, $exp( \u2212 r / \xi )$[55,56]. With this decay function, the probability that the primary aggregate fractal does not extend to length scales of three correlation lengths and does not intrude on the fractal structure of the agglomerate, is 95%, making this length scale sufficient to be used as the lower cutoff size for the agglomerate.

^{−1}and fit to the shear-dependent data using the sasview software package [57]. The model contains ten parameters of which eight have been previously determined and are considered fixed parameters [42], leaving two fit parameters: the fractal dimension of agglomerates, $ D f , agg$, and the correlation length of agglomerates, $ \xi agg$. These two fit parameters are used to calculate the shear-dependent radius of gyration of an agglomerate, $ R g , agg$, using Eq. (3) [55]:

## III. RESULTS AND DISCUSSION

### A. Self-similar structure flow curves and yield stress

The self-similar structure flow curves measured for the eight suspensions studied are shown in Fig. 2 along with the fits to the HB model as the corresponding lines. The stress response exhibits a typical power law slope at high shear rates and trends to a yield stress plateau at low shear rates. The dynamic HB yield stress, critical shear rate, and power law index for each fit are in Table S.1 [81]. The trends in these parameters, as discussed in the following, qualitatively follow expected behavior with volume fraction, suspending fluid properties, and carbon properties. However, it is important to realize that the quantitative values of these parameters also depend on the sample preparation and preshear protocol [37,40,41]. As will be shown shortly, this dependence on sample history can be rationally understood when viewed with knowledge of the agglomerate structure. In addition to the care required in defining the specific shear history, the rheological protocol used here also avoids other undesirable processes that may affect rheological measurements, including wall slip, shear banding, and sedimentation [21,44,58,59].

The family of flow curves shown in Figs. 2(a)–2(c) are conveniently collapsed when plotted on coordinates defined by the HB model, as observed in Fig. 2(d). This dynamic similarity suggests that the physical processes responsible for the development of the yield stress as well as those responsible for the yielding and flow under shear are the same across these variations in sample chemistry and physical properties. A unifying understanding of these physical processes is afforded by hierarchical microstructural measurements, as shown in Sec. III B and III C.

*a*is the radius of particles comprising the network [10]. Thus, the yield stress increases with volume fraction, as observed in Fig. 3(a). The yield stress also increases with the cohesive strength of particle-particle bonds, $ F cohesive$, which is calculated by taking the maximum in the first derivative of the interaction potential [5,10,60]. This trend is observed in Fig. 3(a) for the two families of Vulcan XC-72 suspensions, where the Hamaker constant (Table S.2) [81] in propylene carbonate ( $\u223c17.5 k BT$) is higher than in light mineral oil ( $\u223c15.4 k BT$), leading to stronger interparticle bonds. The combination of these two effects is summarized in a model for the yield stress [Eq. (4)] [10,24],

*C*, used here is system-specific and has been hypothesized to depend on the relationship between bond breaking and yielding [10]. Equation (4) enables estimation of the cohesive force between primary aggregates in the suspension from the measured yield stress values. It should be noted, however, that this estimated cohesive force lacks information concerning the effects of thermal energy, rate of force loading, and the distribution of cohesive forces [62,63].

The critical shear rate from the HB fits, $ \gamma \u02d9 c H B$, is often used to describe the inverse of a characteristic time of the suspension [64–66]. For soft colloids, Nordstrom *et al*. observed a decrease in this characteristic time (increase in critical shear rate) with increasing volume fraction [66], which is the general trend observed in Fig. 3(b). We also observe that increasing the suspending medium viscosity decreases the critical shear rate. Finally, as shown in Fig. 3(b), the power law index, $ n H B$, varies systematically between 0.50 and 0.75, indicating shear-thinning, where larger values are observed for higher volume fractions and for the suspensions in propylene carbonate. The fact that $ n H B$ appears to be a complex function of volume fraction, suspending medium viscosity, and dielectric constant motivates direct quantification of the microstructure to gain insight into the shear-thinning behavior.

### B. Rheo-VSANS measurements of shear-induced primary aggregate structure and anisotropy

To understand the shear-thinning behavior observed for these suspensions, very small angle scattering experiments were performed over length scales of 10 nm to 2 *μ*m, which corresponds to the primary particle, primary aggregate, and internal agglomerate structure. Note that the accessible *q* range measured here is larger than that corresponding to the agglomerate size. The results of these measurements are shown in Fig. 4 as a function of applied shear rate for two representative suspensions of Vulcan XC-72 at $ \varphi e f f=0.20$ in light mineral oil [Fig. 4(a)] and in propylene carbonate [Fig. 4(b)]. Two key observations are evident from these measurements. The first has been observed in previous works, where the scattered intensity at high *q* remains constant with increasing shear rate for all suspensions studied [41]. This confirms that the primary particle and primary aggregate structures are shear-independent and the parameters describing these two levels of structure can be fixed in the hierarchical fractal model. This result is a consequence of sample preparation and preshear protocols, which enable the study of reversible thixotropy in suspensions as has been discussed by Dullaert and Mewis [45] and confirmed by Armstrong *et al*. [67].

The second observation comes from the two-dimensional scattering data measured in the flow-vorticity (1–3) plane shown in the inset in Figs. 4(a) and 4(b). The images capture the *q* range from 1.5 × 10^{−3} to 3.3 × 10^{−4} Å^{−1}, which corresponds to the internal agglomerate structure. These 2-D patterns are isotropic and are representative of the behavior for all suspensions measured. This result is contrary to prior studies on the shear-dependent structure of colloidal gels. In these studies, formulation and shear-dependent anisotropic scattering patterns are observed in the flow-vorticity (1–3) plane as well as the flow-gradient (1–2) plane [24–28,30,32,68]. For example, a concentrated suspension of spherical, 26.8 nm diameter octadecyl silica particles studied by Hoekstra *et al*. showed anisotropy on all length scales greater than the particle size, which was attributed to breakage and recombination of structures along the extension and compression axes, respectively [26]. Similar observations of anisotropy along the compression axis have been made for colloidal gels that exhibit short-range attractions [24,30,68]. Additionally, the isotropic scattering patterns in Fig. 4 suggest that the vorticity-aligned structures commonly observed in carbon black suspensions [25,27,28] are not observable in the range of shear rates studied and in the length scales probed. Such shear-induced heterogeneities are not anticipated for these measurements given that the shear rates and gap width used are consistent with conditions identified by Grenard *et al*. where vorticity-aligned flocs do not form [25]. The shear-independence of the scattering curves at high *q* and the absence of anisotropy in the two-dimensional scattering data emphasize that the relevant shear-induced structural changes are to be observed at lower scattering angles, as afforded by Rheo-USANS measurements.

### C. Rheo-USANS measurements of agglomerate structure

Scattering measurements corresponding to larger length scales from approximately 0.6 to 16 *μ*m are shown in Fig. 5 for the eight suspensions studied at a range of shear rates. For each suspension, the scattering curves were acquired during steady shear and show a similar qualitative behavior to what was described in a previous study of self-similar structural break up of carbon black agglomerates suspended in propylene carbonate [41]. At high *q* corresponding to length scales smaller than the agglomerate size, a shear-independent power law slope is observed. This indicates that the internal structure of the agglomerates is not changing with the shear rate. The slope of this power law yields a mass fractal dimension where $I(q)\u223c q \u2212 D f$ [55], however, because of the slit-smearing of the USANS instrument, accurate determination of the fractal dimension requires the inclusion of the instrument resolution [49,50]. The slit-collimation also prevents the determination of the existence of anisotropy at this length scale. Nonetheless, we observe a significant and systematic reduction in the intensity of the low *q* plateau, characteristic of shear-induced reduction in the size of scattering objects with increasing shear rate for all samples. To quantify the evolution in agglomerate size, the scattering intensity was fit to the slit-smeared hierarchical fractal model as is shown in (black) lines in Fig. 5.

The fitted values of $ R g , agg$ and $ D f , agg$ are extracted at each condition and are summarized in Tables S.3 and S.4 [81]. The fractal dimension of the agglomerates varies depending on the type of carbon black where Vulcan XC-72 agglomerates have a fractal dimension of 2.5 and KetjenBlack EC-600JD have a volume fraction-dependent fractal dimension of 2.6 to 2.7. Similar fractal dimensions have been observed for sheared flocculated suspensions by several authors [12,16,22]. In these studies, small angle light scattering results show that flocs formed under static conditions have fractal dimensions ranging from 1.4 to 2.2 and densify under shear to fractal dimensions of 2.4 or 2.5 [12,16,22]. The self-similar break up observed here has been observed by Sonntag and Russel for dilute polystyrene particle suspensions [12] and is an important assumption made in population balance modeling for thixotropic suspensions [69].

The agglomerate radius of gyration is plotted in Fig. 6(a) as a function of shear rate where the qualitative observations of the scattering curves are confirmed by the reduction of $ R g , agg$ with increasing shear rate for all samples. This is consistent with the traditional picture of agglomerate break up that increasing the shear rate leads to a smaller agglomerate size. This trend is also evident in the decrease of $ R g , agg$ with increasing suspending medium viscosity, $ \eta f$, as observed by comparing suspensions of Vulcan XC-72 in light mineral oil and propylene carbonate at the same shear rate. The combined effect of the medium viscosity and the shear rate is often evaluated using a fluid shear stress, $ \sigma f= \eta f \gamma \u02d9$, as it is assumed that the hydrodynamic forces of the fluid act to break up agglomerates [12,16,70]. An alternate version of Fig. 6(a) showing the dependence of the radius of gyration on the $ \sigma f$ is provided in the supplementary material (Fig. S.5) [81]. The radius of gyration is inversely dependent on the volume fraction, which suggests that the particle stress may also contribute to the force an agglomerate experiences in the shear flow. Another observable trend is the effect of interparticle attractions on the agglomerate size. As shown in Table S.2 [81], for a suspension of Vulcan XC-72 in propylene carbonate, particles have a higher attractive strength and therefore a stronger cohesive force preventing agglomerate break up than Vulcan XC-72 particles suspended in light mineral oil. This increase in attractive strength is observed as an overall larger agglomerate size in propylene carbonate. Again, this trend is similar to that observed by Sonntag and Russel who reported an increase in the floc size with increasing salt concentration and a decrease in electrostatic repulsive strength for dilute polystyrene suspensions [12]. Importantly, the suspension viscosity that arises due to the agglomerate structure was simultaneously measured during the Rheo-USANS experiments at each applied shear rate and is reported in Fig. 6(b). The viscosity measured at these high shear rates is shear-thinning and the measured values align closely with the stresses measured in Fig. 2. This shear-thinning behavior is a signature of reversible thixotropy and occurs due to the self-similar break up of agglomerates and consequential reduction of the hydrodynamic volume fraction with increasing shear intensity [69].

### D. Scaling for the agglomerate size and suspension viscosity

The individual effects of shear rate, suspending medium viscosity, interparticle interactions, and volume fraction on the agglomerate size are apparent in Fig. 6(a). Numerous experimental and simulation studies have previously demonstrated that the Mason number, generally defined as the ratio of shear forces acting to pull particles apart, $ F shear$, to the cohesive forces holding particles together, $ F cohesive$, should govern the break up of agglomerates [12,16,24,31,71,72]. For complex systems such as carbon black, which exhibit polydispersity, surface roughness, and shape irregularity, the details of the interparticle interactions and $ F cohesive$ are challenging to calculate *a priori*. Therefore, in this study, following the approach of Eberle *et al*. [24], $ F cohesive$ is estimated using Eq. (4) in combination with the yield stress from the HB fits presented in Fig. 3(a).

A dimensionless plot of agglomerate size against the Mason number in Eq. (5) is shown in Fig. 7. For each of the three families of suspensions, *C* is constant and has a value of 2 for all suspensions of Vulcan XC-72 in light mineral oil and 1 for all suspensions of Vulcan XC-72 in propylene carbonate and KetjenBlack EC-600JD in light mineral oil. The value of C is empirical and reflects unknown differences between carbon types. As shown in Fig. 7, this form of the Mason number successfully collapses the microstructural information. Further, it appears to correctly capture the magnitudes of shear and cohesive forces as break up is observed for $Mn>1$, which is the regime where shear forces dominate. Additionally, the agglomerate size decreases with increasing Mason number following the power law $ R g , agg\u223cM n \u2212 1.0$. In comparison to previous studies, this power law scaling of the agglomerate size is more dramatic than the $M n \u2212 0.5$ scaling determined by Varga and Swan using simulations, where the Mason number was calculated with only the fluid contribution to the hydrodynamic stress[31]. An arguably more relevant comparison, due to the heavy weighting of scattering measurements by the largest agglomerate size, is the scaling reported for the dependence of the largest aggregate size on the Mason number, $ N g\u223cM n \u2212 1.1$ [31], which compares well to that presented in Fig. 7. Notwithstanding numerical differences, the common scaling behavior observed across the eight suspensions studied shows that at these high shear rates, where $Pe\u226b1$, the underlying physical processes of shear-induced break up and agglomeration are common across this range of carbon black types and suspending fluids.

Of interest for the work presented in this manuscript is the use of microstructural measurements to inform on the modeling of thixotropic suspensions. A signature of reversible thixotropy in these systems is that the shear-thinning behavior arises as a consequence of the self-similar break up of carbon black agglomerates. Consequently, even in the absence of any macroscopic phase separation [28,41], any constitutive model for this class of thixotropic suspensions will necessarily require auxiliary microstructural information, such as typical in semi-empirical structure-kinetic models [43,76,77], or more recent, mechanistic population balance models [69,78]. Whether the findings that we present here extend to other thixotropic systems, e.g., fumed silica [67,79], rouleaux in blood [80], and flocculated suspensions [68] remains to be determined.

## IV. CONCLUSIONS

This paper extends the detailed study of one particular chemistry of carbon black suspensions [41] to demonstrate a common behavior for the hierarchical microstructure, yield stress, and shear-thinning viscosity for a range of carbon black primary particle structures, suspending fluids, and concentrations. A combination of Rheo-VSANS and Rheo-USANS measurements provide a critical understanding of the self-similarity of the underlying physical processes that govern the suspension microstructure and their dependence on the chemical nature of the suspension. Neutron scattering results show self-similar break up of agglomerates where the fractal dimension is unchanging with shear rate and the size decreases with increasing shear rate. The dependence of this agglomerate break up on the suspension properties and flow strength is quantitatively understood in terms of a dimensionless Mason number, $Mn$, which compares the volume fraction-dependent shear force driving agglomerate break up to the cohesive forces holding the agglomerates together. This cohesive force was estimated using a HB yield stress, which was obtained by fitting self-similar structure flow curves to the HB model. The analysis presented here reconciles many observations in the literature concerning the shear-dependent floc structure and expands upon these studies by considering the effects encountered at high volume fractions where considerations of shear-induced microstructural rearrangements provide a rational means to unify observations across a broad range of suspension chemistries and physical properties. Whether the structure-property relationships determined here can be extended to other suspension properties, such as conductivity, remains an active research direction. However, we anticipate these results to be of value for the design and formulation of thixotropic suspensions with a yield stress across a broad range of practical applications.

**ACKNOWLEDGMENTS**

The authors would like to acknowledge the NIST Center for Neutron Research CNS Cooperative Agreement No. 70NANB12H239 grant for partial funding and the National Research Council for support. Access to the BT5 USANS instrument and the NG3 VSANS instrument was provided by the Center for High Resolution Neutron Scattering, a partnership between the National Institute of Standards and Technology and the National Science Foundation under Agreement No. DMR-1508249. The authors also acknowledge the support of the National Institute of Standards and Technology, U.S. Department of Commerce, in providing the neutron research facilities used in this work. This work benefited from the use of the SASVIEW application, originally developed under NSF Award No. DMR-0520547. SASVIEW also contains a code developed with funding from the EU Horizon 2020 program under the SINE2020 Project Grant No. 654000. Certain commercial equipment, instruments, or materials are identified in this paper to foster understanding. Such identification does not imply recommendation or endorsement by the National Institute of Standards and Technology nor does it imply that the materials or equipment identified are necessarily the best available for the purpose.

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