Lithium-ion battery cathode slurries have a microstructure that depends sensitively on how they are processed due to carbon black's (CB) evolving structure when subjected coating flows. While polyvinylidene difluoride (PVDF), one of the main components of the cathode slurry, plays an important role in modifying the structure and rheology of CB, a quantitative understanding is lacking. In this work, we explore the role of PVDF in determining the structural evolution of Super C65 CB in N-methyl-2-pyrrolidinone (NMP) with rheo-electric measurements. We find that PVDF enhances the viscosity of NMP resulting in a more extensive structural erosion of CB agglomerates with increasing polymer concentration and molecular weight. We also show that the relative viscosity of all suspensions can be collapsed by the fluid Mason number (Mnf), which compares the hydrodynamic forces imposed by the medium to cohesive forces holding CB agglomerates together. Using simultaneous rheo-electric measurements, we find at high Mnf, the dielectric strength (Δε) scales with Mnf, and the power-law scaling can be quantitatively predicted by considering the self-similar break up of CB agglomerates. The collapse of the relative viscosity and scaling of Δε both suggest that PVDF increases the hydrodynamic force of the suspending medium without directly changing the CB agglomerate structure. These findings are valuable for optimizing the rheology of lithium ion battery cathode slurries. We also anticipate that these findings can be extended to understand the microstructure of similar systems under flow.

Carbon black (CB) serves as an important material given its industrial applications such as pigment [1], fuel cell catalyst inks [2], flow battery conductive additives [3–5], and lithium-ion battery slurries [6]. Understanding the rheology of these suspensions is, therefore, critical to control their processing. It is known that when formulated into suspensions, CB shows complicated rheological behaviors originating from the evolving microstructure of CB particles. Due to finite van der Waal's forces acting between particles, the primary aggregates of CB physically associate to form agglomerates [7], whose size and orientation can change in shear flow. This evolving microstructure is responsible for complex rheological behaviors both at rest and under flow. At rest, suspensions of CB at high volume fraction solidify. This fluid-to-solid transition occurs as a result of agglomerates forming a system-spanning and interconnecting network whose yield stress and elasticity are determined by the topology and the strength of stress-bearing bonds linking agglomerates together [8]. When the stress experienced exceeds the yield stress, CB suspensions are fluidized and this yielding transition can show heterogeneity [9]. When subjected to strong shear flow, agglomerates show a self-similar break up behavior [10]. The reduction in agglomerate size at constant fractal dimension results in a thixotropic response [11–14]. In contrast, when the suspensions are subjected to low shear rates, the agglomerates grow, densify, and sediment over long time scales [15]. This results in a rheopectic response and long stress transients in the shear flow. Furthermore, when these densified agglomerates are confined, vorticity-aligned flocs have been observed to form [16–18]. In addition to rheological behaviors under shear, CB suspensions show a complex dielectric response that depends on the structure of CB suspensions, and their dielectric properties are important for energy storage system applications [4,6,19]. Helal et al. [20] hypothesized that electrons are transported through the same stress-bearing network formed by CB gel, and they discovered a power-law scaling between the stress-bearing network strength and conductivity, while Richards et al. [21] showed that electron hopping is the dominant mechanism to achieve electron transport.

While much is known about the rheology of CB suspensions in simple fluids, the rheology of CB in complex media needs to be investigated as CB suspensions are usually added with other components when formulated into industrial products [22–24]. For example, cathode slurries of carbon black used in lithium-ion battery (LIB) manufacturing incorporate dissolved polymers. These polymer chains serve both as a mechanical binder in the dried state and as a rheology modifier in the slurry state. As the rheology of these slurries and the way they are processed can be engineered to control the microstructure of the porous electrode, it is of significant interest to understand how the physical properties of the dissolved polymer modify the rheological response of CB suspensions. Therefore, we focus on suspensions of CB that contain polyvinylidene difluoride (PVDF) dissolved in N-methyl-2-pyrrolidinone (NMP). Such suspensions, when processed with an active material, are used to manufacture LIB cathodes. Prior work to understand the rheology of CB/PVDF suspensions has assumed that PVDF was adsorbed to the surface of CB and this mechanism enhanced stabilization [25,26] and reduced agglomeration [27]. However, Sung et al. [28] have recently reported the shear rheology of PVDF/CB suspensions, which suggests that PVDF is not adsorbed to the CB surface and acts only to modify the matrix viscosity [28]. While the findings of Sung et al. [26] showed the qualitative role of PVDF, a quantitative understanding, such as the degree of enhancement in the matrix viscosity and modification of the CB structure under shear, has not been established. This understanding will not only be important to understand fundamentally how non-absorbing polymers contribute to the structural evolution of attractive colloidal suspensions, but also be instructive in manufacturing better electrodes as the LIB cathode has a complex microstructural hierarchy that depends sensitively on how it is processed [29–32].

To this end, we have undertaken a comprehensive study of the suspension rheology of PVDF and CB mixtures as a function of PVDF molecular weight, concentration, and CB content. We first characterized the solution properties of PVDF and showed through rheology and small-angle x-ray scattering that PVDF is unentangled and in a good solvent. We then performed rheological testing on PVDF/CB suspensions and showed that the presence of PVDF mainly enhanced the matrix viscosity, resulting in an apparent reduction in shear thinning at high shear intensities. To quantitatively probe the role of PVDF in this process, we showed that the relative viscosity can be collapsed with the fluid Mason number (Mnf). This indicates that the self-similar break up of CB agglomerates is not significantly altered by the presence of PVDF, but instead adding PVDF to the suspension acts to increase Mnf resulting in smaller agglomerates at the same shear rate as the polymer concentration and molecular weight increase. To probe the microstructure of the carbon black agglomerates, we performed rheo-electric measurements and find the dielectric strength (Δε), extracted from the frequency-dependent permittivity, is a function of Mnf. By performing these measurements as a function of polymer concentration and molecular weight, we were able to develop an empirical collapse of Δε with Mnf after a vertical shift factor of order unity. The collapsed curves showed a plateau at low Mnf and a power-law decay, ΔεMnfm. The observed scaling parameter m was successfully predicted by assuming the self-similar break up of CB agglomerates, consistent with the assumption that the break up of CB agglomerates remains self-similar.

Polyvinylidene difluoride (PVDF), Kynar 741 (batch 20C6072), and Kynar 761 (batch 20C3014) were provided by Arkema (King of Prussia, PA). N-methyl-2-pyrrolidinone (ACS reagent, ≥99.0%, density = 1.028 g/ml) was purchased from Sigma-Aldrich (St. Louis, MO). Super-C65 carbon black (batch 29719AA) was purchased from MSE Supplies (Tucson, AZ).

PVDF solutions were prepared by dissolving a known mass of PVDF powder into a known volume of NMP. The solutions were heated on a hot plate at 70 °C overnight and the dissolution of PVDF was confirmed visually. To prepare CB/PVDF suspensions, PVDF solutions of known concentration and molecular weight were added to vials containing dry CB powder. The resulting suspensions were vortexed for at least 20 s and then sonicated for at least 30 min. The resulting suspensions appeared homogeneous and, in most cases, exhibited a weak yield stress.

Small-angle x-ray scattering (SAXS) experiments were performed at the 5-ID-D beamline of the DuPont-Northwestern-Dow Collaborative Access Team (DND-CAT) located at Sector 5 of the Advanced Photon Source (APS) with an x-ray energy of 17 keV. The wave vector Q was defined as 4πsin(θ/2)/λ, where θ is the scattering angle. The accessible Q-range of the instrument was 0.00254–0.1376 Å−1. Samples were introduced into a capillary flow cell, and scattering patterns were acquired using a fixed exposure time of 10 s. The 2D scattering data were reduced to 1D data (intensity versus Q) in GSAS-II software. The reduced 1D data were then corrected with appropriate background and solvent subtractions in Python.

Ultrasmall-angle x-ray scattering (USAXS) data were collected at the 9-ID-C beamline at APS. The x-ray energy was 21 keV, and the wavelength was 0.5904 Å−1. The accessible Q-range of the instrument was 8 × 10−5–0.1 Å−1. Samples were mounted using Scotch Magic Tape (3M) on an instrument-provided sample holder and USAXS data were collected for 90 s. USAXS [33] data were then reduced using instrument data reduction software provided by beamline and corrected for instrumental background as well as appropriate sample container scattering. Data were placed on absolute intensity and desmeared to remove instrumental slit smearing.

Transmission electron microscopy (TEM) measurements were made using a JEOL JEM-2100 FasTEM microscope operating at 200 kV. Samples were mounted on a carbon film supported copper grid purchased from Sigma-Aldrich (St. Louis, MO).

Polymer molecular weights and dispersity were obtained from size exclusion chromatography (SEC) equipped with a multiangle light-scattering detector (DAWN-HELIOS II, Wyatt Technology) and a refractive index detector (Wyatt Optilab TrEX).

Dynamic light scattering (DLS) measurements were carried out using a Zetasizer Nano (Malvern Panalytical, Cambridge, United Kingdom). The light source was a He-Ne laser whose wavelength was 633 nm. DLS measurements were conducted on 0.001 wt. % CB in NMP right after sonication for 10 min.

Rheological measurements of polymer solutions were carried out using a stress-controlled rheometer (DHR-2, TA Instruments, New Castle, DE) at 25 °C with a cone-plate geometry (diameter = 40 mm, 1°, truncation gap = 25 μm) and a Peltier system for temperature control. A solvent trap filled with room-temperature nitrogen was equipped to avoid solvent evaporation and moisture uptake. The stress response was recorded as a function of shear rate over the range of 1–100 s−1.

Mixture suspensions were centrifuged at 25 °C and 18 213 × g using Eppendorf Centrifuge 5427 R (Eppendorf, Enfield, CT) for 60 min.

Rheo-electric measurements of CB and CB/PVDF mixture suspensions were carried out using a strain-controlled rheometer (ARES-G2, TA Instruments, New Castle, DE) at 25 °C with a dielectric Couette geometry [5,34,35] (Inner diameter = 26 mm, outer diameter = 27 mm, gap = 0.5 mm) and a forced-convection oven for temperature control. The geometry was also equipped with a solvent trap to avoid solvent evaporation. The complex impedance was measured with an impedance analyzer (E4990A, Keysight Technologies, Santa Rosa, CA) with a voltage amplitude of 500 mV over the frequency, f, range of 20 Hz to 10 MHz. We followed the procedure described in the previous publication by Hipp et al. [10] to study the shear rheology of CB and CB/PVDF mixture suspensions. The sample was conditioned in the rheometer using a preshear of 2500 s−1 for 600 s to erase any structural memory from loading. For each shear rate tested, a preshear at 2500 s−1 was applied for 90 s, and the transient stress response was recorded at the desired shear rate for 90 s. (An example of transient stress response at desired shear rates is presented in Fig. S6 [70].) We observe that at low shear rates, no steady state was reached, which is associated with densification and sedimentation of CB agglomerates [11]. At high shear rates, a steady state was rapidly reached with no further stress evolution throughout the test. This behavior is consistent with the observation of Hipp et al. [10] and Wang et al. [14]. To construct self-similar structure flow curves, we followed the protocol described by Hipp et al. [10] and Wang et al. [14]. We recorded the maximum stress value at each rate as the true stress value, and the stress response at the end of each desired shear rate to show the extent of decay in the stress response. The flow curves, only composed of the true stress response, were fit to the modified form of the Herschel–Bulkley model [36,37], which is

σ=σy(1+(γ˙γ˙c)n),
(1)

where σ is the stress, γ˙ is the shear rate, σy is the yield stress, γ˙c is the critical shear rate, and n is the power-law index. We chose this modified form of the Herschel–Bulkley model because the fitted γ˙c [37] can be compared across different formulations while the consistency index used in the original form [36] depends on the power-law index n. For the electric measurements made under shear, we considered only the electric data if the deviation of the stress response is less than 5% of the true stress value measured. We corrected the measured sample impedance with the short and open circuit measurements using standard corrections for the capacitance of the open cell and the resistance and stray inductance of the cables [34]. The corrected impedance, Z, was then converted into permittivity using the following equation: ε(ω)=Ccell/(2πiZfε0)=εiε, where Ccell is the cell constant that has a value of 0.149 1/m and ε0 is the vacuum permittivity.

To quantify the solution state properties of PVDF in NMP, we first performed SEC on PVDF polymers with different nominal molecular weights: the low MW PVDF (LP), Kynar 741, and the high MW PVDF (HP), Kynar 761. The results showed MW = 354.3 kg/mol with a polydispersity (Đ) of 1.95 and MW = 681 kg/mol with a Đ of 1.88 for LP and HP, respectively. To quantify the conformation of PVDF in NMP, we performed SAXS and rheological measurements for both LP and HP polymers as a function of concentration ranging from 0.5 to around 30 g/L. For the most dilute polymer concentration tested (0.5 g/L for both polymers), the scattering intensity versus Q is presented in Fig. 1(a) offset by a factor of 10. We fit these profiles to the polymer excluded volume model [38] [Eq. (S1)] and extracted the radius of gyration, Rg,p, and the excluded volume parameter, v. Consistent with GPC measurements, Rg,p increases with increasing molecular weight, and v>0.5, indicating that both LP and HP polymers are in a good solvent at room temperature [39]. At higher concentrations, our scattering results indicate a concentration-dependent structure factor whose zero-Q intercept decreases with increasing polymer concentration (Figs. S1 and S2) [70]. From this decrease, we obtained S(Q) and calculated the second virial coefficient MWA2 [40]. These parameters are summarized in Table I.

FIG. 1.

Solution properties of PVDF in NMP. (a) Form factors of LP and HP offset by a factor of 10. Solid lines are fits to the excluded volume model. (b) Specific viscosity at different polymer concentrations normalized to c. A power-law scaling of 1.60 is obtained by fitting data points at c/c>3. The solid line shows power-law scaling. (b, inset): Specific viscosity at low concentrations. The Einstein equation model fits are overlaid with data points. c is indicated for both polymers with dashed lines.

FIG. 1.

Solution properties of PVDF in NMP. (a) Form factors of LP and HP offset by a factor of 10. Solid lines are fits to the excluded volume model. (b) Specific viscosity at different polymer concentrations normalized to c. A power-law scaling of 1.60 is obtained by fitting data points at c/c>3. The solid line shows power-law scaling. (b, inset): Specific viscosity at low concentrations. The Einstein equation model fits are overlaid with data points. c is indicated for both polymers with dashed lines.

Close modal
TABLE I.

Summarized characterization results of LP and HP PVDF polymers.

Mw (kg/mol)aĐaRg,p (nm)bυbMwA2(L/g)b[η]0(l/g)cc(g/L)cRv (nm)c
LP 354.3 1.95 22.6 0.56 0.213 0.139 ± 0.003 7.21 19.8 
HP 681.0 1.88 35.6 0.58 0.317 0.189 ± 0.007 5.32 27.3 
Mw (kg/mol)aĐaRg,p (nm)bυbMwA2(L/g)b[η]0(l/g)cc(g/L)cRv (nm)c
LP 354.3 1.95 22.6 0.56 0.213 0.139 ± 0.003 7.21 19.8 
HP 681.0 1.88 35.6 0.58 0.317 0.189 ± 0.007 5.32 27.3 
a

Determined using size exclusion chromatography.

b

Determined using SAXS.

c

Determined using rheology.

For these same concentrations, we measured the solution viscosity (η) and find that all solutions tested are Newtonian over the range of 1–100 s−1 with small deviations (<1%) (Fig. S4) [70], indicating that both polymers are not entangled at those concentrations. To further ensure that entanglement is absent at the concentrations tested, we conducted small amplitude oscillatory shear tests on both polymers at the highest concentration prepared. The measurements were not definitive as the measured torque was below the instrument resolution at most frequencies tested. To calculate the intrinsic viscosity, [η]0, of both polymers, we fitted the viscosity measured in the dilute region, identified as 0.2MwNARg,p3, where NA is Avogadro's number. Using the intrinsic viscosity, we determined the overlap concentration as c=1/[η]0. The dilute region was confirmed because c was larger than the cutoff concentration that was 7.1 and 5.1 g/L for LP and HP, respectively. The ηsp defined as ηsp=η/ηs1 was then calculated and shown in the inset to Fig. 1(b) [41] versus polymer concentration normalized to c. As shown in Fig 1(b), we observe collapse of the specific viscosity of both polymers into one single master curve with a nonlinear scaling of ηsp(c/c)1.60 for c>3c, consistent with power-law scaling observed in semidilute solutions of unentangled polymers in a good solvent [42,43]. To compare the chain conformation determined using SAXS to that determined using rheology, we calculated υ from semidilute scaling ηsp(c/c)p, where p=1/(3v1) [39,42,43]. υ calculated from rheology data is 0.54, close to the values determined from SAXS, which are 0.56 and 0.58 for LP and HP, respectively. This further confirms that both polymers remain unentangled at all concentrations tested, and their rheology is consistent with the structural information obtained from SAXS. Finally, we used [η]0 and Mw to determine the viscometric radius (Rv) with the following equation:

Rv=(3[η]0Mw10πNA)1/3.
(2)

The resulting Rv value is smaller than the Rg,p, which is consistent with the swollen state of the polymer having a pervaded volume smaller than the volume occupied by the coil [40]. Both Rg,p, the radius of gyration, and Rv, the viscometric radius, are important parameters to characterize the size of a polymer coil, as Rg,p indicates the space physically occupied by the polymer chain while Rv is the radius of an effective hard sphere representation of this polymer chain [40]. In summary, our findings are consistent with prior measurements that showed that PVDF solutions are Newtonian, and NMP is a good solvent for PVDF [44–46].

To confirm that suspensions of Super C65 exhibit structural characteristics consistent with other high-structured carbon blacks, we performed TEM, USAXS, and rheological experiments. A representative TEM image is shown in Fig. 2(a). The TEM images show the presence of fractal-like primary aggregates built from fused primary particles. The average primary particle radius (Rp) from TEM images is 17.8 ± 4.1 nm. The scattering intensity versus Q of a 0.1 wt. % suspension of Super C65 in NMP is shown in Fig. 2(b). From the profile, the radius of the primary aggregate (Rp,agg) can be identified at Q0.004Å1, which corresponds to a value of ∼150 nm [Eq. (S6)]. This is consistent with the apparent hydrodynamic size of aggregates whose mean is 156.2 nm measured from a dilute 0.001 wt. % suspension immediately after sonication (Fig. S5) [70]. At low Q, we observe a strong upturn in scattering intensity that is evidence of the formation of large agglomerates within suspension whose size exceeds the length scale probed by the instrument. The power-law scaling at low Q, which has a slope value of −2.8, is comparable to other high-structured CBs characterized in other solvents [21]. For CB suspensions of four different weight fractions, we measured the shear stress as a function of shear rate with a preshear step between each measurement to rejuvenate the sample. At high shear rates, we observe a transient stress response where the shear stress reaches a steady value immediately after applying the desired shear rate at high shear rates, and at low shear rates, the shear stress continues to decrease with time (Fig. S6) [70]. This bifurcation of rheological behaviors in stress response is the same as observed in carbon black suspensions in various solvents [10,11,14], where rheopexy occurs at low intensities as a result of agglomerate densification and sedimentation.

FIG. 2.

(a) TEM image of Super C65 CB. The image shows a primary aggregate composed of primary particles. The primary particle length scale is shown. (b) USAXS profile of 0.1 wt. % Super C65 CB in NMP. The solid line indicates the power-law slope in the low-Q region arising from the formation of mass fractals. The hierarchical fractal structure of carbon black is shown, and the corresponding length scale of the primary aggregate is indicated with an arrow.

FIG. 2.

(a) TEM image of Super C65 CB. The image shows a primary aggregate composed of primary particles. The primary particle length scale is shown. (b) USAXS profile of 0.1 wt. % Super C65 CB in NMP. The solid line indicates the power-law slope in the low-Q region arising from the formation of mass fractals. The hierarchical fractal structure of carbon black is shown, and the corresponding length scale of the primary aggregate is indicated with an arrow.

Close modal

The self-similar flow curves are obtained with a protocol similar to that developed by Hipp et al. [11] and Wang et al. [14], and the protocol is described in detail in the Materials and Method section. The resulting self-similar flow curves are shown for each CB weight percent in Fig. 3(a). The closed symbols are the true stress values at each shear rate that constitute the flow curve, and the open symbols are the measured stress values at the end of each desired shear rate to show the extent of decay in the stress response. The corresponding effective volume fraction ϕeff shown in the legend of Fig. 3(a) was calculated as ϕeff=ϕCB,dry/ΦCB,pagg, where ϕCB,dry is the dry volume fraction, and ΦCB,pagg is the porosity of primary aggregates. ΦCB,pagg was calculated from the oil absorption number [Eq. (S7)] and has a value of 0.079. Qualitatively, we observe from the flow curves that shear stress increases at all shear rates with increasing CB weight percent and that each suspension exhibits a yield stress with strong shear thinning at high shear rates. Furthermore, at high shear rates, the stress response shows no decay while at low shear rates, significant decay in the stress response is observed. We fit each flow curve to the Herschel–Bulkley model [36,37] and observe good agreement at all shear rates. The fitted parameters, σy, γ˙c, and n, are defined as the yield stress, the critical shear rate, and the power-law index, respectively, and are presented in Table S2 [70]. As σy reflects the cohesive force between primary aggregates, this should increase with the weight percent [41,47], and our measurements are consistent with this expected trend. As shown in the visual evidence of Fig. 3(b), CB samples at higher volume fraction can retain their deformed volume more after being transported out of the vials due to their higher yield stress. We also observe that γ˙c decreases with increasing weight percent, consistent with the observations by Hipp et al. [10] and Nordstorm et al. [48]. Finally, n ranges from 0.55 to 0.7 for our measurements without showing a monotonic trend, which agrees with the observation by Hipp et al. [10]. As a smaller value of n indicates a more extensive shear-thinning behavior, we observe that 4 wt. % CB shows the most shear-thinning behavior.

FIG. 3.

Rheology of Super C65 CB in NMP. (a) Measured flow curves are overlaid with fits to the Herschel–Bulkley model. Closed symbols are the true stress values that constitute the flow curves, and open symbols are the stress values measured at the end of each shear rate. The fitted parameters are listed in Table S2 [70]. (b) Photos of CB samples at various weight percents 30 s after being transported out of the vials. Samples at higher weight percent retain volume deformation more due to their higher yield stress.

FIG. 3.

Rheology of Super C65 CB in NMP. (a) Measured flow curves are overlaid with fits to the Herschel–Bulkley model. Closed symbols are the true stress values that constitute the flow curves, and open symbols are the stress values measured at the end of each shear rate. The fitted parameters are listed in Table S2 [70]. (b) Photos of CB samples at various weight percents 30 s after being transported out of the vials. Samples at higher weight percent retain volume deformation more due to their higher yield stress.

Close modal

We then measured the flow curves of LP/CB and HP/CB suspensions at 3 and 4 wt. % CB, as presented in Fig. 4. These two weight percents are chosen due to their relevance to battery research (most battery slurries contain about 3–4 wt. % of CB [27]). We observe that the stress response is enhanced by the addition of polymer, and a higher loading of polymer leads to a further increase in the stress response. Furthermore, a decay in the stress response appears at lower shear rates for samples with higher polymer loading due to a higher stress response achieved at equivalent shear rates. To quantitatively compare different flow curves, we fit the flow curves (closed symbols) to the Herschel–Bulkley model, shown as solid lines in Fig. 4. The Herschel–Bulkley model fits the data well, and we summarized the values of those fitted parameters in Table S2 [70]. First, we observe that the fitted σy is higher for suspensions containing polymers compared to neat CB suspensions. Furthermore, the fitted γ˙c decreases with higher polymer loadings for the same type of polymer, indicating a higher characteristic time of suspensions. We also observe that suspensions containing more polymers have a higher n, indicating a more Newtonian behavior, and this trend is not consistent with our expectations. We expect a more extensive shear-thinning behavior due to the increase in matrix viscosity with higher polymer loadings. However, this may indicate the high matrix viscosity endowed by the added polymer dominates the stress response at high shear rates. Nonetheless, the qualitative character of the flow curves is only modestly changed by the addition of polymer. This is similar to the observation by Sung et al. [28] that PVDF only modifies the matrix viscosity without changing the structure of CB.

FIG. 4.

Self-similar flow curves of Super C65 CB/PVDF mixture suspensions in NMP as a function of polymer concentration. (a) LP/CB suspensions of 3 wt. % CB. (b) HP/CB suspensions of 3 wt. % CB. (c) LP/CB suspensions of 4 wt. % CB. (d) HP/CB suspensions of 4 wt. % CB. Closed symbols are the true stress values that constitute the flow curves, and open symbols are the stress value measured at the end of each shear rate.

FIG. 4.

Self-similar flow curves of Super C65 CB/PVDF mixture suspensions in NMP as a function of polymer concentration. (a) LP/CB suspensions of 3 wt. % CB. (b) HP/CB suspensions of 3 wt. % CB. (c) LP/CB suspensions of 4 wt. % CB. (d) HP/CB suspensions of 4 wt. % CB. Closed symbols are the true stress values that constitute the flow curves, and open symbols are the stress value measured at the end of each shear rate.

Close modal

To confirm that PVDF is nonadsorbing to CB, we conducted centrifugation experiments. We prepared LP and HP stock solutions at 31.9 g/L and followed the sample preparation protocol in the Materials and Methods section to prepare 1 and 3 wt. % CB suspensions in both stock solutions. The measured viscosity of stock solutions and purified polymer solutions obtained from supernatants of centrifuged samples is presented in Table S1 [70]. We observe that for polymer solutions obtained from the supernatants, the viscosity is the same, indicating no physical adsorption of PVDF to CB. This finding is again consistent with the observation by Sung et al. [28]. Furthermore, the result shows that PVDF can access the mesopores formed by CB primary aggregates and primary particles, and PVDF is present at all length scales where NMP is present. If PVDF chains were not able to access those mesopores, there would be an enhanced concentration in the supernatant as a significant amount of solvent would remain in CB mesopores, and consequently, a higher viscosity would be observed in the supernatant.

It is also known that depletion force [49], originating from the presence of polymers, can enhance attractive interactions between particles in suspensions. To quantify the effect of depletion force, we calculated and compared the magnitude of van der Waals force [50] and depletion force [51], as shown in Figs. S7 [70]. The result shows that due to the strong attraction between CB primary aggregates, the effect of polymer depletion force is negligible, and therefore, the interactions between CB primary aggregates remain unchanged. To decouple matrix viscosity effects from microstructural changes impacting suspension viscosity, we calculated the relative viscosity, ηr=η/ηm, where ηm is the suspending medium viscosity, including the polymer contribution. We further scaled the shear rate using the fluid Mason number (Mnf), defined as the ratio of suspending medium hydrodynamic force, Fshear, over the cohesive force holding agglomerates together, Fcohesive. Prior work has shown that Mnf determines the microstructure of agglomerated suspensions including both anisotropy [52–54] and agglomerate size [53,55–57]. The solvent hydrodynamic force Fshear is calculated as 6πRp,agg2ηmγ˙ using Stokes' drag equation. To calculate Fcohesive, we follow the approach of Eberle et al. [10,52] where Fcohesive=Rp,agg2σy/ϕ2. In this expression, the yield stress is scaled with the area density of contact by ϕ2 to approximate the cohesive force. By taking the ratio of Fshear and Fcohesive, Mnf is given as

Mnf=6πϕeff2ηmγ˙σy,
(3)

where ϕ is replaced as ϕeff, which is the effective volume fraction of CB considering the porous nature of CB. Furthermore, when calculating Mnf, we assume that Fcohesive is the same for suspensions containing the same weight percent of CB and the value of Fcohesive is calculated from the fitted σy of neat CB suspensions. While we acknowledge that the fitted yield stress increases for suspensions containing polymers, both our centrifugation experiment and depletion force calculation show that PVDF is not absorbed to CB, and we assume, therefore, that Fcohesive, is not modified. We suspect that the change in the fitted σy comes from the hydrodynamic contribution from added polymers at low shear rates. The relative viscosity of all samples as a function of Mnf is presented in Fig. 5. To aid visualization, we shifted the relative viscosity of suspensions of all 4 wt. % curves by a factor of 5. The original relative viscosity is presented in Fig. S8 [70]. We observe that by using Mnf, the relative viscosity of samples containing the same weight percent of CB is successfully collapsed for both CB weight fractions. Furthermore, because of the high medium viscosity at high polymer concentrations, higher values of Mnf are achieved in these suspensions, inducing a further breakdown of CB agglomerates, as indicated by the lower relative viscosity. Because relative viscosity is directly related to agglomerate size when the effective volume fraction of CB remains the same [58], this result suggests that Mnf predicts the microstructure of CB agglomerates for CB/PVDF suspensions. This finding confirms that the shear-thinning index at high polymer loading is a result of the large matrix viscosity and suggests that the polymer acts to enhance the hydrodynamic force acting to break up carbon black agglomerates. To verify that the polymer/solvent medium can be conceptualized as an effective Newtonian medium, we calculated the longest relaxation time (τ) [59,60] of polymer solutions to estimate polymer elasticity at shear rates tested (SI S8 [70]). As shown in Table S3 [70], even at the highest polymer concentration tested in CB/PVDF mixture suspensions (32 g/L), the onset of shear thinning, calculated as 1/τ, is 88 616 and 19 224 s−1 for LP and HP, respectively. We further ensured that polymer viscoelasticity is negligible by conducting rheological tests on polymer solutions at very high shear rates (up to 2500 s−1), which is shown in Fig. S14 [70]. We observed that at high shear rates, for HP polymer solutions, the decrease in viscosity at 2500 s−1 is noticeable but very small (9.2%, 5.1%, and 3.3% for 32, 20, and 10 g/L, respectively), and for LP polymer solutions, the decrease is negligible (all < 1.5%). Therefore, we conclude that for suspensions tested here polymer viscoelasticity plays a negligible role, supporting the idea that the polymer/solvent medium can be regarded as an effective Newtonian fluid.

FIG. 5.

Viscosity normalized to the medium viscosity plotted against the fluid Mason number. Samples are 3 wt. % CB and 4 wt. % CB added with LP and HP at various concentrations. Data are scaled for clarity. The relative viscosity of suspensions of the same CB weight percent is successfully collapsed.

FIG. 5.

Viscosity normalized to the medium viscosity plotted against the fluid Mason number. Samples are 3 wt. % CB and 4 wt. % CB added with LP and HP at various concentrations. Data are scaled for clarity. The relative viscosity of suspensions of the same CB weight percent is successfully collapsed.

Close modal

To confirm that the collapse of the specific viscosity indicates that similar structures are formed at the same Mnf, we analyzed the frequency-dependent permittivity at every experimental condition including the shear rate, polymer concentration, CB content, and PVDF molecular weight. Representative spectra for a CB/PVDF mixture at three shear rates are shown in Fig. 6(a). In all spectra, we observe signatures of electrode polarization, a power-law upturn that appears in the real component of the permittivity ε [61]. At f ∼ 105 Hz, a characteristic low-frequency plateau (εs) of the charge carrier relaxation process is observed [62,63]. The permittivity decreases at higher frequencies although the high-frequency plateau (εinf) is not reached due to instrument limitations. For the imaginary part of permittivity ε, we observe at low frequencies a power-law scaling of εAfB, coming from electronic and ionic conduction [64,65], and the corresponding relaxation peak at frequencies similar to where ε decays. The height of this peak is sensitive to the shear rate, as evidenced in the decreasing εs in ε and the reducing peak intensity in ε. To quantify the diminishment of this feature as a function of shear rate, we adopted an analysis approach that does not rely on equivalent circuit modeling. We elected to analyze the data in this way as model-free approaches do not require assumptions about the nature of charge transport in flowing CB suspensions, which is complex [21,66]. We first fit a power-law model in the frequency range of 3 –7 × 104 Hz, indicated by the shaded region in Fig. 6(b), and then subtracted the contribution from ε as shown in Fig. 6(c). The resulting difference spectrum reveals the shear rate dependence of the relaxation process that decreases in intensity with increasing shear rate. To quantify this decrease, we calculated the dielectric strength, Δε,

Δε=2πε(f)dln(2πf),
(4)

using direct numerical integration of the difference spectra over the frequency range of 7 × 104–7 × 106 Hz [62]. Δε of this suspension at all Mnf is plotted in Fig. 6(d). We observe a plateau region at low Mnf, followed by a decrease in Δε at high Mnf. We repeated this procedure at all experimental conditions as shown in Fig. 5. The resulting Δε is shown versus Mnf in Fig. 7 for all conditions.

FIG. 6.

Illustration of shear-dependent dielectric response and calculation of Δε. The sample is 3.0 wt. % CB, 21.1 g/L LP suspension. (a) Shear-dependent dielectric response. Open symbols are ε, and closed symbols are ε. (b) The imaginary part of relative permittivity at high frequencies and the power-law fit. The power fit has the form of A×fB, where A and B are fitted parameters. The power-law fit was fitted to the frequency range of 3−7 × 104 Hz, as indicated by the shaded region. (c) The relaxation peak obtained from the difference between the data and the power-law fit. Δε was calculated by using Δε=εsε=2πε(f)dln(2πf) over the range of 7 × 104–7 × 106 Hz [62]. (d) Δε plotted against the fluid Mason number at all shear rates. The error bar was calculated from error propagation of the power-law fit.

FIG. 6.

Illustration of shear-dependent dielectric response and calculation of Δε. The sample is 3.0 wt. % CB, 21.1 g/L LP suspension. (a) Shear-dependent dielectric response. Open symbols are ε, and closed symbols are ε. (b) The imaginary part of relative permittivity at high frequencies and the power-law fit. The power fit has the form of A×fB, where A and B are fitted parameters. The power-law fit was fitted to the frequency range of 3−7 × 104 Hz, as indicated by the shaded region. (c) The relaxation peak obtained from the difference between the data and the power-law fit. Δε was calculated by using Δε=εsε=2πε(f)dln(2πf) over the range of 7 × 104–7 × 106 Hz [62]. (d) Δε plotted against the fluid Mason number at all shear rates. The error bar was calculated from error propagation of the power-law fit.

Close modal
FIG. 7.

Dielectric strength (Δε) calculated from the permittivity loss plotted against the fluid Mason number (Mnf) for 3 and 4 wt. % CB suspensions added with LP and HP at various concentrations.

FIG. 7.

Dielectric strength (Δε) calculated from the permittivity loss plotted against the fluid Mason number (Mnf) for 3 and 4 wt. % CB suspensions added with LP and HP at various concentrations.

Close modal

To ensure that polymers or the solvent do not contribute to the changes observed in Δε, we also measured the dielectric behavior of NMP and high-concentration PVDF solutions, as shown in Fig. S10 [70]. All PVDF and pure NMP solutions showed minimal to no changes in their dielectric response at all tested shear rates, confirming that the changes in dielectric response of CB/PVDF do not explain our experimental observations. Furthermore, this confirms that the origin of this enhanced conductivity is a direct result of the electrical transport between carbon black agglomerates [21], as the conductivity of the suspension far exceeds the conductivity of PVDF solutions. For samples containing 4 wt. % CB, Δε is larger than that of samples of 3 wt. % CB at equivalent Mnf. In the frequency-dependent permittivity of some samples, especially 4 wt. % CB samples, we observed significant inductance (as indicated by the Nyquist plot in Fig. S11 [70]). The calculation of the dielectric strength via our method is not significantly influenced by the inductance as it primarily affects the real component of the permittivity.

Based on the decreased relative viscosity and dielectric strength with increasing fluid Mason number, we speculate that the dielectric strength observed in these suspensions originates from the changing agglomerate size under shear. Specifically, prior work has established that charge hopping produces a dipole moment [62] within a conductive cluster whose magnitude is proportional to the distance over which electrons are polarized [67]. While in general, a conductive cluster need not be associated with any specific microstructure within the sample, in semiconductors and conducting polymer composites, it is widely known that charges are strongly localized to conducting domains within the sample, and they must hop from site to site to transport over macroscopic length scales [63]. In the context of these suspensions, we assume this distance to be associated with the length scale of conductive clusters within the sample [63,66]. Specifically, when the electric field is applied, the charge carriers migrate in the direction of the field. Migration is a diffusive process where charges undergo a random walk between conductive sites and on long time scales, and they lose the memory of their previous position. These sites can include primary aggregates, closely connect agglomerates, or even hop between primary particles. At lower carbon weight loadings, our results indicate that the conductive cluster size is dependent on the agglomerate size. This is supported by the decreasing dielectric strength and decreasing relative viscosity with Mnf. The dependence of Δε on Mnf across all samples suggests that there is a universal relationship between the conductive cluster size and Mnf, which we interrogate by shifting the curves in Fig. 7 vertically by a shift factor α. This empirical shift collapses all samples as a function of Mnf at a given carbon black loading, as shown in Fig. 8(a). The shift factor increases monotonically as a function of polymer concentration for the same CB weight percent and polymer type [Fig. 8(b)], which is consistent with the increased polymer concentration leading to a modification of the relationship between the conductive cluster size and the agglomerate size.

FIG. 8.

(a) Dielectric strength (Δε) shifted by a shift factor α plotted against the fluid Mason number (Mnf) for 3 and 4 wt. % CB suspensions added with LP and HP at various concentrations. Mnf,c corresponding to the onset of the decrease in Δε is indicated by red dashed lines. Power-law scaling, indicated with a solid line, is observed for the 3 wt. % sample group, and the power-law value is −0.76. (b) The shift factor (α) as a function of polymer concentration for all samples. For the same CB weight percent and the same polymer type, the trend is monotonic.

FIG. 8.

(a) Dielectric strength (Δε) shifted by a shift factor α plotted against the fluid Mason number (Mnf) for 3 and 4 wt. % CB suspensions added with LP and HP at various concentrations. Mnf,c corresponding to the onset of the decrease in Δε is indicated by red dashed lines. Power-law scaling, indicated with a solid line, is observed for the 3 wt. % sample group, and the power-law value is −0.76. (b) The shift factor (α) as a function of polymer concentration for all samples. For the same CB weight percent and the same polymer type, the trend is monotonic.

Close modal

To understand the collapse of the dielectric strength with Mnf, we first identified the onset of the decline of Δε or the critical fluid Mason number Mnf,c with dashed red lines in Fig. 8(a). We observe that Mnf,c of the 3 wt. % CB group is lower than that of the 4 wt. % CB group. Varga et al. [53] showed that shifting Mnf with Mnf,c is necessary to collapse the agglomerate size across sample of different volume fractions, and the value of Mnf,c increases with increasing volume fraction, consistent with the trend observed here. We further speculate that the plateau at low Mnf is correlated with anisotropy that develops in these suspensions at low shear intensity. Eberle et al. [52] showed that flow heterogeneity along the flow direction is present until the relative viscosity decreases to smaller than 10. In our data, the plateau region ends when the relative viscosity drops below 10, as shown in Fig. S13 [70]. Moreover, when MnfMnf,c, we observe that ΔεMnfm, where m is −0.76 ± 0.01 for samples of 3 wt. % CB. The value of m for samples of 4 wt. % CB is difficult to determine due to the limited range of Mnf probed. Therefore, we focus on the power-law determined for the 3 wt. % sample. To understand and predict m, we expect a relationship between the dielectric strength and the microstructure of suspension under flow. This can be seen more clearly from the definition of the dielectric strength [62],

Δε=ρμ23ε0kT,
(5)

where ρ is the number density of charge carriers, k is the Boltzmann constant, μ2 is the dipole moment, and T is the temperature. Due to the fractal nature of CB agglomerates and their breakup in the shear flow, we anticipate both the size of the conductive cluster and their number density to change as a function of Mnf. To quantify the latter effect, we adopt an empirical scaling relationship linking the number density of conductive clusters to their fractal character ρ(Rg/Rp,agg)(3Df), where Df is the fractal dimension of primary aggregates within an agglomerate and Rg is the radius of gyration of CB agglomerate. To estimate μ2, we assume that a unit charge is delocalized over a conductive cluster whose size is proportional to the agglomerate size such that μ2(Rg/Rp,agg)2. Furthermore, we assume the charge on an aggregate is insensitive to the shear rate. Combining these two relationships, the dielectric strength can be written as Δε(Rg/Rp,agg)(3Df)(Rg/Rp,agg)2(Rg/Rp,agg)Df1. To develop a link between agglomerate size and Mnf, Varga et al. [53] observed that (Rg/Rp,agg)Mnf0.5 and Hipp et al. [10] observed that carbon black suspensions showed a scaling of (Rg/Rp,agg)Mnf0.33to0.43 (Fig. S9) [70]. Using these relationships, we anticipate the scaling between Δε and Mnf is ΔεMnf0.50to0.85, where Df has a value between 2.5 and 2.7, which is similar to values observed in CB suspensions [10]. The scaling observed in Fig. 8(a) is consistent with the predicted power-law scaling, lending support to the idea that the conductive cluster size is linked to the agglomerate size through the dielectric strength. As this finding relies on the self-similar nature of the agglomerate break up (i.e., constant fractal dimension), we conclude that PVDF has no impact on the steady-shear agglomerate structure other than to enhance the magnitude of the Mnf.

Our finding here shows that the agglomerate size of carbon black is further decreased by the presence of nonabsorbing polymers. This originates from the enhanced suspending medium viscosity that acts to increase the Mnf without changing the self-similar breakup process. This finding has significance in understanding the processing of lithium-ion battery cathode slurries as the effect of PVDF on CB steady-shear rheology can be predicted by intrinsic polymer properties alone. Furthermore, this work serves to pave the way for linking battery performance to the slurry processing conditions as the dispersion state of CB has an important role in affecting both ionic [68] and electronic conductivities [31,69] of porous electrodes processed from the slurries. Therefore, we anticipate that the molecular weight and the concentration of PVDF can be optimized in slurry formulation to achieve an adequate breakup of CB agglomerates without compromising the battery performance.

This study quantifies the role of PVDF in modifying the structure of Super C65 CB in suspensions of NMP with simultaneous rheo-electric measurements. The structure and dynamics of PVDF and CB are characterized individually first with SAXS and rheology. For CB suspensions of the same weight percent containing PVDF, the relative viscosity is successfully collapsed with the fluid Mason number, Mnf, which is the ratio of the medium hydrodynamic force over the cohesive force between CB primary aggregates. This suggests that PVDF only enhances the medium viscosity, and Mnf is an accurate descriptor for predicting CB agglomerate size even for CB/PVDF mixture suspensions. We further confirm that the CB agglomerate break up is self-similar at high Mnf by analyzing rheo-electric measurements. We identify that the dielectric strength Δε, accounting for polarization of charge carriers delocalized within a conducting cluster, changes dramatically as a function of shear rate and can be used as a proxy for agglomerate size. We identify two regions for the dielectric strength as a function of Mnf, a plateau at low Mnf and a power-law scaling at high Mnf. We speculate that the low-Mnf plateau can be explained by anisotropy, and we successfully predict the power-law scaling at high Mnf by assuming a self-similar break up of CB agglomerates. The combination of rheology and electric measurements shows that PVDF modifies the steady-shear structure of CB agglomerates indirectly via enhancing the medium viscosity and the breakup of CB agglomerates is still self-similar in these suspensions. However, time-dependent structural changes may depend on the addition of PVDF, which was not explored in this work. We also anticipate that to completely correlate the dielectric response and the structural information, simultaneous measurements of the structure under flow and dielectric response will be necessary. In this work, we show that simultaneous electric measurements have the potential to be an effective structural probe for suspensions containing conducting particles under flow. Furthermore, these findings suggest that the structure and the rheological response of CB added with nonabsorbing polymers can be simply predicted by their intrinsic polymer properties. These findings will be valuable to energy storage systems containing CB as the rheology of CB can be predicted and controlled by engineering the polymer properties and the processing conditions.

This material was based upon work supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences Energy Frontier Research Centers program under Award No. DE-SC-0022119. SAXS measurements were performed at the DuPont-Northwestern-Dow Collaborative Access Team (DND-CAT) located at Sector 5 of the Advanced Photon Source (APS). DND-CAT is supported by Northwestern University, The Dow Chemical Company, and DuPont de Nemours, Inc. Data were collected using an instrument funded by the National Science Foundation (NSF) under Award No. 0960140. USAXS measurements were performed at 9-ID. This research used resources of the Advanced Photon Source, a U.S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by the Argonne National Laboratory under Contract No. DE-AC02-06CH11357. Extraordinary facility operations were supported in part by the DOE Office of Science through the National Virtual Biotechnology Laboratory, a consortium of DOE national laboratories focused on the response to COVID-19, with funding provided by the Coronavirus CARES Act. This work made use of the NUFAB facility of Northwestern University's NUANCE Center, which has received support from the SHyNE Resource (No. NSF ECCS-2025633), the IIN, and Northwestern’s MRSEC Program (No. NSF DMR-1720139). The authors also acknowledge Arkema for providing PVDF samples. The authors would like to thank Jan Ilavsky and Steven Weigand for their assistance in remote experiments at APS. The authors would like to thank Matt Thompson and the Gianneschi Group at Northwestern University for running SEC measurements.

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon request.

1.
Spinelli
,
H. J.
, “
Polymeric dispersants in ink jet technology
,”
Adv. Mater.
10
,
1215
1218
(
1998
).
2.
Khandavalli
,
S.
,
J. H.
Park
,
N. N.
Kariuki
,
D. J.
Myers
,
J. J.
Stickel
,
K.
Hurst
,
K. C.
Neyerlin
,
M.
Ulsh
, and
S. A.
Mauger
, “
Rheological investigation on the microstructure of fuel cell catalyst inks
,”
ACS Appl. Mater. Interfaces
10
,
43610
43622
(
2018
).
3.
Duduta
,
M.
,
B.
Ho
,
V. C.
Wood
,
P.
Limthongkul
,
V. E.
Brunini
,
W. C.
Carter
, and
Y.-M.
Chiang
, “
Semi-solid lithium rechargeable flow battery
,”
Adv. Energy Mater.
1
,
511
516
(
2011
).
4.
Narayanan
,
A.
,
F.
Mugele
, and
M. H. G.
Duits
, “
Mechanical history dependence in carbon black suspensions for flow batteries: A rheo-impedance study
,”
Langmuir
33
,
1629
1638
(
2017
).
5.
Ramos
,
P. Z.
,
C. C.
Call
,
L. V.
Simitz
, and
J. J.
Richards
, “
Evaluating the rheo-electric performance of aqueous suspensions of oxidized carbon black
,”
J. Colloid Interface Sci.
634
,
379
387
(
2023
).
6.
Spahr
,
M. E.
,
D.
Goers
,
A.
Leone
,
S.
Stallone
, and
E.
Grivei
, “
Development of carbon conductive additives for advanced lithium ion batteries
,”
J. Power Sources
196
,
3404
3413
(
2011
).
7.
International Carbon Black Association
,
Carbon Black User’s Guide
(International Carbon Black Association, New Orleans, LA,
2016
).
8.
Ovarlez
,
G.
,
L.
Tocquer
,
F.
Bertrand
, and
P.
Coussot
, “
Rheopexy and tunable yield stress of carbon black suspensions
,”
Soft Matter
9
,
5540
5549
(
2013
).
9.
Gibaud
,
T.
,
D.
Frelat
, and
S.
Manneville
, “
Heterogeneous yielding dynamics in a colloidal gel
,”
Soft Matter
6
,
3482
3488
(
2010
).
10.
Hipp
,
J. B.
,
J. J.
Richards
, and
N. J.
Wagner
, “
Direct measurements of the microstructural origin of shear-thinning in carbon black suspensions
,”
J. Rheol.
65
,
145
157
(
2021
).
11.
Hipp
,
J. B.
,
J. J.
Richards
, and
N. J.
Wagner
, “
Structure-property relationships of sheared carbon black suspensions determined by simultaneous rheological and neutron scattering measurements
,”
J. Rheol.
63
,
423
436
(
2019
).
12.
Dullaert
,
K.
, and
J.
Mewis
, “
A structural kinetics model for thixotropy
,”
J. Non-Newton. Fluid Mech.
139
,
21
30
(
2006
).
13.
Armstrong
,
M. J.
,
A. N.
Beris
,
S. A.
Rogers
, and
N. J.
Wagner
, “
Dynamic shear rheology and structure kinetics modeling of a thixotropic carbon black suspension
,”
Rheol. Acta
56
,
811
824
(
2017
).
14.
Wang
,
Y.
, and
R. H.
Ewoldt
, “
New insights on carbon black suspension rheology—Anisotropic thixotropy and antithixotropy
,”
J. Rheol.
66
,
937
953
(
2022
).
15.
Hipp
,
J. B.
,
Structure, Rheology, and Electrical Conductivity of High-Structured Carbon Black Suspensions
(
University of Delaware
, Newark, DE,
2020
).
16.
Negi
,
A. S.
, and
C. O.
Osuji
, “
New insights on fumed colloidal rheology–Shear thickening and vorticity-aligned structures in flocculating dispersions
,”
Rheol. Acta
48
,
871
881
(
2009
).
17.
Osuji
,
C. O.
, and
D. A.
Weitz
, “
Highly anisotropic vorticity aligned structures in a shear thickening attractive colloidal system
,”
Soft Matter
4
,
1388
1392
(
2008
).
18.
Grenard
,
V.
,
N.
Taberlet
, and
S.
Manneville
, “
Shear-induced structuration of confined carbon black gels: Steady-state features of vorticity-aligned flocs
,”
Soft Matter
7
,
3920
3928
(
2011
).
19.
Wang
,
Z.
,
T.
Zhao
, and
M.
Takei
, “
Clarification of particle dispersion behaviors based on the dielectric characteristics of cathode slurry in lithium-ion battery (LIB)
,”
J. Electrochem. Soc.
166
,
A35
A46
(
2019
).
20.
Helal
,
A.
,
T.
Divoux
, and
G. H.
McKinley
, “
Simultaneous rheoelectric measurements of strongly conductive complex fluids
,”
Phys. Rev. Appl.
6
,
1
19
(
2016
).
21.
Richards
,
J. J.
,
J. B.
Hipp
,
J. K.
Riley
,
N. J.
Wagner
, and
P. D.
Butler
, “
Clustering and percolation in suspensions of carbon black
,”
Langmuir
33
,
12260
12266
(
2017
).
22.
Iijima
,
M.
,
M.
Yamazaki
,
Y.
Nomura
, and
H.
Kamiya
, “
Effect of structure of cationic dispersants on stability of carbon black nanoparticles and further processability through layer-by-layer surface modification
,”
Chem. Eng. Sci.
85
,
30
37
(
2013
).
23.
Subramanian
,
S.
, and
G.
Øye
, “
Aqueous carbon black dispersions stabilized by sodium lignosulfonates
,”
Colloid Polym. Sci.
299
,
1223
1236
(
2021
).
24.
Yasin
,
S.
, and
P. F.
Luckham
, “
Investigating the effectiveness of PEO/PPO based copolymers as dispersing agents for graphitic carbon black aqueous dispersions
,”
Colloids Surf. A
404
,
25
35
(
2012
).
25.
Bauer
,
W.
, and
D.
Nötzel
, “
Rheological properties and stability of NMP based cathode slurries for lithium ion batteries
,”
Ceram. Int.
40
,
4591
4598
(
2014
).
26.
Ma
,
F.
,
Y.
Fu
,
V.
Battaglia
, and
R.
Prasher
, “
Microrheological modeling of lithium ion battery anode slurry
,”
J. Power Sources
438
,
226994
(
2019
).
27.
Kraytsberg
,
A.
, and
Y.
Ein-Eli
, “Conveying advanced Li-Ion battery materials into practice the impact of electrode slurry preparation skills,”
Adv. Energy Mater.
6, (2016).
28.
Sung
,
S. H.
,
S.
Kim
,
J. H.
Park
,
J. D.
Park
, and
K. H.
Ahn
, “
Role of PVDF in rheology and microstructure of NCM cathode slurries for lithium-ion battery
,”
Materials
13
,
1
11
(
2020
).
29.
Higa
,
K.
,
H.
Zhao
,
D. Y.
Parkinson
,
H.
Barnard
,
M.
Ling
,
G.
Liu
, and
V.
Srinivasan
, “
Electrode slurry particle density mapping using x-ray radiography
,”
J. Electrochem. Soc.
164
,
A380
A388
(
2017
).
30.
Wang
,
M.
,
D.
Dang
,
A.
Meyer
,
R.
Arsenault
, and
Y.-T.
Cheng
, “
Effects of the mixing sequence on making lithium ion battery electrodes
,”
J. Electrochem. Soc.
167
,
100518
(
2020
).
31.
Saraka
,
R. M.
,
S. L.
Morelly
,
M. H.
Tang
, and
N. J.
Alvarez
, “
Correlating processing conditions to short- and long-range order in coating and drying lithium-ion batteries
,”
ACS Appl. Energy Mater.
3
,
11681
11689
(
2020
).
32.
Font
,
F.
,
B.
Protas
,
G.
Richardson
, and
J. M.
Foster
, “
Binder migration during drying of lithium-ion battery electrodes: Modelling and comparison to experiment
,”
J. Power Sources
393
,
177
185
(
2018
).
33.
Ilavsky
,
J.
,
F.
Zhang
,
R. N.
Andrews
,
I.
Kuzmenko
,
P. R.
Jemian
,
L. E.
Levine
, and
A. J.
Allen
, “
Development of combined microstructure and structure characterization facility for in situ and operando studies at the advanced photon source
,”
J. Appl. Crystallogr.
51
,
867
882
(
2018
).
34.
Richards
,
J. J.
, and
J. K.
Riley
, “
Dielectric RheoSANS: A mutual electrical and rheological characterization technique using small-angle neutron scattering
,”
Curr. Opin. Colloid Interface Sci.
42
,
110
120
(
2019
).
35.
Riley
,
J. K.
,
J. J.
Richards
,
N. J.
Wagner
, and
P. D.
Butler
, “
Branching and alignment in reverse worm-like micelles studied with simultaneous dielectric spectroscopy and RheoSANS
,”
Soft Matter
14
,
5344
5355
(
2018
).
36.
Herschel
,
W. H.
, and
R.
Bulkley
, “
Konsistenzmessungen von Gummi-Benzollösungen.: Kolloid-Zeitschrift
,”
Kolloid-Zeitschrift
39
,
291
300
(
1926
).
37.
Nelson
,
A. Z.
, and
R. H.
Ewoldt
, “
Design of yield-stress fluids: A rheology-to-structure inverse problem
,”
Soft Matter
13
,
7578
7594
(
2017
).
38.
Hammouda
,
B.
, “
SANS from homogeneous polymer mixtures: A unified overview
,” in
Polymer Characteristics. Advances in Polymer Science
(Springer, Berlin, Heidelberg, 1993), Vol. 106.
39.
Rubinstein
,
M.
, and
R. H.
Colby
,
Polymer Physics
(
OUP
,
Oxford
,
2003
).
40.
Heo
,
Y.
, and
R. G.
Larson
, “
The scaling of zero-shear viscosities of semidilute polymer solutions with concentration
,”
J. Rheol.
49
,
1117
1128
(
2005
).
41.
Larson
,
R. G.
,
The Structure and Rheology of Complex Fluids
(
Oxford University
,
New York
,
1999
).
42.
Huang
,
C.-C.
,
R. G.
Winkler
,
G.
Sutmann
, and
G.
Gompper
, “
Semidilute polymer solutions at equilibrium and under shear flow
,”
Macromolecules
43
,
10107
10116
(
2010
).
43.
Raspaud
,
E.
,
D.
Lairez
, and
M.
Adam
, “
On the number of blobs per entanglement in semidilute and good solvent solution: Melt influence
,”
Macromolecules
28
,
927
933
(
1995
).
44.
Ali
,
S.
, and
A. K.
Raina
, “
Dilute solution behavior of poly(viny1idene fluoride).: Intrinsic viscosity and light scattering studies
,”
Die Makromolekulare Chemie
179,
2925
2930
(
1978
).
45.
Damdinov
,
B. B.
,
V. A.
Danilova
,
A. V.
Minakov
, and
M. I.
Pryazhnikov
, “
Rheological properties of PVDF solutions
,”
J. Sib. Fed. Univ. Math. Phys.
14
,
265
272
(
2021
).
46.
Kutringer
,
G.
, and
G.
Weill
, “
Solution properties of poly(vinylidene fluoride): 1.: Macromolecular characterization of soluble samples
,”
Polymer
32
,
877
883
(
1991
).
47.
Mewis
,
J.
, and
N. J.
Wagner
,
Colloidal Suspension Rheology
(
Cambridge University
, Cambridge, United Kingdom,
2011
).
48.
Nordstrom
,
K. N.
,
E.
Verneuil
,
P. E.
Arratia
,
A.
Basu
,
Z.
Zhang
,
A. G.
Yodh
,
J. P.
Gollub
, and
D. J.
Durian
, “
Microfluidic rheology of soft colloids above and below jamming
,”
Phys. Rev. Lett.
105
,
1
4
(
2010
).
49.
Poon
,
W. C. K.
, “
The physics of a model colloid-polymer mixture
,”
J. Phys.: Condens. Matter
14
, R859–R880 (
2002
).
50.
Hamaker
,
H. C.
, “
The London-van Der waals attraction between spherical particles
,”
Physica
4
,
1058
1072
(
1937
).
51.
Joanny
,
J. F.
,
L.
Leibler
, and
P. G.
De Gennes
, “
Effects of polymer solutions on colloid stability
,”
J. Polym. Sci.: Polym. Phys. Ed.
17
,
1073
1084
(
1979
).
52.
Eberle
,
A. P. R.
,
N.
Martys
,
L.
Porcar
,
S. R.
Kline
,
W. L.
George
,
J. M.
Kim
,
P. D.
Butler
, and
N. J.
Wagner
, “
Shear viscosity and structural scalings in model adhesive hard-sphere gels
,”
Phys. Rev. E
89
,
050302
(
2014
).
53.
Varga
,
Z.
, and
J. W.
Swan
, “
Large scale anisotropies in sheared colloidal gels
,”
J. Rheol.
62
,
405
418
(
2018
).
54.
Varga
,
Z.
,
V.
Grenard
,
S.
Pecorario
,
N.
Taberlet
,
V.
Dolique
,
S.
Manneville
,
T.
Divoux
,
G. H.
McKinley
, and
J. W.
Swan
, “
Hydrodynamics control shear-induced pattern formation in attractive suspensions
,”
Proc. Natl. Acad. Sci. U.S.A.
116
,
12193
12198
(
2019
).
55.
Mwasame
,
P. M.
,
A. N.
Beris
,
R. B.
Diemer
, and
N. J.
Wagner
, “
A constitutive equation for thixotropic suspensions with yield stress by coarse-graining a population balance model
,”
AIChE J.
63
,
517
531
(
2017
).
56.
Harshe
,
Y. M.
,
M.
Lattuada
, and
M.
Soos
, “
Experimental and modeling study of breakage and restructuring of open and dense colloidal aggregates
,”
Langmuir
27
,
5739
5752
(
2011
).
57.
Chen
,
D.
, and
M.
Doi
, “
Microstructure and viscosity of aggregating colloids under strong shearing force
,”
J. Colloid Interface Sci.
212
,
286
292
(
1999
).
58.
Deepak Selvakumar
,
R.
, and
S.
Dhinakaran
, “
Effective viscosity of nanofluids—A modified Krieger–Dougherty model based on particle size distribution (PSD) analysis
,”
J. Mol. Liq.
225
,
20
27
(
2017
).
59.
Doi
,
M.
, and
S. F.
Edwards
,
The Theory of Polymer Dynamics
(
Clarendon
, Oxford, United Kingdom,
1988
).
60.
Liu
,
Y.
,
Y.
Jun
, and
V.
Steinberg
, “
Concentration dependence of the longest relaxation times of dilute and semi-dilute polymer solutions
,”
J. Rheol.
53
,
1069
1085
(
2009
).
61.
Ishai
,
P. B.
,
M. S.
Talary
,
A.
Caduff
,
E.
Levy
, and
Y.
Feldman
, “
Electrode polarization in dielectric measurements: A review
,”
Meas. Sci. Technol.
24
,
102001
(
2013
).
62.
Kremer
,
F.
and
A.
Schönhals
,
Broadband Dielectric Spectroscopy
(Springer Science & Business Media, Berlin, Germany,
2003
).
63.
Pelster
,
R.
, and
U.
Simon
, “
Nanodispersions of conducting particles: Preparation, microstructure and dielectric properties
,”
Colloid Polym. Sci.
277
,
2
14
(
1999
).
64.
Chakrabarty
,
R. K.
,
K. K.
Bardhan
, and
A.
Basu
, “
Measurement of AC conductance, and minima in loss tangent, of random conductor-insulator mixtures
,”
J. Phys.: Condens. Matter
5
,
2377
2388
(
1993
).
65.
Shalaev
,
V. M.
, “
Electromagnetic properties of small-particle composites
,”
Phys. Rep.
272
,
61
137
(
1996
).
66.
Lin
,
H.
,
M. V.
Majji
,
N.
Cho
,
J. R.
Zeeman
,
J. W.
Swan
, and
J. J.
Richards
, “
Quantifying the hydrodynamic contribution to electrical transport in non-Brownian suspensions
,”
Proc. Natl. Acad. Sci. U.S.A.
119
,
1
7
(
2022
).
67.
Stoylov
,
S. P.
, “
Electro-optical investigations of the dipole moments of nanoparticles
,”
Colloids Surf. B
56
,
50
58
(
2007
).
68.
Stephenson
,
D. E.
,
B. C.
Walker
,
C. B.
Skelton
,
E. P.
Gorzkowski
,
D. J.
Rowenhorst
, and
D. R.
Wheeler
, “
Modeling 3D microstructure and ion transport in porous Li-ion battery electrodes
,”
J. Electrochem. Soc.
158
,
A781
(
2011
).
69.
Morelly
,
S. L.
,
N. J.
Alvarez
, and
M. H.
Tang
, “
Short-range contacts govern the performance of industry-relevant battery cathodes
,”
J. Power Sources
387
,
49
56
(
2018
).
70.
See supplementary material at https://www.scitation.org/doi/suppl/10.1122/8.0000615 for detailed calculations and data manipulation. Polymer information includes SAXS and rheology measurements. CB information consists of DLS and rheology measurements. CB/PVDF mixture information includes the depletion force calculation and rheo-electric measurements. Additional information includes the polymer longest relaxation time calculation.

Supplementary Material