Binder migration across film consolidation and pore emptying in electrode drying: a continuous model validated against drying-rate, thickness, and schedule experiments

Nomenclature

• Binder concentration cb is measured per unit total film volume, and ˆc is its normalization by the initial value cb0.Source: Nomenclature · p. 2
• Transport notation distinguishes binder and liquid fluxes and velocities, effective diffusivity, and the Péclet number Pe = γH/D.Source: Nomenclature · pp. 2–3
• The symbols also name film thickness, liquid saturation, porosity, the accumulation-layer thickness δ, and the surface and interface concentrations.Source: Nomenclature · pp. 2–4

1. Introduction

• One parameter set must respond with the correct direction and magnitude to drying intensity, coating amount, and oven-zone scheduling to support drying-condition design.Source: §1 · p. 5
• Published data provide three validation cases: drying rates spanning 5.6-fold, coatings six times thicker, and interface-binder response to switch timing.The scheduling data also include a three-stage schedule reported to restore adhesion while shortening drying by 40%.Source: §1 · p. 5
• Earlier binder models stop convection at consolidation’s end, while pore-emptying models omit binder or assume its initial field, leaving no model to follow a schedule across both stages.Source: §1 · pp. 5–6
• The study solves one continuous binder-concentration equation across both stages using prescribed evaporative flux, then compares one parameter set against all three data sets.The flux is prescribed because the literature experiments’ air-side conditions are unpublished.Source: §1 · p. 6
• The study is limited to the electrode side; oven operating variables enter through prescribed evaporative flux, with oven-level coupling left for future work.Source: §1 · p. 7

2.1. Governing equations and the stage transition

• During consolidation, the saturated film shrinks as surface evaporation drives liquid and binder toward the receding surface.The base binder flux follows the superficial liquid flux, and the diffusivity uses a Bruggeman closure.Source: §2.1 · pp. 7–9
• Binder is transported by advection and diffusion in both stages, with its concentration defined per unit total film volume and its conserved inventory per unit area.The model retains the same concentration definition at the transition.Source: §2.1 · pp. 7–8
• At the receding surface, the no-outflow condition means binder cannot leave the film and can only redistribute.Source: §2.1 · p. 9
• During pore emptying, the film height is fixed and liquid flow and binder transport weaken as saturation falls; binder diffusivity also declines with liquid-network connectivity.This connectivity-based diffusivity prevents finite diffusion where the liquid no longer forms a connected path.Source: §2.1 · p. 9
• The transition passes its time, film height, and end-of-consolidation binder and temperature profiles into pore emptying without resetting the enriched binder field.Source: §2.1 · p. 10

2.2. Nondimensionalization

• Using ξ = x/H places the current collector and surface at ξ = 0 and ξ = 1 in both stages, and the scaled binder equation retains no-flux boundaries.Source: §2.2 · p. 10
• The scaled-coordinate binder velocity combines liquid-driven advection with apparent drift caused by the shrinking film coordinate.As H falls, a material point at fixed physical height moves toward larger ξ in the scaled coordinate.Source: §2.2 · p. 10
• The model’s surface enrichment depends on drying rate and thickness through Pe, which is the ratio of advective to diffusive transport rates.The paper defines enrichment as the logarithm of the surface-to-interface concentration ratio.Source: §2.2 · p. 11
• The predicted enrichment follows one curve over the measured range, with model evaluations spanning Pe = 0.19–6.5.At fixed material properties and microstructure, the paper reports that changing thickness fourfold while holding ˙mH fixed leaves enrichment identical to five decimal places.Source: §2.2 · pp. 11–12

2.3. Corrective physics terms

• The corrections address two base-model mismatches: depth enrichment profiles and binder transport stopping at the same saturation in every drying case.All five terms act on binder flux or mobility and leave the liquid, solid, and energy balances unchanged.Source: §2.3 · p. 13

2.3.1. Localized pumping profile

• A nearly depth-uniform enrichment rate supports plug-like advection above the pumping region, unlike the base model’s strongly depth-varying profile.For the 400 µm films, measured enrichment-rate ratios are 0.74 and 1.12, versus 3.0 and 3.8 in the base model.Source: §2.3.1 · p. 15

2.3.2. Carrier coupling

• Binder velocity is set to 0.58 times liquid velocity in both stages, reducing upward binder transport and raising the fast-dried 400 µm film’s interface concentration to 0.687.That value is at the lower edge of the measured 0.69–0.78 band.Source: §2.3.2 · pp. 15–16

2.3.3. Dual-mode shrinkage

• Dual-mode shrinkage concentrates consolidation in the lower film and surface packing layer while the middle moves as a block, preserving flat interior enrichment and restoring the measured surface level.The packing layer is an effective description, not an observed microstructure.Source: §2.3.3 · pp. 16, 18

2.3.4. Dual-porosity pumping-failure threshold

• The dual-porosity closure makes the carrier-failure threshold case-dependent: it is near S ≈1 for fast-dried thick films and S = 0.16–0.30 for the 70 µm series.Concentration-dependent surface viscosity drives the case dependence; the closure is not a first-principles result.Source: §2.3.4 · pp. 19–20

2.3.5. Binder immobilization function

• The immobilization function reduces binder mobility to zero at Sc = 0.65, with inferred fixation at S = 0.55–0.64; its arrest point is fitted to switch-time response.Removing this term leaves endpoint predictions essentially intact but removes the modeled switching-time response.Source: §2.3.5 · pp. 20–21

2.3.6. Two arrest mechanisms

• The arrest mechanisms are complementary: immobilization stops binder transport first in five measured conditions, while carrier failure acts first only in the fast-dried 450 µm film.The five immobilization-limited cases include the 70 µm series and the slowly dried 430 µm film; the fast 450 µm case stops near S ≈1.Source: §2.3.6 · pp. 21–22

2.4. Observation operators

• The observation corrections are applied only when comparing model output with measurements; they do not change the simulated binder field.Calendering is represented as a measurement operator, not as part of the transport physics.Source: §2.4 · p. 23
• After binder transport stops, remaining solvent evaporates without further binder motion, so the model’s dissolved-binder field is compared with the measured solid distribution.The model treats the binder as precipitating in place.Source: §2.4 · pp. 22–23
• Averaging the simulated profile over the calendered surface layer corrects surface-dependent comparisons; without it, the drying-rate ratio comparison shifts from 1.01 to 1.51.The averaging depth is set to half the compression depth; interior-profile RMS difference remains nearly unchanged when the operator is switched off and D0 is refitted.Source: §2.4 · pp. 23–24
• EDS surface and interface readings come from opposite faces, and the method changes their relative enrichment: EDS/XPS ratios are 0.75–0.89, while rate sensitivities remain 0.20–0.25.The methods’ different information depths prevent comparing their absolute concentrations; the series is therefore compared by rate sensitivity, not intercept.Source: §2.4 · p. 24

2.5. Numerics

• 500 uniform ξ cells preserve binder inventory to machine precision in the conservative finite-volume scheme.Halving or doubling the cell count changes every Table 3 model value by less than 0.3% and the response midpoint by less than 0.1 s.Source: §2.5 · p. 24
• The COMSOL and screening implementations agree to relative parity of 2 × 10^-9, and the end-to-end chain reproduces screening-solver comparison values within ±3%.Source: §2.5 · pp. 24–25
• Changing the grid from the adopted 500 cells to half or twice that count changes the response midpoint by less than 0.1 s.Source: §2.5 · p. 24

3.1. Datasets

• The four experiments use public conditions on graphite/PVDF/NMP electrodes and vary drying-oven operating variables one at a time.Source: §3.1 · p. 25
• The A70 series measures top- and bottom-face binder in ~70 µm films at four rates; A400 measures depth-block profiles in 430–450 µm films at 1.0 and 2.8 g m−2 s−1.The A400 films are six times thicker, and both series were calendered before measurement.Source: §3.1 · p. 25
• The switch-time experiment lowers drying rate from 1.19 to 0.52 g m−2 s−1 and measures final interface binder and peel strength across switch times.The reported adhesion transition times are 28 s and 44 s.Source: §3.1 · p. 25
• The experiments are driven by prescribed evaporative flux because air-side conditions are unpublished; that flux also connects the electrode model to the process model.Source: §3.1 · p. 27

3.2. Calibration summary

• Dried-film measurements and the switch-time midpoint set fitted parameters, but the schedule-experiment drying times and three-stage outcome are predicted without fitting.The three-stage experiment is not used in any calibration.Source: §3.2 · pp. 27–28
• D0 is an effective diffusivity, not a PVDF-in-NMP material constant; α belongs to the observation operator and changes no simulated field.Source: §3.2 · p. 27
• The immobilization parameters Sc and q are not individually determined; the switch-time experiment constrains the saturation at which binder motion ends.Table 2 lists the measurement setting each model degree of freedom.Source: §3.2 · p. 27, Table 2 · p. 28

4.1. Final binder distribution across drying rate and film thickness

• 1.44 was the model’s fast-film enrichment-rate ratio, within the measured 1.12–1.49; the model also matched the thick-film surface enrichments and interface concentrations closely.For films about six times thicker, the model’s thickness exponent was 0.317 versus 0.314 measured, and its rate-sensitivity ratio was 1.76 versus 1.75 measured; the slow-film enrichment-rate ratio was not reproduced (1.20 versus 0.74).Source: §4.1 · pp. 31, 33, 34, Figure 6 · p. 32

4.2. Interface binder against the time of the drying-rate switch

• 35.6 s was the model’s switch-response midpoint, close to the measured 35.9 s, while its response window was narrower than measured.The measured interface level fell from 1 to 0 over 28–46 s, versus 31–39 s in the model; without the immobilization function, the midpoint shifted to 59.1 s. During pore emptying, slower drying allows diffusion to return binder toward the interface before it stops moving.Source: §4.2 · pp. 34, 36, Figure 7 · p. 35

4.3. The arrest point

• The calibrated immobilization function stops binder motion at saturation Sc = 0.65, consistent with the inferred fixation range of 0.55–0.64, though the agreement does not prove the mechanism.The switch-time response constrains one effective saturation because the arrest point and immobilization exponent can substitute for one another; the model halves binder mobility near S = 0.8 and stops it by about S = 0.55–0.65.Source: §4.3 · p. 37

4.4. Removal of each arrest mechanism

• Removing the immobilization function shifts the switch-response midpoint from 35.6 s to 59.1 s, while removing the pumping-failure threshold leaves that response unchanged but moves thick-film endpoints outside measured ranges.The threshold governs arrest near pore-emptying onset in the fast-dried thick film, whereas immobilization governs the switch-time response at lower saturation; dried-film endpoints alone cannot determine when binder stops.Source: §4.4 · pp. 38–39

4.5. The three-stage experiment

• Without fitting to the three-stage experiment, the model reproduced drying times across all four scenarios and full interface recovery after an early rate reduction, but predicted the interface drop after a later switch too late.The schedule applies a high rate before binder transport creates a damaging gradient and after the binder is fixed, with a low-rate interval between; the model uses the prescribed evaporative flux.Source: §4.5 · p. 39
• 1.00: early-switch scenarios S1 and S2 recover the interface fully in the model, versus measured levels of 1.04 and 0.92.For later switches, S3 and S4 give model levels of 0.97 and 0.85 versus measured levels of 0.67 and 0.74; the model's drop comes later, while trial scatter is comparable to the discrepancy.Source: Figure 8 · p. 40, §4.5 · p. 41
• About 40%: scenario S2 shortens drying versus the slow-throughout film in both model and source, whose schedule drying times agree within 2 s each.The slow-throughout film dries in 157 s in the model versus 162 s in the source; the four scenarios take 100.7, 93.3, 88.9, and 85.0 s versus 101.6, 94.8, 90.7, and 86.8 s.Source: §4.5 · pp. 40–41
• The return to the high rate at ϕt2 = 0.45 does not change the interface because saturation is 0.57, below Sc = 0.65, where binder mobility is zero.Each three-stage scenario therefore matches a single high-to-low switch at the same time to within 10^-4 in normalized interface level.Source: §4.5 · p. 41

5.1. Placement of the low-rate stage

• The interface responds to drying rate only between the end of consolidation and binder arrest, so returning to a high rate after arrest does not change it.At ϕt1 = 0.60, the interface remains above 0.98 when the high-rate return moves to about ϕt2 = 0.50.Source: §5.1 · pp. 42–44
• During consolidation, the shrinking film’s binder profile rises while its inventory remains conserved; during pore emptying, low-rate stages let diffusion return surface enrichment toward the interface.The consolidation profile is passed unchanged into pore emptying; constant high-rate drying nearly preserves the inherited enrichment.Source: §5.1 · p. 42
• The map’s interface level of at least 0.9 is bounded mainly by the first switch near ϕt1 ≈ 0.57 and depends little on the second switch below ϕt2 ≈ 0.50.Because the modeled response window is narrower and later than measured, the first switch should be earlier than the mapped edge, around the end of consolidation.Source: §5.1 · pp. 43–44
• 81 s is the shortest mapped schedule retaining an interface level of at least 0.9, versus 157 s for slow-rate drying throughout.The map overlays interface level and total drying time across 710 schedules.Source: §5.1 · pp. 43–44, Figure 10 · p. 43

5.2. Limitations and future work

• The model’s switch-response window is about half as wide as measured, and its fast-dried film loses less interface binder than measured, making its process-design predictions non-conservative.The present data cannot distinguish proposed causes; depth-resolved measurements during drying are needed.Source: §5.2 · p. 44
• The framework could connect to coater-scale oven settings for joint energy and interface-quality evaluation; dissolved CMC is a candidate extension, while SBR latex needs separate particle-transport treatment.Pore-scale models are candidates to replace the corrective terms with resolved physics.Source: §5.2 · pp. 44–45

6. Conclusions

• With one parameter set, the continuous simulation reproduces drying-rate, thickness, depth-profile, switch-time, and three-stage schedule measurements.The three-stage drying times are within 2 s of the source calculations, and early rate reduction gives interface level 1.00 versus measured 0.92–1.04.Source: §6 · p. 45
• The calibrated arrest point S = 0.65 lies just above the independently inferred binder-fixation range of 0.55–0.64.The switch-time experiment determines arrest timing; dried-film measurements alone cannot.Source: §6 · pp. 45–46
• The model can serve as the electrode-side component for NMP-based schedule design and, with a coater-scale process model, evaluate oven energy and interface quality together.Extending the approach to water-based anodes is identified as a next step.Source: §6 · p. 46
• The model’s switch-time response window is about half as wide as measured, and it underestimates interface-binder loss for films dried fast throughout.The authors state that identifying the causes requires depth-resolved measurements during drying.Source: §6 · p. 46

• Younghwan Joo is credited with conceptualization, methodology, software, validation, formal analysis, investigation, data curation, writing, and visualization.Source: CRediT authorship contribution statement · p. 46
• Sanghyun Oh is credited with resources and writing–review and editing.Source: CRediT authorship contribution statement · p. 46
• Sung-il Kim is credited with supervision, project administration, funding acquisition, and writing–review and editing.Source: CRediT authorship contribution statement · p. 46