A Review of Dispersion Film Drying Research

Wet film drying is crucially relevant in coating applications including painting [1] and fabrication of electrodes [2], solar cells [3], and luminescent organic films [4]. Often, the wet film is composed of a volatile liquid and one or two types of particles or polymers. The drying processes involve complicated underlying phenomena ranging from particle dynamics to heat and mass transfer. Key physical concepts in the drying process, including internal capillary flow, coffee-ring formation, stratification, and crack formation, were discussed in a recent review article [5]. Other researchers have produced detailed summaries of work on coffee rings or capillary flow during droplet drying and stratification of multi-particle dispersions [6–10]. In addition to reviewing the literature on the modeling of dilute dispersions, the present review puts significant emphasis on concentrated dispersions, which are common in industrial applications. Furthermore, most engineering systems contain multiple components; in particular, we review how polymer components are involved in dispersion drying processes. We also construct a composite drying map that summarizes the physical phenomena that are dominant in different processing regimes.

In this review, we have addressed three important aspects of drying processes: First, we divide drying processes into two periods with distinct heat and mass transfer physics and as described in Secs. 2–4 examine first-stage drying and second-stage drying, respectively. Second, drying dynamics are different for dilute and dense dispersions, due to distinct interactions among sparse and compact components (particles, polymers, and ionomers) [11–13]. Third, the wet films may contain one or more kinds of particles with different chemical compositions, shapes, and sizes, which lead to different drying behaviors; we first discuss single-component dispersions and later address multi-component dispersions.

Finally, we have also highlighted the fabrication of electrochemical device electrodes in this review. Fabrication processes for Li-ion battery electrodes and fuel-cell catalyst layers [14,15] represent important examples of drying processes of dense dispersions. The Li-ion battery electrode precursor is a mixture of solvent/suspension fluid, microparticles, nanoparticles, and polymers while the fuel-cell ink includes nanoparticles and ionomer. Previous studies have demonstrated that the drying process has a profound effect on the final electrode properties [16,17]. This review concludes with a summary and comments on issues that need to be resolved with future work.

Figure 1 shows the process of film drying, which is typically divided into two conceptual stages, which we will call the first and second drying stages. In the first stage, the suspension fluid (which we will call “solvent”) evaporation leads to a loss in volume, and suspended particles are drawn together by capillary forces. Relative movement among different kinds of particles as well as an evaporation-driven solvent flow toward the surface of the coating changes the spatial distribution of the particles. This stage might also be called the film shrinkage stage. In the second stage, further solvent evaporation drives the particles to form a porous network, and the solvent interface recedes into the porous structure. The film thickness changes only slightly during the second drying stage [10].

Many mechanisms are at play during dispersion drying. During the first stage, the distribution of particles is mainly influenced by three basic factors: solvent evaporation, particle diffusion, and particle sedimentation. Initially, the particles are dispersed and suspended in the solvents in which they undergo Brownian motion. With the solvent molecules escaping from the liquid phase to the atmosphere, the air–liquid interface recedes and particles under the interface accumulate and become concentrated. However, the diffusion of the particles could alleviate the concentration gradient. Additionally, if the particles are denser than the solvent, the particles will settle toward the bottom of the film. Furthermore, particles may interact with each other. The quasi-stable state of colloids is broken by drying indicating particles' attraction or repulsion contributes to the diffusion process and the final film microstructure. Furthermore, the addition of polymer can induce other interactions like steric repulsion and depletion forces which further complicate particle motion, and hydrodynamic interactions can be significant. The influence of other experimental parameters like evaporation rate, volume fraction, and temperature gradient will be discussed in the following sections.

The methods used to simulate the drying of dispersions can be divided into two categories: continuum modeling and component-level models. Continuum models treat the entire dispersion as a continuum and use differential equations to describe drying physics, while component-level models mainly focus on individual particle dynamics and interparticle interactions, such as electrostatic interaction and Van der Waals forces. Continuum modeling and component-level modeling complement each other as they are used to investigate phenomena occurring on different scales.

The references we examined in this review paper included a variety of component-level simulation methods, which we now briefly describe. These are summarized in Table 1.

Monte Carlo methods are a broad class of computational methods based on random sampling that can be used to simulate the statistical behavior of a large number of particles. Molecular dynamics, Langevin dynamics, Brownian dynamics, and Stokesian dynamics solve Newton’s equations of motion numerically and aim to simulate the movement molecules/particles within a period of time. We will briefly describe each of these terms as used in this article. In Molecular dynamics, the interparticle interactions are typically represented with a simplified pair potential called the Lennard–Jones potential, which is only applicable in vacuum. Therefore, molecular dynamics uses the dielectric constant of vacuum for calculating the electrostatic and van der Waals interactions without considering the dielectric properties of the environment. In addition, molecular dynamics also has limits in considering the underlying molecular force field, such as molecule collision and temperature effects, as it assumes a vacuum media. To describe realistic molecular systems, Langevin dynamics builds on Molecular dynamics by considering perturbations caused by additional phenomena, such as solvent-molecule friction and high-velocity collisions between molecules. Brownian dynamics (BD) is a simplified version of Langevin dynamics that neglects inertial effects. Both Langevin dynamics and Brownian dynamics use highly simplified hydrodynamic interactions between fluid and particles. Stokesian dynamics is an extension of Langevin dynamics that considers the hydrodynamic interactions among solvents and particles. The strengths and weaknesses of these simulation methods are summarized in Table 1.

Additionally, some mesoscale models have included both particle interactions and continuous physical quantities like volume fraction and viscosities. Liu et al. [14] presented a two-dimensional coarse-grained lattice-gas model for Li-ion battery electrode drying. In this model, the spatial domain is divided into numerous 1 nm lattice cells, with only one phase (solvent, active material, binder, etc.) assigned to each lattice cell. The subsequent system evolution is determined using a modified Kinetic Monte Carlo algorithm. This model was also extended to investigate the slurry mixing sequence [40]. Forouzan et al. [41] built a mesoscale particle dynamic simulation with carbon and binder particles lumped together to be computationally feasible. The required interaction parameters of potential functions were regressed to experimental properties. Nikpour et al. [42] developed a multi-phase smoothed particle model by integrating smoothed particle hydrodynamics (SPH) in a discrete element method. Macroscopic properties of density, viscosity, and Young’s modulus were used to parameterize the coarse-grained mesh-free model. More mesoscale models are needed to investigate the connection between macroscopic properties like viscosity, packing fraction with the microscale particle, or solvent interactions.

Besides the simulation methods mentioned earlier, emerging machine learning methods are starting to reveal more details that are difficult to obtain through experiment and traditional simulation methods. For example, Howard et al. [23] used a recently developed machine learning method named neighborhood graph analysis (NGA) to characterize the local structure of the growing crystal. Their results showed that the machine learning method could show unprecedented microscopic detail of the crystal such as the defects and particles at the interface, which are challenging for traditional simulation methods. Although there is not a lot of work on the application of machine learning in studying dispersion drying dynamics now, we believe machine learning is very promising in such applications and will provide complementary information experiments and traditional simulation methods.

Drying map: In the earliest study referenced in the drying map shown in Fig. 2(b), Routh et al. [43] modeled a single-component (monodisperse particles in liquid) evaporating film using a 1D convection-diffusion equation. A direct numerical solution, a standard asymptotic analysis, and a patched asymptotic analysis were performed. They defined an evaporation Péclet number, Pe, as H₀E/D where D is the Stokes–Einstein diffusion coefficient, H₀ is the initial film thickness, and E is the evaporation rate, expressed as the downward speed of the evaporation surface and usually obtained from the thermal gravimetric analysis. For large Pe, a sharp spatial discontinuity in volume fraction was observed. One important inference from the model was that the spatial gradient in volume fraction scales with Pe^0.5.

Cardinal et al. defined the sedimentation Péclet number Pe_sed as U₀H₀/D, or, equivalently, N_s = Pe_sed/Pe. Analytical approximations of the dependence of the critical dimensionless number on the initial volume fraction in limiting cases were determined. Additionally, lower initial concentration and film height were found to enlarge the diffusion-dominated region of the drying regime map. Finally, a bimodal mixture of silica particles with 200 nm and 1-µm diameters was dried and cryo-SEM experiments were used to validate the regime map. For the silica mixture, Cardinal et al. observed a stratified film where small particles accumulated at the top interface and both large and small particles accumulated in the bottom layer, as shown in Fig. 2(c).

Consolidation layer: In the latter period of the first drying stage, the concentration under the air–liquid interface approaches the random maximum packing fraction ϕ_m = 0.64. The diffusion equation in the previous regime map model [44] becomes invalid due to the divergence of the compressibility factor. The suspended particulate solids gel into a porous network and form the consolidation layer. In the porous consolidation layer, one might describe fluid transport with Darcy’s law and stress with poroelasticity theory. In the solution phase under the consolidation layer, one can continue to describe the particle motion with the previous diffusion equations [44]. Holl et al. [45] investigated polymer colloid film drying by three distinct drying modes: homogeneous drying, drying normal to the surface (skin layer growth), and lateral drying. In “homogeneous drying,” corresponding to small Pe, the concentration remains uniform across the film while drying. “Drying normal to the surface,” corresponding to large Pe, involves the formation of a consolidation layer just under the air–liquid interface. “Lateral drying” describes lateral propagation of a drying front. Holl et al. [45] emphasized lateral drying and combined lateral and normal drying modes. The drying rate was found proportional to the inverse of the square root of time for normal drying.

Style et al. [46] modeled the transition process from suspension to porous medium, crust, or gel formation. Both in solution and gel phase, the advection-diffusion equations were derived from permeability and pressure. The stress in the gel phase was discussed. While the boundary condition and different diffusion coefficients in advection-diffusion equations of sol and gel phase require further attention, the model provided some insight into how stress constraints influence diffusion in the gel phase.

Roy et al. [47] account for the drying process at the particle level in terms of both elastic strain in particles and plastic strain due to particle re-arrangement. In case of unstable or flocculated dispersions, particle networks form below the random close packing concentration. The model integrated interparticle forces, particle and contact deformation, and plastic events such as rolling/sliding during the deformation process. The main finding was that the dimensionless yield stress only depends on particle packing characteristics (coordination number and volume fraction) and the particle contact parameters. They concluded that during consolidation, only pure elastic deformation happens below a critical strain, above which both elastic and plastic deformation occur in the particle network.

Polymer solution drying: Using a mutual diffusion coefficient derived from free volume theory [48] and solvent vapor pressure and evaporation rate derived from Flory–Huggins theory, Arya [49] presented a heat and mass transfer drying model for binary polymer solutions. Parameters of a binary polymer–solvent system and pure substance properties were determined from experiments. Using confocal Raman spectroscopy, the concentration profile of a polymer thin film was measured from Raman spectra peaks. A model predicting concentration profile with free volume parameters to be fit was built based on experimental results at a low evaporation rate. For a highly volatile solvent, the model prediction was in good agreement with the experiment, but the model deviated from experiments with less volatile solvents.

In the majority of the continuum drying model, the drying region with a moving boundary representing the drying surface is described in shrinking dimensionless coordinates. Gromer et al. [50] instead performed a simulation in stationary lab coordinates. Their simulation considered the particle distribution, deformation, and different drying stages including dispersion, colloidal solid, and wet and dry gel. They concluded that repulsive force between polymers and capillary deformation favors more homogeneous distributions, while a receding air–liquid front leads toward heterogeneity. Polymer distributions varying in time under different interaction potentials and drying conditions were simulated.

When the evaporation rate is much higher than the diffusion rate, corresponding to Pe > 1, the particles will accumulate around the evaporating interface and form a “skin” layer. When the evaporation rate is much smaller than the diffusion rate, that is Pe < 1, the diffusion is more dominant and the particles tend to distribute uniformly during the drying process. The particle distribution in the direction normal to the substrate is influenced by different drying parameters and interparticle forces such as electrostatic forces. We now discuss the influences of these factors on particle distribution.

Evaporation rate: Evaporation rate directly influences the value of Pe, which describes the dominant influences on particle motion. The influences of evaporation rate on particle distribution and drying thin-film microstructure were investigated through simulation [18,19,22]. Rabani et al. [19] used a two-dimensional lattice-gas model to study the influence of evaporation uniformity on the thin-film structure and found that under uniform evaporation rate, the microstructure of nanoparticle assembly would consist of disk-like or ribbon-like domains that continue growing as long as the aggregates can flow, while under a nonuniform evaporation process, network structures would form. For uniform evaporation rate, the rate of evaporation would influence the particle distribution and drying film microstructure. For a polymer colloid dispersion, it was shown that at low evaporation rates, the drying film would form hexagonal and tetragonal structures [22]. It was also demonstrated via simulation [18] that for nanoparticle solutions when dimensionless evaporation rate or Pe is sufficiently high, nanoparticles will accumulate at the interface and form a densely packed skin layer as shown in Fig. 3(a), while at a low evaporation rate or Pe, diffusion is dominant and particles will agglomerate, resulting in a highly porous structure as shown in Fig. 3(b). Figure 3(c) shows the final microstructure of the drying thin film under different values of Pe.

Particle interaction: Routh et al. [43] did not consider particle interactions in their initial model. As the colloidal forces are important in determining particle distribution, Cheng and Grest [24] produced a drying model that included repulsive particle interactions. They used a large-scale molecular dynamics simulation considering purely repulsive particle interactions and found an optimal evaporation rate at which the crystal quality of the assembled nanoparticle layer had the fewest defects and grain boundaries. The evaporation rate needed to be sufficiently high to initiate nanoparticle assembly and crystallization under the liquid–air evaporating interface. However, at excessively high evaporation rates, the nanoparticles assembled into the layer without sufficient time to diffuse and rearrange into ordered grains. This led to the growth of grains in different orientations, and grain boundaries and point defects would form when grains in different orientations grew into contact with each other.

Evaporating interface movement: Wang and Brady [37] investigated the microstructure formation of the drying film under different interface conditions. They used BD based on the modified energy minimization potential-free algorithm (EMPF) to study the suspension drying process. Different interface drying dynamics were considered: a constant interface velocity or a constant external normal stress on the evaporating interface. The interface movement was shown to influence the microstructure and mechanics of the drying thin film. For the case of constant interface velocity, the microstructure was homogeneous at small Pe and heterogeneous at large Pe, and moderate Pe induced the most amorphous structure. When the evaporating interface moved under a constant external normal stress, the interface movement was hindered by pressure build-up in the suspension, causing the interface to stop moving and resulting in weaker influence on the microstructure of the drying thin film. They concluded that the structural and mechanical properties of the drying film can be controlled by the evaporating interface movement.

Hydrodynamic interactions: Hydrodynamic interactions between colloids are usually neglected to reduce computational cost [30,43]; only a few articles included the hydrodynamic interactions in their models [23,24]. To fully investigate the effects and importance of hydrodynamic interactions on the three-dimensional crystal structure of the dried film, Howard et al. [23] combined massive-scale non-equilibrium molecular dynamics simulation with machine learning to characterize local structure using both implicit and explicit solvent models. The explicit solvent model included solvent molecules and considered hydrodynamic interactions, and the implicit solvent model used a continuum representation and did not consider hydrodynamic interactions. NGA, based on machine learning, was used to characterize the crystalline character of the films. By comparing the simulation results of implicit and explicit models, they found that though the two models gave a similar equilibrium crystal structure, the crystallization dynamics were different. The accumulation of particles at the liquid–air interface and beginning of crystallization was faster for the explicit solvent model than for the implicit solvent model. In addition, the crystal only grew downward from the liquid–air interface in the explicit solvent model, while the crystal also grew upward from the substrate in the implicit solvent model. Furthermore, the explicit solvent model could capture the effects of temperature gradients and evaporative cooling induced by the hydrodynamic interactions during drying. They concluded that to observe crystallization or drying dynamics, hydrodynamic interactions should be considered, but that if one only needs to investigate the microstructure of the assembled nanoparticle layer at equilibrium and needs to simulate larger length and time scales, one should neglect hydrodynamic interactions to simplify the model and save computation time.

Heat transfer: Heating will influence evaporation rate and defect formation in drying coatings. Common ways for heating in industry are forced hot-air convection and infrared radiation. Avco et al. [52] presented a simple heat and mass transfer model for the drying process of thin films. They assumed that the vapor pressure of solvent remains at the quasi-saturated value at a given temperature during first-stage drying. They calculated a mass transfer coefficient from a heat transfer coefficient and Lewis number based on a heat and mass transfer analogy. They argued that the effective surface area for evaporation decreases with increasing concentration during drying, as an explanation for the evaporation rate falling at the end of the drying process. Evaporation rate and film temperature under different air temperatures and velocities were predicted and shown to agree quantitatively with experiments. From the same research group, Turkan et al. constructed a more complete model for first-stage drying based on wet bulb theory and for second-stage drying with a falling evaporation rate [53]. They used this model to optimize the heating system.

Price et al. [54] optimized operating conditions such as heat transfer coefficients to achieve a balance between minimizing the residual solvent and avoiding defects or blistering for a drying polymer solution. Solvent concentration, substrate, and coating film temperature distributions were solved. Mutual diffusivity was determined based on free volume theory. The authors automated the optimization of the heat transfer coefficients under the constraint that the film temperature remains below the boiling point temperature and then determined parameter sensitivities. The authors concluded that the optimal coating-side heat transfer coefficients should not be lower than the optimal substrate-side heat transfer coefficients.

At this point, we will shift our focus to the drying of multi-component mixtures (containing two or more types of particles/polymers). First, we will discuss the physics of drying mixtures containing two types of particles, focusing on the mechanisms behind stratification phenomena. We will summarize the influences of different parameters on stratification in the form of a drying map. Second, we will include the soft matter in our system and highlight what additional effects polymers bring in. Additionally, we will specifically highlight recent research on battery slurry drying as a complex technical example of multi-component mixture drying.

In a dispersion containing more than one type of particle, the average motions of different types of particles during a drying process might differ due to differences in properties such as size. The resulting stratification is sometimes called “autostratification.” Fortini et al. [36] observed an “inverted” stratification structure for a dispersion film containing particles of two sizes, with smaller particles accumulating at the top and bigger particles accumulating at the bottom, as shown in Fig. 4. The stratification may be influenced by a wide variety of parameters, including evaporation rate, volume fraction, temperature gradient, pH, and strength of hydrodynamic interaction. The effects of these parameters on autostratification were not discussed in the work of Fortini et al. [36], but subsequent work has improved the understanding of these influences, as we discuss next.

Evaporation rate: Fortini et al. [36] did not investigate how evaporation rate affects stratification in the regime where Péclet numbers for both small and large particles are much larger than 1. Inverted stratification has been studied using both molecular dynamics [27] and Langevin dynamics models [33,35] that considered the effects of different evaporation rates on stratification.

Howard et al. [27] used implicit solvent molecular dynamics simulations to systematically study inverted stratification dynamics at different drying rates and particle size ratios for bimodal particle dispersions. They found that small-on-top inverted stratification arrangements appeared at both moderate (the Pe for small particle Pe_s ≪ 1, the Pe for large particle $Pel∼1$⁠) and large (Pe_s ≫ 1, Pe_l ≫ 1) film Pe. When evaporation rate or Pe was decreased, the extent of inverted stratification also decreased. In addition, larger size ratios resulted in thicker stratified layers. Tatsumi et al. [35] and Cusola et al. [33] obtained similar results using Langevin dynamics simulations. Tatsumi et al. [35] explained that the inverted stratification was caused by larger particles being pushed away from the evaporating interface by local accumulation of smaller particles. Larger values of Pe tended to enhance particle segregation, and both Tatsumi et al. [35] and Cusola et al. [33] found that there was an optimal Pe value which maximized segregation.

In earlier work, for the regime in which Péclet numbers are around 1, Trueman et al. [55] developed a two-component drying film model for hard spheres from entropy-determined chemical potential, the Gibbs-Duhem equation, and principles of classical diffusion mechanics. The larger particles have a lower diffusivity and hence a higher Péclet number, and so tend to accumulate at the top surface. These authors found that autostratification occurred with Pe < 1 for smaller particles in conjunction with Pe > 1 for larger particles. Stratification caused by this diffusive mechanism corresponds to the “Evaporation-large-on-top” region in Fig. 7. In addition, Trueman et al. found that a large particle size ratio tends to increase stratification, but that an extreme size difference allowed small particles to migrate through the voids between large particles, mitigating stratification.

Particle interactions: Atmuri et al. [56] investigated the vertical drying/segregation of mixtures of particles differing in size or charge. Building on their previous diffusive model, chemical potentials were used to provide a simple representation of particle interactions. They found in the drying of binary colloidal mixtures that, all else being equal, evaporation tends to cause all particles to accumulate near the surface, as self-attractive particles tend to accumulate without difficulty at the top while self-repulsive particles tend to resist this accumulation. This effect could reverse the stratification that would otherwise be induced only by diffusivity differences resulting from relative particle sizes. When charged particles were mixed with neutral particles of the same size, they found that segregation occurred because the charged particles repelled each other during drying, pushing them away from the air interface, while neutral particles do not experience this effect. Moreover, increasing the Péclet number decreases segregation since the time for interactions to take effect is reduced.

Osmotic pressure: Fortini et al. [36] observed small-on-top inverted stratification through both experiments and Langevin dynamics simulations. They developed a physical model and explained that the phenomenon was due to the gradients of density and osmotic pressure. When Pe ≫ 1, the moving evaporating interface would generate a density gradient of particles which induced a negative pressure gradient. The osmotic pressure different between the top and bottom of a particle was different for particles with different sizes and pushed larger particles downward faster than smaller particles; larger particles had larger diffusiophoretic velocities and moved downward more quickly than smaller particles, inducing small-on-top stratification. Fortini and Sear [31] later demonstrated that the small-on-top stratification can also occur in ternary or polydisperse suspensions.

Hydrodynamic interactions: To determine if hydrodynamic interactions affect stratification, Tang et al. [25] used large-scale molecular dynamics models with implicit solvents and with explicit solvents to study binary nanoparticle suspensions during drying processes. The explicit solvent model treated solvent molecules explicitly and considered hydrodynamic interactions, and the implicit solvent model treated solvent as a continuum medium and neglected hydrodynamic interactions. They observed inverted stratification in both explicit and implicit solvent models and concluded that the implicit solvent model was sufficient to simulate the drying of colloidal suspensions. However, Tang et al. [26] subsequently found that the drift velocity of large particles was overestimated by the implicit solvent model due to the neglect of the back flow of solvent, suggesting that the inverted stratification should be less prominent than predicted by this model.

Sear and Warren [58] discussed the diffusiophoretic drift of large colloidal particles in response to a gradient in polymer molecules as an alternative explanation for stratification. From the Asakura–Oosawa model for depletion forces, the increase in surface tension due to the presence of the polymer was found to force colloidal particles towards regions of lower polymer concentration. They argued that the neglect of hydrodynamic interactions in other models [36,57] resulted in unbalanced osmotic forces. After the solvent backflow is taken into account, the correct drift velocity is independent of particle size instead of being proportional to the square of radius (and accordingly, much weaker).

Temperature gradient: Most models for suspension drying employ a constant temperature background rather than considering the temperature gradient along the direction normal to the substrate that one might expect from evaporative cooling. To investigate the influences of temperature gradient on stratification for a suspension containing multiple types of particles, Tang et al. used explicit solvent molecular dynamics to simulate the colloid drying process, and considered the effects of both back flow and evaporative cooling [26,28]. They observed inverted stratification due to diffusiophoresis when the product of Péclet number and volume fraction of small particles was larger than a certain threshold. However, for ultrafast evaporation, the temperature gradient caused by evaporative cooling was found to induce thermophoresis effects that would drive larger particles more quickly to regions with cooler solvent, which would tend to promote a large-on-top structure. They then proposed that temperature gradient might be used to control the stratification structure [28]. They found that a negative temperature gradient (with the temperature of the interface being lower than that at the bottom of the film) induced by evaporative cooling would facilitate large-on-top stratification, while positive temperature gradient would result in a stronger small-on-top stratification.

Second, the diffusiophoretic velocity U of large particles must be higher than v_jam to push large particles away from top jammed layer. If the initial volume fraction ϕ_0,s is too high, the jammed layer may grow too fast, with v_jam exceeding the diffusiophoretic velocity U, slowing the large particles in the small particle jammed layer and preventing stratification. The condition obtained for non-stratification due to quick jamming layer growth is ϕ_0,s > ϕ_jam − 4/9. This scenario is illustrated in Fig. 6(b) and denoted as the jammed region in Fig. 7. For the randomly-packed spherical particles, the jamming fraction ϕ_jam is taken as 0.64. Then the upper limit of small particle volume fraction ϕ_s for small-on-top stratification is predicted to be 0.2. Note that in this model, the diffusion and interactions of large particles were neglected and the size ratio is very large.

Sear suggested that crystallization should be considered in subsequent work. Wang et al. [37] investigated crystallization using Brownian dynamics for monodisperse colloids. The model was built on the assumption that Péclet numbers and initial volume fractions of small particles are sufficiently high. When Pe ≪ 1, crystallization happens homogeneously when the volume fraction was above the equilibrium boundary near the end of drying. When Pe ≫ 1, local crystallization occurs near the drying front even during the early drying period.

The experimental data in the literature are plotted for comparison with both the Sear and Sear-Warren model in Fig. 6(a) [9]. In general, both models have underestimated stratification. We note that few experiments in the literature satisfy the very high Péclet number and size ratio these models assume. For battery slurry drying, in particular, the volume fraction of small particles like carbon black and binder is relatively low, so these models may not apply for the delamination of inactive materials.

Sedimentation: When the densities of the suspended particles are higher than that of the surrounding liquid, then similar to the single-component dispersion drying, sedimentation modeling must be considered. Sui [60] modeled sedimentation-driven stratification in mixtures containing platelet and spherical particles that were denser than the surrounding liquid. The sedimentation–diffusion equations were derived using the Onsager variational principle. For liquid crystals, a nematic phase refers to a state in which the platelets self-align to have long-range directional order with their long axes roughly parallel. The nematic phase configuration of platelets and stratification structures were investigated and correlations between the two were discussed. Sui is used α to represent the size ratio of the spherical particle radius and the platelet discotic plane radius. Sphere-on-top stratification accompanied by a nematic-bottom phase was observed for size ratios of α < 0.3 and the platelet-on-top stratification accompanied by a floating nematic phase was observed for size ratios of α ≥ 0.4. State diagrams for stratification and phase configuration were plotted.

Based on the literature discussed in the previous section, we constructed a drying regime map for multi-component colloids, shown in Fig. 7. In the context of multi-component suspension, we used the subscript s to denote quantities associated with small particles and l for those of large particles. For a mixture of small and large particles, we plotted how the evaporation rate E and the initial volume fractions of small and large particles ϕ_0,s, ϕ_0,l are expected to affect the final dried film structure. We assumed that the initial volume fraction of large particles ϕ_0,l is comparable with that of small particles ϕ_0,s, allowing both to be represented by the vertical axis on the map. The map regions are divided as follows:

• When both the evaporation rate and the initial concentration are low, both sizes of particles have enough time to diffuse, relaxing the concentration gradients generated during drying and resulting in a final dried film that is homogeneous.

• At low evaporation rates, the small particles remain uniformly distributed during drying due to their large diffusivity and stratification regimes simply reflect the distribution of large particles. If the initial concentration of large particles is large, the settling of the large particles dominates. From Cardinal’s calculation [44], the critical sedimentation Péclet number, which defines the boundary between the sedimentation-dominated and diffusion-dominated regimes, is Pe_sed,cr,l = 9/ϕ_0,l. In the sedimentation regime, the large particles tend to accumulate at the bottom and the film ultimately forms a small-on-top structure.

• As the evaporation rate increases, particles concentrate under the air–liquid interface if they do not have sufficient time to diffuse away from the receding surface. Large particles are more likely to stay at the top as they diffuse more slowly. If the size ratio of large and small particles is chosen carefully, the film can stratify, as discussed by Trueman et al. [55]. We take the critical Péclet number of large particles, Pe_cr,l depicted in Fig. 9 of [47], as the boundary between evaporation-dominated and diffusion-dominated regions in Fig. 7.

• The boundary between sedimentation and evaporation-dominated region is described by the sedimentation number of large particles N_sed,l defined in Eq. (1). Cardinal et al. [44] also derived the corresponding condition as N_sed,cr,l = 1/(1 − ϕ_0,l)^6.55.

• Recently, Fortini et al. [36] reported inverted stratification (small-on-top) at a very high evaporation rate. They incorporated colloidal osmotic pressure to model this phenomenon. Different formulations and mechanisms were proposed, but generally, the condition for small-on-top stratification is that the product of the Péclet number and initial volume fraction of small particles is larger than a constant [9,57,58]. That constant summarizes other factors like size ratio or maximum packing fraction. And the Péclet number of small particles is the Péclet number of large particles divided by the size ratio, Pe_s = Pe_l/α.

• However, if the initial volume fraction of small particles is too high, a jammed layer forms under the air–liquid interface and grows so quickly that the large particles are arrested by the jammed layer and do not stratify [59], leading to a homogeneous dried film.

In Sec. 3.2.1, we mainly discussed the drying process of dispersions containing solid nanoparticles without polymers. We now highlight the role in the dispersion drying of polymer particles that are soft and not perfect spheres. Howard et al. [13] used implicit solvent Langevin dynamics simulations to simulate the drying of polymer–polymer (two polymer types) and polymer–colloid mixtures, examining the effects of polymer chains on stratification. They found that both polymer–polymer and polymer–colloid mixtures could also develop small-on-top stratification, as observed in pure colloid dispersions by others. For polymer–polymer mixtures, polymers with shorter chains would accumulate at the top surface, while for polymer–colloid mixture, the relatively smaller particles accumulated at the top surface. A theoretical diffusion equation for the mixtures was also developed. The chemical potential expression consisted of ideal solution terms and excess potential terms including volume exclusion and chain formation. The small-on-top stratification was understood to develop because the downward migration velocity increases with particle size or chain length. The concentration density was in quantitative agreement with Langevin dynamics calculations for polymer blends and in qualitative agreement for colloid–polymer mixtures.

Unlike rigid nanoparticles, the sizes of polymer chains may change with the properties of the solvent, such as solvent pH. Fabiani et al. [32] demonstrated that pH can be used to control the stratification process in mixtures in which the polymer chains are responsive to changes in pH. These authors used Langevin dynamics modeling to simulate the drying process of a binary dispersion containing polymers of two different sizes. The smaller polymer chain was pH-responsive and would swell as pH increased as shown in Fig. 8(a). When the pH increased, the effective diameter of the smaller polymer would increase. In simulation results for size ratios of 7:1 and 4:1, shown in Figs. 8(b) and 8(c), respectively, the stratification was more obvious at the larger size ratio. These authors also experimentally verified that the small-on-top stratification appeared at lower pH and disappeared at higher pH, as shown in Figs. 8(d)–8(g).

To study the influence of hydrodynamic interaction on the stratification of polymer suspensions, Statt et al. [29] used corresponding explicit and implicit solvent molecular dynamics models to simulate the drying process of polymer mixtures with two different types of polymer chains. Unlike the explicit solvent model, the implicit solvent model neglected the hydrodynamic interactions between polymers. Statt et al. [29] observed stratification when using the implicit solvent model but did not observe this when using their explicit solvent model. They concluded that the hydrodynamic interaction played an important role in the stratification of polymer suspension and could not be neglected, unlike in the case of the pure nanoparticle suspensions studied by Tang et al. [25].

Furthermore, concentration distributions might be controllable by tuning the interactions among nanoparticles and polymers. Cheng et al. [12] developed a large-scale molecular dynamics simulation to simulate nanoparticle dispersion in a polymer solution film. The wettability of nanoparticles by the polymer and solvent was tuned. For relatively strong interactions between nanoparticles and polymer solvent, that is, when the nanoparticle/polymer interaction was more favorable than that between nanoparticles and solvent, the nanoparticles would accumulate at the top of the film, rapidly forming a polymer skin layer during the evaporation process. For relatively weak interactions between nanoparticles and polymer, the nanoparticles were more uniformly distributed in the solvent.

From a continuum modeling perspective, various roles for polymers in dispersion drying have been considered. Some authors have represented the polymers as interacting particles [61], some considered their effect on rheology [62–65], and others have considered their effect on evaporation rate [63]. Nikiforow et al. [61] investigated the self-stratification of mixtures of neutral and charged latex particles both experimentally and theoretically. They found that while the solvent is evaporating from the free surface, both types of particles become more concentrated under the air–film interface than in the bulk dispersion. However, the charged particles had higher chemical potential and collective diffusivity and faster transport due to their mutual electrostatic repulsion. Accordingly, the evaporation-induced concentration gradient of charged particles equilibrated faster than for neutral particles. Therefore, at the end of the drying process, if the charged particles had time to diffuse but neutral particles did not, the top of the film will be enriched in neutral components. Additionally, these authors created a model in which the collective diffusivity was expressed as the weighted average of that at dilute limit and concentrated limit. Qualitative agreement with experiments was achieved.

Buss et al. [63] discussed the effects of soluble polymer binders on particle drying regimes, including sedimentation, diffusion, and evaporation. These authors used free volume theory to model the diffusion of polymer binder and found that the distribution of polymer affects the evaporation rate and local solution viscosity. The Stefan–Maxwell equations were used to describe diffusion in the binary system, and the solvent activity coefficient was derived from Flory–Huggins theory. The dependence of relative viscosity on polymer concentration was estimated from experimental data. In both model calculations and experimental results, the evaporation rate was hardly affected by the presence of polymer in the earliest stage of drying, in which liquid evaporates at a constant rate. The authors claimed that the increase of polymer concentration during drying increases the solution viscosity and slows the diffusion and sedimentation processes, leaving the evolution of particle distribution to be controlled primarily by evaporation.

More recently from the same research group, Baesch et al. [66] presented an improved 1D ternary model including particle sedimentation and diffusion, soluble polymer diffusion, and solvent phase evaporation. With this model, outcomes corresponding to different drying regimes (sedimentation, evaporation, and diffusion) in these ternary systems could be produced using only data from binary polymer–solvent and particle–solvent systems. However, no component interactions were included and the model was not applied to drying at high Péclet numbers. Their most recent follow-on work [67] discussed the influence of plate-like particle shapes on the drying regime map for particle–polymer composites.

Romermann [65] found that the addition of crosslinking agent or thickener slowed diffusion and induced composition gradients of small polymer particles in the films, both vertically and across the plane, as inferred from dried film structures. To explain the inward flow from the edges of a serum, the so-called anti-coffee-ring effect, Romermann proposed a collapse of porous networks at the edges, driven by capillary forces.

Lee and Choi [62] studied how the polymer accumulation under the top interface during drying alters stratification. First, they defined two characteristic times: the time at which stratification starts $(ts*)$ and the time at which the accumulated polymer under the air–dispersion interface reaches a concentration at which it can arrest colloid movement, $(tc*)$⁠; they refer to the latter as the “viscosity transition.” The time at which stratification starts is taken as the time at which the diffusiophoretic velocity exceeds the velocity of the air–solution interface $U(ts*)=E$⁠. As with Sear’s earlier concept [59], the condition for stratification to occur is that the large particles begin to stratify before the polymer becomes a dense jammed layer, trapping the colloidal particles $(tc*>ts*)$⁠, and before the film dries completely $(ts*<1)$⁠. The authors proposed that if Pe ≫ 1, the criteria for stratification may not depend on evaporation rate or Pe, but on the polymer viscosity transition concentration. Their state diagram, which includes theoretical predictions and experimental results, is shown in Fig. 6(c).

Battery slurries are colloidal mixtures that include active material, conductive additives, binders, and solvents. These components range from 50 nanometers in size (carbon black) to tens of micrometers (active material). The models described in previous sections provide simplified descriptions of very complicated slurry drying processes.

It is important to understand these complex drying processes, as they influence final electrode performance in battery manufacturing, through the final spatial distributions of the slurry components. These influence characteristics such as adhesion, capacity, resistance, and, impedance [2,16,68–72]. Modeling efforts in this specialized field of colloidal dispersion drying precede recent reports on autostratification. Some simulations [14,41,73,74] were developed to understand the microstructures of Li-ion battery electrode fabrication. Liu et al. [14] built a 2D coarse-grained kinetic Monte Carlo simulation to investigate the impact of mixing sequence, drying conditions, and component interactions on electrode morphology. They demonstrated the effect of solvent evaporation rate on agglomeration of conductive additives. However, particle-based simulations usually include restrictive physical assumptions and parameters. Continuum models have been used to investigate phenomena such as binder/carbon black migration. However, since binder polymers are typically soluble in the solvents and account for low volume fractions (typically 0.01 ∼ 0.05), many continuum drying studies model these mixtures as monodisperse and ignore the interactions among different colloidal components.

Li et al. [68] developed a model of binder migration. The binder was assumed to be dissolved in the solvent and to migrate upward due to solvent evaporation, creating a nonuniform distribution. They assumed that the shrinkage of the electrode coating was uniform. The adsorption isotherm of polymers on the LiCoO₂ particles was considered, and the rheology of slurries using water or organic solvents was compared. In this model, there were no interactions between active material particles and binder or conductive graphite additive, and the settling of active material particles was not considered. From experiments, they found that water-based electrodes have a more uniform binder distribution and that the nonuniform distribution of binder leads to weaker adhesion and higher electrical resistance. They reported that electrodes fabricated from water-based slurries yielded better cell performance with lower IR drop than those based on organic solvents, especially before electrode compression. We note that they were fabricating unusual thick (1500 µm) electrodes.

Stein et al. [76] developed a diffusion model for secondary phase components (conductive additive and binder) during the second, high-temperature stage of a two-stage drying process. They employed similar diffusivity expressions as in single-component suspension drying [43] with hydrodynamic interactions and concentrated solution effects included through the sedimentation coefficient and compressibility factor. Figure 9(c) shows that the secondary phase distribution becomes less uniform as the initial volume fraction and the Péclet number increase. While the tendency shown is reasonable, their models are oversimplified and more variables like viscosity and film height need to be considered. Figure 9(b) demonstrates the typical two stages of battery slurry drying. Consistent with our analysis in the Drying periods section, a film shrinks with a constant drying rate in the first drying stage (regime I). Then, the solvent recedes into the porous particle network, leading to a lower evaporation rate and stable film height in the second drying stage (regime II). They verified that poor electrochemical performance could be caused by the nonuniform distribution of conductive additives and binders seen experimentally, highlighting the critical role that the solvent evaporation process plays in the final electrode structure and cell performance.

As in the other binder migration theories previously discussed, Font et al. [77] developed a model of binder migration without interactions between the binder and other components. Two 1D mass transport equations for active particles and solvent were solved; the authors justified independent treatment of the dissolved binder (with a 1D advection-diffusion model) by the low binder volume concentration. The authors argued that the solvent flow at a high evaporation rate brings the dissolved binder to the top surface of the film and that their prediction of binder migration agrees qualitatively with experimental results. Based on this model, they proposed a strategy of using an initially high drying rate and decreasing it to make the total drying time short while also keeping binder distribution uniform.

However, the in-situ characterizations of dynamic information such as drying rate, particle and binder distribution, and stress evolution are very limited, and none of the existing models are valid for all drying processes [78,79]. The optimal drying protocols proposed by Susarla et al. [80], Font et al. [77] and Jaiser et al. [81] require validation through more in situ and advanced metrology observations. An example of such a strategy is to use lower solvent content to decrease the drying time and binder migration, provided that the components are well-dispersed after mixing, and to optimize the drying temperature and air flow based on electrode morphology and in situ observations [78]. Then, near the end of drying, one can increase the drying power to remove the residual solvent [80]. However, in general, many parameters influence the coating process [82] and particle/binder transport during drying [83]. Insufficiently high viscosity, for example, leads to particle aggregation [64] and sedimentation/segregation [84]. Due to this complexity, much future research, including in situ measurements and more comprehensive models, is needed to explore drying processes for battery slurries.

The drying of Perovskite solar cell inks has similarities with the drying of lithium-ion battery slurries. Ternes et al. [85] developed a model that considered the thickness of thin perovskite films and evaporation rate during drying. The authors used a laser reflectometer to monitor the drying process in situ. As typical in battery drying models, they divided the drying stage into two stages, the first one with a constant drying rate and the second with an exponentially decreasing drying rate. The accurate calculations of mass transfer coefficient β_ij could be a reference for battery slurry drying kinetics modeling. However, the perovskite film is several microns thick and much thinner than battery electrodes, and the transport of particles along the vertical direction was not investigated. The authors demonstrated that from the drying process model a thin-film morphology is predictable. And a homogeneous blade-coated perovskite thin film with good performance could potentially be fabricated on a large scale.

Fuel-cell catalyst layer manufacturing is another important application of dispersion drying. These catalyst layers are usually made up of carbon-supported platinum particles, ionomer, and solvents. Hatzell et al. [86] compared the similarities and differences of the ideal structure between battery and fuel-cell catalysts. With more charged ionomers incorporated in fuel cells, the morphology of polymers and interactions among particles and polymers played a more important role. Shukla et al. [87] analyzed the stability of ink dispersion and aggregate size in four different non-aqueous solvents. The interaction energy depends largely on the solvent dielectric constant and the particle surface potentials. The addition of ionomer (Nafion) strongly increased the ink stability and decreased the size of carbon agglomerates, with the stabilizing effect being independent of Nafion concentration. Interactions among particles, ionomer, and solvents need more investigation and the modeling of the ink drying process also requires additional investigation.

In the second stage, further solvent evaporation drives the particles to form a porous particle network, and the solvent interface recedes into the porous structure. Solvent removal from the porous film is sometimes accompanied by crack formation. These two phenomena have been intensively studied in many drying applications, such as desiccated soil, concrete casting, ceramics drying, and food drying [88]. Less attention has been paid to solvent-processed coatings for batteries. Here, we mainly review the research proceedings related to the drying processes of battery slurries or electrode inks.

Solvent removal represents a key step in the development of electrode structure and its associated properties. The need for experimental techniques that allow for investigation into solvent transport in microstructures is evident. Jaiser et al. [89] used fluorescence microscopy to observe drying processes. They found that pore emptying happened in both first and second drying stages and that some of the surface pores remained filled in the second-stage drying (Fig. 10). This finding indicates that pore emptying, or solvent removal from the film, is not taking place in the form of a horizontally receding liquid front, but in a very heterogeneous way. However, these microscope observations only provided a top-view of drying processes. They [90] further made side-view observations by applying a cryogenic scanning electron microscope and verified the results that surface pores remain saturated beyond the instant at which air intrudes into the porous network. They reasoned that small pores maintain a high level of saturation at the expense of larger pores due to the transport of the liquid phase from larger to smaller pores that are driven by capillary pressure. One main challenge in investigating the second drying stage is the difficulty of directly observing the porous structure and solvents inside. Additionally, in-situ observation of pore-scale drying processes is still lacking.

Solvent removal is typically viewed as a problem of drying from porous media, which has been widely modeled in past decades. Lots of methods have been used to describe the drying of porous media. At the macroscale, continuum effective medium descriptions were generally used; at the mesoscale, discrete pore networks have been developed extensively in recent years. However, little attention has been paid to the drying of battery colloids. Susarla et al. [80] used a continuum-level mathematical model to describe the physical phenomenon of cathode drying involving coupled simultaneous heat and mass transfer with phase change and studied the effect of varying temperature and air velocity on the second-stage drying process. They showed that second-stage electrode drying is not a heat transfer but rather a mass transfer-controlled process and that 90% of drying is completed in nearly half of the total time. Based on these findings, they introduced a three-zone approach to electrode drying to reduce drying time. They also compared the drying behavior of electrodes using an aqueous solvent instead of N-Methyl-2-pyrrolidone (NMP) and found that these could be dried 4.5 times faster than those using NMP. As with experimental studies, few studies have been done to model the pore-scale solvent removal during the drying process of battery colloids. However, similar studies are essential, since as Jaiser et al. [89] experimentally observed, solvent removal proceeds in a very heterogeneous way.

When a wet coating containing suspended particles is dried, the meniscus of the air–solvent interface between particles generates a capillary pressure that increases as the solvent evaporates and exerts compressive force on the particles. Eventually, the coating may crack at certain critical points to release the drying stresses.

Aqueous processed electrodes, such as Li-ion battery cathodes and fuel-cell catalyst layers, have been found to crack during drying. Loeffler et al. [91] observed crack formation during battery slurry drying processes but attributed this to the limited capabilities of lab-scale equipment for mixing and coating slurries with higher viscosity. Lim et al. [92] experimentally investigated the development of drying stress and the microstructure in dried anode slurry films. Based on drying stress measurements, they obtained a processing window map that shows the effect of the binder on the mechanical strength of the film. Kumano et al. [17] used critical crack thickness to characterize cracking behavior and influenced the arrangement of Pt/carbon (Pt on carbon support) through the presence or absence of an absorbed ionomer on Pt/carbon particles. They showed that the crack behavior can be controlled by ionomer adsorption into the particles in catalyst ink, as shown in Fig. 11. Uemura et al. [93] investigated ink degradation and its effects on crack formation in fuel-cell catalyst layers using gas chromatography-mass spectrometry. They attributed the crack formation to the agglomeration of Pt/carbon particles caused by the formation of hydrophobic compounds in the catalysis process. Note that most of the studies only demonstrated novel experimental findings and proposed possible mechanisms behind them; no systematic modeling work specifically on crack formation in battery electrode fabrication has been reported.

Many fabrication parameters affect crack formation. Du et al. [94] studied the cracking of thick, aqueous processed electrodes. They found that the critical cracking thickness of the coating increased with increasing isopropyl alcohol content due to decreased surface tension. Rollag et al. [95] investigated the influence of drying temperature and thickness on crack formation. Experiments revealed that the cracking worsens with increased electrode thickness and elevated drying temperatures. The combination of strong evaporation and weak diffusion played a critical role in the nonuniform distribution of the inactive materials, which caused capillary-driven motion and crack formation.

Drying is a complex process involving physics on multiple spaces and time scales. In this review, we discussed recent advances in understanding film drying processes and highlighted applications in Li-ion battery electrodes and fuel-cell ink catalyst layer fabrication.

Despite the recent progress that has been made, more research in this field is still needed. Unsolved issues and opportunities include the following:

1. Models rarely attempt to consider the complex interactions among components in dense dispersions.

2. Non-invasive operando or in-situ experimental methods and analytical techniques are needed to track component states during drying. These kinds of methods are especially important for battery or fuel-cell dispersions which are difficult to probe with visible light.

3. Polymer or ionomer dispersions should be further investigated in experiments and models, especially when the polymer is mixed with the particles. It is necessary to know the states of polymers inside dispersions and how the polymers interact with rigid particle surfaces.

4. Mesoscale models that link microscale models like molecular dynamics and macroscale heat and mass transfer models should be developed to fill in the major disconnection between the molecular level and continuum level.

• This work was supported by the Advanced Manufacturing Office, Office of Energy Efficiency and Renewable Energy, of the U.S. Department of Energy (DOE) under Contract No. DE-AC02-05CH11231.

There are no conflicts of interest.

The data sets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

D =

diffusivity, m²/s

E =

evaporation rate, m/s

H =

coated film height, m

R =

radius, m

U =

settling velocity, m/s

V =

volume

K_c =

consolidation factor

α =

large-to-small particle size ratio

β =

mass transfer coefficient, m/s

η =

dynamic viscosity, Pa s

ρ =

density, kg/m³

τ =

time

ϕ =

volume fraction

Pe =

Péclet number

N_s =

sedimentation number

cr =

critical

diff =

diffusion

jam =

jamming

L =

liquid or large particles

m =

maximum

p =

particle

s =

small particles

sed =

sedimentation

tot =

total

0 =

initial value