# A computational study of the effect of particle migration on the low-flow limit in slot coating of particle suspensions

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## Abstract

Slot coating of particle suspensions is commonly used in the manufacturing of a wide variety of products. An important limit in this process is known as low-flow limit, which refers to the minimum wet film thickness that can be coated at a given substrate velocity. Recent studies have shown that shear-induced particle migration leads to a highly non-uniform particle distribution in the coating bead, playing an important role in the flow dynamics in slot coating of particulate systems. In this work, we extend the previous analyses to investigate the effects of particle migration on the low-flow limit in slot coating of particle suspensions at both dilute and concentrated conditions. As a first approximation, the suspension is modeled as a Newtonian liquid with a concentration-dependent viscosity, and shear-induced particle migration is described according to the diffusive flux model. The resulting set of governing equations is solved with a stabilized finite element method coupled with the elliptic mesh generation method for the free-boundary problem. The results show that particle migration changes the liquid viscosity near the downstream meniscus and strongly affects the force balance that sets the critical operating conditions at the low-flow limit. Remarkably, it was found that particle migration enlarges the operating window of the process when the suspensions are compared to a Newtonian liquid with the same bulk properties, especially at high concentrations.

## Keywords

Slot coating Suspensions Particle migration Low-flow limit Finite element method## Introduction

*q*is the prescribed flow rate (per unit width) in the feed slot,

*h*is the feed slot channel height,

*V*is the substrate velocity,

*H*is the gap between the slot dies and the substrate, and

*t*is the wet film thickness. In slot coating, the thickness of the liquid layer deposited on the substrate is defined by the ratio between the flow rate in the feed slot and the web speed, i.e., \(t = q/V\). Remarkably, it is independent of other process variables, such that slot coating belongs to a class of coating methods known as pre-metered methods, which are ideal for high-precision coatings.

The coating bead region is defined by the liquid confined between the slot dies, the substrate and two air–liquid interfaces (or *menisci*). The free surfaces determine two static contact lines at the die walls and one dynamic contact line at the substrate, and each contact line defines a contact angle with the adjacent solid boundary. Here, \(\theta _{\mathrm{s}}\) is the static contact angle between the upstream free surface and the upstream slot die lip, \(\theta _{\mathrm{d}}\) is the dynamic contact angle between the upstream free surface and the moving substrate, and \(\theta\) is the static contact angle between the downstream free surface and the downstream slot die lip. To maintain a stable coating bead at very high substrate velocities and/or to produce very thin films without defects, the gas pressure near the upstream meniscus is usually made lower than the ambient pressure by using a vacuum chamber. The flow under the die lips is close to rectilinear and can be decomposed into a drag flow due to the web movement (planar Couette flow) and a pressure-driven flow between two parallel plates (planar Poiseuille flow). The velocity profile in this region intrinsically depends on the relative contribution of each one of these mechanisms, which, in turn, depends on the operating conditions of the process, such as the film thickness and the substrate velocity.

The required condition to obtain a uniform coated film in both down-web and cross-web directions is a steady, two-dimensional flow. This condition depends on a complex force balance in the coating bead, which might include viscous, capillary and pressure forces, and, in some cases, inertial and elastic effects as well. The competition between these mechanisms sets the range of operating parameters at which the quality of the delivered liquid layer is acceptable and defines what is known as the coating window of the process. Coating windows are bounded by coating defects in the space of operating parameters and can be constructed from experimental observations and/or theoretical models. Therefore, previous knowledge of the coating window of a given coating method is essential to predict if the process can be used to coat a substrate at a prescribed production rate. A broader review on the operability limits and coating windows of slot coating processes was recently presented by Ding et al.1

An important limit in slot coating is usually referred to as low-flow limit, which corresponds to the minimum film thickness at a given substrate velocity or, similarly, the maximum substrate velocity at a given film thickness at which the coating bead remains stable. The low-flow limit is characterized by the receding action of the downstream meniscus as the web velocity increases and/or the film thickness falls. The onset of the low-flow limit is dictated by a force balance at the interface and defines a critical pair of web speed and film thickness (or capillary number and thickness-over-gap ratio, in a dimensionless form) at which the flow is stable. At unstable conditions, the meniscus becomes so curved that the coating bead breaks, leading to stripes of coated and uncoated layers on the substrate, a coating defect known as rivulets.

Carvalho and Kheshgi2 presented an analysis in which the low-flow limit in slot coating flows of Newtonian fluids was studied by theory, experiments and numerical simulations. They conducted a broad investigation on the effects of both capillary and Reynolds numbers on the flow dynamics and stability of the coating bead. Remarkably, it was found that the minimum wet film thickness that can be coated considerably decreases when the substrate velocity is raised, such that the coating window of the process is much larger than the one predicted by the classical viscocapillary model of Ruschak3 and Higgins and Scriven.4 Other experimental reports concerning the low-flow limit in slot coating flows of Newtonian liquids were presented by Lee et al.,5 Chang et al.6 and, more recently, by Benkreira and Ikin.7

Notwithstanding, most of the liquids used in the coating industry are not ordinary Newtonian fluids, but are polymer solutions, particle suspensions, or both. Therefore, it is expected that the non-Newtonian behavior of such fluids in complex coating flows changes the force balance in the coating bead, affecting the operating limits of the process. The low-flow limit in slot coating of viscoelastic polymer solutions was extensively studied by Romero et al.8^{,}9 and Bajaj et al.10 On the one hand, it was found that when the fluid elasticity is very small, it may stabilize the flow near the downstream meniscus, leading to a smaller wet film thickness when compared to a Newtonian fluid under the same conditions. On the other hand, when the fluid elasticity is high, a strong polymeric stress boundary layer arises near the downstream meniscus, destabilizing the flow and increasing the thickness of the deposited liquid layer. A number of other experimental investigations found similar results concerning viscoelastic effects on slot coating flows.11^{–}14

In the case of coatings of particle suspensions, the functionality of the final product depends on the inner microstructure of the coated layer, which, in turn, is directly affected by both deposition and drying methods. Therefore, the optimization of particulate coating processes relies on the fundamental understanding of the flow dynamics as the suspension is deposited on the substrate. Recently, Carvalho and coworkers have presented a number of different studies on slot coating flows of particle suspensions.15^{–}18 The focus was on the non-uniform particle concentration distribution caused by shear-induced particle migration in the flow, and the results show that the operating conditions play a key role in the particle distribution in the flow and across the deposited liquid film. As a matter of fact, previous knowledge about particle distribution in the film is fundamental for the drying process of particulate coatings, as addressed by Francis et al.19^{–}23

In this paper, we extend previous studies on slot coating flows of particle suspensions to investigate the effect of shear-induced particle migration on the low-flow limit of the process. The analysis considers both dilute and concentrated regimes for a suspension of spherical particles in a Newtonian liquid. The suspension is described as a Newtonian fluid with a concentration-dependent viscosity that is coupled with shear-induced particle migration according to the diffusive flux model. The model is solved with a stabilized finite element method together with the elliptic mesh generation method for the free surfaces. The results show how particle migration affects the force balance near the downstream meniscus and its effect on the low-flow limit of the process. The remainder of this article is organized as follows. The mathematical formulation of the problem is described in “Mathematical modeling” section, and the numerical method is reviewed in “Computational solution” section. Results and discussions are presented in “Results” section, and concluding remarks are given in “Conclusions” section.

## Mathematical modeling

### Conservation equations

*p*is the pressure field, \(\varvec{I}\) is the identity tensor, and \(\varvec{\tau }\) is the extra-stress tensor field.

- 1.
At solid walls, the no-slip and no-penetration conditions are used. So, \(\varvec{u}= \varvec{0}\) at the die walls and \(\varvec{u}= V\hat{\varvec{e}}_1\) at the substrate, where \(\hat{\varvec{e}}_1\) is the unit vector in the \(x_1\)-direction.

- 2.Along the synthetic inflow plane in the feed slot region, the velocity field is prescribed as a fully developed, parabolic profile given by the classical Hagen–Poiseuille solution for channel flows:$$\begin{aligned} \varvec{u}= -\frac{q}{6h}\left[ \left( \frac{x_1}{h}\right) - \left( \frac{x_1}{h}\right) ^2\right] \hat{\varvec{e}}_2. \end{aligned}$$(4)
- 3.
A the synthetic outflow plane, the flow is fully developed with a fixed pressure, such that \(\varvec{n}\cdot \nabla \varvec{u}= \varvec{0}\) and \(p = p_0\), where \(p_0 = 0\) is the ambient gas pressure.

- 4.
At the free surfaces, the shear stress vanishes, i.e., \(\varvec{t}\cdot (\varvec{n}\cdot \varvec{T}) = 0\), and the liquid traction should balance the pressure in the external gas and the capillary pressure induced by the curvature of the air–liquid interface. The stress jump along the free surface is given by the Young–Laplace equation, \(\varvec{n}\cdot \varvec{T}= (-p_{\mathrm{g}} + \sigma \kappa )\varvec{n}\), where \(p_{\mathrm{g}}\) is the external gas pressure, \(\sigma\) is the surface tension of the coating liquid, and \(\kappa\) is the curvature of the free surface. Note that \(p_{\mathrm{g}} = p_0\) at the downstream free surface and \(p_{\mathrm{g}} = p_{\mathrm{v}}\) at the upstream free surface, where \(p_{\mathrm{v}} < p_0\) is the vacuum pressure. Moreover, there is no liquid flux across the interfaces, \(\varvec{n}\cdot \varvec{u}= 0\).

- 5.
At the dynamic contact line near the upstream meniscus, the no-slip condition is replaced by the Navier’s slip equation, such that \(\beta ^{-1} \varvec{t}\cdot (\varvec{u}-V\hat{\varvec{e}}_1) = \varvec{t}\cdot (\varvec{n}\cdot \varvec{T})\), where \(\beta\) is the slip coefficient.

### Shear-induced particle migration

*a*is the particle radius, and \(\dot{\gamma }\) is the local shear rate (i.e., the second invariant of the rate of strain tensor). Particle migration induced by curved streamlines and sedimentation due to gravity are both neglected here, so that \(\varvec{N}_\phi = \varvec{N}_{\mathrm{c}} + \varvec{N}_\eta\) in equation (5). For the interested reader, we explored the effects of flow curved streamlines and particle sedimentation on the dynamics of particle migration in slot coating flows in other works.16

^{–}18

- 1.
At the synthetic inflow plane, the particle concentration distribution is assumed to be equal to the suspension bulk concentration, i.e., \(\phi = \bar{\phi }\).

- 2.
At the solid walls, free surfaces and synthetic outflow plane, there is no diffusive flux of particles, so that \(\varvec{n}\cdot \varvec{N}_\phi = 0\).

^{,}16

^{,}18

^{,}26

### Dimensionless numbers

*V*and

*H*as characteristic scales of velocity and length, respectively. These parameters are:

- 1.
The Reynolds number, \(Re = \rho VH/\bar{\eta }\).

- 2.
The capillary number, \(Ca = \bar{\eta }V/\sigma\).

- 3.
The dimensionless flow rate, \(Q = t/H\).

- 4.
The dimensionless vacuum pressure, \(P_{\mathrm{v}} = p_{\mathrm{v}} H/\sigma\).

- 5.
The dimensionless feed slot height, \(S = h/H\).

- 6.
The dimensionless particle size, \(\xi = a/H\).

*Q*and

*Ca*. The vacuum pressure was adjusted to obtain no recirculation near the upstream meniscus keeping the dynamic contact line fixed at approximately the same position in all cases.

## Computational solution

Numerical approximations for the free-surface slot coating flow were obtained with a stabilized finite element method. The free surfaces were computed by a mapping between the unknown physical domain and a known computational/reference domain according to the elliptic mesh generation method.27 The mesh equation and transport equations, i.e., mass conservation, momentum conservation, and particle transport, were discretized by using the so-called DEVSS-TG/SUPG finite element method.28 This formulation deals with the velocity gradient as an independent variable with a traceless, continuous interpolation in order to compute higher-order derivatives of the velocity across element boundaries. The independent variables of the problem are written as a linear combination of a finite number of basis functions: Lagrangian biquadratic functions represent the velocity field, particle concentration, and mesh position; Lagrangian bilinear functions are used for the interpolated velocity gradient; and linear discontinuous functions are used for the pressure field. Galerkin’s weighting functions are applied in the residual equation of mass conservation, momentum conservation, mesh position and interpolated velocity gradient, and Streamline-Upwinding Petrov–Galerkin weighting functions are used in the residual equation of particle migration. After discretization, one obtains a large, sparse set of fully coupled, nonlinear algebraic equations, which was solved by Newton’s Method together with a frontal solver based on the lower-upper (LU) factorization method. This numerical methodology has been successfully used to study the effects of shear-induced particle migration in free-surface flows of particle suspensions and further details can be found elsewhere.15^{,}16^{,}18^{,}26

Meshes used in the grid-independence test

Mesh | Elements | Nodes | Degrees of freedom |
---|---|---|---|

1 | 411 | 1785 | 12,086 |

2 | 1080 | 4549 | 30,555 |

3 | 2160 | 8913 | 60,233 |

## Results

First, we investigate the effect of particle migration in slot coating flows and analyze the particle concentration distribution at the coated film. As discussed in previous works,15^{,}16^{,}18 the flow in the coating bead plays a key role in the particle dynamics, such that the pattern of particle distribution at the wet film is a strong function of the dimensionless flow rate, i.e., the film thickness over coating gap ratio. Here, we extend the analysis for suspensions at both dilute and concentrated conditions (i.e., low and high concentrations, respectively) and then investigate the effect of particle migration on the low-flow limit in slot coating.

As the film thickness decreases, the downstream meniscus becomes more curved and the pressure-driven contribution to the flow dynamics in the coating bead increases. For Newtonian fluids at \(Q < 0.50\), an adverse pressure-driven contribution arises; at \(Q = 1/3\), the adverse pressure-driven contribution is strong enough to create a point of zero shear rate at the downstream die wall; and at \(Q < 1/3\), the adverse pressure-driven contribution is so strong that a recirculation zone appears under the downstream die lip, creating a layer of negative velocity and leading to a region of vanishingly small shear rate near the wall. As the recirculation region drastically changes both velocity and shear rate fields in the flow, it is expected that it has a strong effect on shear-induced particle migration in slot coating of particle suspensions.

It is worth noting that particle migration also has important implications in the flow dynamics upstream the coating bead. The curved flow creates a zone of low shear rate under the upstream die lip, such that a small region of high particle concentration appears near the upstream static contact line. Again, this point might be a source of undesirable particle agglomeration in the coating process. Moreover, particle migration strongly changes both viscosity and velocity fields in the upstream region and thereby affects the vacuum pressure required to counteract the drag by the substrate and control the position of the dynamic contact line. The vacuum pressure was adjusted to keep the dynamic contact line fixed at the same position with no recirculation upstream the coating bead for all cases. For instance, at \(Q = 0.60\) and \(Ca = 0.10\), \(P_{\mathrm{v}} = -0.91\) for the equivalent Newtonian liquid, \(P_{\mathrm{v}} = -0.94\) for suspensions at \(\bar{\phi } = 0.10\), and \(P_{\mathrm{v}} = -1.08\) for suspensions at \(\bar{\phi } = 0.40\); at \(Q = 0.20\) and \(Ca = 0.10\), \(P_{\mathrm{v}} = -5\) for the Newtonian liquid, \(P_{\mathrm{v}} = -4.97\) for suspensions at \(\bar{\phi } = 0.10\), and \(P_{\mathrm{v}} = -4.88\) for suspensions at \(\bar{\phi } = 0.40\). Although it is not the main focus of this work, these results show that the shear-induced particle migration might have a substantial effect on the vacuum pressure operability limits in the coating window of slot coating of particle suspensions.

^{,}29 These results indicate that, at a fixed substrate velocity, concentrated suspensions can be used to produce thinner films than a Newtonian liquid with the same average bulk properties. Similarly, for a given film thickness, the maximum web speed is higher for suspensions at high concentrations, which can increase the production rate of the process.

*x*-axis expresses the coating gap-to-film thickness ratio, which is actually the inverse of the dimensionless flow rate

*Q*. As a reference, we also show the results of the viscocapillary model of Ruschak3 and Higgins and Scriven,4 in which \(Ca = 0.65[2/(H/t - 1)]^{3/2}\). In the case of dilute suspensions, particle migration is not strong enough to lead to a considerable difference with respect to a Newtonian liquid. For a fixed coating gap-to-film thickness ratio within the range analyzed here, the critical capillary number at which the low-flow limit occurs for dilute suspensions at \(\bar{\phi } = 0.10\) is only 0.5–4.5% higher than the one obtained for a Newtonian liquid. The outcome is very different for suspensions at high concentrations. In this case, particle migration strongly affects the viscosity field close to the downstream meniscus and thereby leads to remarkable differences in the low-flow limit conditions in relation to a Newtonian liquid. The smaller differences in the critical capillary number at a fixed film thickness occur when the flow is dominated by surface-tension forces, i.e., at very small capillary number and very thin films. Even though, for a fixed film thickness such that \(H/t > 6.5\), the critical capillary number for a concentrated suspension at \(\bar{\phi } = 0.40\) is at least 10% higher than the one for a Newtonian liquid. This difference changes as viscous forces become more important to the force balance in the coating bead, i.e., as the capillary number grows and the film thickness increases. For instance, it is of 35% at \(H/t = 5\); reaches 65% at \(H/t = 3.7\); and falls to 40% at \(H/t = 2.9\). We believe that the non-monotonic behavior of the difference between the critical capillary number of concentrated suspensions and Newtonian liquids is due to the recirculation region under the downstream die wall, which plays an important role in the particle migration in the coating bead and near the free surface. As a matter of fact, for coatings at \(H/t = 2.5\) there is no recirculation in the downstream coating bead and the relative difference between the critical capillary number for the concentrated suspension and the Newtonian liquid falls to 23%. Nevertheless, for the range of capillary number and film thickness studied here, the numerical results show that particle migration is responsible for increasing the maximum production rate of slot coating process for concentrated suspensions when compared to Newtonian liquids.

## Conclusions

Slot coating is one of the preferred methods for high-precision coatings. One important operability limit of this process is known as low-flow limit, which is especially relevant during the production of very thin films at relatively high speeds. The operating conditions at the low-flow limit in slot coating of Newtonian liquids and non-Newtonian polymeric solutions have been determined by previous researchers. However, most of the liquids typically used in industrial applications are particle suspensions, which can show very complex properties because of the physical phenomena occurring at the particle level, such as shear-induced particle migration.

In this work, we presented a numerical investigation on the effect of particle migration on the low-flow limit in slot coating of particle suspensions. The model considered a suspension of non-colloidal, spherical particles, which was described as a Newtonian liquid with a concentration-dependent viscosity. Shear-induced particle migration was computed according to the diffusive flux model. The resulting set of equations governing the steady, two-dimensional slot coating flow was solved with a stabilized finite element method together with the elliptic mesh generation method to capture the free surfaces.

The results show that shear-induced particle migration leads to a highly non-uniform particle distribution in the coating bead and deposited liquid film. The particle dynamics in the coating bead and the pattern of particle distribution at the coated film strongly depend on the film thickness over coating gap ratio. When the wet film is relatively thick, the particles become more concentrated near the middle of the film. In turn, as the film thickness falls, the particles tend to migrate to the recirculation zone under the downstream die lip. The trends observed here are in good agreement with previous reports on particle migration in slot coating flows. Moreover, particle migration changes the viscosity field near the downstream free surface, affecting the force balance that defines the onset of the low-flow limit in slot coating process. The numerical results suggest that particle migration considerably enlarges the coating window of the slot coating process when the suspensions are compared to a Newtonian liquid with the same average bulk properties, especially at high concentrations of the dispersed phase.

## Notes

### Acknowledgments

This work was supported by the Brazilian Research Council (CNPq) and the Industrial Partnership for Research in Interfacial & Materials Engineering of the University of Minnesota (IPRIME).

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