# Numerical simulation of the dip-coating process with wall effects on the coating film thickness.

Abstract The dip-coating process for liquid film deposition on cylindrical substrates is numerically simulated, and the effects of the coating bath walls proximity to the substrate are investigated for the deposited film thickness. In the present work, the hydrodynamics of nonNewtonian liquid films are studied, applying Carreau and power law models. The free surface position is determined by the volume of fluid technique in a three-dimensional system, while the impacts of density, viscosity, and surface tension are taken into account. The momentum and mass balance, alongside the constitutive equations, were solved for dip-coating process using numerical simulations in an open source CFD software package of OpenFOAM. Numerical outcomes are validated with experimental data over a large range of withdrawal velocities up to 6 m/s. Also good agreement is obtained for numerical simulation results with experimental data for coating thickness, considering the proximity effects of bath walls to the cylindrical substrate being withdrawn from a coating bath.Keywords Coating, Thin film, Carreau model, Power law model, Numerical simulation

Introduction

Vertical withdrawal of a substrate from a Newtonian or non-Newtonian pool of liquid, known as dip-coating or free coating, is an inexpensive method of depositing a thin, uniform layer of materials in the liquid state onto a substrate. (1) In this process, the surface to be coated is initially immersed in the coating fluid and then withdrawn. One of the advantages of this process is the ability to coat irregularly shaped substrates. (1,2) Dip-coating is a popular post-metered coating method, and nowadays dipping and withdrawal are not only used to apply the coating, but also to control the final microstructure of the coated film. (3) Free coating is essential for the surface engineering of high quality products and it is one of the better developed ways to enhance and alter the physical and mechanical surface characteristics. For instance, the performance properties of optical fibers, including abrasion resistance and strength, are strongly influenced by adding a coated layer. (4) In recent years, electroactive polymers have received attention because of their distinct potential as actuators and sensors, and the cylindrical shape may bring explicit advantages to actuator design. (4,5) Therefore, cylindrical microfabrication is of interest, particularly due to the advantages conferred by geometry of substrate. Unlike planar substrates, cylindrical substrates offer axisymmetric flexibility, complete surface area utilization, and radial symmetry. These exclusive properties offer opportunities such as catheter-based electronics in the medical industry, fiber optic modulation for medical devices, or equipment for heat and mass transfer in gas-liquid systems to provide a large interfacial area. (6) The thickness of the liquid layer that remains on the surface of substrate depends on the viscosity, surface tension of the coating fluid, gravity, and the speed of withdrawal. The principle complicating feature of dip-coating problems is the strong role played by the free surface in controlling the coating dynamics, particularly in the region, where the object leaves the surface of the coating fluid bath. (7)

The calculation of coating film thickness started with the work of Landau and Levich (8) and Derjaguin. (9) Guttfinger and Tallmadge (10) estimated coating film thickness for non-Newtonian fluids on flat plates. However, considering a film on a highly curved surface, where the radius of curvature of the coated surface is in the order of magnitude of the film thickness, the film flow can no longer be approximated by a planar coating film. A significant improvement in film thickness assessment was attained by White and Tallmadge (11) with the gravity-corrected model for cylindrical substrates. There are a smaller number of studies on cylindrical substrates compared to flat plate substrates due to the difficulties in handling the curvature in analytical solutions. Furthermore, handling both curvature effects and non-Newtonian coating fluids, which are more realistic in the industries, makes analytical calculation very complicated for dipcoating process of cylindrical geometries. Accordingly, for handling the substrate geometries and interfacial phenomena, sound knowledge regarding the relationship between the final coating thickness, withdrawal speed, and fluid properties is essential for effective design, scale up, optimal control, and efficient operation of the free coating process.

Theoretical procedure

The dip-coating process is characterized by the physical conditions of liquid and gas and the topology of the interface. (12) Development of an approach that predicts substantial details with sufficient accuracy in the entire flow field is essential for free coating system. In this study, dip-coating process numerically studied by an open source computational fluid dynamics package (OpenFOAM--Open Source Field Operation and Manipulation) (13) and three-dimensional simulations are developed in order to capture all of the physically important features of the free surface flows. In addition, the volume of fluid method (VOF) is applied with the benefit of handling the evolution of free surface. (14) Discretization of equations is obtained by the Finite Volume methods (FVM) applying a collocated variable arrangement. (13,15)

In a VOF method, the free surface is identified by a marker function such as the volume fraction of a tracking variable, unlike surface-fitting methods which treat the free surface as a moving boundary of the computational domain. Surface-fitting methods are very efficient for free surface problems, but the validity is tested by the skewness of the resulting grid. If the free surface becomes highly distorted, a new mesh may have to be generated to maintain the accuracy of the solution. There are other limitations in the cases of wave breaking and fluid-fluid interactions, such as the trapping of one fluid within another, as in this case no explicit boundary conditions can be specified at the interface. (16,17) An advantage of the VOF technique and FVM applied in this study is the ease of code implementation and working with parallel simulation of three-dimensional systems in OpenFOAM. The main disadvantage of the VOF technique is the limited accuracy of results in numerical simulation with large mesh elements. Therefore, to overcome the disadvantage of VOF method, the mesh is densified with reasonably small elements around the cylindrical substrate surface.

Mathematical formulation

Incompressible flows in free coating process are governed by the Navier-Stokes and continuity equations. In OpenFOAM, the individual terms of the transport equation are treated separately.

The approach used here is to consider the density and viscosity as variables over the entire domain, but constant in each specific fluid. The conservation of momentum, equation (1) and the conservation of mass, equation (2), govern the fluid flow over the whole domain, where u and [rho] represent the velocity field and density, respectively, shared by the two fluids throughout the flow domain, [tau] is the stress tensor, and [F.sub.b] denotes the body force.

[partial derivative][rho]u/[partial derivative]t + [nabla] x ([rho]uu) = -[nabla]P + [rho]g + [nabla] x ([tau]) + [F.sub.b], (1)

[nabla] x (u) = 0. (2)

In the solution process, piezometric pressure [P.sub.rgh] = P - [rho]g x x is implemented instead of the pressure P, where x is the coordinate vector and g is the gravitational acceleration. Accordingly, -[nabla]P + [rho]g has to be rearranged as -[nabla][P.sub.rgh] - (g x x)[nabla][rho], and the momentum equation is developed into equation (3).

[partial derivative][rho]u/[partial derivative]t + [nabla] x ([rho]uu) = -[nabla][P.sub.rgh] - (g x x)[nabla][rho] + [nabla] x ([tau]) + [F.sub.b]. (3)

The momentum equation is modified in order to account for the surface tension effects. At the liquid--gas interface, surface tension generates an additional pressure gradient resulting in a force, which is evaluated per unit volume using the continuum surface force (CSF) model of Brackbill et al. (18) The numerical solution procedure for the two-fluid methodology is based on the Pressure Implicit method for pressure-linked equations (PIMPLE) algorithm, which is a combination of Pressure Implicit Split Operator (PISO) algorithm proposed by Issa (19) and Semi Implicit Methods Pressure-Linked Equations (SIMPLE) algorithm to handle the pressure-velocity coupling. (20,21)

Free surface implementation

The VOF method is a technique to capture the interface, proposed by Hirt and Nicholls. (14) The VOF method accommodates free surfaces by an indicator function with large deformation on non-uniform fixed grid system. The phase fraction [[alpha].sub.1] can take values within the range 0 [less than or equal to] [[alpha].sub.1] [less than or equal to] 1, with the values of zero and one corresponding to regions accommodating only one phase and for the interface where the value of [[alpha].sub.1] lies between 0 and 1. Movement of free surface is accomplished by calculating net mass flux through control surfaces and by updating the fluid volume fraction in each control volume. Free surface can be predicted by solving continuity equation for the indicator function, represented in equation (4).

[partial derivative][[alpha].sub.1]/[partial derivative]t + [nabla] x ([[alpha].sub.1]u) = 0. (4)

Conservation of the phase fraction is a critical issue in numerical simulation of free surface flows using the VOF model. This is especially the case in flows with high density ratios, where small errors in volume fraction may lead to significant errors in the calculations of physical properties. Moreover, accurate calculation of the phase fraction distribution is crucial for a proper evaluation of surface curvature, which is required for the determination of surface tension forces and the corresponding pressure gradient across the free surface. (22) An artificial surface compression term is added to the interface continuity equation to overcome the problem of a smeared-free surface. This technique is developed by Rusche (12) and defined in equation (5).

[partial derivative][[alpha].sub.1]/[partial derivative]t + [nabla] x ([[alpha].sub.1]u) - [nabla] x ([[alpha].sub.1](1 - [[alpha].sub.1])[u.sub.r[alpha]]) = 0. (5)

An artificial surface compression term leads to a sharper interface between the two different fluids, where [u.sub.r[alpha]], is the compression velocity expressed by equation (6). [C.sub.[alpha]] is the compression factor, which yields a conservative compression if the value is one and enhances compression for values greater than one. The compression factor provides no contribution if it is set to zero.

[u.sub.r[alpha]] = min ([C.sub.[alpha]][absolute value of u], [max.sub.G] [absolute value of u] [partial derivative][[alpha].sub.1]/[absolute value of [partial derivative][[alpha].sub.1]]]) (6)

The artificial surface compression term affects only the liquid surface due to [[alpha].sub.1] (1 - [[alpha].sub.1]]), whereby [max.sub.G][absolute value of u] is the global maximum value of the velocity field. According to the VOF method, calculation of physical properties is illustrated in equations (7) and (8) as a weighted average, based on the distribution of the liquid volume fraction, where [[rho].sub.1] and [[rho].sub.g] are densities of liquid and gas, respectively. The average viscosity is represented by [[mu].sub.f], that is shared by the two fluids throughout the flow domain. Correspondingly, [[mu].sub.1] and [[mu].sub.g] are the viscosities of liquid and gas, respectively.

[rho] = [[alpha].sub.1][[rho].sub.1] + (1 - [[alpha].sub.1]) [[rho].sub.g] (7)

[[mu].sub.f] = [[alpha].sub.1][[mu].sub.1] + (1 - [[alpha].sub.1]) [[rho].sub.g] (8)

Hence, the momentum continuity, equation (1), for the mixture of the two phases remains the same, but is treated with the mixture velocity and surface tension considered as an additional body force which develops as equation (9).

[partial derivative][rho]u/[partial derivative]t + [nabla] x ([rho]uu) = - [nabla][P.sub.rgh] - (g x x)[nabla][rho] + [nabla] x ([tau]) + [sigma]k [partial derivative][[alpha].sub.1]/[absolute value of [partial derivative][[alpha].sub.1]]], (9)

whereby k = -[nabla] x ([partial derivative][[alpha].sub.1]/[absolute value of [partial derivative][[alpha].sub.1]]]) is the curvature of the interface and a is the surface tension. (18)

Constitutive equations

Numerical simulation of free coating process for non-Newtonian fluids has been prepared by implementing the power law and Carreau models in the calculations. In this system, for Newtonian fluids, the rate of strain tensor is linearly related to the stress tensor, [tau], given by equations (10) and (1). The viscosity of fluid is characterized by [mu] which is a constant value for Newtonian fluids and superscript T denotes the transpose of the matrix.

[??] = (u + [([nabla]u).sup.T]), (10)

[tau] = -[mu][??]. (11)

The viscosity and shear rate expression for power law fluids is defined by equation (12). In this equation, [??] is the shear rate, K is the viscosity at shear rate of one reciprocal of second, and n, is the power law index.

[mu] = K[[??].sup.n-1]. (12)

For Carreau fluids, the viscosity and shear rate relationship is given by equation (13), in which [[mu].sub.0] is the zero shear rate viscosity and [[mu].sub.[infinity]] is the viscosity at the shear rate of infinity. [lambda] and n are two other parameters of Carreau constitutive equation.

[mu] = [[mu].sub.[infinity]] + ([[mu].sub.0] - [[mu].sub.[infinity]])[[1 + [([lambda][??]).sup.2].sup.(n - 1)/2]. (13)

Parameters of the power law and Carreau models are calculated by fitting the experimental data of rheological measurements.

Numerical procedure

In the current study, OpenFOAM is employed with the great advantage of parallel computing. The availability of parallel computing provides the quick simulation of complex problems, with great accuracy. In order to ensure stability of the solution procedure, the calculations are performed using a self-adapting time step which is adjusted at the beginning of the time-iteration loop based on the Courant number, defined as Co = (u/ [increment of x])[DELTA]t, where [DELTA]t is the time step and [increment of x] is the cell size in the direction of the velocity. Using values for velocity and [DELTA][t.sup.0] from previous time step, a maximum local Courant number [Co.sup.0] is calculated and the new time step is evaluated from equation (14).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (14)

where [DELTA][t.sub.max] and [Co.sup.max] are corresponding prescribed values for the time step and Courant number at the run time, and for this study the [Co.sup.max] is set to 0.5 and the value of [DELTA][t.sub.max] is 1. To avoid time step oscillations that may lead to instability, the increase of the time step is restrained using [[beta].sub.1] and [[beta].sub.1] as damping factors according to the conditions in equation (14). Considering the impact of the meshing on Courant number, it can be concluded that for small grid sizes, small time steps are needed to keep the Courant number below its maximum value.

In numerical simulation, the discretization process is usually carried out as a first step toward numerical evaluation and it proceeds by defining the boundary conditions of the simulation domain.

Grid generation

CFD problems usually require discretization of the problem into a large number of cells/grid points. In this work, simulation domains for cylindrical coating baths are considered with a radius of 4-65 times the radius of the substrate and height of 20-80 times the substrate radius. The cylindrical substrate was centered in the coating bath. Three-dimensional meshes have been created for finite volume analysis as discrete representation of the flow domain.

Mesh quality is an essential factor for an accurate solution. Therefore, the mesh around the cylindrical substrate is refined by adding 40-60 layers of mesh to increase the mesh density, because it is required to be sufficiently high in order to capture all the flow features and coating film thickness. Alternatively, to optimize the simulation time, a coarse mesh is used for areas far from the substrate. In the densified mesh region around the cylinder, the first layer of mesh is approximately 1 [micro]m and to ensure smoothness in mesh, a 2-5% increase in subsequent mesh layer thickness has been applied relative to the first layer. This percentage of increase in element size provides a smooth transition to the coarse mesh area. Figure 1 shows the cross section of grid with densified mesh layers around the cylindrical substrate. All meshes consist of hexahedral elements around the cylinder to generate body-fitted grids.

For the mesh independency study, the simulation was run on the initial mesh, and residuals were below [10.sup.-8] for variables, which was acceptable in the defined tolerance. When the convergence criteria were met for the first simulation, the mesh was refined globally and finer cells were generated throughout the domain with 1.5 times the initial mesh size. The simulation was carried out with globally refined mesh and residual error dropped below [10.sup.-8] with 2% difference with the first mesh results. Achieving relatively the same results for the film thickness and residual errors with the refined mesh indicated that the described mesh resolution was accurate enough to capture the coating film thickness on different cylindrical substrate sizes.

Boundary conditions

Boundary conditions need to be specified following the discretization of the simulation domain. Boundary conditions are defined for boundaries of coating bath and the cylindrical substrate and shown in Fig. 2. In these simulations, boundaries are set as coating bath wall, bath base, coating bath top, and cylindrical substrate. For each boundary, the pressure, velocity, and volume faction of [[alpha].sub.1] are defined.

Coating bath wall

A fixed flux pressure with zero value is set for pressure on the coating bath wall. This condition adjusts the pressure gradient on the boundary by zero value velocity and consequently zero flux. The zero gradient for volume fraction of [[alpha].sub.1] with a constant value for contact angle and zero value for velocity is set for this boundary.

Base of coating bath

For base region of the flow domain, the symmetry plane boundary has been set for velocity, pressure, and volume fraction of [[alpha].sub.1]. In OpenFOAM, this boundary condition is different from that of a zero gradient condition. Zero gradient sets the boundary value to the near-wall cell value whereas a symmetry plane has the same values for scalars but for vectors all components parallel to the patch are mirrored, while the normal components of vectors are set to zero.

Coating bath top

The upper part of the simulation domain, which contains the gas phase, is open to the atmosphere. For this region, total pressure condition with the value of zero is set for the pressure. Total pressure boundary condition is defined as P = [P.sub.0] + 1/2 [rho] [[absolute value of u].sup.2], where [P.sub.0] is the atmospheric pressure and when u changes, P is adjusted accordingly.

Pressure-inlet-outlet-velocity is the velocity boundary condition, applied for this region. This boundary condition can be employed, where the pressure is specified and reverse flow is possible or expected. When the flow is inward, the inflow velocity is set as a fixed value of zero and for the outflow, it is a zero gradient velocity condition.

For volume fraction of [[alpha].sub.1], the inlet-outlet boundary condition is set in this region, which switches u and P between fixed value and zero gradient, depending on the direction of velocity. This boundary condition provides a generic outflow situation by applying zero gradient condition and specific inflow for the case of return flow with setting specified value for velocity, which has been set to zero in this study.

Cylindrical substrate

Boundary conditions for cylindrical substrate are considered as fixed value velocity of the withdrawal speed, zero flux for pressure and zero gradient for volume fraction of [[alpha].sub.1].

Results and discussion

Validation of simulation results with previous experimental data

Middleman (23) reported film thickness data of 0.75% Polyox 301 solution as non-Newtonian coating fluid, and a cylinder with radius of 262 [micro]m was used for coating substrate. In addition, rheological properties of Polyox 301 solution are presented by Middleman (23) in the shear range of 1-1000 [s.sup.-1]. In the present work, non-Newtonian effects on the coating film thickness have been investigated by power law and Carreau constitutive equations in the simulations. The power law and Carreau models are fitted to the reported rheological data of Middleman (23) as illustrated in Fig. 3.

In addition, according to the rheological data of viscosity as a function of shear rate for 0.75% solution of Polyox 301 in water, associated parameters of power law and Carreau models are correlated to the reported rheological data and values are presented in Tables 1 and 2, respectively. The results presented in Fig. 4 are the numerical solution for coated film thickness for a wide range of withdrawal velocities for the cylindrical substrate with the radius of 262 [micro]m in the non-Newtonian fluid.

Carreau model has the capability of coating film thickness prediction in withdrawal velocities up to around 6 m/s when the coating layer is in the same order of substrate diameter, and close agreement was achieved between simulation results and experimental data.

Roy and Dutt (24) reported film thickness data for mineral oil as Newtonian fluid and data are presented for two cylindrical substrates with radii of 317 and 445 [micro]m. Physical properties of mineral oil are presented in Table 3, which were specified by Roy and Dutt. (24)

Simulation data obtained for the final film thickness through numerical solution are compared with both experimental data and analytical solution of Roy and Dutt. (24) This analytical solution is only used for the estimation of final film thickness for Newtonian fluid and was not accurate for the prediction of non-Newtonian coating film thickness for 0.75% solution of Polyox 301 for high withdrawal velocities. Figure 5 illustrates the results of coating mineral oil onto the cylinders of radius 317 and 445 [micro]m.

Capillary number is defined as Ca = [[mu].sub.u]/[sigma] and as a dimension less number, plays an important role in dipcoating processes. Increasing the capillary number can be associated with high withdrawal speed or high viscosity of coating fluid increasing the film thickness. These results are expected, and film thickness growth with increasing the withdrawal velocity is shown in Figs. 4 and 5. The analytical solution of Roy and Dutt (24) as mentioned in their publications is restricted to small capillary numbers, which for mineral oil is valid up to a capillary number of 0.7 and limited to withdrawal velocities less than 0.8 m/s as shown in Fig. 5. The simulation results of the mineral oil coating film thickness approach a capillary number of about 1.1. Considering the viscosity at a shear rate of 1 [s.sup.-1] for the solution of Polyox 301, the capillary number can be up to 60 for non-Newtonian simulations.

Wall effects

Wall effects in dip-coating process are considered as additional factors for studying the deposited film thickness. Understanding the role played by this parameter facilitates controlling the coated film in a more precise manner.

Experimental

Coated film thickness is measured above the coating bath surface at the point that the film thickness reaches a constant value and this point is considered to be far from the coating meniscus. Images are captured using a high-speed, high-resolution camera, adjusted to capture 15 frames per second (IO Industries camera model Flare 4M180-CL with resolution of 2048 x 2048 pixels). The camera is positioned about 30 cm above the bath surface and perpendicular to the cylindrical substrate. A black piece of paper was placed in the background and light sources were provided around the coated film, which helped to ensure a clear view of the deposited film. Film thickness images have been captured by repeating the measurements eight times for each velocity and individual coating bath size. The precision of the measurements of film thickness is estimated to be within [+ or -]10 [micro]m over most of the range, without taking the parallax effect into account.

Different velocities are applied by changing the speed of dip coater apparatus motor, with velocities limited to 15 cm/s. Acrylic tubes with different radius sizes are used as the coating baths for measuring the wall effects in the system. Food grade mineral oil (FG WO 35 White Mineral Oil, Petro-Canada) is selected as the Newtonian fluid, and 0.75% solution of polyethylene oxide (POLYOX WSR-301, Dow Chemical) with the molecular weight of 4,000,000 has been chosen as the non-Newtonian fluid for the wall effects study. The cylindrical substrates used in this work have radii of 785 [micro]m and 1.59 mm, chosen based on the radii to be small enough for considering the substrate curvature in the same order of magnitude of coating film thickness and yet large enough to be able to capture the deposited film thickness in a photograph. Figure 6 illustrates a schematic of the dip-coating apparatus used in experiments and an actual picture of the coated substrate used for the film thickness measurement is shown as a subfigure.

Rheological data of polyethylene oxide solution differ based on the preparation method of solution and some of the effective factors are mixing time and intensity of mixing in producing the solution. Therefore, the rheological data are attained using a rheometer (Rheometrics model RDS-II) for 0.75% POLYOX WSR-301 solution. Rheological tests have been conducted in a parallel plate geometry (diameter of 25 mm) and five samples are taken from various parts of the solution to ensure the homogeneity of the prepared solution. Measurements for each sample were repeated three times to ensure the accuracy of the rheological quantifications. All rheological measurements were conducted at 22[degrees]C, through setting the temperature of the rheometer at 22[degrees]C and homogeneity of the samples temperature was confirmed by measuring it before the rheological tests. In addition, total time of rheological tests for each sample was arranged from 2 to 4 min; thus, the effect of drying of the sample at the edges was not observed for 0.75% POLYOX WSR-301 solution. Measured rheological data for 0.75% POLYOX WSR-301 solution are presented in Fig. 7, and the power law and Carreau models are fitted to the measured rheological data, similar to what has been employed for the rheological data of Middleman. (23) Parameters of the power law and Carreau models are calculated based on the rheological data of Fig. 7, and corresponding values are reported in Tables 4 and 5, respectively. Physical properties of food grade mineral oil are presented in Table 6.

The contact angle must be applied in the numerical calculations to recognize the effects of the coating bath wall proximity to the substrate, on the film thickness. In particular, the angle formed by the free surface with the solid substrate when the three phase contact line is moving is described as the dynamic contact angle. The dynamic contact angle depends on the contact line speed and the entire flow field in the vicinity of the moving contact line. (25) Based on the images taken from the contact angle during the experimental measurements, the contact angle remains relatively constant, while the substrate is being withdrawn and coating liquid level declines gradually in the coating bath. The relatively constant value of the contact angle can be due to the low withdrawal velocities applied in the experimental process and the steady-state condition that develops in the flow field in the surrounding area of the contact angle, in which the contact line is moving, but the angle does not change significantly. A constant value assumed for the dynamic contact angle and behavior of the dynamic contact angle is approximated with a static contact angle ([theta]) between the fluids and an acrylic surface which measured with a contact angle goniometer (Rame-hart model 100).

The static contact angle is implemented in the numerical calculations to recognize its effects in the coating process and final coating film thickness. In the study of wall effects in dip-coating process, the ratio of bath radius over cylinder substrate radius (R/r) is considered in the range of 4-60. The practical minimum value for R/r is 4 to avoid oscillation of the cylindrical substrate during withdrawal and maintain the concentric position of the substrate in the coating bath.

Simulation

Food grade mineral oil and 0.75% solution of POLYOX WSR-301 in water were selected for the investigation of Newtonian and non-Newtonian fluids, respectively. The coating film thickness was measured at various withdrawal velocities. The wall effect was considered as a function of (R/r), which is a coating bath radius over the cylindrical substrate radius. The simulation domain, bath radius (R), and cylinder substrate radius (r) are illustrated in Fig. 8.

The film thickness is calculated for infinite bath length by applying symmetry plane boundary condition at the base of the bath. The three-dimensional, free surface numerical simulation result shown in Fig. 8 demonstrates the parameters for simulation. In simulation of dip-coating, the level of coating fluid in the bath decreases gradually by withdrawing the substrate from bath, and contact angle at the bath wall remains constant, similar to the images captured through the experimental measurements. Numerical results are shown in Fig. 9 for the Newtonian, power law, and Carreau models. The simulation and experimental data in Figs. 9a, 9b, 9c are for the cylindrical substrate of radius 785 pm, and results in Figs. 9d, 9e, 9f are for a substrate with radius of 1.58 mm.

Enhancing the R/r ratio increases the deposited film thickness on the cylindrical substrate. Although the power law model under predicts the experimental data, it does capture the rising trend of film thickness with increasing R/r as shown in Figs. 9b and 9e. The Carreau constitutive equation has been applied in the simulation of coating thickness prediction, with results presented in Figs. 9c and 9f indicating very good agreement between simulations and experiments.

The numerical and experimental results reveal that there is a plateau value of the coating thickness as a function of the ratio of bath radius over cylinder radius. Below this value of R/r, the deposited film thickness increases with increasing R/r at a constant withdrawal velocity. The results for the substrate with the radii of 785 [micro]m demonstrate that for R/r less than 30, wall effects need to be considered in the coating process; and for R/r > 30, a plateau coating thickness is reached for film thickness values at constant withdrawal velocity. Similarly, for a substrate radius of 1.58 mm, coating thickness increases with increasing R/r up to approximately 25.

Analysis of the flow field of the zone beneath the bath surface helped to understand the wall effects of the coating bath proximity on the development of flow streams and consequently on the coating film thickness. Flow streams reveal the coating liquid circulation in the bath. The results indicate, reducing the distance between the coating bath wall and the substrate forces the coating liquid in the bath to circulate in a narrow zone. The circulation of flow in a more constricted region affects the location of the stagnation point. In the flow field, stagnation point is near the meniscus region, where part of the flow engages in the coating and adheres to the substrate and part of it returns back to the coating bath. Results verify that the stagnation point moves closer to the substrate surface with decreasing distance of the bath wall to the cylindrical substrate, resulting in decreasing coating film thickness. In Fig. 10, the flow field in the free coating bath is displayed and flow streams indicate the coating liquid circulation and the stagnation points.

Conclusion

Numerical solution for two-phase flow in dip-coating process is investigated and the simulation results indicate significantly better estimation of the coated film thickness compared to analytical models over a wide range of withdrawal velocities.

Three constitutive equations have been considered for numerical solution of both Newtonian and non-Newtonian fluids and compared to experimental data in the literature. Wall effects in dip-coating show that the coating film thickness depends on the coating bath dimension. The rheological behavior of fluid and calculation of coating film thickness for non-Newtonian fluids indicate very good agreement using the Carreau model as compared with experimental data. Based on experimental and simulation results, the wall effect needs to be considered as an extra factor when R/r is less than a plateau value and this value decreases with increasing cylindrical substrate radius. For both Newtonian and non-Newtonian fluids, the final film thickness increases with raising the ratio of R/r, before achieving a plateau. The desired final thickness is affected by controlling the withdrawal velocity, physical properties of coating fluid, substrate geometry, and wall proximity. Simulation of the dip-coating process, regarding the wall effects has been completed in this work and confirms that the wall influences the deposited film thickness and stagnation point.

DOI 10.1007/s11998-015-9699-7

M. Javidi, A. N. Hrymak ([mail])

Department of Chemical and Biochemical Engineering, University of Western Ontario, London, ON, Canada

e-mail: ahrymak@uwo.ca

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Table 1: Physical properties and parameters of Polyox 301 (0.75%) solution for power law fluid [sigma] [rho] n K 0.062 N/m 1000 kg/[m.sup.3] 0.64 0.66 (kg/m s) [s.sup.n-1] Table 2: Physical properties and parameters of Polyox 301 (0.75%) solution for Carreau fluid [[mu].sub. [sigma] [rho] n [lambda] [[mu].sub.0] [infinity]] 0.062 N/m 1000 kg/ 0.47 1.04 0.92 kg/ms 0.01 kg/ms [m.sup.3] Table 3: Physical properties of mineral oil (24) [sigma] [rho] [mu] 0.0284 N/m 876 kg/[m.sup.3] 0.025 kg/ms Table 4: Physical properties and parameters of POLYOX WSR-301 (0.75%) solution for power law fluid [sigma] [rho] n K [theta] 0.062 N/m 1003 kg/[m.sup.3] 0.62 0.76 (kg/ms) 65[degrees] [s.sup.n-1] Table 5: Physical properties and parameters of POLYOX WSR-301 (0.75%) solution for Carreau fluid [sigma] [rho] n [lambda] 0.062 N/m 1003 kg/ 0.44 1.224 [m.sup.3] [[mu].sub. [mu].sub.0] [infinity]] [theta] 1.03 kg/ms 0.012 kg/ms 65[degrees] Table 6: Physical properties of food grade mineral oil for Newtonian fluid [sigma] [rho] [mu] [theta] 0.028 N/m 864 kg/[m.sup.3] 0.036 kg/ms 11[degrees]

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Author: | Javidi, Mahyar; Hrymak, Andrew N. |
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Publication: | Journal of Coatings Technology and Research |

Geographic Code: | 1USA |

Date: | Sep 1, 2015 |

Words: | 6236 |

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