Numerical simulation of the water-based ink transfer process in roll-to-roll gravure printing based on fluid–solid interactions

As one of the four major printing methods, gravure printing is mostly applied in the publishing, packaging, building materials decoration, and clothing printing fields. Gravure products have rich and clear layers and bright and vivid ink colors and truly reflect the original image, which have made it increasingly popular.^1–3 Numerous studies on gravure printing have been performed, and it has been extensively used in the production of organic solar cells,⁴ radio frequency identification (RFID) tags,⁵ quantum dot light-emitting diodes (QLEDs),⁶ and so on.

At present, continuous-running roll-to-roll gravure printing technology has attracted attention for its simple mechanical process and few control variables and has seen a wide-ranging use in the field of electronic component printing.^7,8 To improve its reliability, researchers have investigated the matter from a variety of perspectives. Among several technical issues, the ink transfer process plays an important role in enhancing printing quality and saving on costs. The realization of the ink transfer process relies on the extrusion and mutual rotation between the impression roller and the plate roller. The extrusion between the impression roller and the plate roller exerts a force on the ink and changes the fluid flow state; at the same time, the ink also has a reaction force on the impression roller so that the rubber layer of the impression roller is deformed to a certain extent, which, in turn, causes the ink channel width between the two rollers to change and affects the ink transfer. To achieve high-quality printing, it is necessary to ensure the stability, uniformity, and proper amount of ink transfer. It can be seen that the study of the ink transfer is of great significance for the control of the ink quantity in the gravure printing.

In early computational work, Powell et al. used a Lagrangian finite element algorithm with time-varying free surface flow to simulate the liquid-transfer process by inverting the gravure cavity and moving the substrate in a downward speed.⁹ They found that increasing the speed of the printing substrate increased the break time of the liquid filament between the gravure printing cavity and the substrate, which promoted more liquid removal from the cavity. Dodds et al. solved the Navier–Stokes equation under appropriate boundary conditions to study the transfer of Newtonian liquid from the cavity to a moving plate during the stretching process of an axisymmetric liquid bridge.¹⁰ As the Reynolds number increased, the breaking point of the liquid filament gradually approached the bottom of the cavity, effectively improving the emptying of the liquid. Nevertheless, due to limitations with numerical simulation, those studies did not simulate the fracture of liquid filament after stretching. To understand the liquid-transfer process more intuitively, Sankaran and Rothstein used a modified version of a capillary breakup rheometer to simulate the breaking and stretching of the liquid under a magnified ideal gravure cavity to study the influence of fluid viscoelasticity; the flow dynamics in the cavity were recorded using high-speed photography.¹¹ Although experimental studies can visually describe the liquid-transfer process, it is difficult to obtain specific data on the transfer process. With the rapid development of computer technology, computer simulation has been fully applied in various fields, and its accuracy has been confirmed.^12–14 Huang et al. used the volume of fluid (VOF) method to simulate the liquid-transfer process between a trapezoidal cavity and a substrate.¹⁵ They considered different contact angles, cavity shapes, and initial distances between the substrate and the cavity, as well as the effect of the vertical velocity. Huang et al. used Newtonian fluids and shear-thinning non-Newtonian fluids. Dong et al. first used shear-thickening non-Newtonian fluids to conduct a similar study on the ink transfer process, and the effect of ink properties was fully considered.¹⁶ However, all of them ignored the effect of the horizontal velocity. Kim et al. used a phase field model based on a finite element formulation to study the influence of the vertical and horizontal velocity caused by cycloid movement on the ink transfer ratio in gravure offset printing.¹⁷ Unfortunately, the liquid used in the experiment was Newtonian fluid, where the real ink is a non-Newtonian fluid.

From a review of the literature, we found that there are still two gaps in the research of the ink transfer process. First, the liquids used in most simulations are Newtonian and shear-thinning liquids, while there are very few works on transfer simulation of shear-thickening liquids. Therefore, we decide to use a shear-thickening water-based ink to further study and hope this paper is helpful for filling this gap. Second, printing pressure and velocity are two important factors that affect printing quality.^18,19 Most studies just discuss the influence of the printing velocity on ink transfer, and the effect of printing pressure has been ignored. Actually, printing pressure has a great influence on the printing quality. Under suitable printing pressure, the ink can be transferred from a plate roller to substrate as much as possible; when the printing pressure is insufficient, the amount of ink transferred is reduced and the satisfactory original reproduction products cannot be obtained; in addition, excessive printing pressure will lead to partial overfilling of the ink, which will cause serious deformation of some graphic and text dots and uncoordinated color tone of the printed matter. This paper adds printing pressure and velocity to the modeling to make the simulation process of roll-to-roll gravure printing more realistic than previous studies.

Thus, herein, we introduce a power-law fluid model to describe the shear-thickening properties of the water-based ink. We take the roll-to-roll gravure printing system as the research object and apply static analysis and the fluid–solid interaction (FSI) method to simulate the whole process of gravure printing, as well as the effect of ink layer thickness (ILT) on ink transfer. The remainder of this paper is organized as follows. In Sec. II, mathematical models describing the properties of fluids and solids are described. Section III introduces the fluid governing, solid governing, and FSI governing equations. In Sec. IV, the actual gravure printing process is simulated and the results are analyzed in detail. Finally, Sec. V summarizes our conclusions.

Water-based ink is a kind of environment friendly ink, mainly composed of water-soluble resin, pigment, solvent, and related additives.²⁰ It is also a common non-Newtonian fluid. Some non-Newtonian fluids exhibit shear thinning and shear thickening, which are described by using the power-law fluid model, in general,²¹ 

where η is the apparent viscosity, K is the consistency coefficient, ε is the shear rate, and h is the power-law index. In this study, a Brookfield RST rheometer was used to measure the rheological properties of the water-based ink, and the power-law fluid model was fitted to obtain Κ = 0.04 Pa s^−h, h = 1.4. The relationship between the apparent viscosity (η) and the shear rate (ε) is shown in Fig. 1. It can be seen that the viscosity of the water-based ink increased with an increase in the shear rate. In addition, the change in the ink shear stress (τ) and shear rate (ε) is described in Fig. 2. We found that the shear stress and shear rate of the water-based ink showed a significant nonlinear positive correlation, which is consistent with the characteristics of the shear-thickening fluid. That is, the ink used in this experiment has the physical characteristics of shear thickening. For such liquids, the apparent viscosity depends on more than just the shear rate.²² Therefore, the power-law fluid model simply provides a qualitative prediction of liquid shear thickening.

A roll-type gravure printing machine was used as the experimental model, using ANSYS software to simulate the transfer process of the water-based ink. The machine has the advantages of high printing velocity, accurate overprint, and stable output, among others and is suitable for printing various common plastic films, such as PE, PP, and PVC. The radius of the plate roller R₁ and the radius of the impression roller R₂ are both 75 mm, and the roller length L is 300 mm. The plate roller material is No. 45 steel, the elastic modulus E₁ is 2.09 GPa, the Poisson’s ratio v₁ is 0.269, and the density ρ is 7890 kg/m³. The impression roller is made up of an inner steel core and an outer rubber layer. Because the elastic modulus of the steel core is much larger than that of rubber and the internal steel core hardly deforms, the middle hard roller part of the model is omitted. Only the rubber layer part is constructed; it is NBR rubber with a shore hardness (HS) 60 and thickness δ of 12.5 mm.

As a typical nonlinear material, the elastic mechanical properties of rubber depend on various factors such as material hardness, applied load frequency, and magnitude. It is extremely difficult to describe its stress–strain relationship with an accurate mathematical model. At the moment, constitutive models are mostly used to describe it.²³ Among them, the two-parameter Mooney–Rivlin model has high accuracy in characterizing the stress–strain relationship of rubber material under small strain, which can fully reflect the mechanical properties of the material, so it is the most widely used in engineering applications.²⁴ It is defined as^25,26

where W is the strain potential energy, C₁₀ and C₀₁ are material constants representing the shear deformation of the material, I₁ and I₂ are the first deformation tensor and the second deformation tensor, d is the material incompressibility coefficient, and J is the elastic volume ratio.

For incompressible rubber materials, the elastic modulus E₀ and shear modulus G have the following relationship with the material coefficient under small deformation:²⁷ 

HS can be converted into E₀ using the following equation:

The establishment of the two-parameter Mooney–Rivlin model can be completed referring to the approximate relationship curve between C₀₁/C₁₀ and HS measured by Wang et al.²⁸ The relevant material parameters of NBR with an HS of 60 were obtained using the following values in the calculation: C₁₀ = 0.579 928 MPa, C₀₁ = 0.023 197 MPa, and d = 0.016 580 MPa⁻¹. The hardness is very easy to measure using a durometer in practical applications, so this method can conveniently and economically determine the coefficient of rubber material and facilitate subsequent engineering analysis.

The ink is considered an incompressible fluid, and the movement characteristics of viscous incompressible fluid are commonly described using by the mass conservation equation and the momentum conservation equation,²⁹ 

where ρ is the fluid density, S_m is the source term, p is the pressure on the fluid micro-unit, τ_ij is the stress tensor, and g_i and F_i are the gravitational volume force and external volume force in the i direction, respectively. F_i contains other model-related source items.

The rubber is a nonlinear elastic solid material. In large deformation problems, six basic strain components can be used to describe the deformation at any point in a nonlinear elastic body,^30,31 which are expressed as

where ɛ_xx, ɛ_yy, ɛ_zz, γ_xy, γ_yz, and γ_zx are the strain components describing any point of the elastic solid.

When an elastic deformable body undergoes a large deformation due to an external force, the balance condition of the external force and the internal force acting on the object is very complicated.³² In the space rectangular coordinate system, (x, y, z) represents the position of the center of gravity of the object. Then, the force balance equation is given as follows:

where σ_x, σ_y, and σ_z represent the corresponding normal stress; τ_xy, τ_yz, and τ_zx represent the corresponding shear stress; and F_x, F_y, and F_z represent the force in the corresponding direction.

The constitutive relationship of material stress and strain is given as

where E is the elastic modulus, v is the Poisson ratio, and θ is the first strain invariant.

FSI transfers data through the interface between a fluid and a solid and obtains the flow field and solid change characteristics by solving the fluid governing, solid governing, and FSI governing equations. At the interface, the displacement of the fluid element must match that of the solid element. Based on Newton’s third law, the force exerted by the fluid domain and the solid domain at the interface is equal, and the governing equation of the interface boundary should be satisfied,³³ 

where d_f and d_s are the displacements of the fluid domain and the solid domain at the interface, respectively, τ_f and τ_s are the stresses of the two domains at the interface, and n_f and n_s are the normal vectors, respectively. Equation (10) represents the basic governing equation in the application of FSI. In actual analysis, a general governing equation can be established, on which various given parameters and boundary conditions can be added to solve together.³⁴ 

This research chiefly included the following processes. First, the contact surface of the two rollers is defined as an asymmetric binding contact in the static structural module, and the augmented Lagrange method is adopted for static analysis of the solid deformation under different printing pressures. The results are compared to theoretical calculations to determine whether the simulation results meet the needs of subsequent analysis. Second, the pressure distribution at the FSI is obtained by solving the fluid governing equation on the ANSYS Workbench, and the rubber layer variable of the impression roller under the pressure of the flow field is further calculated to obtain the width of the ink flow channel. Finally, based on the obtained solid deformation, the flow field model is constructed in AutoCAD and imported into the Fluent module; the semi-implicit method for pressure-linked equation (SIMPLE) algorithm is adopted for the numerical simulation. Considering the stability of the pressure correction equation, the second-order upwind equation is used to discretize the governing equation. Finally, the effective ink layer thickness (ET) before and after the ink passes through the gap between the two rollers is extracted in the CFD-post module of ANSYS, and the effective ink transfer ratio (ER) is obtained after processing the results.

In the actual printing process, both the plate roll and the impression roll are in a state of rotating and pressing each other. For the convenience of study, the process can be divided into three typical stages: the static compression stage, where the two rollers are extruded with each other to form a static contact; the initial moment of ink transfer, when the two rollers press against each other and the printing plate roller begins to rotate at a certain speed. Under the action of the ink pressure, the rubber layer will incur a certain deformation, forming an ink flow channel. The third stage is the process of ink passing through the gap between two rollers and the separation of ink layer with the rotation of two rollers. The rollers rotate at a stable velocity, and the ink transfer process is complete. The three stages are analyzed, in turn, below.

The static compression stage mainly involves the static effects of materials and structures under a fixed load. In general, the effects of inertia and damping are not taken into account. Because this experiment primarily focuses on the stamping process of gravure printing, only the impact of printing pressure on the rubber layer is considered; the influence of the cavity micro-structure caused by the pressure of the squeegee and the weight of the plate roller are ignored. In addition, the substrates are generally relatively thin and have a small absolute deformation;³⁵ therefore, our model does not consider the substrate. To facilitate finite element analysis and reduce computational cost, the roll-to-roll gravure printing system is appropriately simplified, ignoring the millions of micron-sized cavities on the plate roller. Figure 3 represents a schematic diagram of roll-to-roll extrusion in the static compression stage, where F is the printing pressure.

Generally speaking, printing pressure refers to the pressure that must be applied to transfer the words and images coated with the ink to the substrate. The printing pressure is controlled by the pressure-regulating valve,³⁶ and the actual pressure is calculated as follows:

where F is the printing pressure, D₀ is the roll diameter, and P is the dial reading.

Under static conditions, according to the empirical formula, the relationship between the printing pressure at the midpoint of the contact width and the deformation of the rubber is given as

where Q is the printing pressure at the midpoint of the contact width, m is the material coefficient, E₁ is the elastic modulus of the rubber layer, ζ is the relative deformation of the rubber layer, λ is the absolute deformation of the rubber layer, and δ is the thickness of the rubber layer.

We find that the contact surface of the two rollers is locally deformed under the action of the printing pressure, forming a contact surface that is approximately rectangular. The contact width between the two rollers (b) can be obtained using Hertz’s elastic contact theory,^37,38

where L is the length of the roller, v₁ and v₂ are the Poisson ratios of the two roller materials, E₁ and E₂ are the elastic modulus of the two roller materials, and R₁ and R₂ are the radii of the two rollers.

Because the elastic modulus of the plate roll material is much greater than that of the rubber and R₁ = R₂ = R, Eq. (13) can be simplified as³⁹ 

Through the contact width between the two rollers and the radius of the two rollers, it is easy to obtain the maximum deformation of the rubber layer λmax as follows:

The boundary conditions are defined, and the load constraints are applied using the ANSYS static structural module. Next, λmax and b were calculated for different printing pressures, and the results are shown in Fig. 4. Both parameters increased with an increase in printing pressure. The results from simulations showed good agreement with the theoretical calculations, verifying their robustness, indicating that the simulation results met the needs for subsequent analysis.

Previous experimental studies have reported that the ideal printing pressure range is 0.2–0.4 MPa.^36,40 We can also see from Fig. 3 that when the printing pressure is greater than 0.5 MPa, the simulated and theoretical values of λmax and b begin to deviate, and the deviation increases with an increase in the printing pressure. To reduce the abrasion of printing machine components and ensure efficient transfer of ink, a pressure that causes only slight deformation should be selected. Consequently, a printing pressure F = 0.2 MPa is selected for subsequent experiments and analysis.

Figure 5 represents a schematic diagram of the initial moment of ink transfer, at which time the two rollers press against each other and the ink coats the left side of the plate roller. As the plate roller rotates, the ink is transferred to the rubber layer of the impression roller. Under the pressure of the ink flow field, the rubber deforms to a certain extent, separating the two rollers along these ink flow channels. To obtain the ink flow channel width s, the fluid and solid models and the FSI method are used to simulate the initial moment of ink transfer under different ILT α. When the ink is about to pass through the gap between the two rollers, only the ink layer and the rubber layer are modeled because the deformation of the plate roller in this process is negligible. Furthermore, in order to reduce the computational cost, only a local physical model is established.

Figure 6 describes the movement of the fluid in the flow field. The ink velocity increases as the distance to the plate roller decreases. This is because the viscosity of the ink itself leads to a kind of viscous resistance between different flow layers, which affects the flow state of ink. When the ILT is less than 72 µm, the internal flow velocity of the ink layer changes continuously from high to low; when ILT is increased to 73 µm, a wide range of velocity vortices appear in the ink layer, which disrupts the continuity and causes the ink backflow. We refer to this here as the “ink backflow” phenomenon. This phenomenon might be an important factor affecting ink transfer.

To obtain the ink flow channel width (s) under different ILT α, FSI experiments were conducted under a printing pressure F = 0.2 MPa and printing velocity n = 200 rpm (Fig. 7). The ink flow channel width gradually increases as the ink volume increases when the ink layer is thin. This is because as the ILT increases, the contact area between the ink and the rubber also increases, and the pressure generated by the ink continues to act on the rubber, resulting in a larger deformation of the rubber. When the ILT is 25–70 µm, the ink flow channel width is maintained at about 18 µm, and the deformation of the rubber layer is relatively stable; when the ILT increases from 70 to 75 µm, the ink flow channel width rises sharply with a maximum at 22.066 µm; then, the ink flow channel width decreases considerably with a further increase in ILT. The reproduction of tonal gradations in gravure products is dependent on the depth of the cavity, and in general, ILTs below 50 µm are sufficient for most printing applications. The simulation results also show that, when there is no ink backflow phenomenon, the ink flow channel becomes wider (30–50 µm ILT) and the ink passes more easily through the gap between the two rollers; the largest ink flow channel width is 19.899 µm under an ILT of 50 µm.

According to the velocity nephogram in Fig. 5, when the ILT exceeds 73 µm, velocity vortices inside the ink layer change the flow state of the ink. According to Fig. 6, when the ILT is 70–75 µm, the width of the ink flow channel increases rapidly and then gradually decreases with an increase in ILT. Hence, we infer that the internal flow state of the ink changes abruptly when the ILT exceeds 73 µm; at this point, the rubber layer deforms, changing the width of the ink flow channel between the two rollers.

To understand the flow characteristics of the ink in the stage of ink passing through the gap between two rollers and the separation of ink layer with the rotation of two rollers, a fluid model was established in AutoCAD based on static deformation, as shown in Fig. 8. The rubber layer deformation has been in a stable state and no longer changes during this process, and this stage is to study the effect of previous solid deformation on the fluid flow state. Therefore, the solid model is ignored. Flow field analysis under different ILT α is carried out using the Fluent module.

Figure 9 shows a velocity nephogram of ink flow at an ink transfer ratio of 50%, printing pressure F = 0.2 MPa, and printing velocity n = 200 rpm. The ink velocity increases with a decrease in the distance from the two rollers. As the ink passes through the gap, the internal flow velocity continuously decreases. Inside the gap, the shearing effect introduces a small number of extremely small vortices, which lead to sudden spikes in the local ink velocity. However, the effect on ink transfer is negligible due to their small number and very small areas.

Assuming that the ILT on the plate roller before passing through the gap is α, the ILT remaining after the gap is β, and that transferred to the rubber layer is γ; then, the ink transfer ratio f can be calculated as follows:

The ink transfer ratio f is usually determined by the ILT on the two rollers before and after ink transfer, while the ER f* is calculated by determining the ET before and after the ink passes through the gap between the two rollers in the CFD-post module according to whether the ink is flowing in the flow field.⁴¹ Compared to the ink transfer ratio, the ER can more truly reflect the ink flow state, so this study uses the ER f* to evaluate printing quality. Assuming that the ET on the plate roller before passing through the gap is α*, the ET on the plate after ink transfer is β*, and the ET transferred to the rubber layer is γ*; then, the ER f* can be calculated as

Next, we obtain the ER under different thicknesses at an ink transfer ratio of 50%, a printing pressure of 0.2 MPa, and a printing velocity of 200 rpm (Table I). When ILT α is 25–50 µm, the ET α* increases approximately linearly with an increase in ILT. The same is true at 50–70 µm, but the increase is less than before. However, at an ILT exceeding 70 µm, the ET begins to show irregularities. In addition, because part of the ink does not flow during the ink transfer process, the ET is generally lower than the ILT.

The effect of ILT α on ER f* is depicted in Fig. 10. When the ILT is 25–50 µm, the ER is stable at 50%; when it is 50–70 µm, the ER gradually increases with an increase in ILT; and when it exceeds 70 µm, the ER begins to show irregularities. This segmental change with an increase in ILT is consistent with the changes in ET before passing through the gap. It can be known that the ink backflow phenomenon caused by the increase in ILT worsens the ink flow state, resulting in an irregular ER.

In summary, under the printing conditions considered herein, the critical value of ILT is 70 µm; thus, the actual ILT should not exceed this critical value to ensure the stability of ink transfer. In fact, the maximum depth of the current electronic engraving gravure cavity is 60–70 µm, so the water-based ink can be printed under the plate to avoid ink backflow. When the ILT is in the range of 30–50 µm, there is no ink backflow phenomenon and the width of the ink flow channel is relatively large, and the ER is almost stable at 50%.

In order to achieve high-quality printing, it is necessary to ensure the stability and uniformity of ink transfer. The understanding of the roll-to-roll gravure printing ink transfer process is helpful for enhancing printing quality and saving on costs. This work took the roll-to-roll gravure printing system as the research object. Based on actual measurements of the rheological properties of the water-based ink, static analysis and the FSI method were combined for the first time to simulate the dynamic extrusion and rotation process of gravure printing using ANSYS software. The process was divided into three main stages: static compression, the initial moment of ink transfer, and the process of the ink passing through the gap between two rollers and the separation of the ink layer with the rotation of two rollers. The coupling between the fluid flow state and the solid deformation in the whole process of roll-to-roll gravure printing is fully studied.

The effect of ink layer thickness on ink transfer was studied in detail. In the first stage, the maximum deformation of the rubber layer, the width of the contact zone, and printing pressure all had nonlinear and positive correlations. The simulation results were basically consistent with the theoretical values. In the second stage, an ink backflow phenomenon was noted: the ink thickness increased to a critical value, and a wide range of velocity vortices appeared in the ink layer. In the third stage, velocity vortexes appeared in the liquid as it passed through the gap between the two rollers, which made the local ink velocity suddenly increase.

It is worth emphasizing that the ink backflow phenomenon was found to significantly worsen the ink flow state and destabilize the ER. The critical ILT at which this occurred was 70 µm (at a printing pressure of 0.2 MPa and a printing speed of 200 rpm). Hence, the ILT, in practice, should not exceed this critical value to ensure the stability of ink transfer. When the ILT was 30–50 µm, there was no ink backflow phenomenon and the ink flow channel width was relatively large, and the ER is almost stable at 50%.

The critical thickness of the ink layer can be determined by simulation in this paper, and it is consistent with the actual maximum depth of the current electronic engraving gravure cavity. Compared to traditional electronic engraving gravure printing, laser engraving gravure printing can engrave U-shaped cavities with deeper cavities and larger ink storage capacity, which greatly improves the ink transfer ratio. However, as we have shown, an ILT that is too large will destabilize the ER. The ILT of gravure products is basically determined by the cavity volume and the amount of ink released from the cavity in the printing process and cannot be adjusted during printing. As a consequence, when making plates, whether the ink backflow phenomenon will occur under the current ILT should be carefully considered. In addition to printing, the numerical model can be extended to all industrial study fields involving roll-to-roll structures (such as coating, textiles, film-manufacturing, and papermaking). For example, by determining the critical coating layer thickness value, the amount of coating liquid can be effectively reduced and the cost can be saved under the premise of ensuring the stability of coating quality.

In fact, there are millions of micron-sized cavities in the plate roller. Due to the limitations related to computational power and the ANSYS software, we simplified the printing plate model herein, ignoring the existence of walls and regarding the ink layer as a uniform flow field. Therefore, there are shortcomings to this paper. In the following research, the ink transfer mechanism in a single cavity structure can be further studied. In addition, due to the limitation of objective conditions, the influence of vortex in ink backflow has not been deeply analyzed in this paper. It is hoped that later researchers can continue to study in this field and reveal its essence and mechanism from a theoretical point of view.

The authors have no conflicts to disclose.

X.G. and J.X. contributed equally to this work and should be considered first authors.

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