Introduction

With the development of science and technology, flexibility and wearability have become important characteristics for electronic products1,2,3, such as flexible displays4,5, solar cells6,7, RFID systems8,9,10, and wearable health-care sensors11,12,13. Printing technologies for printing electronics, such as screen printing14,15, inkjet printing16,17, and transfer printing18,19,20, have been developed and studied extensively because of their low cost and large areas21. Inkjet printing, as a mature digital controlled technology, has features of noncontact printing22 and pattern designability23 and is widely used for the printing of conductive24,25,26, semiconductive27,28 and dielectric29,30 materials. In inkjet printing technologies, continuous inkjets always waste ink31, and advanced inkjet printers such as those with aerosol jets32 and electrohydrodynamic jets33 are specialized for small-scale refined patterns. Considering the printing scale, efficiency, and cost, a conventional drop-on-demand (DOD) inkjet printer is very suitable for printed electronics34. However, there are also several problems associated with fabricating patterns by a DOD inkjet. Individual droplets cannot be exactly arranged along the bevel outlines, and uneven edges will cause anisotropic patterns and other potential problems35. There is also a conflict between printability and pattern resolution; a relatively low contact angle improves the printability of the ink, but an increased droplet area affects the pattern resolution36,37.

The problems mentioned above have limited the wide usage of inkjet printing electronics. Moreover, the physical properties of the ink such as viscosity and surface tension also influence the formation of patterns38. Many types of solvent may be considered for a given functional material, and solvent selection is an important issue. To solve the above problems, the morphology formation rules of droplets should be studied systematically, and numerical simulation is a suitable method to research this subject.

Numerical simulations of inkjet technologies have been conducted by several researchers. Stringer and Derby studied the deposition and formation of multiple droplets by theoretical analysis of experimental data, and they found that there are maximum and minimum values of drop spacing to allow printing of stable lines39, but they did not change the solvent or printing frequency. Lee et al. numerically investigated droplet impingement and coalescence dynamics using the sharp-interface level-set method40 and illustrated that advancing and receding contact angles play important roles in this process. Zhang et al. simulated the morphology of a line of multiple deposited droplets with a lattice Boltzmann model41, they obtained all types of printed lines but did not study the influence of printing frequency. From existing studies, both properties of ink (impact velocity, surface tension, viscosity) and printing parameters (drop spacing) have an influence on printed patterns, but these studies always only study part of the factors in the printing process, an overview of this question is still not clear. Therefore we try to systematically research the influence of properties of ink and printing parameters in this study and gave several guidelines for fabricating inkjet-printed electronics.

A suitable model is necessary to achieve reliable simulated results. Numerical studies of one droplet or multiple droplets impacting a solid surface have been conducted by many researchers42,43,44,45. In these studies, the VOF method is the most popular multiphase model, as it is a convenient way to define the initial state of droplets and construct interfaces46. The fundamental VOF method uses the static contact angle for boundary conditions; however, because of the friction between the fluid and the solid surface, the contact angle is not a single value in the real world47. The maximum value of the quasi-static contact angle is the advancing contact angle (θadv), the minimum value is the receding contact angle (θrec), and the difference between θadv and θrec is the contact angle hysteresis (CAH)48. These parameters obviously influence the morphology of tiny droplets and should be considered in the simulation model.

With the help of user-defined files (UDFs), researchers can apply variable contact angles as boundary conditions. In previous studies, the most promising formula for the dynamic contact angle was the Hoffman function, which is summarized by Kilster49 from Hoffman’s experiments50 as follows:

$$\begin{array}{l}f_{\mathrm{{Hoff}}}\left( x \right) = {{{\mathrm{arccos}}}}\left\{ {1 - 2\tanh \left[ {5.16\left( {\frac{x}{{1 + 1.31x^{0.99}}}} \right)^{0.706}} \right]} \right\}\\ \theta _d = f_{\mathrm{{Hoff}}}\left[ {{\mathrm{Ca}} + f_{\mathrm{{Hoff}}}^{ - 1}\left( {\theta _s} \right)} \right]\end{array}$$
(1)

where θd is the dynamic contact angle, θs is the static contact angle and Ca is the capillary number, which is a dimensionless number equal to vcontline·η/γ49. In many studies, the Hoffman function is used in UDFs to calculate θd, which is then used as a boundary condition51. θd is calculated from the empirical equation based on the observed results; however, the contact angle set by the boundary condition in the simulation is similar to an interaction parameter between a fluid and a solid surface. A rational model should yield results that are consistent with experimental data; thus, the Hoffman function should be a criterion rather than a boundary condition.

For the reasons above, this paper establishes a model for morphology simulation of droplets that uses CAH UDFs based on force analysis at the contact line and discusses the rationality of models with CAH UDFs, with Hoff UDFs, or without UDFs. Then, single droplets and multiple droplets are simulated to study the influence of different factors, and finally, some principles to improve printing quality are proposed.

Results and discussion

Overview of the model

Geometry and meshes are built by ICEM software. As Fig. 1a shows, the computational domain is a cuboid divided by hexahedral meshes, and every mesh has dimensions on the 1 µm × 1 µm × 1 µm scale.

Fig. 1: Overview of the model.
figure 1

a Geometry of computational domain and meshes. b Deposited droplets and definition of the spherical droplet. a Is an overview of the computational domain and meshes drawn from ICEM, it is a cuboid divided by hexahedral meshes, and every mesh has dimensions on the 1 µm × 1 µm × 1 µm scale. Every side of the cuboid is named respectively in order to be defined as different types of boundaries in Fluent. b Is drawn from Fluent, which shows the deposited droplets and newly defined droplet region on air with an initial velocity.

Full size image

Multiphase is described by the VOF method, which defines the volume fraction (color function) of one of the fluids in each cell. The following conservation equations are solved in the computational domain:

$$\begin{array}{l}\partial \left( {\rho {{{\mathbf{u}}}}} \right)/\partial t + \nabla \cdot \left( {\rho {{{\mathbf{u}}}} \times {{{\mathbf{u}}}}} \right) = - \nabla p + \nabla \cdot (2\mu {{{\mathbf{D}}}}) + {{{\mathbf{F}}}}_{{\mathrm{st}}} + \rho {{{\mathbf{g}}}}\\ \partial \rho /\partial t + \nabla \cdot \left( {\rho {{{\mathbf{u}}}}} \right) = 0\end{array}$$
(2)

where ρ is density, u is velocity vector, t is time, p is pressure, μ is dynamic viscosity, D is the rate of deformation, Fst is body force and g is gravity46.

In this study, the first phase is air, and the second phase is ink. The viscosity type is laminar. Gravitational acceleration is −9.81 m s−2 along the z direction. Parameters of ink droplets are set as Table 1 shows, which are in accord with the real conditions.

Table 1 Parameters of ink droplet set in simulation.
Full size table

The droplets are generated by patching a spherical region filled with a second phase and an initial velocity along the z direction, as shown in Fig. 1b. The patched regions determine the drop spacing, and the interval time between droplets determines the printing frequency.

Friction effect and UDF

The CAH can be explained by the friction effect from Wang’s study52. As Fig. 2a shows, a friction tension term f is added to Young’s equation, and the value is positive if f and γSL have the same direction. It is assumed that the maximum value of f is fmax and f ∈ [−fmax,fmax], similar to the static friction force. Some relations can be obtained as follows:

$$\begin{array}{l}\gamma _{{{{\mathrm{SG}}}}} = \gamma _{{{{\mathrm{LG}}}}}\cos \theta _{{{\mathrm{s}}}} + \gamma _{{{{\mathrm{SL}}}}}\\ \gamma _{{{{\mathrm{SG}}}}} = \gamma _{{{{\mathrm{LG}}}}}\cos \theta _{{{{\mathrm{adv}}}}} + \gamma _{{{{\mathrm{SL}}}}} + f_{{\mathrm{max}}}\\ \gamma _{{{{\mathrm{SG}}}}} = \gamma _{{{{\mathrm{LG}}}}}\cos \theta _{{{{\mathrm{rec}}}}} + \gamma _{{{{\mathrm{SL}}}}} - f_{{\mathrm{max}}}\end{array}$$
(3)
Fig. 2: Friction effect and designation of UDF.
figure 2

a Schematic diagram of friction tension and contact angle hysteresis. b The gradient of the VOF function from different views. c Structure of the CAH UDF. a Shows the relationship between friction effect and contact angle hysteresis, the red arrows represent the friction tension. b Shows the VOF gradients on the surface of ink fluid, the left figure is the vertical view and the right figure is lateral view.

Full size image

It can be concluded that cos θs is the average of cos θadv and cos θrec, and experimental data also support this relation52,53. However, this relation is not enough to conform to a certain value of cos θadv and cos θrec.

Combining the experimental data and Cheng’s theoretical analysis54, it is assumed that for the contact angle of ink droplets (20–50°), CAH is linearly related to θs:

$$\begin{array}{l}{{{\mathrm{CAH}}}} = \theta _{{{{\mathrm{adv}}}}} - \theta _{{{{\mathrm{rec}}}}} = a\theta _{{{\mathrm{s}}}} + b\\ 2\cos \theta _{{{\mathrm{s}}}} = \cos \theta _{{{{\mathrm{adv}}}}} + \cos \theta _{{{{\mathrm{rec}}}}}\end{array}$$
(4)

The kinetic friction force is similar to the maximum static friction force; similarly, the friction effect is fmax for a contact line moving forward and −fmax for a contact line moving backward.

First, θadv and θrec are calculated. Then, the following formula is defined:

$$f\left( x \right) = {{{\mathrm{arccos}}}}(\cos \theta _{\mathrm{s}} - x) - {{{\mathrm{arccos}}}}(\cos \theta _{\mathrm{s}} + x) - a\theta _{\mathrm{s}} - b$$
(5)

In this study, a is 0.5, and b is −5 (in degrees), based on the experimental data of the Ra0.1 surface53. The zero of this formula, x0, can be found by the bisection method, and the UDFs θadv and θrec are defined as follows:

$$\begin{array}{l}\theta _{{{{\mathrm{adv}}}}} = {{{\mathrm{arccos}}}}(\cos \theta _{\mathrm{s}} - x_0)\\ \theta _{{{{\mathrm{rec}}}}} = {{{\mathrm{arccos}}}}(\cos \theta _{\mathrm{s}} + x_0)\end{array}$$
(6)

Second, the moving direction (forward or back) of the contact lines and θd is defined. The VOF function gradient (gVOF) of every cell in the computational domain is necessary to derive the parameters. Here, we use mature code to obtain the gVOF values of three coordinate directions (gVOF,x, gVOF,y, and gVOF,z)55. Then, some parameters are defined:

$$\begin{array}{l}{\mathrm{UDMI}}_0 = g_{{\mathrm{VOF}},{x}}/\sqrt {{{g_{{\mathrm{VOF}},{x}}}}^2 + {g_{{{\mathrm{VOF}}},{y}}}^2} \\ {\mathrm{UDMI}}_1 = g_{{\mathrm{VOF}},{y}}/\sqrt {{g_{{{{\mathrm{VOF}}}},{{{x}}}}}^2 + {g_{{{{\mathrm{VOF}}}},{{{y}}}}}^2} \\ {{{\mathrm{UDMI}}}}_2 = g_{{{{\mathrm{VOF}}}},{{{z}}}}/\sqrt {{g_{{{{\mathrm{VOF}}}},{{{x}}}}}^2 + {g_{{{{\mathrm{VOF}}}},{{{y}}}}}^2 + {g_{{{{\mathrm{VOF}}}},{{{z}}}}}^2} \end{array}$$
(7)

UDMIi are user-defined parameters that can be saved and read in user-defined memory spaces for every cell, which are renewed in every calculation time step. Here, only cells at the interface of air and ink have useful UDMIi values. As Fig. 2b shows, gVOF is perpendicular to the interface, the contact line moves only along the x–y plane, and the unit normal vector of the contact line is

$${{{\mathbf{n}}}}_{{{{\mathrm{contline}}}}} = \left( {{{{\mathrm{UDMI}}}}_0,{{{\mathrm{UDMI}}}}_1} \right)$$
(8)

In FLUENT, the velocity of a fluid phase (ink) v can be directly read, and its velocity along the normal direction is

$$v_{{\mathrm{contline}}} = {{{\mathbf{v}}}} \cdot {{{\mathbf{n}}}}_{{{{\mathrm{contline}}}}} = v_x{{{\mathrm{UDMI}}}}_0 + v_y{{{\mathrm{UDMI}}}}_1$$
(9)

If vcontline > 0, this piece of contact line moves forward, and if vcontline < 0, the contact line moves backward. Then, the dynamic contact angle θd is calculated by

$$\theta _{\mathrm{d}} = {{{\mathrm{arccos}}}}({{{\mathrm{UDMI}}}}_2)$$
(10)

The forces at the contact line are directly reflected in quasi-static cases, as discussed. As a result, for contact lines moving forward, the boundary condition is θadv, and for contact lines moving backward, the boundary condition is θrec. If the contact line is not moving but it may move, θd is needed. When θd > θadv, the contact line should move forward, and the boundary condition is θadv. Similarly, if θd < θrec, the boundary condition is θrec. If θrec ≤ θd ≤ θadv, the force at the contact line is wetted and maintains its quasi-static state, and the boundary condition is θd. The frame of the UDF is shown in Fig. 2c, and the UDF works on every cell and is renewed every time step.

Single droplet impact simulation

To compare the agreement between the Hoffman function of different models and experimental data, θd and vcontline should be derived from simulation results. θd is directly estimated from the profile image, and vcontline is calculated as UDMIi and read from the results. In addition to vcontline, the ratio of surface tension and viscosity γ/η determines the capillary number Ca. The simulated results and curves of the Hoffman function with different γ/η and θs = 30° values are shown in Fig. 3a–c. For cases without UDFs and with the CAH UDF presented in this study, the simulated points are in good agreement with the curves of the Hoffman function. In contrast, for cases with the Hoff UDF, the simulated points have a higher θd than the theoretical value. These results indicate that the basic VOF method in FLUENT is able to reproduce the Hoffman function. For cases with the CAH UDF, the applied boundary condition is not too different from 30° to cause obvious differences when Ca is dominant. For cases with Hoff UDF, the applied boundary condition is much higher than 30°, so the simulated points have much higher θd values. Figure 4d shows the diameter of a single droplet in different cases. Because the Hoff UDF applies a greater friction effect at the contact line, the contact line moves more slowly, and the diameter is smaller than in the other two cases.

Fig. 3: Results of single droplet impact simulation.
figure 3

a Simulated results of cases without UDFs. b Simulated results of cases with the CAH UDF. c Simulated results of cases with the Hoff UDF. d Simulated time-dependent diameter of different cases. e Comparison of the simulated droplet and real droplet in ref. 56. a–c Shows the relationship between derived simulated dynamic contact angle and moving velocity of the contact line. These figures show that results from cases without UDFs and cases with CAH UDF are in accord with the Hoffman function. In e, every picture is composed of photographs from ref. 56 (left part) and simulated shape from the case with CAH UDF (right part).

Full size image
Fig. 4: Results of two droplets fusion simulation.
figure 4

a Simulated two-droplet fusion morphology of cases without a UDF and with the CAH UDF. b Simulated two-droplet fusion morphology of cases with different densities and impact velocities. c Simulated two-droplet fusion morphology of cases with different surface tensions and viscosities. All pictures are top views of simulated two-droplet fusion morphology. a Shows the difference between cases without UDFs and with CAH UDF. b Results reveal that the influence of impact velocity and density mainly worked in the early stage(<0.05 s) and can be ignored in the whole printing process. c Shows that the same value of surface tension/viscosity is the major factor that determines the simulated morphology at different times.

Full size image

Further simulation is carried out to verify the function of the CAH UDF. The simulation comes from ref. 56, the diameter of the water droplet is 2.5 µm, the impact velocity is 0.16 m s−1, θadv is 120° and θrec is 65°. Figure 3e shows the results. The left half of every image is a real photograph of the droplet. The right half shows the morphology simulated with the CAH UDF, and all results agree with the photograph better than the results in refs. 56,57, in terms of both the shapes and the dynamic contact angles, so the CAH UDF is useful to obtain a relatively accurate morphology of droplets.

Two droplets fusion simulation

Although cases without UDFs or with the CAH UDF have similar single droplet behavior, the aim of this study is to simulate the formation process of multiple deposited droplets. Some quasistatic states and contact lines moving backward will be different when using different models. The two-droplet fusion process is simulated for comparing cases without a UDF or with the CAH UDF, and the results are shown in Fig. 4a. In the early stage, contact lines move forward and mix together to form a smooth transition, and two droplets become one large elliptical droplet, and the shapes of the two cases are similar. When the velocity of the contact line is decreased as a result of viscous dissipation, the effect of applied θadv in the CAH UDF case becomes important and decelerates the moving of contact lines.

With two-droplet fusion simulations, the influence of factors can be studied. Impact velocity is difficult to measure and control in real printing processes58, and it is related to the impact kinetic energy, so the influence of the impact velocity and the density of droplets are studied together first. The impact velocity in inkjet printing is 3–15 m s−1 59,60. Figure 4b shows the simulated results for droplets with different densities and an impact velocity of 6 m s−1. No apparent differences can be found in these cases after 0.5 ms, indicating that the influence of density or impact kinetic energy mainly exists at a very early time after the droplet impacts the solid surface. This is because the volume of the droplet is tiny and the impact velocity is low, and the kinetic energy will be exhausted by viscous dissipation for a short time. Kinetic energy will not have a long-term influence. The same simulated results in the right part of Fig. 4b indicate that momentum determines the early morphology of the droplet. To summarize, the density and impact velocity of droplets have no important influence on the final simulated morphology, and they can be fixed in the following simulations.

In the single-droplet simulation, γ and η are considered together because γ/η determines the shape of the curve of the Hoffman function. In the two-droplet simulation, a series of cases with different γ and η values are simulated, and their results are classified as γ/η. Figure 4c shows the results. Cases with the same values have a similar morphology, which means that the value of γ/η is also a decisive factor in the two-droplet simulation. A higher γ/η results in a faster fusion process. The fusion process of two droplets is also a moving process of contact lines and should be in accordance with the Hoffman function. It is assumed that the final static shapes of all cases are the same, so the same transition state should have the same θd and Ca from the Hoffman function. As Ca = vcontline·η/γ, a higher γ/η requires a higher vcontline in the same transition state. The fusion process in cases with higher γ/η values is equivalent to a film of the fusion process occurring at a higher speed.

However, real droplets are not exactly 10 µm in radius, so the scale of droplets should also be taken into consideration. Assuming that the radius of a real droplet is fR = 10 μm, then the moved distance of the contact line should also be fR times that of the simulated case. Because the dynamic contact angle corresponds to vcontline, the average vcontline of droplets with the same γ/η and θs should also be the same. As a result, for the same initial and final state of the droplet, the moving time of the contact line is in direct proportion to fR. To verify this thought, simultaneously enlarge the radius of droplets, drop spacing, the computational domain, and mesh scale in order to use the same mesh file, as Fig. 6a shows. Results are shown in Fig. 6b, in which the time is proportional to fR, and the shape of fused droplets is almost the same in three cases except at the early time when impact kinetic energy works.

Multi-droplet deposition simulation

In multi-droplet simulations, γ/η is also regarded as a determined property of the ink fluid, and printing parameters are mainly studied in this part. The drop spacing S should be smaller than the diameter of a single droplet, so the scale ratio of the deposited droplet to the falling droplet (R/r) should be calculated. By equalizing the volume formula of the sphere and spherical segment, R/r can be calculated:

$$R/r = \left[ {4/\left( {2/{{{\mathrm{sin}}}}\theta _{\mathrm {s}} + {{{\mathrm{cot}}}}\theta _{\mathrm{s}}} \right)/\left( {1/{{{\mathrm{sin}}}}\theta _{\mathrm{s}} - {{{\mathrm{cot}}}}\theta _{\mathrm {s}}} \right)^2} \right]^{1/3}$$
(11)

Figure 6a shows a diagram of Eq. (11). It can be observed that R/r increases with decreasing contact angle. It is assumed that the cross-section of the smooth continuous line is a circular segment, so the width of line DL can be calculated:

$$D_{\mathrm {L}}/r = 2\left[ {4\pi /(3\theta _{\mathrm {s}}{{{\mathrm{sin}}}}^2\theta _{\mathrm {s}} - {{{\mathrm{cot}}}}\theta _{\mathrm {s}})/(S/R \ast R/r)} \right]^{1/2}$$
(12)

Figure 5b shows the curves of Eq. (12) with different S values, and a multidroplet simulation can test whether a pair of θs and S values can form a continuous line.

Fig. 5: Influence of droplet size in simulation.
figure 5

a Simulated morphology of three cases with the same measuring scale. b Simulated morphology of three cases at different calculation times. a shows the scale difference of droplets and meshes in three cases with the same measuring scale. b Shows the simulated morphology of these cases, but the time is chosen by the same proportion of their scale. Results verify that it takes more time for larger droplets to form a certain morphology.

Full size image

In the multi-droplet simulation, the aggregation of droplets at the starting point may affect the final result. To avoid this, as Fig. 6c shows, several droplets constitute the initial line, which is limited in a region with the width calculated by Eq. (12), and the left boundary is symmetric to further avoid the aggregation of droplets. Then, droplets are deposited individually after the initial line. The interval time between droplets is 0.5 ms, γ/η is 2–4, θs is 20–50°, and S decreases from 2R until a continuous line is generated; in this way, the maximum S and minimum DL can be found.

Fig. 6: Preparation of multi-droplet deposition simulation.
figure 6

a Scale ratio of deposited droplets to falling droplets with different static contact angles. b Width of a printed continuous line with different drop spacing and a static contact angle. c Schematic diagram of the method to avoid aggregation of droplets. In a R/r means the radius ratio between deposited droplets and falling droplets, and the ratio rationally decreases with increased contact angle. In b, DL is the width of the printed line, R and r are the radii of deposited droplets and falling droplets, and the figure shows that both larger drop spacing and larger contact angle induce thinner lines. c Is the top view of the initial settings to avoid aggregation of droplets, circles with the number 0 represent initial patched droplets, and regions with θs = 180° are dewetting regions, the DL is set as the value in (b).

Full size image

Figure 7a shows the maximum S/r for different conditions. At a relatively low θs, the maximum S/r values for different γ/η values are similar, but the difference increases with the contact angle. A lower γ/η results in a larger S and smaller DL, which can be explained by the relationship between γ/η and vcontline. Figure 7b shows the minimum DL/r of different cases. When θs < 30°, a larger R results in a larger DL. When θs > 40°, droplets easily aggregate, and a smaller S is necessary to form a continuous line; thus, DL becomes larger. Therefore, 30° < θs < 40° is appropriate for fabricating thin single lines. Figure 7c draws the points in Fig. 7a and b together. Because the minimum DL/r is larger than the maximum S/r, these printing parameters can be used for fabricating patterns with multirow lines.

Fig. 7: Results of multi-droplet deposition simulation.
figure 7

a Simulated maximum drop spacing with different γ/η values. b Calculated minimum line width with different γ/η values. c Comparison of the maximum drop spacing and minimum line width. d Simulated maximum drop spacing with different printing frequencies. e Calculated minimum line width with different printing frequencies. f Real printed droplets and corresponding simulated results. In a, points represent maximum drop spacing to form continuous printed line for different conditions, and shows that small γ/η results in larger maximum spacing. b Shows the minimum width of lines in different conditions, this figure shows that 30–40° are proper contact angle for fabricating thinner lines. c Is a mixture of a and b, this figure is made to see whether the maximum spacing can be used to print sheet patterns. d and e are just like a and b, but the γ/η is fixed (γ/η = 2) and printing frequency is variables.

Full size image

Figure 7d and e show the simulated results of cases with γ/η = 2 and different fp, and the maximum S/r increases with fp. Because contact lines move a shorter distance during the interval time with higher fp values, droplets have more chance to contact neighboring droplets and form continuous lines. The distinction of the results between fp = 2 kHz and fp = 4 kHz is not very obvious, so fp = 2 kHz is sufficient for most conditions. The simulation results of cases with γ/η = 2 and fp = 1 kHz are extremely similar to those of cases with γ/η = 4 and fp = 2 kHz, which can be explained by the moving distance of the contact line. As discussed above, vcontline with γ/η = 4 is twice that with γ/η = 2 at every transition state, so it takes half the time for cases with γ/η = 4 to obtain a similar shape. In summary, (γ/η)/fp is decisive in multi-droplets simulation, and a lower (γ/η)/fp is beneficial for improving the pattern quality. Consider the scale factor, real required fp could be 1/fR in the simulated case.

To verify the simulation results, real printed lines were compared with the simulated morphology. When S decreases from 50 to 20 µm, the simulated results include isolated, scalloped, uniform and bulging formations, which are the same as the real printed formations shown in Fig. 7f.

Selective treatment on substrates is a method to improve the quality of patterns61,62,63, and the effect can be simulated using models in this paper by applying different boundary conditions on different surface regions in the simulation, as shown in Fig. 8a (θs2 > θs1). When θd < θs2, the dynamic contact angle tends to be higher and contact line tends to move backwards, as discussed. As a result, high θs2 makes droplets thinner and longer.

Fig. 8: Simulation of selective treatment.
figure 8

a Schematic diagram of printing with selective treatment. b Simulated single droplet scales of different cases. c Macroscopic contact angle and cross-section shape of different cases. d Simulated multidroplet morphology of different cases. a Is the top view of the computational region to realize selective treament, the contact angle of different regions are shown in the figure and droplets are deposited by the numerical order. In b, the number/number mean θs1/θs2 and this figure shows the simulated width and length of single droplet in different selective treatment conditions. In c, the number/number-number of different cases mean θs1/θs2-width of θs1 region, and the vertical coordinate is the macroscopic contact angle, and d shows the top view of simulated morphology.

Full size image

In this simulation, θs1 is 20°, 30° and 40° (CAH UDF); θs2 is defined as 60°, 90° and 120° (static contact angle); the widths of the line region are 20, 22, 24, 26, and 30 μm; γ/η is 2; and the time is 2 ms. Figure 8b shows the simulated results of a single droplet. The width of the droplet decreases with the width of the line region, but the droplet is always 1–7 μm wider than the line region. The difference between θs1 and θs2 plays an important role in the shape of the droplet: a low θs1 corresponds to an elongated droplet, while a high θs2 compresses droplets into the line region. The simulation results of cases with θs2 = 90° and θs2 = 120° are very similar, so θs2 = 90° is sufficient to have an obvious effect. Because droplets deposit on both treated and untreated regions, they are influenced by both θs1 and θs2 and ultimately remain in a balanced state in which the macroscopic contact angle is θs. Figure 8c shows θs and the cross-section of continuous lines, and the drop spacing is 0.9 times the length of a single droplet. The results show that θs is between θs1 and θs2, and the height of cross-section H is relatively stable for a certain θs1; thus, θs decreases with increasing line region width. Moreover, both θs and H increase with θs1, but they are very similar for cases with different θs2, indicating that θs2 has an unimportant influence. Figure 8d shows the shape of the simulated continuous line of cases with γ/η = 2 and fp = 2 kHz, whose shapes are the same as those of cases with γ/η = 4 and fp = 1 kHz (not shown in this paper), indicating that γ/η and fp are not important in selective treatment assistant printing. The line with selective treatment is thin, and the edge is smooth, which is ideal for printing. However, it should be noted that the treated line region limits the movement of the contact line; for the planar treated region where contact lines move freely, γ/η and fp also matter. In summary, selective treatment is helpful to form thinner lines and smoother edges, θs1 should be lower than 30°, and θs2 should be higher than 90° to obtain an obvious effect.

In summary, the friction effect plays an important role in the formation of tiny ink droplets, and the UDF proposed in this study could help to properly simulate the friction effect in Ansys Fluent. From the simulation results, the shape of the droplets is controlled by the movement of the contact line. The value of (γ/η)/fp is a decisive factor, and a small value facilitates the formation of thin lines. Overall, 30–40° is a suitable contact angle range for printing a thin single line. Selective treatment helps to print thinner lines with smoother edges. The transitional region is found for the balance of contact lines, and a higher difference is necessary to fabricate fine patterns. The inner contact angle should be lower than 30°, and the outer contact angle should be higher than 90°.