Coarse-grained molecular dynamics simulation of wettability and bleed out of capillary underfill

The wettability of curable resin materials is significant in the development of various materials because it affects several different processes and ultimately the final products. One such material, capillary underfill (CUF), is a sealing resin material used in flip-chip semiconductor packages (Fig. 1)^1,2 and it typically consists of epoxy resin, a curing agent, other additives, and fillers. It is used to fill the space in a semiconductor package in a liquid state, so knowing its properties and functions before curing is important.

When manufacturing a flip-chip semiconductor package, a liquid state CUF is placed next to the gap between a semiconductor chip and a substrate, where those components are aligned as parallel planes. Then, the CUF flows naturally by means of the capillary action.^3–9 During the capillary flow, the CUF touches various components in the package such as the solder resist, the conductive materials of solder pastes, and the pillars. The capillary flow ends when the CUF reaches the edge of the semiconductor chip, and packages are then sent on to subsequent processes such as a thermal curing. One of the biggest issues in terms of the capillary process is finding a way to shorten the flow time so that the processing cost can be reduced.³ We also need to make sure that the CUF stops flowing at the edge of the semiconductor chip and does not bleed out,⁴ as such bleeding would interfere with the other components.⁵ To address these issues, clarifying the capillary flow and the wettability between the CUF and each component is essential.

Several studies have examined redesigned capillary processes to improve the flow time. Schwiebert and Leong constructed an experimental capillary flow system for a CUF composed of a ceramic plate and a glass plate⁶ and found that the flow time is affected by the absolute viscosity, the surface tension, the wetting angle, and the configuration parameters of the flow length and the separation distance. The separation distance between the ceramic plate and the glass plate, i.e., the size of the gap between the two, has a threshold that influences whether filler particles can flow smoothly. Lin et al. studied the mathematical processes behind capillary flow by using an original system to control the pressure of CUF⁷ and clarified the macroscopic phenomenon and the mechanisms of the capillary flow under different pressure conditions. In recent developments, the gap of the capillary flow has been miniaturized, which means that the filler diameters are not much smaller than the gap, e.g., 10–250 μm for a general gap⁷ and a couple of μm or less for fillers.^1,2,8 This means that analysis using a non-equable liquid model is now required. At the very least, the dynamics of fillers should be considered separately from those of resin.

As for improving the bleed out, Faucher-Courchesne et al. investigated using ablation at depths of 5 µm.⁴ They found that this had no effect on the bleed spread, although the underfill spread could be contained. Several other studies have examined bleed out,^5,9 but to the best of our knowledge, none of them focus on prevention by established theory because the mechanisms and constituents of bleed out during capillary flow remain unclear. Some studies have examined a related phenomenon involving a precursor thin film in which liquid constituents are present in the molecular scale when a droplet becomes wet.^10–12 Ueno et al. detected the precursor thin film of silicon oils on a glass substrate by using the Brewster angle microscope system.^10,11 They separated the precursor part from the contact line of a droplet and found that the precursor part is a couple of nm thick. Shiomoto et al. observed the precursor thin film of water on line-patterned substrates with polymer brushes by using an upright transmitted light microscope and cameras¹² and revealed the mechanisms of growing and stopping of the precursor thin film stemming from the interfacial energies. These studies on the precursor thin film contributed to our understanding of nm-scale liquid behaviors. When considering the precursor thin film, it is reasonable to predict that the bleed of the CUF is caused not by fillers but by molecules that spread ahead. However, it is difficult to know precisely how the fillers and molecules themselves behave. To clarify this, the wettability and the capillary flow should be ideally investigated on the molecular scale.

To investigate the wettability and related phenomena for a liquid material on the molecular scale, computational simulation is a promising approach. The wettability of a liquid material has been calculated extensively by using molecular dynamics approaches.¹³ Andrews et al. performed wetting simulations of a water nanodroplet and graphene and observed molecular spreading on the graphene surface as well as the time evolution of the contact angle. They concluded that the spreading obeys the universal inertial spreading regime r ∼ t^1/2, where r and t are the spreading radius of the droplet and time, respectively. Lu et al. observed the wetting kinetics of a water droplet containing Au nanoparticles on an Au(100) surface¹⁴ and found that the contact line velocity decreases when the Au nanoparticles and particle–water interactions are increased. This is caused by the increase in surface tension and solid–liquid friction and by the absence of nanoparticle ordering in the vicinity of the contact line. In addition to the wettability calculation, capillary phenomena have been calculated by methods based on molecular dynamics. Dimitrov et al. performed coarse-grained molecular dynamics (CGMD) simulations to represent the capillary phenomenon of a Lennard-Jones (LJ) liquid in a cylinder.¹⁵ They compared the simulation results with the Lucus–Washburn equation, which is the macroscopic law predicting fluid meniscus depending on time.^16,17 As for the polymer melt, they suggested a small modification of the Lucus–Washburn equation to consider the effect of slip flow. CGMD has also been applied to systems containing fillers. Hagita et al. constructed a filler model and calculated the mechanical properties of the nanocomposite material.¹⁸ They utilized a spherical shell model whose surface was covered by beads as the filler and found that they could reduce the beads and prevent slipping, although a large sphere created by just one bead tended to slide on other beads. Thus, the molecular dynamics approach is suitable for representing the wettability and related phenomena for a liquid material, and the CGMD simulation can be adapted to show the dynamic behaviors of fillers. However, to our knowledge, these have not been applied to the capillary system of a CUF. Different behaviors of resin and fillers in a CUF are still unclear, and effects of material composition on the flow time and bleed out of the capillary flow are also unrevealed in the molecular scale.

In this study, we use the CGMD simulation to elucidate the capillary flow of a CUF. First, we perform a wettability simulation to reveal the basic behaviors of a monomer molecule and fillers in a droplet of CUF. Next, we perform a capillary flow simulation to detect the flow time. We also try to define the bleed length in the simulation model. We focus on clarifying the effects of the filler compositions on the flow time and bleed length in this simulation. The purpose of this study is to develop a method that can clarify the capillary flow of a complicated constitute resin to aid in material development. In addition, material compositions are suggested for decreasing the flow time and bleed length.

We use the CGMD method^15,18,19 to calculate the wettability and the capillary flow. The bead–spring model is utilized under a three-dimensional periodic boundary condition. The 12-6 Lennard-Jones (LJ) potential is used in non-bonding bead pairs, as

where r_ij is the distance between beads i and j and r_cut is the cutoff distance. ε and σ are parameters related to energy and distance, respectively. In addition, l is a parameter related to distance, used only to make filler models. The finitely extensible nonlinear elastic (FENE) potential is used in bonding bead pairs as

where R₀ is the parameter corresponding to maximum extension length and k is the spring constant. Each bead obeys the stochastic dynamics described by the Langevin equation,

where F is the force calculated by the aforementioned potentials U^LJ and U^FENE. The Brownian force W satisfies ⟨W_i(t)W_j(t′)⟩ = 2k_BTm_iξΔ_ijΔ(t – t′), where k_B is the Boltzmann constant and T is the absolute temperature. In the wettability and capillary flow simulations performed in this study, we basically use ε_ij = 1.0 ε, σ_ij = 1.0 σ, l = 0.0 σ, r_cut = 2.5 σ, k = 30.0 ε/σ², R₀ = 1.5 σ, k_BT = 1.0 ε, m_i = 1.0 m, and ξ = 0.5/τ. These parameters were determined, referencing to the CGMD model of filled polymer nanocomposites in which the bead–spring model successfully investigated mechanical properties such as a stress–strain curve.¹⁸ When special parameters are used in the following, those are mentioned. Each CGMD simulation is performed using an open-source package, LAMMPS.²⁰ 

To represent a liquid state CUF, a monomer molecule and fillers are described using computational models (Fig. 2 and Table I). In the simulation model, the resin part of a CUF is simplified and consists of only monomers since we want to focus on the effects of the filler characteristics. We modeled a monomer as an epoxy molecule by exampling a bisphenol-F epoxy, which is one of the typical molecules utilized in CUFs. Figures 2(a) and 2(b) show a coarse-grained representation of bisphenol-F epoxy. In this study, a monomer model is treated as a simplified model, which satisfies a relative density to fillers as described later, and then, we do not intend to represent a more realistic bisphenol-F molecule. We separate this monomer into four parts [as shown in (b)], so it is represented by four beads and three bonds (Table I). Figures 2(c)–2(e) show the creation of the filler model, which is done using a method with an icosahedron structure.¹⁸ First, each regular triangle face of the icosahedron structure is covered by beads arranged as a triangular lattice [Figs. 2(c) and 2(d)]. Then, adjacent pair beads are registered as having a bond, even if those beads are on different faces. A bead is also located at the center of a three-dimensional icosahedron structure. After that, we perform a relaxation CGMD calculation under the specified potentials. The beads on the icosahedron surface then move to equilibrium positions and a spherical shell structure is generated [Fig. 2(e)]. In this study, we created two filler models. One is a large filler consisting of 1281 beads and 1920 bonds, and the other is a small filler consisting of 61 beads and 90 bonds (Table I). The conditions of the relaxation calculation are designed so that the distance between a bead on the icosahedron surface and the bead next to it is 1.0 σ. Although there can be several conditions that satisfy this, we utilize the following ones in this study. For the LJ potential between the center bead and others, ε_ij = 10.0 ε, σ_ij = 20.0 σ, l = −11.0 σ, and r_cut = 40.0 σ are used for the large filler, and ε_ij = 0.1 ε, σ_ij = 10.0 σ, l = −8.8 σ, and r_cut = 20.0 σ are used for the small filler. In the LJ potential between beads on the icosahedron surface, ε_ij = 1.0 ε, σ_ij = 1.0 σ, and r_cut = 0.01 σ are used for both the large and small fillers. In the FENE potential, k = 30.0 ε/σ², R₀ = 30.0 σ, m_i = 1.0 m, and ξ = 1.0/τ are used. For performing the molecular dynamics, k_BT = 1.0 × 10⁻⁷ ε, the time step dt = 1.0 × 10⁻³ τ, and the total simulation steps of 1.0 × 10⁵ are used. These parameters were used to obtain surface beads positions of the fillers with the short simulation steps under a less effect of a bead temperature fluctuation. On the basis of the relaxation calculations, we set the average distances between a bead on the icosahedron surface and the bead next to it to 0.99 σ and 1.00 σ for the large and small fillers, respectively. Radii of the large and small fillers are 11.5 σ and 2.5 σ (Table I). During the wettability and capillary flow simulations, the masses of large and small fillers are 9.8 × 10³ and 1.0 × 10² m, respectively. These settings correspond to the density ratio of typical resin (about 1.2 g/cm³) and silica-based filler (about 2.2 g/cm³). Moreover, the allocation of masses in each filler model (spherical shell) is decided to fit the inertia moment to that of a real filler whose inside is fully filled. Masses of the center and surface beads are 9.9 × 10² and 6.9 m for the large filler and 10 and 1.5 m for the small filler. Fillers are treated as a rigid state during the wettability and capillary flow simulations.

We first performed the wettability simulation. Figure 3 shows the simulation model. The size of the simulation cell is (x) 350 σ × (y) 40 σ × (z) 350 σ. We use a droplet model of the CUF in which the monomer and fillers are mixed as the initial model. A droplet CUF is created on the basis of precalculations. We first perform a CGMD simulation under a shrinking condition using a model in which monomers and fillers are randomly located in a large simulation cell. A three-dimensional block of the CUF model is then created. After that, we perform a relaxation calculation with a new vacuum phase. When the substrate is put in place, any vaporing monomers that leave the droplet by a molecular movement due to the temperature satisfying k_BT = 1.0 ε are then removed from the model, as they may interfere. Figure 3(a) shows an overview of the droplet, which is cylindrical in shape and has its axis on the y direction. The created droplet is initially placed 1.0 σ above the substrate model and the substrate is then created by beads arranged as a triangular lattice whose constant is 1.12 σ. This substrate plane is placed vertically to the z axis and is fixed during the simulations. To investigate the wetting behavior, we calculate the height a and width b of the droplet. Those are used to calculate a contact angle θ as θ = 2 tan⁻¹(2a/b). A calculation cell is utilized to detect height a and width b [Figs. 3(b) and 3(c)]. Each calculation cell is (x) 1.0 σ × (y) 40.0 σ × (z) 1.0 σ. When a calculation cell includes ten or more beads, it is defined as occupied, and when it includes less than ten beads, it is defined as unoccupied. Height a and width b correspond to the numbers of occupied cells along the x and z directions, respectively. In the wettability simulations, we utilize the time step dt = 2.0 × 10⁻³ τ and the total simulation steps of 3.0 × 10⁷.

The model of the capillary flow simulations is shown in Fig. 4. The CUF and wall models are initially located in the simulation cell of (x) 370 σ × (y) 30 σ × (z) 180 σ under the three-dimensional condition. Figure 4 also shows the parallel plane system to be filled by the CUF. The initial CUF position is prepared on the basis of precalculations, the same as the wettability simulation model. A relaxation calculation is performed after the walls are prepared and then walls S and T consisting of triangular lattice beads are put in line. Wall T is tentatively used to create an initial model and is then removed after precalculations. To represent the capillary system, a CUF channel is created using walls B, U, and S. Wall B forms the bottom surface of the channel and wall U covers the upper surface and side wall. Walls B and U also consist of triangular lattice beads and correspond to a solder resist and a semiconductor chip in a real package. Therefore, the space sandwiched by walls B and U is used to observe the capillary flow. Wall U constructs the edge shape of a semiconductor chip, which makes it possible to observe the CUF behavior at the edge. Wall S is used to form the CUF pool. The CUF is initially located at the left-hand side of wall U and then flows to the gap between walls U and B by means of the capillary phenomenon. The lengths of the walls are as follows: length I—40 σ, length II—110 σ, length III—60 σ, and length IV—220 σ. The triangular lattice constant of each wall is the same as the wettability simulation. In the capillary flow simulations, we use the time step dt = 6.0 × 10⁻³ τ and the total simulation steps of 4.78 × 10⁷. To investigate the capillary flow, we measure the flow time and bleed length. The flow time is calculated as the time steps taken to fill the space sandwiched by walls B and U, whose size is (x) 40 σ × (y) 30 σ × (z) 60 σ. The filling rate is calculated every 1 × 10⁶ steps. The filling rate data are approximated as a second-order polynomial to detect the time step to 75% fill, v = αu² + βu + γ, where u and v are the time step and the number of beads in the sandwiched space, respectively. Then, α, β, and γ are fitting parameters. The bleed length is calculated as a wetting range for only the monomer by using the calculation cell defined in the wettability simulation. A new calculation cell is defined for the capillary flow simulation cell as (x) 1.0 σ × (y) 30 σ × (z) 1.0 σ, and the occupation of that cell is then judged. Wetting ranges of the monomer and fillers are separately calculated along the x axis, and differences of them are defined as the bleed length.

Table II shows the characteristics of nine wettability simulation models (A–I) having different content ratios of total fillers/total components and of small fillers/total fillers. The total fillers/total components ratios of models A–C, D–F, and G–I are roughly 46%, 41%, and 36%, respectively. The small fillers/total fillers ratios of models A, D, and G are 0%, those of models B, E, and H are 10%, and those of models C, F, and I are 20%. The numbers of beads and the volumes are listed in Table II. We calculated the volumes of the monomer molecules using the density of the bead–spring model of 0.85/σ³.¹⁸ Table III shows the characteristics of capillary simulation models D′, E′, and F′, which correspond to the compositions of models D, E, and F in Table II. These models have about double the number of components as models D, E, and F. The substrate in the wettability simulation has 13 083 beads, and the substrate and walls in the capillary flow simulation have 20 444 beads.

First, we discuss the wettability simulations performed to observe the basic behaviors of monomers and fillers. Snapshots of the wettability simulations of models D, E, and F are shown in Fig. 5. Model D included just monomers and large fillers. The droplet was a cylinder shape at the first stage [see (D1) of Fig. 5]. After CGMD simulation started, monomers spread across the substrate, and large fillers also moved close to the substrate [see (D2) of Fig. 5]. After 30 × 10⁶ steps, the droplet became wet on the substrate [see (D3) of Fig. 5]. Large fillers showed less movement than monomers during this process. Large fillers piled up vertically in the droplet, and after the simulation, the droplet was slightly lower than it had been at the start. In the simulation of model E, which included monomers, large fillers, and small fillers [see (E1)–(E3) of Fig. 5], the behaviors of the monomers and large fillers were almost the same as those of model D [see (D1)–(D3) of Fig. 5]. Just after the simulation started [see (E1) of Fig. 5], monomers spreaded across the substrate and large fillers moved close to the substrate [see (E2) of Fig. 5]. The small fillers could move faster than large fillers in the droplet and easily followed the monomer spread across the substrate. Small fillers encouraged wetting dynamics, which was mainly caused by monomers. After 30 × 10⁶ steps, the droplet of model E became wet on the substrate at a lower height than model D [see (D3) and (E3) of Fig. 5]. Model F included twice the number of small fillers compared to model E [see (F1)–(F3) of Fig. 5]. In this simulation, each component spread out across the substrate, as in the previous two simulations. However, at 10 × 10⁶ steps, the droplet showed a constricted shape [see (F2) of Fig. 5]. After 30 × 10⁶ steps, the droplet became wet on the substrate, but its height was higher than that of model E [see (E3) and (F3) of Fig. 5]. In these simulations, we found that the dynamics of the monomer caused the wetting, and the fillers followed the spreading monomers. Each component had different dynamics.

We clarified the dynamics of the monomer, the large filler, and the small filler in a relatively short time by mean square deviation (MSD). MSD is the average of the squared distances between two coordinates and is calculated here by

where N is the total number of beads. In our case, r_nk and r_n′k are the three-dimensional coordinates of a bead number k at simulation steps n and n′. The MSD is calculated every 2.0 × 10⁵ steps, and then pairs of given n and n′ (= n − 2.0 × 10⁵) are set in each MSD calculation. The MSD of fillers is calculated by using a bead located at the center of a three-dimensional filler structure. Figure 6 shows the MSD time evolutions of (a) the monomer, (b) the large filler, and (c) the small filler during the wettability simulations with models D, E, and F (Fig. 5). Each MSD was gradually converged. Therefore, MSDs were used for the discussion of the dynamics especially at 300 × 10⁶ steps. When comparing the components, we can see that the monomer showed almost 25–45 σ², which was higher than that of both the large and small fillers. Moreover, the MSD of the small fillers (0.4–1.4 σ² in most simulation steps) was larger than that of the large fillers (0.02–0.2 σ²). These results indicate that each component had a different speed in the spreading process. The monomer could spread fast and led the contact line of droplets on the substrate. Spreading of the fillers was shorter. In particular, small MSDs of large fillers are presumably due to the low diffusivity stemming from their large size. While the monomer could become wet very fast and the large filler was easily left behind in model D [see (D1)–(D3) of Fig. 5], the small filler used in models E and F was more likely to follow the fast movement of the monomer. Thus, left large fillers disturb wetting of a CUF droplet, and a small filler contributes to accelerated wetting. Here, the Stokes–Einstein equation, a simple relationship between the diffusion coefficient D and the spherical particle radius r_a as D = k_BT/Cπμr_a, where μ and C are the dynamics viscosity and a constant, is used to compare the large and small fillers. When μ is the same in each simulation, radii of the large and small fillers indicate a ratio of the diffusion coefficients of the large filler D_L and that of the small filler D_S, as D_L: D_S = 1/11.47: 1/2.48 ≓ 1.0: 4.6. In Fig. 6, an average ratio between root MSDs of the large and small fillers during the latter 100 × 10⁶ steps is (the large filler) 1.0: (the small filler) 3.0–3.8 σ. This ratio is similar to the ratio of the diffusion coefficients if the behaviors of fillers are sufficiently converged to compare them. Thus, the small filler is more likely to diffuse in the droplets than the large fillers. Next, when we compare models D, E, and F in Fig. 6, differences are evident in the monomer and small filler [Figs. 6(a) and 6(c)]. As shown in (a) and (c), the MSDs of the monomer and the small filler decreased with an increase in the small filler. We assume there are two reasons for these behaviors. The first is the increase in interactions between the monomer and the small fillers. The large filler has a surface area of 413.1 σ², while that of the small filler is just 19.3 σ². When a large filler is replaced with a 100 small fillers totaling the same volume, e.g., when switching from model D to E and from E to F, the total surface area of the fillers increases and the monomers thus interact with many filler surface beads. The monomers feel the effect of these increased interactions from the filler surfaces and it slows them down. The other reason is the slowdown of small fillers by the increases in their number. There is suddenly less space for the small fillers to move around in, and this slows down their speed. This behavior is different from that of the large filler, which exhibited almost the same MSD in models D, E, and F. In the discussion of the diffusion coefficient and the average ratio between root MSDs, it is observed that the small filler in the simulation is slightly slower than that of the theory. This is mainly because bead interaction slows down the filler. In addition, effective dynamics viscosity μ might change although an exact estimation of that is difficult here. These results enable us to clarify that the monomer, large filler, and small filler move at different speeds in the droplets.

We also performed wettability simulations with models A, B, C, G, H, and I. Snapshots of these at 30 × 10⁶ steps are shown in Fig. 7. Considering Fig. 6, it is conceivable that the dynamics at 30 × 10⁶ steps would not dramatically change in a latter process, and then, they were approximately converged although they would not indicate static droplets. In other words, the time at 30 × 10⁶ steps is sufficient to discuss the dynamics of the droplets. Models A, B, and C had high filler ratios (Table II) and consequently exhibited constricted shapes [see (A)–(C) of Fig. 7]. While there was a certain amount of monomer spread on the substrate, many fillers were left inside the droplet. When a CUF included remaining fillers, it did not look like a liquid anymore. This result agrees well with the finding in Fig. 6 that the monomer had a higher MSD than the others. From the macroscopic point of view, the CUF seemed to become wet on the substrate uniformly. However, when we look at it from the microscopic point of view, i.e., like a CGMD level, it seems that the monomer could spread fast just above the substrate surface, and it was possible for the fillers to agglomerate. This non-uniformity could potentially change the macroscopic properties of the CUF, such as the viscosity. When we compare models A, B, and C, model B had the lowest height in the droplet. Snapshots of models G, H, and I, which have low filler ratios (Table II), are shown in Fig. 7 [(G)–(I)]. The models in these cases had smooth wetting without constricted shapes, which is very different from models A, B, and C. Thus, we conclude that a high concentration of fillers makes the CUF hard and slows down its wetting. It is also clear that the ratio of total fillers/total components has a larger effect on the droplet behaviors than the ratio of small fillers/total fillers. In a comparison between models G, H, and I, model H, which had a medium amount of small fillers, showed the lowest height. This tendency is the same as that of models B and E among models A–F.

Next, we calculated the contact angles. Figure 8 shows (a) height a, (b) width b, and (c) contact angles θ in each simulation at the final step of 30 × 10⁶. Height a and width b were used to calculate contact angles θ. In Figs. 8(a) and 8(b), we observed that a low ratio of total fillers/total components showed a high width and a low height, and a medium ratio of small fillers/total fillers showed a high width and a low height. These tendencies are in good agreement with the observed droplet, such as the smooth wetting of models G–I and lowering the heights of models B, E, and H (Figs. 5 and 7). In Fig. 8(c), we also observed that height a and width b affect contact angles θ so that contact angles θ show two similar tendencies. The first is that a low ratio of total fillers/total components showed a low contact angle and the second is that a medium ratio of small fillers/total fillers showed a low contact angle. As stated in Sec. II, we calculated the contact angles in a basic way in this study, so the width and height of the droplet decide the contact angle. The tendency of the contact angle depending on an increase in fillers agrees well with a previous study using a water droplet containing Au nanoparticles,¹⁴ although the sizes and constitutions are different in our simulations. We also found that the ratio of total fillers/total components affected how much the width changed. As shown in Fig. 8(b), at 0% in the ratio of small fillers/total fillers, the calculations of models A, D, and G ranged from 69.9 to 81.7 σ. This range increased with the ratio of total fillers/total components: to 75.2–121.6 σ at 10% (models B, E, and H) and to 58.0–103.2 σ at 20% (models C, F, and I). The tendency was also found at the contact angle in Fig. 8(c) as 73.1°–89.2° at 0% (models A, D, and G), 44.3°–78.8° at 10% (models B, E, and H), and 57.1°–100.1° at 20% (models C, F, and I). Adjusting the amount of small fillers enables the droplet behavior to be controlled robustly. For example, although some dispersion at a filler composition amount cannot be avoided in a manufacturing process, a lesser amount of small fillers is better to decrease in a dispersion of the width and contact angle. We conclude that the results demonstrate that the wettability simulation is effective for showing the basic behaviors of CUF components.

Next, to investigate the flow time, we performed capillary flow simulations with models D′, E′, and F′ (Fig. 9). The simulation of model D′, which had no small fillers, is shown in Fig. 9 [(D′1)–(D′5)]. Just after the simulation started [see (D′1) of Fig. 9], monomers and large fillers moved to the gap between walls U and B. The CUF created an arc-shaped liquid–gas interface [see (D′2) of Fig. 9]. Both monomers and large fillers existed near this liquid–gas interface, although the leading surface on each wall consisted of monomers. The CUF flowed by means of the capillary action, so the fillers also continuously moved [see (D′3) of Fig. 9]. When the liquid–gas interface reached the edge on wall U, the flow on that stopped [see (D′4) of Fig. 9]. In contrast, it continued on wall B. Finally, at 47.8 × 10⁶ steps, the liquid–gas interface formed a ridgeline from wall U to wall B, as shown in (D′5) of Fig. 9. During this process, large fillers seemed to ride on a flow of monomers. This is understandable from the snapshots showing a number of monomers moving to the gap from the CUF pool, which is a space to the left of wall S (Fig. 4). We also performed this simulation with models E′ and F′ [see (E′1)–(E′5) and (F′1)–(F′5) of Fig. 9]. Models E′ and F′ had 10% and 20% of small fillers, respectively. In these cases, the macroscopic behaviors of the CUF were similar to that of model D′: a liquid–gas interface was observed at the first stages [see (E′2) and (F′2) of Fig. 9], the CUF flowed in the gap between walls U and B, and the flow stopped at the edge of wall U [see (E′2)–(E′4) and (F′2)–(F′4) of Fig. 9]. Finally, the CUF made ridgelines, as shown in (E′5) and (F′5) of Fig. 9. Here, in models E′ and F′, the flows moved faster than the case of model D′. This is understandable by observing the ridgelines of models E′ and F′, which are farther from the edge of wall U than the case of model D′. In addition, the CUF pools kept their bulk height after the simulations ended [see (E′5) and (F′5) of Fig. 9]. In model D′, large fillers did not move smoothly in the pool [see (D′5) of Fig. 9]. The usage of small fillers makes it possible for the CUF flow smoothly. This effect should be considered when redesigning the processes. When we compare models E′ and F′ [see (E′) and (F′) of Fig. 9], there is no significant difference in the snapshots.

In order to compare the processes of models D′, E′, and F′ in more detail, we calculated the bleed lengths and time steps to fill the capillary space corresponding to the flow time, as shown in Fig. 10. The bleed lengths were calculated, as described in Sec. II. In this study, the bleed length is defined as a difference in wetting ranges between monomer and fillers. Wetting ranges of only monomers and that of only fillers were calculated at the times when a capillary space was filled and at the end of the simulation. Each plot is averaged for 5 × 10⁶ steps, just before a capillary space filling and the end of simulation, respectively. In Fig. 10, error bars indicate the standard deviation of five values corresponding to data detected every million step for 5 × 10⁶ steps. The results show that the bleed length decreased with the increase in the ratio of small fillers/total fillers (Fig. 10). In Fig. 9, the simulation of model D′, which had no small fillers, showed a larger area where only the monomer wets than that of models E′ and F′. This is in agreement with the results in Fig. 10. Different bleed lengths do not seem to appear in (E′5) and (F′5) of Fig. 9. This is because of the effect of fillers inside CUFs on the bleed length. The tendency of the bleed length to decrease depending on an increase in small fillers was observed at both the measuring times. As such, using small fillers should prevent monomers from spreading ahead on the substrate. In a manufactural capillary process, we should actually consider a later curing process. In that case, monomer molecules are possible to cure during wetting because of a manufacturing thermal effect. Therefore, it is conceivable that the dynamics of the droplet at the first stage, like just after the bleed occurrence, is important to discuss a capillary process.

Here, we thought that monomers having no interaction with the filler may bleed out and then simply counted beads of those monomers as free beads in the capillary space. The number of monomer beads in the capillary space is 36 108. This is calculated by the volume of the capillary space of 30.0 × 40.0 × 60.0 σ³, the monomers/total components ratios of 59 vol. %, and the density of monomer beads of 0.85/σ³.¹⁸ In the same way, the numbers of surface filler beads are 5963, 8177, and 10 362 in models D′, E′, and F′, respectively. These are calculated by the volume of the capillary space, the total fillers/total components ratios of 41 vol. %, and the densities of fillers of (D′) 0.202, (E′) 0.277, and (F′) 0.351/σ³, which correspond to Tables I and III. If a surface filler bead has an interaction with a monomer bead, numbers of the free beads are (D′) 30 145, (E′) 27 931, and (F′) 25 746, which are calculated by the number of monomer beads and the surface filler beads. If a surface filler bead has interactions with two monomer beads, numbers of the free beads are (D′) 24 182, (E′) 19 754, and (F′) 15 384. In the latter case, the rate is normalized as (D′) 1.0, (E′) 0.82, and (F′) 0.63. Besides, the bleed lengths in Fig. 10 were (D′) 29.4, (E′) 25.7, and (F′) 18.2 σ, and these are normalized as (D′) 1.0, (E′) 0.87, and (F′) 0.62. In this case, numbers of the free beads in the capillary space show the similar tendency to the bleed length for models D′, E′, and F′. This means that when small fillers are used, the total area of the filler surface increases, and then, more monomers can be captured. In the above calculation, it is conceivable that the surface filler beads captured almost two monomer beads. Capturing monomer beads by the filler increases the apparent filler size, and therefore, using small fillers is possible to reduce a space in which fillers and monomer move.

The time steps to fill the capillary space, which correspond to the flow time of the CUF, are shown on the right vertical axis in Fig. 10. In comparison between models D′ and E′, model E′ that includes small fillers showed a shorter time. In the experimental study, the fine filler shows the improvement on the flow time: Ho et al. detected that 2.0 and 0.6 μm fillers took 7 and 4.5 min, respectively, to fill the 19.7 × 19.7 mm² die.⁸ The simulation results of models D′ and E′ represent the experimental tendency in which the small filler contributes to the decrease in the flow time. However, model F′ took more time than model E′ in Fig. 10. Interestingly, the medium ratio of small fillers/total fillers had the lowest value and, as such, is the best ratio to reduce the flow time. This is very similar to the tendency of contact angles shown in Fig. 8(c). As we pointed out in the above discussion, small fillers move faster (Fig. 6). However, if too many small fillers are used, they can slow themselves down because of an increase in monomer interactions and less space in which to move. This is why the time steps to fill the capillary space increased from 10% to 20% in the ratio of small fillers/total fillers. When designing the capillary process of a CUF, we conclude it is best to use small fillers to both improve the flow time and prevent bleeding especially. There is an optimum amount for the flow time.

In this study, we used CGMD simulations to elucidate the capillary flow of a CUF. First, we performed a wettability simulation with droplets of a 41% ratio of total fillers/total components and 0%–20% ratio of small fillers/total fillers. Results showed that a certain amount of the monomer is spread on the substrate, and many fillers were left inside the droplet. This was numerically confirmed by the MSD calculation showing that monomers had a higher MSD, 25–45 σ², than those of small and large fillers, which were 0.4–1.4 σ² and 0.02–0.2 σ², respectively. We conclude that monomers and fillers have different effects on the droplet wetting. The fact that the small fillers had a larger MSD than the large fillers means that they were more likely to follow the fast movement of a monomer. Thus, small fillers contribute to accelerated wetting dynamics (e.g., in the case of a 10% ratio of total fillers/total components in one of the models we examined). However, when the ratio of small fillers/total fillers got too high, the MSD of both the monomer and the small filler decreased. This means that the monomer and the small fillers tend to slow down during wetting. This is caused by an increase in the interactions between the monomer and small fillers, and, especially, a slowdown of small fillers is due to a decrease in the space in which they can move. In the case of a 10% ratio of total fillers/total components, wetting was smoother than with the ratios of 0% and 20%. We also calculated the wettability in ratios of total fillers/total components ranging from 36% to 46%. The higher ratios made the CUF wetting slow down. In contrast, when a lower ratio was utilized, smoother wetting was observed. We conclude that a small ratio of total fillers/total components makes the contact angles low.

Second, we performed capillary flow simulations with the models having 0%–20% ratios of small fillers/total fillers. During these simulations, the monomer and fillers flowed between parallel walls. After the simulations, the CUF formed a ridgeline at the edge of the upper wall. We found that using small fillers contributed to the decrease in the flow time, as the small fillers move at faster speeds than the large ones. However, in the case of a 20% ratio of total fillers/total components, the flow time increased. A similar tendency was observed with the contact angle. We conclude that using too many small fillers can slow down the flow due to the increase in monomer–filler interactions and the decreased space in which the small fillers can move. We also calculated the bleed length and confirmed that it decreased as the ratio of small fillers/total fillers increased.

These results demonstrate that the CGMD method is effective for simulating the wettability and capillary flow of the CUF. The typical epoxy model was sufficient to investigate them from the viewpoint of this study such as a filler size and a mixture ratio between the filler and monomer, which are important factors to design the CUF material. We were able to clarify that, when designing the CUF, the use of small fillers has the potential to improve both the bleed length and the flow time. However, the optimum amount of small fillers will be different depending on the use case. To predict this quantitatively, in future work, we will examine the effects of different constitutions and size combinations of fillers and thermal curing during flow. In addition to that, we will try to construct more optimal models to represent realistic material components of CUF.

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