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Contents lists available at ScienceDirect

Composites Part A
journal homepage: www.elsevier.com/locate/compositesa

Particle simulation of dual-scale ﬂow in resin transfer molding for process
analysis

7⁎

Shigeki Yashiroa, , Daichi Nakashimab, Yutaka Oyac, Tomonaga Okabec,f, Ryosuke Matsuzakid,eaDepartment of Aeronautics and Astronautics, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan
Graduate School of Integrated Science and Technology, Shizuoka University, 3-5-1 Johoku, Naka-ku, Hamamatsu 432-8561, Japanc
Department of Aerospace Engineering, Tohoku University, 6-6-01 Aoba-yama, Aoba-ku, Sendai 980-8579, Japand
Department of Mechanical Engineering, Tokyo University of Science, Yamazaki 2641, Noda, Chiba 278-8510, Japane
Institute of Fluid Science, Tohoku University, 2-2-1 Katahira, Aoba-ku, Sendai 980-8577, Japanf
Department of Materials Science and Engineering, University of Washington, BOX 352120, Seattle, WA 98195, USAb

A R T I C LE I N FO

A B S T R A C T

Keywords:
A. Polymer-matrix composites (PMCs)
B. Wettability
C. Process simulation
E. Resin ﬂow

Modeling the inhomogeneous microstructures of ﬁbrous tows is important for analyzing the process of resintransfer molding because dual-scale pores in a preform can lead to void formation. This study focused on thedevelopment of a microscopic ﬂow analysis method to predict the impregnation of ﬁber bundles. The moving
particle semi-implicit method was adopted to model the microstructure of a ﬁber bundle explicitly and interparticle potential force was introduced into the numerical model to take account for the capillary eﬀect. The
predicted process of impregnation and void formation agreed with empirical observations. The developed approach was applied to predict the relationship between the modiﬁed capillary number and void content to
identify the optimal molding conditions to reduce microvoids. The obtained relationship reproduced the trendsof a reported experiment, which indicates that the proposed approach will provide information about optimal
conditions for minimizing void content.

1. Introduction
Because ﬁber-reinforced plastics are lightweight and have desirable
mechanical properties, they have been widely adopted in aerospace andautomobile structures to improve fuel eﬃciency. More cost-eﬀective
molding processes compared to prepreg-based autoclave molding are
required for the mass production of automobiles and next-generation
aircraft. Resin transfer molding (RTM) is a promising method based on
its high productivity and the excellent mechanical properties of manufactured composites. However, voids may be formed in or near ﬁber
tows during the impregnation process, which deteriorates the mechanical properties of the composites [1]. The formation of such defects
is an obstacle to the widespread use of liquid molding processes.
The voids formed during RTM are classiﬁed depending on theirscale. Macrovoids (also called dry spots) are large air pockets acrossﬁber bundles, which are caused by incomplete resin ﬁlling. Mesovoids
and microvoids are formed based on molding conditions and the microstructure of the ﬁbrous preform. Dual-scale pores in the preform leadto diﬀerent permeability values inside and between ﬁber tows, whichleads to air entrapment during the resin ﬁlling process. Mesovoids are⁎

air bubbles between ﬁber bundles that are generated when the ﬂow rate
inside the ﬁber bundles is higher than that outside the bundles.
Microvoids are tiny air residuals inside ﬁber bundles that are formed
when the ﬂow rate outside the ﬁber bundles is higher than that inside
the bundles. The balance of viscous ﬂow and capillary ﬂow governs
void formation and is important for the reduction of void content.
There have been numerous studies focusing on the experiment,
analytical modeling and numerical simulation of void formation andthe optimization of the RTM process [2]. Sadiq et al. [3] experimentally
determined that low local permeability in a dense bundle causes void
formation in ﬁber bundles. Patel and Rohatgi [4,5] observed the impregnation of unidirectional stitched ﬁberglass mats and determinedthat the fraction of mesovoids and microvoids is correlated with themodiﬁed capillary number, which is the ratio of viscous force to capillary force. Microscopy analysis of a chopped-strand randomly oriented mat revealed signiﬁcant void variation in the through-thickness(impregnation) direction [6]. In-situ observations of the ﬁlling process
of woven fabrics have revealed that mesovoids are formed by thedominant capillary ﬂow and lead-lag ﬂow front [7]. Mathematicalmodels for analyzing void formation have been developed by

Corresponding author.
E-mail address: yashiro@aero.kyushu-u.ac.jp (S. Yashiro).

https://doi.org/10.1016/j.compositesa.2019.03.038
Received 12 December 2018; Received in revised form 21 February 2019; Accepted 25 March 2019
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S. Yashiro, et al.

considering microscale ﬂow into inhomogeneous preforms based on
Darcy’s law. Kang et al. [8] predicted the relationship between the void
fraction and modiﬁed capillary number by analyzing time diﬀerences
when ﬁlling a bundle and open channel. Matsuzaki et al. [9,10] developed a similar mathematical model using a mathematical permeability model [11] and their prediction of the fraction of mesovoids
based on ﬂow velocity agreed with experimental measurements. Lebel
et al. [12] applied a mechanistic imbibition model to fabrics to characterize the impregnation process and determined that two key parameters, namely the modiﬁed capillary number and penetrativity of theﬂuid, should be optimized to reduce voids. They also proposed an experimental method to obtain the optimal conditions for a given resinfabric pair.
RTM simulations based on the ﬁnite-element/control-volume (FE/CV) method [13–15] have been developed over the past few decades topredict void content. In the standard approach, the governing equation
for single-phase ﬂow is solved on the macroscopic scale, which essentially neglects microscale pores in the bundle. This approach has been
extended to involve the microscale intra-tow ﬂow and to predict microvoids and mesovoids [16–19]. Although the optimal value of themodiﬁed capillary number has been predicted using FE/CV methods[18], such methods always require a measured value of local permeability or model of permeability as input data. Proper calibration may
be required to apply these permeability models [11,20,21] to a real
preform.
To overcome this diﬃculty, this study presents an impregnation
analysis method from a microscopic viewpoint. In recent years, microscopic ﬂow simulation in a ﬁber array has been investigated bysolving the momentum equation using the ﬁnite-element method andﬁnite-volume method, and the eﬀective permeability have been analyzed based on the intra-tow ﬂow [22–26]. A particle method is employed in this study to explicitly model the microstructures of ﬁbroustows and analyze the void formation process based on the ﬂow throughopen channels and intra-tow paths. The surface tension eﬀect can beeasily introduced into particle simulations, and this characteristic enables us to predict the entrapment of air without using permeability.
Okabe et al. [27] proposed a method for analyzing microscopic ﬂows
through ﬁbrous tow based on the moving particle semi-implicit (MPS)
method [28] and considered the capillary eﬀect as a potential force.
However, the results did not reproduce the mechanisms of air entrapment during RTM [4,5] accurately because the open channels between
bundles were neglected. Lu et al. [29] presented a ﬂow analysis of the
RTM process using smoothed particle hydrodynamics, but they neglected the capillary eﬀect, which is essential to estimating optimal
RTM process conditions.
Therefore, this study focused on the development of a numericalmethod for analyzing microscale ﬂows during the RTM process. To thisend, we adopted the MPS method to explicitly model the microstructures of ﬁber bundles and incorporated inter-particle potential
force into the MPS method to account for the capillary eﬀect. This remainder of this paper is organized as follows. Section 2 describes experiments on liquid ﬂow using a transparent mold with simulated ﬁber
bundles. Section 3 brieﬂy introduces the MPS method and an interparticle potential force related to surface tension and wettability. In
Section 4, ﬂow analysis corresponding to the experiments is performed
to examine the process of microvoid formation. A numerical study ispresented in Section 5, where we estimate optimal molding conditions
to reduce the void fraction without using permeability.

Fig. 1. Transparent mold with simulated ﬁber bundles. (For interpretation of
the references to colour in this ﬁgure legend, the reader is referred to the web
version of this article.)

was 0.4 mm. The space between adjacent bundles (blocks with densely
arranged rods) was 2.5 mm in the width direction. The local volume
fraction of the rods in each bundle was 50% and the nominal rod volume fraction in the rods arranged area was 27%.
Liquid was injected from one end of the mold (left side in Fig. 1a)and the impregnation process was observed. In this study, water andolive oil were used as test liquids, which were chosen by considering
their diﬀerent viscosities and resultant capillary eﬀects. Their properties are listed in Table 1. The liquids were injected by a micro-syringe
pump at a constant ﬂow rate of 500 ml/h. This ﬂow rate corresponds toa ﬂow-front velocity of 6.3 mm/s in the empty space. A high-speed
digital microscope (VW-9000, Keyence) was used to observe the ﬂow.
The impregnation of the bundle shown in Fig. 1a was observed in detailand the void fraction was determined as the area of the voids divided by
the observed rectangle area, in which the nominal rod volume fraction
was 30%.
Fig. 2 presents snapshots of the observed water ﬂow into the bundle,
where the dotted line is the ﬂow front of the liquid. Water impregnatedonly the outermost layer of bundles and then ﬂowed between bundles
without further impregnation. The ﬂow velocity in the open channels
was higher than that in the bundles and the bundles were surroundedby water before complete saturation occurred. As a result, air remained
Table 1
Material properties of liquids.

Density (kg/m3)
Kinematic viscosity(m2/s)
Surface tension (mN/m)
Contact angle to quartz(°)

2. Experiment
To observe the process of void formation in a ﬁber bundle, atransparent polymethyl methacrylate mold with simulated ﬁber bundles (Fig. 1) was prepared. The mold cavity space was 22.0 mm wide
and 1.0 mm thick. Quartz rods with a diameter of 1.0 mm were embedded into the mold and the minimum spacing between adjacent rods

(a)



Water

Olive oil

Measurement method

1000(a)1.0 × 10−6(a)914
8.37 × 10−572.8(a)34.6

Archimedes method
Capillary viscometer
methodPendant drop method

39.9

27.4

A half-angle method

Values from literature.

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S. Yashiro, et al.

them, this study employs the MPS method for the analysis of inter- andintra-tow ﬂow, because it is suitable for solving incompressible ﬂow
and easy to introduce the capillary force into the analysis framework.
3.1. MPS method
The MPS method [28] is a particle-based method in which a continuum is expressed as an assembly of particles. Each particle is assumed to have an eﬀective radius for particle interaction and diﬀerential models are prepared based on particles within the eﬀective
volume. When a particle i has a scalar variable ϕi and a vector variableui, the corresponding diﬀerential models are deﬁned as follows:〈∇ϕ〉i =〈∇ ·u〉i =〈∇2 ϕ〉i =Fig. 2. Snapshots of the water ﬂow into a bundle. The dotted line indicates theﬂow front. (For interpretation of the references to colour in this ﬁgure legend,
the reader is referred to the web version of this article.)λ=Nd⎡ ϕj − ϕi (r − r ) w (|r − r |) ⎤
∑
j
i
j
i ⎥
0
2
n j≠i ⎢
⎣ |rj − ri|⎦(uj − ui)·(rj − ri)dw (|rj − ri|) ⎤
∑⎡
2
⎥
|
|
r
r
n0 j ≠ i ⎢
−
j
i
⎣
⎦

(1)

N

2d
∑ [(ϕj − ϕi ) w (|rj − ri|)]
λn0 j ≠ i(2)

N∫r r 2 ⎛⎝ rre − 1⎞⎠ 2πr dr / ∫r ⎛⎝ rre − 1⎞⎠ 2πr dr
re

re

0

0

(3)
(4)

Here, d is the dimensional number, n0 is the constant particle
number density under incompressible conditions, N is the number ofparticles within the eﬀective radius re, r is the position vector, and w(=re/r − 1) is the weight function.
The governing equations for incompressible ﬂow are the massconservation and the Navier-Stokes equation:

Dρ
= 0,
Dt

Du
1
= − ∇P + ν∇2 u + g ,
Dt
ρ

(5)
(6)

where t is time, ρ is density, u is velocity, P is pressure, ν is kinematic
viscosity, and g is acceleration caused by an external force, such as
gravity. These values are discretized by diﬀerential operator models. In
the standard MPS method, the viscous (second) and external force(third) terms on the right of Eq. (6) are solved explicitly and the Poisson’s equation for pressure derived from the pressure (ﬁrst) term andEq. (5) is solved implicitly. However, analyzed pressure distributions
frequently include numerical oscillations. This study employed an improved solving scheme [33] to obtain a smoother pressure distribution.

Fig. 3. Impregnation process of a bundle with oil. The dotted line indicates theﬂow front. (For interpretation of the references to colour in this ﬁgure legend,
the reader is referred to the web version of this article.)

3.2. Inter-particle potential force
To analyze the capillary eﬀect, an inter-particle potential force wasintroduced into the MPS method. The inter-particle potential force Fs isdeﬁned as the gradient of the potential Π. This study employed theMorse potential, which is an interatomic interaction model.

inside the bundles and the void fraction of the observed area was
10.8%. This mechanism of microvoid formation agrees with previous
experiments [3,5]. Fig. 3 presents snapshots of the oil injection experiment. In contrast to the results of water injection, the oil ﬂowed
into the bundles. Although the ﬂow front in the open channels still
moved faster than that inside the bundles, the velocity diﬀerence wassmall. Therefore, the bundles were ﬁlled completely, and no voids wereformed. The capillary eﬀect is the main cause for this diﬀerence between the two liquids [30].

Fs = ∑

rj − ri
∂Π
·
∂
(|
r
−
r
|)
|
r
j
i
j − ri |j≠iNΠ = C {e−2(r / l0 − 1) − 2e−(r / l0 − 1) },

(7)
(8)

where C is the coeﬃcient, r is the distance between two particles, and l0
is the equilibrium distance (i.e., initial particle spacing). The Morsepotential, Eq. (8), was adopted with consideration for the stability ofanalysis.
Wettability is calculated using the same potential model. Wettability
can be represented by using two speciﬁc potential coeﬃcients for theinteraction between two ﬂuid particles and for the interaction between
a ﬂuid particle and solid particle, respectively. This study used the

3. Analysis
Microscale ﬂow in ﬁbrous media as well as ﬂow of ﬁber suspension
have been studied using particle simulation methods owing to theircapability to represent microstructures [31]. In particular, the lattice
Boltzmann method [32], the smoothed particle hydrodynamics [29]and the MPS method [27] have been used for those analyses. Among

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S. Yashiro, et al.

Table 2
Parameters of the inter-particle potential model.

Cﬀ
αEﬀective radiusInjection wall

Water

Oil

20
0.73
3.9 l0

200
1.215

Fiber

8.5 mmFig. 4. Schematic of the estimation of inter-particle potential force.

following relationship between the two coeﬃcients:

Cfs = α Cff ,

(9)

Air

where Cﬀ and Cfs are the coeﬃcients for ﬂuid–ﬂuid interaction (i.e.,surface tension) and ﬂuid-solid interaction, respectively. Cﬀ is only usedin Eq. (8) if the particle of interest i and its neighboring particle j areboth ﬂuid particles. Cfs is used when either i or j are solid (Fig. 4). The
arbitrary positive constant α was calibrated based on droplet collapse
simulations to obtain an appropriate contact angle. The inter-particle
potential force was adopted as the external force term in the NavierStokes equation.
The potential model was veriﬁed through droplet collapse. The
contact angles between the liquids and a quartz plate were measured byan automatic contact angle meter (DMe-201, Kyowa Interface Science)and found to be 39.9° and 27.4° for water and olive oil, respectively.
The collapse of a droplet with a diameter of 2 mm was then analyzed
using 0.125-mm particles. The contact angles for analysis were determined using the half-angle method and were averaged during theperiod of 0.2–0.5 s, during which the motion of the droplet was mostly
converged. Fig. 5 depicts the observed droplet shape and the predicted
result with adjusted parameters. The predicted contact angles (38.3°
and 30.7° for water and oil) were matched to the measured angles byadjusting the parameters Cﬀ and α (Table 2). However, the potential
model needs to be improved in order to reproduce the observed droplet
shape.

LiquidArea to measure the void fraction

Fig. 6. Analytical model of the transverse ﬂow into a ﬁber bundle. (For interpretation of the references to colour in this ﬁgure legend, the reader is referredto the web version of this article.)

4. Liquid ﬂow in a ﬁber bundle
Two-dimensional analysis of liquid ﬂow into a ﬁber bundle was
performed to predict the void formation process. Fig. 6 depicts theanalytical model that corresponds to this experiment. The particle size
was 0.125 mm and a single rod was represented by 52 particles. The
total number of particles in the initial condition was 8364. Liquid
particles were injected with a velocity of 6.3 mm/s from the left side of
the model. Speciﬁcally, the left wall moved in the injection direction at
the prescribed velocity. When the travel distance reached the size of aparticle, the wall returned its original position and new ﬂuid particles
were generated in the empty space. A periodic boundary condition wasapplied to the top and bottom ends of the model. An outlet boundary
condition was imposed on the right end of the model, and particles
passing through the boundary were excluded from the analysis. Waterand olive oil were used as test liquids again with the properties listed inTable 1. The inter-particle potential model was considered and theparameters listed in Table 2 were used. For the sake of simplicity andstability of analysis, the density of air was set to the same value as thatof water and its kinetic viscosity was set to 1.515 × 10−5 m2/s. Theinter-particle force was not applied to the air particles. The void fraction was calculated as the number of air particles divided by the initial
number of particles in the designated region.
Fig. 7 depicts snapshots of the predicted water ﬂow into the bundle.
When the water reached the left end of the bundle, it impregnated up to
one layer inside the outermost layer of the bundle (Fig. 7a). The bundle
was hardly immersed in the injection direction and water ﬂowed
mainly through the open channels, as depicted in Fig. 7b and c. The
bundle was ﬁnally surrounded by water and air remained inside thebundle (Fig. 7d). This process of microvoid formation agrees well withthe experimental results (Fig. 2). The predicted void fraction in the ﬁnal
state was 9.4%, which also agrees with the experimental results.
Fig. 8 presents snapshots of the predicted ﬂow of oil into the bundle.
Similar to the experimental results, impregnation proceeded from theleft end of the bundle until 0.6 s (Fig. 8a and b). The bundle was thenimmersed in the direction perpendicular to injection because of thecapillary eﬀect and a small amount of air was entrapped between theﬁbers (Fig. 8c). The ﬂow front velocity in the open channels was
slightly higher than the impregnation speed and the bundle wasFig. 5. Final shape of the droplet on a quartz plate. (For interpretation of thereferences to colour in this ﬁgure legend, the reader is referred to the web
version of this article.)&RPSRVLWHV3DUW$  ²

S. Yashiro, et al.

Fig. 7. Predicted process of impregnation of a bundle with water. (For interpretation of the references to colour in this ﬁgure legend, the reader is referredto the web version of this article.)

Fig. 9. Analytical model for predicting the void fraction with a transverse ﬂow.
(For interpretation of the references to colour in this ﬁgure legend, the reader isreferred to the web version of this article.)

speciﬁed after a number of molding experiments and observations foreach combination of ﬁber preform and resin. Microscale ﬂow analysis
can be used as a substitute for such laborious experiments. The remainder of this section will demonstrate such an application of the
proposed approach.
The process of air entrapment varies based on molding conditions.
The void fraction in an RTM process can be determined based on themodiﬁed capillary number Ca* as follows:

Ca∗ =

µu
,
σ cos θ(10)

where μ is the viscosity, u is the superﬁcial velocity (injection ﬂow
velocity in this study), σ is the surface tension, and θ is the contact
angle.
A numerical study was performed to predict the optimal conditions
from the perspective of microvoid formation. Fig. 9 depicts the analytical model. The size of each particle was 2 μm and seven particles werearranged in a hexagonal shape to approximate a ﬁber with a diameterof 6 μm. There were 3942 particles in the initial state. Fluid particles
were injected from the left end of the model and the periodic boundary
condition was applied to the top and bottom of the model. The void
fraction in the ﬁber bundles was predicted under 27 diﬀerent conditions with varying injection speeds (1, 10, and 100 mm/s), viscosities(0.1, 0.5, and 1 Pa·s), and contact angles (40°, 60°, and 80°). The density
and surface tension of the ﬂuid were constant (1000 kg/m3 and7.28 × 10−2 N/m, respectively). These conditions correspond to a
modiﬁed capillary number range of 1.8 × 10−2–7.9. The number of air
particles remaining in the region shown in Fig. 9 was counted after the
major ﬂow front reached the right end of the model.
Fig. 10a presents an example of the ﬁnal particle distribution; air
was entrapped in the bundles under the condition Ca* = 7.9, similar tothe observation (Fig. 2). The predicted relationship between the void
fraction and modiﬁed capillary number is plotted in Fig. 10b. The void
fraction was zero at Ca* values less than 0.01 and increased monotonically in with Ca* values greater than 0.02. This trend in the void
fraction agrees with reported experimental data on microvoids [5].
Therefore, ﬂow analysis considering the microstructure of the preformand capillary eﬀect has possibility of predicting optimal molding conditions to minimize microvoids. However, this analysis could not reproduce the mesovoids observed in past experiments [4]. This diﬀerence is the result of the analytical model disregarding the woven
structures of bundles. Warp yarns should be included in the model topredict the relationship between the void fraction and Ca* within a low
capillary number range. It should be noted that the optimal Ca* value

Fig. 8. Predicted process of impregnation of a bundle with oil. (For interpretation of the references to colour in this ﬁgure legend, the reader is referredto the web version of this article.)surrounded by the oil before all air was ejected (Fig. 8d). The amount ofentrapped air diﬀered from the experimental results, but most of thebundle was ﬁlled and the void fraction was 2.9%.
Although there was a small error in the void fraction, the overall
impregnation process agreed with the experimental results in Section 2.
Therefore, ﬂow analysis with an adequate capillary eﬀect model can
predict change in the fraction of microvoids in ﬁber bundles undervarying molding conditions. Dimensionless analysis to address the difference in scale from an actual ﬁber bundle and veriﬁcation of thesimpliﬁed air viscosity are our future work.

5. Discussion
Although the proposed approach reproduced the process of microvoid formation successfully, it can analyze only a small portion of a realmold based on its high calculation cost. The FE/CV method [13–15] isrealistic for predicting the molding process of a structural member fromthe viewpoint of the calculation cost, and the present approach can beused as a convenient auxiliary tool. The FE/CV method requires thepermeability of a ﬁber preform as input data, which is typically measured experimentally. Furthermore, optimal molding conditions can be&RPSRVLWHV3DUW$  ²

S. Yashiro, et al.

Acknowledgement
This work was supported by the Cross-ministerial Strategic
Innovation Promotion (SIP) Program, “Structural Materials forInnovation.”
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Fig. 10. Analytical results of a transverse ﬂow in ﬁber arrays. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the
web version of this article.)

will depend on the number of ﬁbers in a tow.
6. Conclusions
This study focused on the development of a ﬂow analysis method topredict void formation during the RTM process using the MPS method,which enabled us to explicitly model the microstructures of ﬁber bundles. An inter-particle potential force was introduced into the MPS
method to account for the capillary eﬀect. The impregnation of simulated ﬁber bundles was observed in a transparent mold to verify the
proposed analytical method. Our conclusions are summarized below.
1. The developed ﬂow analysis method with microstructure models ofﬁber preforms reproduced the process of microvoid formation in
ﬁber bundles observed in experiments successfully. The predicted
impregnation and void formation processes agreed with our experimental observations.
2. The relationship between the void fraction and modiﬁed capillary
number was predicted by the developed analysis method successfully. The predicted results reproduced the trends observed during
past experiments on microvoids [5]. This demonstrates the ability ofthe proposed method to predict optimal molding conditions tominimize the void content in RTM processes.
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