# Numerical model of fiber wetting with finite resin volume

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## Abstract

The partial wetting of cylindrical surfaces is encountered in many industrial applications such as composites manufacturing, MEMS, hair care products, and textile engineering. Understanding the impact of key parameters such as resin and fiber surface interaction properties and the geometric arrangement of the fibers on wetting would lead to tailoring a desired interface between the resin and the fiber surface. A three-dimensional model of resin wetting a single fiber is presented. This model is then extended to study a finite volume of resin wetting fibers in square and triangular packing arrangements. The impact of changing wetting properties and fiber volume fraction is examined for each packing arrangement.

## Keywords

Wetting Contact angle Computational multiphase flow Composites## Background

The partial wetting of cylindrical surfaces by a finite volume of resin is an important phenomenon in many industrial applications such as composites manufacturing, MEMS, and textile engineering. A constitutive equation governing the partial wetting of a finite volume of liquid on a flat plate has been formulated and reported [1]. The equilibrium shape of resin on single fibers has also been studied in depth [2-5]. Carroll [2] was the first to develop an analytical solution for the equilibrium shape of a resin drop on a single fiber. A drop at rest on a fiber will either conform to a “barrel” geometry, where the drop wraps around the fiber, or a “clamshell” geometry in which the fiber rests on the fiber’s surface without wrapping around it. A phase diagram predicting which of these configurations a particular drop will adopt has been constructed [5].

*x*is the location on the axis of the fiber measured from the center of the drop, and

*y*is the height of the drop, measured from the axis of the fiber.

*y*

_{0}is the maximum height of the drop measured from the axis of the fiber, and

*r*is the fiber radius.

*E*(

*k*,

*φ*) and

*F*(

*k*,

*φ*) are Legendre’s elliptical functions of the second and first kind, respectively. Here

*λ*and

*k*are defined as follows:

Here, *θ* is the static contact angle between the fiber and resin.

*L*, which is also defined in Figure 1, can be calculated using the known volume

*V*once

*y*

_{0}is solved for with the above equations and:

*φ*_{0} is found through setting *y* = 0 in Equation 1.

There have also been other investigations with resin spreading within multiple fibers, for example final resin configuration between two parallel fibers has been studied with relation to static contact angle, filament spacing, resin volume, and fiber diameter [6]. The axial wetting of a single fiber from a reservoir of resin has been experimentally examined and constitutive equations have been developed to describe this phenomenon [7]. The dynamics of a finite volume of resin spreading on a single fiber has yet to be explored and is the subject of this paper. Trends seen with the dynamics of resin wetting a single fiber hold true for systems with multiple fibers.

A numerical model is presented using the level set method to study the movement and spreading of a finite volume of resin on any planar or curvilinear surface. The method and accuracy is verified by comparing the model results with experiments conducted of a drop spreading on a flat plate. The method is then used to describe the wetting dynamics of a finite drop of resin on a single fiber. The model is further extended to investigate the flow of finite volume resin within multifiber unit cells representing square and hexagonal fiber packing arrangements, which are commonly used in composites. The use of a finite volume of resin is necessary because there are situations where it is desirable to strategically create microvoids to increase the composite’s energy-absorption capabilities. Each tow would be a porous structure comprised of a series of microstructures represented by the unit cells. The model studies the impact of key composite-processing properties on the fiber-matrix interfacial area. Through examination of the interfacial area instead of the contact length, one is better able to understand the impact of manipulating processing parameters on the resulting composite properties. This investigation should prove useful in tailoring the interface properties between fibers and resin as a function of the resin and fiber surface properties and the fiber arrangement.

## Methods

### Model setup

The numerical models were developed using the COMSOL Multiphysics and the Microfluidics Module to investigate the dynamics of wetting over a single fiber and within unit cells of multiple fibers with a finite volume of resin and are presented below. A model was also constructed of a drop spreading on a flat plate with the goal of experimentally validating the solution method.

#### Axisymmetric single-fiber model

**Baseline properties used for parametric studies for the axisymmetric single-fiber model as well as the three- and four-fiber unit cells**

| |
---|---|

Resin density | 1.17 g/cm |

Resin viscosity | 9.5 Pa∙s |

Static contact angle | 30° |

Resin surface tension | 0.07 J/m |

Slip length ( | 0.1 μm |

#### Three-dimensional single-fiber model

^{3}, and contact angle of 15° were selected to ensure that the final shape is a “barrel” guided by the studies performed by Eral et al. [5]. The other important properties are the same as the baseline values described in Table 1.

#### Unit cell with square and triangular packing arrangements

#### Resin spreading on a flat plate

### Assumptions

The Reynolds number in this problem is on the order of 10^{−8}; thus, it can be assumed that the inertial forces are negligible relative to the viscous forces and the Navier–Stokes equations can be reduced to the Stokes flow equations. The ratio of gravitational to capillary forces, represented by the bond number, is on the order of 10^{−6}, making it acceptable to neglect gravity. For the axisymmetric model, due to the geometry being symmetric about the fiber axis and our assumption of no gravity, the flow is considered axisymmetric about the axis of the fiber. It is assumed that the resin used does not cure during the wetting process, allowing us to maintain a constant viscosity value during the flow. It is also assumed that the fibers are rigid and do not move as the resin flows.

### Governing equations

*φ*), which continuously changes from 0 to 1 across the interface using a smeared out Heaviside function [10]. These equations (Equation 9), modified to account for the stated assumptions, are given by [11]:

*is the velocity vector, the subscript denotes the partial derivative with respect to that variable, μ is the viscosity,*

**u***is the pressure,*

**p**

**F**_{st}is the force due to surface tension,

*σ*is the reinitialization parameter for the interface,

*ε*is the interface thickness, and

*ϕ*is the level set variable. To minimize computational cost, the interface thickness is set to one half of the largest element length [11,12]. The interfacial tension term is implemented using the continuous surface force formulation, given by:

Here, *γ* is the surface tension of the resin-air interface, and *δ* is a Dirac delta function.

### Fiber and resin parameters

The properties of the resin and the fiber-resin interactions play an important role in the wetting of the fibers by the resin. The viscosity of the resin has a large impact on the rate of wetting, but not a significant effect on the final shape of the drop. The bond number is the ratio of surface forces to body forces, providing a good indication if the resin flow is driven by surface forces or gravity. This study focuses on flows with low-bond numbers. The contact angle between the fiber and resin, largely impacted by the surface tension of the resin, represents the principle force driving wetting at the microscale. The fiber diameter and resin droplet size will be important geometrical parameters when investigating drops spreading on the fiber surfaces. When the model is extended to include multiple fibers, the fiber spacing and packing arrangement will influence the wetting dynamics.

#### Static contact angle between fiber and resin

In Young’s equation, *γ*_{ij} represents the surface energy at the i-j interface. As shown in Equation 14, the final static contact angle takes into account both the resin surface tension and the difference in interfacial energies of the solid-vapor and solid–liquid interfaces. The solid-vapor and solid–liquid surface energies can be manipulated by modifying the fiber sizing, which is a coating that is applied to the fiber surface. The final static contact angle of the resin on the fiber surface has been shown to have a direct relationship with the interfacial shear strength of the resulting composite [14].

#### Resin viscosity

The viscosity of the resin does not affect the final position of the resin on the fibers since it is assumed to be constant. As the Stokes solution is linear, the time it takes to wet the fiber surface will be directly proportional to the viscosity of the resin.

#### Slip length

There are stress and velocity singularities at the three-phase contact line when solving the Stokes equations with a no-slip condition at the solid surface [15]. A way to handle this boundary condition is to move the “no-slip” condition to a plane located a distance *β* (slip length) below the solid surface and assume simple shear flow in the region between the wall and the no-slip plane [16]. The frictional force at the wall is scaled with the slip length [11]. Not unlike viscosity, changing the slip length will influence the wetting rate, but not the final distribution and configuration of the resin on the fiber surface. The slip length is a parameter that models the interactions at the fiber-resin interface.

#### Fiber volume fraction and packing

*v*

_{f}), fiber radius (

*r*), and distance between fiber axes, (

*d*), will be:

### Experimental setup and procedure

^{3}. Sample images of the drop spreading are shown in Figure 6. The static contact angle between the glass and glycerin, measured using image analysis software on the drop in equilibrium, is 28.5°.

## Results and discussion

First, experimental and analytical validation of the model used will be provided in the next section before parametric studies are conducted to investigate the dynamics of resin spreading.

### Validation of the numerical model

The physics involved in the preceding models is multiphase fluid flow with a high surface to gravitational force ratio. A model of a drop spreading on a flat plate was developed to experimentally verify that the governing equations could predict an acceptable numerical solution to a multiphase wetting dynamics dominated by surface forces.

#### Comparison between experimental and numerical results

*β*which defines the resin fiber surface characteristics. A

*β*value of zero would correspond to the case where the liquid will not wet the substrate, and an infinite value would describe the scenario where the liquid would reach its final configuration instantaneously. The

*β*value in real systems will fall between the preceding extreme cases and can be determined experimentally by comparing the numerical and experimental solutions using a range of

*β*values. As

*β*is increased or decreased, the wetting rate in the numerical solution will become higher or lower. The

*β*value that describes the liquid-substrate system is found by adjusting the

*β*value until the dimensionless length, defined as the drop length at time

*t*divided by the final length of the drop, matches the experimental results. This experiment used the resin volume of 0.057 mm

^{3}. The

*β*value for slip length from this case was determined to be 0.25 μm. The experimental and COMSOL results for this

*β*value are shown in Figure 7 along with an inset that describes distance

*β*(slip length) below the solid surface and assumes simple shear flow in the region between the wall and the no slip plane.

Having determined the *β* value, the next experiment was conducted with a drop volume of 0.140 mm^{3}. The dimensionless length of the COMSOL simulation is compared to the experimental dimensionless length in Figure 7 using the characterized *β* value. Comparing the results verify the numerical model used to describe the dynamics of resin spreading on a surface for a large surface force to body force ratio.

#### Mesh-refinement study

#### Comparison of final drop shape with an analytical solution

*y*

_{0}. Once

*y*

_{0}is known, it can be substituted back into Equations 1 to 4 to develop a parametric equation for

*x*and

*y. φ*was varied for values corresponding to

*y*>

*r*. The resin-air interface shape at the

*yz*-plane is solved by using this method and is compared to the numerical solution in Figure 9. The closeness of the two solutions provides further validation of the numerical model.

### Parametric study of axisymmetric model

The wetting physics in the axisymmetric model was influenced by the static contact angle, slip length, fiber and resin geometry, and viscosity.

#### Static contact angle between fiber and resin

#### Resin volume

#### Fiber radius

#### Slip length

#### Three-dimensional single-fiber model

### Square and hexagonal packing fiber unit cells

#### Fiber volume fraction

#### Static contact angle

*θ*) makes sense because the final results shown are with a fiber volume fraction of 30%. It is clear that for a given fiber volume fraction, the square packing arrangement is preferred for increasing fiber-resin contact area. For this particular combination of resin volume, fiber volume fraction, and fiber size, the ratio of fiber-resin contact area for the triangular and square packing arrangements ranged between 1.07 and 1.11 for the given static contact angles. The static contact angle did not have as significant of an impact on the resin spreading as the packing arrangement did. Resin reached its equilibrium position inside square-packed fibers in about 0.1 to 0.2 s as compared to the 1.1 to 1.5 s with these initial conditions. This indicates that when the same volume of resin wets fibers in a square packing arrangement, the resulting composite will have a higher fiber-resin contact area and faster processing time when compared to a triangular packing arrangement.

### Limitations of the model

A limitation on this model is imposed by the assumption of a microscopic length scale. This is because when the diameter of the fiber or the volume of the resin is increased by a large amount, the inertial and gravitational forces are no longer considered negligible. This would invalidate the axisymmetric assumption in the axisymmetric fiber model. In the four-fiber model, one would no longer be able to use the symmetry plane orthogonal to the direction of gravity. The trends seen in these models may not hold for models with extremely large contact angles because they only examine the case where the liquid will wet the fiber’s surface.

## Conclusions

Numerical models describing the partial wetting of a finite volume of resin on a single fiber and in triangular- and square-packed unit cells was presented and validated. The static contact angle affected both the rate of axial spreading as well as the final fiber-resin contact area. The volume of resin impacted the final fiber-resin contact area and the wetting rate because larger volumes of resin travel farther. Both the wetting length and final fiber-resin contact area increased with increasing fiber diameter. This claim is only for the case when the resin is in a barrel shape around the fiber as the clamshell shape was not investigated. The slip length had a defined effect on the rate of wetting, but did not impact the final fiber-resin contact area. This indicates that the slip length will not impact the composite properties. Fiber volume fraction had a significant impact on fiber-resin contact area, being more influential at higher fiber volume fractions. The final fiber-resin contact area was larger for square-packed unit cells than triangular-packed unit cells. In unit cells with triangular or square packing arrangements, the static contact angle had a large impact on the final fiber-resin contact area. The effect of static contact angle on wetting rate was small compared to the impact of packing arrangement on wetting rate. These models can be used to predict the impact of manipulating fiber and resin surface properties, interaction, and geometry on the wetting of fibers by a finite volume of resin. By predicting the influence of processing parameters on fiber wetting, one can correlate the resulting microstructure in the unit cell with process and material parameters. The properties of the fibers and matrix can then be used to determine the mechanical properties of a unit cell with the predicted microstructure. The mechanical properties of the unit cells can be used to determine the composite properties.

## Notes

### Acknowledgements

Research was sponsored by the Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF-12-2-0022. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.

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