Analysis of Wettability Model Using Adhesional and Spreading Works

Nobuhiko Mukai

1,2 a

, Takuya Natsume

, Masamichi Oishi

and Marie Oshima

2,3

Graduate School of Integrative Science and Engineering, Tokyo City University, 1-28-1 Tamazutsumi, Setagaya,

Tokyo 158-8557, Japan

Institute of Industrial Science, The University of Tokyo, 4-6-1 Komaba, Meguro, Tokyo 153-8505, Japan

Initiative in Information Industries, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-8654, Japan

Keywords:

Fluid Dynamics, Particle Method, Wettability, Contact Angle, Adhesional Work, Spreading Work.

Abstract:

We have developed a new method of wettability, which is a feature for a liquid to keep the contact angle formed

between a liquid and a solid body. Conventional models required the contact angle in advance for simulations,

which angle can be measured by physical experiments. On the other hand, our new model does not need the

contact angle and forms the shape of liquid on a solid body by considering adhesional and spreading works.

We demonstrated that the proposed method was able to represent wettability by simulations without contact

angles. This paper evaluates the proposed method by investigating the drop time of the liquid extruded from a

thin tube.

1 INTRODUCTION

Liquid simulation is a very challenging issue since

it deforms dynamically and the topology changes all

the time with the separation and integration of many

small molecules. The simulation where one kind of

liquid drops in the air is relatively simple because the

air is usually ignored and the simulation can be per-

formed by considering just one type of liquid. On the

other hand, the simulation where two kinds of liquid

should be treated is very complex and difﬁcult to per-

form.

One example is the simulation of an emboliza-

tion material dropped in the cerebral aneurysm. In

this case, two different kinds of liquid, which are em-

bolization material and blood, should be considered

in the simulation, and the interfacial tension works on

the boundary between the two materials. The inter-

facial tension is different from free surface tension,

which works between liquid and air that can be usu-

ally ignored.

For ﬂuid simulations, two types of methods are

usually used: grid-based Euler method and particle-

based Lagrangian one. Grid-based Euler methods can

perform simulations for large spaces, however, it is

difﬁcult to treat the interfacial tension that works on

the boundary of two different kinds of ﬂuid. On the

other hand, particle-based Lagrangian ones can eas-

https://orcid.org/0000-0001-8909-9454

ily detect the boundary of two different kinds of ﬂuid

and can treat the topological change; however, the cal-

culation accuracy is relatively low since it does not

consider some particles that are outside of a constant

range for the calculation.

For the safety veriﬁcation of a new embolization

method for a cerebral aneurysm, we have been trying

to simulate the behavior of a droplet that is ejected

from a thin tube, which is a real catheter, into a

water tank, which imitates a cerebral aneurysm, us-

ing the MPS (Moving Particle Semi-implicit) method

that is one of the particle methods and was devel-

oped for incompressive ﬂuid. We have also evaluated

our method by the comparison between the simula-

tion results and the physical experiments. The method

considered the effect of liquid-liquid two-phase ﬂow;

however, the droplet that came out of the catheter did

not adhere to the edge of the catheter because we did

not consider wettability.

To simulate wettability, which is a feature for a

liquid to keep the contact angle between the liquid

and the solid body, the contact angle is necessary

and it can be measured by physical experiments. It

means that we cannot perform the simulation unless

the contact angle is known, which is decided depend-

ing on the physical features of two materials: liquid

and solid.

Then, we developed a new wettability method that

does not need the contact angle between a liquid and a

230

Mukai, N., Natsume, T., Oishi, M. and Oshima, M.

Analysis of Wettability Model Using Adhesional and Spreading Works.

DOI: 10.5220/0011710100003417

In Proceedings of the 18th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2023) - Volume 1: GRAPP, pages

230-236

ISBN: 978-989-758-634-7; ISSN: 2184-4321

 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)

solid and succeeded in representing wettability. How-

ever, we have not evaluated the method for the drop

time of liquid by the comparison between the simu-

lation results and the physical experiment. Therefore,

this paper reports the evaluation of the new wettabil-

ity method for the drop time of liquid. The maximal

merit of our method is that it can represent wettability

without contact angles, which must have be measured

before simulations in the conventional methods. The

proposed method can visualize wettability of a liquid

on a solid body not by estimating the contact angle

but by changing the liquid shape with adhesional and

spreading works, which operate between liquid and

solid body.

2 RELATED WORKS

For ﬂuid simulations, mainly two types of methods

are usually used: the grid-based Euler method and the

particle-based Lagrangian one. The grid-based Eu-

ler method calculates physical features such as den-

sity and velocity of ﬂuid at ﬁxed positions. Then,

it requires a lot of memories for simulations and it

is difﬁcult to determine the boundary where multiple

ﬂuids contact. On the other hand, the particle-based

Lagrangian one simulates physical phenomena using

many particles, and it can easily determine the bound-

ary between different types of particles, but, the cal-

culation accuracy is relatively low because it does not

consider some particles outside of a constant range

for the calculation. For the simulation of liquid em-

bolization, particle-based methods are more suitable

since they can treat the topological change and can

calculate the interfacial tension on the boundary be-

tween two kinds of materials.

In general, two major particle methods are used

depending on the purpose of the simulation: One is

the SPH (Smoothed Particle Hydrodynamics) method

that was developed by (Gingold and Monaghan,

1997) for compressive ﬂuid, and the other is the MPS

method that was developed by (Koshizuka and Oka,

1996) for incompressive one.

For the simulation of a droplet that is ejected from

a thin tube, force balance should be considered. The

droplet drops when the gravitational force becomes

more than the attractive force caused by the interfa-

cial tension. Two kinds of models are usually used for

interfacial tension calculation. One is the CSF (Con-

tinuum Surface Force) model developed by (Brack-

bill et al., 1992), and the other is a potential energy

model. (Morris, 2000) applied the CSF to the in-

terfacial model of the SPH method, while (Nomura

et al., 2001) used the CSF to the MPS method. On

the other hand, (Tartakovsky and Meakin, 2005) uti-

lized the potential energy model for the SPH method,

while (Shirakawa et al., 2001) proposed an interfacial

tension model based on potential energy for the MPS

method.

Wettability is caused by the attractive force that

works to support the droplet against the gravitational

force. (Wang et al., 2005) proposed a wettabil-

ity model with a level set method, while (Zhang

et al., 2012) established the model by using a mesh

method. They both represented a droplet with wetta-

bility, where the liquid adheres to a solid body.

In addition, (Akinci et al., 2013) proposed a

method considering adhesional wetting, while (Yang

et al., 2016) investigated another model with poten-

tial energy by considering the interaction between liq-

uid and air. (Hattori and Kohizuka, 2019) also repre-

sented a droplet that slides down on a slope by using

potential energy and the MPS method.

(Natsume et al., 2019b) investigated the droplet

behavior by using the MPS method and the compari-

son between the simulation result and a physical ex-

periment, and (Natsume et al., 2019a) performed a

droplet simulation with a particle method for liquid-

liquid two-phase ﬂow. On the other hand, (Ruan et al.,

2021) proposed a method to model the contact inter-

action using a hybrid Euler-Lagrangian framework,

and (Xing et al., 2022) simulated surface tension ﬂow

with a position-based dynamics (PBD) framework. In

addition, (Natsume et al., 2021a) also simulated the

viscous ﬂuid injection by considering the effect of the

force working between two kinds of liquid. These

methods, however, did not consider wettability. For

the simulation considering wettability, the contact an-

gle, which is the angle formed between a liquid and

a solid body, is necessary. Unless the contact angle

is known, the wettability simulation cannot be per-

formed.

(Kondo and Matsumoto, 2021) proposed a sur-

face tension model and expressed wettability by in-

troducing the interaction ratio between ﬂuid and wall;

however, it required contact angle calculation. On

the other hand, (Natsume et al., 2021b) proposed a

wettability method based on surface free energy be-

tween a liquid and a solid body, not by specifying

the contact angle but by calculating the interfacial

tension based on surface free energy. In addition,

(Natsume et al., 2022) proposed another method to

represent wettability by considering adhesional and

spreading works and (Mukai et al., 2022) performed

a liquid injection simulation with the new wettabil-

ity method. The paper demonstrated that the liquid

ejected from the catheter adhered to the solid body;

however, it did not investigate the drop time of each

Analysis of Wettability Model Using Adhesional and Spreading Works

231

droplet. Then, this paper evaluates the proposed wet-

tability method by the comparison of the drop time

between the simulation results and the physical ex-

periment in the same environment. The biggest merit

of the proposed method is to be able to simulate wet-

tability without contact angles that were necessary for

the simulations using conventional methods.

3 METHODS

3.1 Governing Equations

The purpose of the paper is to simulate the behavior

of the droplet ejected from a thin tube to a water tank

and to investigate the drop time by the comparison be-

tween the simulation results and a real physical exper-

iment. We employ the MPS method for the simulation

because particle methods can easily treat the topolog-

ical change caused by the detachment of the droplet

extruded from a tube, and the embolization material

and water are considered as incompressive ﬂuid.

The governing equations for the liquid behavior

analysis are the equation of continuity and the Navier-

Stokes equations described as follows.

Dρ

∂ρ

∂t

+ ∇ · (ρu

u) = 0, (1)

= −∇p + µ∇

u + ρg

g + f

Tension

, (2)

where, ρ is the density, t is the time, u

u is the velocity,

p is the pressure, µ is the viscosity coefﬁcient, g

g is the

gravitational acceleration, and f

Tension

is the interfa-

cial tension.

For incompressive ﬂuid, ∇ · u

u = 0 is true. Then,

Eq. (1) can be written in the following.

∂ρ

∂t

= 0 (3)

3.2 Wettability Model

Wettability is a feature that liquid on a solid body

keeps the contact angle between them, and there is

the following relationship between the contact angle

and the surface tensions as shown in Fig. 1 and Eq.

(4), which is called Young’ formula.

= γ

cosθ + γ

, (4)

where, θ is the contact angle, γ

, γ

, and γ

are the

surface tensions of solid, liquid, and solid-liquid, re-

spectively.

However, we cannot simulate liquid behavior with

wettability unless the contact angle is known. On the

Figure 1: Relationship between the contact angle and the

surface tensions.

other hand, there is a work that separates the solid

and the liquid from the condition where they are at-

tached, which is called “adhesional work”. In addi-

tion, there is another work that restrains the liquid to

spread out on the solid body, which is called “spread-

ing work”. There are also the following relationships

between these works and the surface free energies ac-

cording to Dupr

e’s formula.

+ E

= E

+ E

, (5)

+ E

= E

− E

, (6)

where, W

and W

are the adhesional and the spread-

ing works, respectively. E

, E

, and E

are the surface

free energies of the solid, the liquid, and the solid-

liquid, respectively. Here, the unit of surface free en-

ergy is J/m

and J = N · m. Then, the unit of surface

free energy becomes N/m, which is the same unit as

surface tension, and Eqs. (5) and (6) can be written as

Eqs. (7) and (8), respectively.

+ γ

= γ

+ γ

, (7)

+ γ

= γ

− γ

. (8)

Finally, the potential force that works at a particle

i is deﬁned in the following by replacing works and

surface free energies with forces.

= f

+ f

− f

, (9)

= f

− f

, (10)

= C

∑

j̸=i

i j

)

− r

i j

, (11)

− T

)

, (12)

∑

i j

)

− r

i j

· n

, (13)

∑

i j

)

− r

i j

·t

, (14)

i j

) =



i j

− l

)(r

i j

− r

) (r

i j

≤ r

)

0 (Otherwise),

(15)

i j

=| r

− r

|, (16)

where, f

and f

are the potential forces of a particle

i for the adhesional and the spreading works, respec-

tively. k means s, l or sl, C

, C l

, and C

) are

GRAPP 2023 - 18th International Conference on Computer Graphics Theory and Applications

232

the potential coefﬁcients of f

( f

, f

, and f

) of a

particle i, respectively, f

is the strength of the force

working between particles, and r

and r

are the posi-

tion vectors of particles i and j, respectively. l

is the

initial distance between particles, A

is the small area

element at the curvature 0, N

and N

are the num-

bers of particles in the normal and the tangential di-

rections, respectively. n

and t

are the normal and

the tangential vectors for the calculation of the po-

tential coefﬁcients C

, C

, and C

) of a particle i,

respectively. r

is the radius of inﬂuence for the calcu-

lation of potential forces. r

is 3.1 times of the initial

distance between particles (l

), and other parameters

are deﬁned in the above equations.

Here, the adhesional force works vertically to sep-

arate the liquid and the solid. Then, the vertical force

of W

is used as the adhesional force, and the vertical

component is described as f

. Finally, f

and f

are connected with the Heaviside function (H), and

Tension

is calculated as the interfacial tension of a

particle i, and applied to f

Tension

in Eq. (2). f

in Eq.

(17) is designed so that the larger surface tension of

a particle i ( f

Tension

) is used for the smaller spreading

work of a particle i ( f

= f

− H f

, (17)

= ( f

· n

Sur f

, (18)

Tension

= S





∑

sinφ

i j

−1

− 1





, (19)

where, n

Sur f

is the normal vector on the boundary

surface. S

is 1 or -1 for the convex or the concave

boundary surfaces of a particle i, respectively. d is the

dimensional factor, V

is the volume of one particle,

is the number of particles within the radius of in-

ﬂuence for a particle i, and φ

i j

is the angle between

the normal vectors of particles i and j.

Fig. 2 and Eq. (20) show the approximate Heavi-

side function.

H =









x < −

∆x



1 −



2x+∆x

∆x

sin

2πx

∆x

 

|x| ≤

∆x





x >

∆x



(20)

The domain and the range of the Heaviside func-

tion are [0,180] for the contact angle and [0,1] for the

value, respectively. The outputs of 1 and 0 correspond

to the inputs of 0 and 180, respectively. The contact

angle, however, cannot be measured during the sim-

ulation. Then, the domain should be decided without

the contact angle, and it is deﬁned by the potential

force instead of the contact angle in our method. The

maximum value of the potential force was 75% of the

Figure 2: Approximate Heaviside function.

basis one, which is measured at the curvature 0. Fi-

nally, the domain becomes [−0.375,+0.375] because

the center of the domain should be 0.

4 SIMULATIONS AND RESULTS

Fig. 3 shows the environment of the physical experi-

ment and the model of the simulation, which are the

same ones for easier comparison. In the physical ex-

periment, SCR780 was used as the liquid that was in-

jected into a water tank from a thin tube that is a real

catheter, which inner and outer diameters are 0.5 [mm]

and 1.0 [mm], respectively.

Tables 1 and 2 show the parameters used in the

simulation and the surface free energies of solid, liq-

uid, and solid-liquid, respectively. In fact, the theoret-

ical contact angle between the embolization material

(SCR780) and the catheter (Teﬂon tube) is 26.2

◦

. The

number of particles dynamically changes during the

simulation because the particles of the embolization

material (SCR780) are injected into the water tank

through the catheter. The detail is shown in Table 3.

Table 4 shows the speciﬁcation of the PC used in the

simulation.

Fig. 4 shows the comparison between the simula-

tion results and the physical experiment in 4 [s] after

the injection starts. Fig. 4 (a) is the simulation image

by the method without wettability, it does not show

the feature of wettability, while Fig. 4 (b) shows the

feature and the liquid adheres to the surface of the

tube, which is similar to the image of the physical ex-

periment shown in Fig. 4 (c). The contact angle (θ)

in Fig. 4 (b) is about 25.0

◦

, which is similar to that in

Fig. 4 (c) that is about 28.0

◦

, although the theoretical

angle is 26.2

◦

. These results show that the proposed

method is effective for the wettability representation.

Fig. 5 shows the comparison between the simula-

Analysis of Wettability Model Using Adhesional and Spreading Works

233

Figure 3: Experimental environment and simulation model.

tion result with the wettability method and the phys-

ical experiment. In Fig. 5, the comparison starts at

1.03[s] from the beginning because there is no liquid

in the tube at 0.0[s] in the physical experiment, while

there should be some particles in the simulation at the

initial state. After one droplet is dropped, there is

some liquid remaining in the tube, which amount cor-

responds to the volume in 1.03[s] after the injection

starts. Then, both initial states can start at 1.03[s] by

setting some particles that correspond to the volume

of the real liquid at 1.03[s] after the injection starts.

In the ﬁgure, both liquid states are almost the

same; however, the liquid in the simulation elongates

vertically more than that in the physical experiment,

and the liquid almost forms a droplet at 10.0[s] in the

simulation, while a droplet has not been formed yet in

the physical experiment. Table 5 shows the drop time

for one droplet.

Table 1: Parameters used in the simulation.

Parameter * Value Unit

Density Water ρ 1.00 × 10

kg/m

Liquid 1.18 × 10

Viscosity

coefﬁ- Water µ 1.00 × 10

−3

Pa · s

cient

Liquid 7.42 × 10

−1

Injection speed u

u 8.50 × 10

−3

m/s

Gravity g

g 9.80 m/s

Particle radius l

1.00 × 10

−4

(=Initial distance)

Time step ∆t 5.00

−5

* Symbol

Table 2: Surface free energies used in the simulation.

[J/m

]

Material γ

Thin tube 10.5 — 0.8

Embolization material — 10.8 —

Table 3: Number of particles used in the simulation.

Particle type Numbers

Water 453,150

Wall 190,804

Catheter 3,384

SCR 780 (Minimum) 1,070

SCR 780 (Maximum) 44,072

Total (Minimum) 648,408

Total (Maximum) 691,410

Table 4: Speciﬁcation of the PC used in the simulation.

CPU Xeon E5-1650 v3 3.5GHz

Main memory 32GB

GPU Tesla K40 with 12GB memory

OS Arch Linux

In Table 5, the drop time in the simulation is

shorter than that in the physical experiment except for

the ﬁrst drop. On the ﬁrst drop, there is the same vol-

ume of the liquid in the tube, and they drop at almost

the same time. However, there are more amounts

of liquid remaining in and on the tube in the sim-

Table 5: Drop time for one droplet.

[s]

Number Experiment Simulation

1 10.69 10.01

2 10.33 5.38

3 10.41 5.88

Average 10.32 7.09

GRAPP 2023 - 18th International Conference on Computer Graphics Theory and Applications

234

Figure 4: Comparison between the simulation results and the physical experiment for wettability.

Figure 5: Comparison between the simulation result and the

physical experiment for the drop time.

ulation by introducing the wettability method. The

volume of the droplet in the physical experiment was

about 1.80 × 10

−8

], while that in the simulation

was about 8.84 × 10

−9

] in average, which is about

the half of the real droplet volume. This means that

the interfacial tension of the particle in the simulation

is weaker than that in the real liquid. Then, the drop

time in the simulation becomes faster than that in the

physical experiment.

5 CONCLUSIONS AND FUTURE

WORKS

The conventional methods needed contact angles to

represent wettability since simulations were not able

to be performed without them. This means that

we could not represent wettability for the materials,

which contact angles are unknown. Then, we have

developed a new wettability method that does not

need to specify the contact angle but change the liquid

shape using adhesional and spreading works, and we

became able to represent wettability of materials even

if the contact angles were not known. In addition,

we have conﬁrmed that the proposed method can rep-

resent wettability that is similar to the physical phe-

nomenon by the comparison between the simulation

results and the physical experiment,

However, the liquid elongated vertically more

than that in the experiment, and the drop time was

shorter. It seems that there are three main reasons

for the difference. One is that more liquid remains in

and on the tube by introducing the wettability method.

The second is that the interfacial tension based on po-

tential energy in the simulation is weaker than the real

force. The last one is that we did not consider the

visco-elastic feature of the liquid. The material is a

visco-elastic ﬂuid so it has both characteristics of vis-

cosity and elasticity. This simulation, however, con-

sidered only viscosity and did not consider elasticity.

Then, we plan to perform the droplet simulation

by reconsidering the wettability method and the in-

terfacial tension model and also by introducing the

visco-elastic feature of the liquid in the future.

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