Article
pubs.acs.org/Langmuir
Experimental Investigation of Dynamic Contact Angle and Capillary
Rise in Tubes with Circular and Noncircular Cross Sections
Mohammad Heshmati* and Mohammad Piri
University of Wyoming, Laramie, Wyoming 82071-2000, United States
ABSTRACT: An extensive experimental study of the kinetics
of capillary rise in borosilicate glass tubes of different sizes and
cross-sectional shapes using various fluid systems and tube tilt
angles is presented. The investigation is focused on the direct
measurement of dynamic contact angle and its variation with
the velocity of the moving meniscus (or capillary number) in
capillary rise experiments. We investigated this relationship for
different invading fluid densities, viscosities, and surface
tensions. For circular tubes, the measured dynamic contact
angles were used to obtain rise-versus-time values that agree
more closely with their experimental counterparts (also
reported in this study) than those predicted by Washburn equation using a fixed value of contact angle. We study the
predictive capabilities of four empirical correlations available in the literature for velocity-dependence of dynamic contact angle
by comparing their predicted trends against our measured values. We also present measurements of rise in noncircular capillary
tubes where rapid advancement of arc menisci in the corners ahead of main terminal meniscus impacts the dynamics of rise.
Using the extensive set of experimental data generated in this study, a new general empirical trend is presented for variation of
normalized rise with dynamic contact angle that can be used in, for instance, dynamic pore-scale models of flow in porous media
to predict multiphase flow behavior.
■
INTRODUCTION
Flow of fluids through porous media and the associated
capillarity phenomenon have long been the focus of physicists,
soil scientists, petroleum and environmental engineers, and
researchers in many other areas of science, technology, and
engineering. This along with the fact that a porous medium is a
complicated system of connected and mostly rough-walled
capillary pores and throats makes the study of fluid/fluid
displacement mechanisms in capillary systems critically
relevant. However, direct pore-level investigation of flow and
transport in random porous mediums is very difficult due to
scale and imaging challenges. Using glass micromodels and
capillary tubes simplifies the study of such systems, thereby
enabling investigation of complicated displacement physics and
parameters such as dynamic contact angle, which are difficult to
probe directly in a naturally-occurring random porous medium.
Experimental and modeling studies of displacement processes
in simplified capillary systems have long been the focus of
authors in different research areas. Insights developed through
these studies coupled with representative pore space topology
maps obtained using, for instance, X-ray imaging technologies
enable development of predictive, physically-based pore-scale
models of flow and transport in porous media. It is therefore
imperative to investigate subtle aspects of dynamic flow in
capillary tubes and thereby enrich and improve the predictive
capabilities of dynamic pore-scale flow models.
Kinetics of liquid rise in single capillary tubes of circular cross
section was formulated almost at the same time by Lucas,1
Washburn,2 and Rideal3 in the early 20th century. Later on,
© 2014 American Chemical Society
many other scientists in different areas of science and
engineering attempted to improve the formulation and
associated analysis.7,9,11,12,14,15,17,20,23,26−28,37−39 Washburn2
modeled the fluid flow in circular capillary tubes using
Poiseuille’s law. The author ignored the changes in contact
angle of fluid meniscus during displacement. This assumption is
one of the main reasons why the rise-versus-time curves
predicted by the proposed equation do not match the
experimental data. Hence, scientists have tried to improve the
predictions through collection of more experimental data and
improvements in the modeling of the displacement process.
Quere26 performed experiments showing that the position of
the meniscus versus time in the early stages of rise, can be
described using a linear relationship. Furthermore, the
oscillations around the equilibrium occur if the liquid viscosity
is low enough. The author along with Hamraoui et al.27 and
Siebold et al.,28 on liquid/air systems, and Mumley et al.,14 on
liquid/liquid systems, emphasized on the importance of
implementing dynamic contact angle in Washburn’s model in
order to obtain a better agreement with the measured riseversus-time data.
There have been several experimental investigations on the
effects of velocity and capillary number on dynamic contact
angle. Some of these studies have led to development of
empirical correlations. Hoffman7 performed experiments in
Received: May 7, 2014
Revised: October 7, 2014
Published: October 16, 2014
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This is also the case in Lee’s35 and Hilpert’s semianalytical
models.37,38 In Lee’s35 work, a simple geometrical model is
utilized to solve the force balance equation on the liquid
meniscus. While Hilpert37 applied a power law and a power
series model for dynamic contact angle in order to generalize
Washburn’s analytical solution for flow in horizontal capillary
tubes. For flow in tilted capillary tubes, the author applied the
power law model and a polynomial.38 The resulting semianalytical models compared well with the numerical solutions.
The second approach does not include the velocity of
contact line to correlate the dynamic contact angle. For
instance, Deganello et al.39 proposed a numerical approach that
does not explicitly include velocity of the contact line to model
the dynamic contact angle. The authors combined an
equilibrium of forces in the contact region near the solid
boundary with a diffuse free fluid interface within a level-set
finite volume numerical framework. The dynamic contact angle
versus capillary number data derived from the model showed an
excellent agreement with the empirical correlations proposed
by Hoffman7 and Jiang et al.9 There have also been studies of
capillary rise performed in microgravity systems.13,24,31−33
Utilizing available experimental data, van Mourik et al.33 tested
some dynamic contact angle models in a numerical simulator.
They found that Blake’s theoretical dynamic contact angle
model23 gives the best agreement with two sets of available
experimental data presented in their paper.
Even though the majority of studies in this domain have
focused on displacements in capillary tubes with circular cross
section, there have been some limited investigations performed
on tubes with angular cross sections as well. Ransohoff and
Radke16 developed a model for the flow of fluids at low
Reynolds numbers in tubes with angular cross section. They
divided the problem into individual corner flows and solved it
numerically. They defined a dimensionless flow resistance
parameter, β, which depended on the corner half angle, degree
of roundedness, surface shear viscosity, and contact angle. Tang
and Tang25 analytically studied the dynamics of fluid flow in
tubes with sharp grooves and proposed that when the diameter
of the tube is smaller than the capillary length, the early-time
rise-versus-time data for main terminal meniscus (MTM) and
late-time data for arc meniscus (AM) follow t1/2 and t1/3
relationships, respectively. This is also reported in the studies
performed by Ponomarenko et al.40 in which they found a
universal relationship for the capillary rise in the corners.
As discussed earlier, dynamic contact angle plays a critically
important role in the description of the dynamic displacements
in capillary tubes with varying cross-sectional geometries and
wettabilities. Therefore, the extent and quality of the
experimental data on this interfacial parameter control the
predictive capabilities of the models that one can develop. To
the best of our knowledge, the experimental data on dynamic
contact angle in capillary rise experiments are scarce and have
been generated under limited range of relevant conditions. In
other words, there are very limited number of experimental
data sets available in the literature that can be used to
characterize the variation of dynamic contact angle for different
capillary tube/fluid systems. For instance, majority of the
experimental studies focused on measuring dynamic contact
angles have been performed in horizontal capillary tubes using a
piston to move the fluid phases, or as in the case of Bracke et
al.,17 a continuous solid strip has been drawn into a large pool
of liquid. In this work, we present, to the best of our knowledge,
the first extensive, well-characterized experimental study of rise-
horizontal circular tubes and presented a general trend showing
that dynamic contact angle correlates with capillary number.
Using Hoffman’s experimental data, Jiang et al.9 suggested a
correlation for the advancing dynamic contact angle measured
through the liquid phase during liquid−gas interface displacement in circular glass capillary tubes. Rillaerts and Joos11 used
mercury to perform displacements in circular glass capillary
tubes and presented a correlation between the dynamic contact
angle and the capillary number. Bracke et al.17 utilized a
continuous solid strip drawn into a large pool of liquid and
measured dynamic contact angle. This resulted in a correlation
between the dynamic contact angle values and the speed by
which the strip was drawn into the liquid. Another study by
Girardo et al.19 focused on the effect of roughness of the walls
of trapezoidal polydimethylsiloxane (PDMS) microchannels on
the dynamics of imbibition by ethanol. The authors concluded
that the roughness of the walls of a microcapillary plays an
important role in the dynamics of advancement of the wetting
front on a solid surface and the values of dynamic contact angle.
It also results in the “stick−slip” motion of the wetting liquid at
the edges of the microcapillary tubes with rough walls. Li et
al.18 measured dynamic contact angle values in horizontal glass
capillary tubes of 100−250 μm diameter using several liquids
ranging from silicone oils with different viscosities to deionized
water and crude oils. They reported the change of dynamic
contact angle with the change of the velocity of the contact line
at low capillary numbers for different fluid/tube sizes. They also
derived a master curve which relates the dynamic contact angle
variation at a specific contact line velocity with the Crispation
number, Cr = (ηα)/(σl), in which Cr is the Crispation number,
η is viscosity, α is thermal diffusivity, σ is the surface tension,
and l is the length scale or the pore radius.
Researchers have used hydrodynamic and the molecular
kinetic theories to explain the reason for the variations in
dynamic contact angle with changes in meniscus velocity. The
hydrodynamic theory divides the distance from the surface of
the solid to the bulk of the moving liquid into three regions of
micro-, meso-, and macroscopic scales. It states that the
bending of the liquid−gas interface due to viscous forces within
the mesoscopic region is the main reason for the changes in the
experimentally observed dynamic contact angle. On the other
hand, the microscopic dynamic contact angle is governed by
intermolecular short-range forces and is equal to the static
contact angle.8,10,21,22 In the molecular kinetic theory, the
dependence of dynamic contact angle to the velocity of contact
line is studied at the molecular level and is related to the
attachment and detachment of liquid molecules to and from the
solid surface. Therefore, the microscopic dynamic contact angle
is considered to be velocity dependent, and the same as the
macroscopically measured dynamic contact angle.4,5,23,34
There are two main approaches that are generally used to
take the effect of dynamic contact angle into account in the
modeling of dynamics of rise in capillary tubes. The first
approach is based on incorporation of the dependence of
dynamic contact angle on velocity of contact line or capillary
number. In other words, one can improve rise-versus-time
models by integrating those relationships with rise equations.
This approach has successfully been utilized in Hoffman’s
molecular model12 and Cox’s theoretical approach.15 It is based
on the solution of Stoke’s equation and the assumption of fluid
slip in the vicinity of the three-phase contact line. This
approach is also used in Joos’ model20 in which the correlation
proposed by Bracke et al.17 is integrated with Poiseuille’s law.
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Table 1. Properties of Liquids, Surface Tensions, and Tube Sizes and Tilt Angles Used in This Study at a Temperature of 25°C
WP
ρ (gr/cm3)
η (cP)
σ (dyn/cm)
NWP
ID (mm)
θtilt
glycerol
Soltrol 170
water
1.26
0.774
0.997
1011.1
2.6
1.1
63.47
24.83
72.8
air
air
air
0.5, 1.0, 2.0
0.75, 1.0, 1.3
0.75, 1.0, 1.3
90°, 45°
90°
90°
Figure 1. Schematic of the experimental setup.
versus-time and the change of dynamic contact angle with
meniscus velocity in capillary rise experiments for different fluid
systems and tilt angles using capillary tubes with circular and
angular cross-sectional shapes. Furthermore, we present a new
general correlation for the variation of dynamic contact angle
versus meniscus rise that is developed based on our
experimental measurements. The correlation can be used to
(1) validate their theoretical counterparts and (2) predict, when
coupled with Washburn’s equation, rise-versus-time trends for
other systems for which measured dynamic contact angle data
may not be available. The data can also be used for validation of
numerical or theoretical solutions of rise as well as development
of new correlation for dependence of dynamic contact angle on
velocity of meniscus. Finally, the observed trends can be
incorporated in, for instance, dynamic pore-scale network
models used to predict multiphase flow functions (e.g., relative
permeabilities and capillary pressures) in porous media.
In this document, we first present the materials and the
experimental setup and procedure used to perform the rise
experiments. Our experimental results are first validated against
dynamic contact angle data available in the literature. Measured
rise data are then compared with those predicted by the
Washburn’s equation with and without experimental values of
dynamic contact angle. Four widely used semiempirical and
theoretical correlations of dynamic contact angle versus velocity
of contact line are validated against our experimental data. A
new correlation is introduced for variation of dynamic contact
angle versus rise for different fluid systems in tubes with varying
internal diameters and tilt angles. We then present experimental
data of rise in noncircular capillary tubes and study the rise of
both MTM and AMs. The paper is then concluded by a set of
final remarks.
■
EXPERIMENTAL SECTION
Materials and Properties. We used water, glycerol, and Soltrol
170 as the wetting fluids and the laboratory air as the nonwetting
phase in the experiments performed under this study. Ultra clean
distilled water was obtained from a water distiller made of glass,
ensuring there were no contaminants present in the water. Certified
ACS Fisher Scientific glycerol was used as received and Soltrol 170 was
supplied by Chevron Phillips Chemical Company, The Woodlands,
TX. Soltrol 170 was purified using a dual-packed column of silica gel
and alumina.
Glass capillary tubes of 0.5, 0.75, 1.0, 1.3, and 2 mm internal
diameter with circular and square cross sections were obtained from
Friedrich & Dimmock Inc., Millville, NJ. The tubes were 50 cm long,
and therefore, they were cut, depending on the tube internal diameter,
to a desired length for each experiment. The tips of the tubes were
straightened using a rotary drill and a very fine sand paper. Each single
tube was used only once to avoid any possible contaminations and to
enhance the accuracy and reproducibility of the results.
Every single capillary tube was thoroughly cleaned to establish
strongly water-wet glass surfaces. Glass capillary tubes were first rinsed
with isopropyl alcohol and then with 150 mL of distilled water. The
tubes were immersed in a mixture of 0.5 L of sulfuric acid (95-98)%
obtained from Sigma-Aldrich and 25 g of Nochromix from Godax
Laboratories Inc., Cabin John, MD. The beaker containing the tubes in
the acid solution was then placed in an ultrasonic bath for 15 min.
They were then left in the same solution to soak overnight. The tubes
were vacuum rinsed thoroughly by flowing 600 cm3 of distilled water
through them using the laboratory’s vacuum line.41 Upon finishing the
cleaning procedure, the tubes used in experiments with Soltrol 170 and
glycerol were vacuum-dried for 1 min and then immediately used to
perform the flow tests. For experiments with water, the tubes were
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partially dried using a piece of paper filter and immediately used to
perform the measurements. Dry tubes were not used in water
experiments because clean dry borosilicate glass surfaces and distilled
water are both very active6 and a small amount of contamination
dramatically affects the values of contact angle and equilibrium rise.
Fluid densities used in the experiments were measured using Anton
Paar DMA 5000 M density meter. Viscosities were obtained using a
Cambridge Viscosity viscometer, and surface tensions were determined with a state-of-the-art IFT measurement system using pendant
drop technique. All measurements were made at ambient temperature
and pressure conditions. The inner diameters of the tubes were
measured optically with 0.01 mm precision using a camera and a highmagnification lens attached to a vertical positioning column. Table 1
lists the measured values as well as the fluid pairs and tube sizes for
each group of rise experiments.
In Table 1, WP stands for wetting phase, ρ is the density, η is the
viscosity, σ is the surface tension, NWP is the nonwetting phase, ID is
the inner diameter of the capillary tube, and θtilt is the tilt angle of the
axis of the capillary tube (along the length) with respect to a horizontal
plane.
Experimental Setup. A precise positioning column was built to
hold and move the capillary tubes in the vertical direction. The column
could be tilted to perform rise experiments with different tilt angles.
Two high-speed cameras, Sony SCD-V60CR and Phantom V310, were
employed to record the position and shape of the moving meniscus.
The cameras were also used to time-stamp the images. Two different
types of lenses with different magnifications were utilized. The one
with the lower magnification was used to detect the position of the
meniscus and time-stamp the images during the rise experiments. The
lens with the higher magnification captured the shape of the meniscus,
providing high-quality, high-magnification images in order to measure
dynamic contact angles during the rise. A Schott fiber optic flat backlight system with an active area of 20 × 20 cm2 was mounted to evenly
illuminate the capillary tubes (Figure 1).
Experimental Procedure. Each capillary tube was placed in the
capillary tube holder attached to a precise vertical positioning column.
The column’s position was controlled manually and accurately (0.01
mm increments) using an adjustment knob. The invading fluid was
poured in a wide Petri dish, 9 cm in diameter, to eliminate any surface
curvature caused by the edges of the dish that could affect the rise
experiment. The tube was lowered slowly toward the surface of the
fluid in the Petri dish. Recording of the images was started a few
seconds before the tube tip touched the fluid surface and continued
until the meniscus finally stopped at the equilibrium height.
The procedure to measure the dynamic contact angle was different
from the one used to measure the rise-versus-time values; that is, to
measure the dynamic contact angle, it was necessary to have a
magnified image of the interface. This in turn meant that the field of
view of the camera/lens system had to be less than the whole height of
rise. Thus, for glycerol with a high viscosity and a low rise velocity, the
camera could be moved up along with the meniscus, while recording.
However, for water and Soltrol 170, having low viscosities and very
high speeds of rise, the tube length had to be divided into several
intervals and imaged separately. For example, for a rise of 20 mm, if
the field of view of the high magnification lens was 5 mm, one would
need at least four separate measurements to cover the whole range of
the dynamic contact angle for one tube/fluid combination.
Furthermore, for reproducibility purposes, each experiment for a
tube/fluid set was repeated at least three times, and if all the results
compared well with each other within experimental error, the tests
were called acceptable. During the contact angle measurement tests,
we also recorded rise-versus-time data. These data agreed very well
with those generated by the experiments mentioned in the first
paragraph of this section.
The dynamic contact angle is the angle that the meniscus formed by
two fluids makes with a contacting solid surface through the denser
phase. In the experiments presented here, one fluid was always air,
while the other was glycerol, Soltrol 170, or water. The solid surface
was the inner surface of the glass capillary tubes. In order to measure
dynamic contact angle using the magnified images recorded during the
flow tests, we, similar to the approach used by Siebold et al.,28 assumed
that the meniscus was part of a circle and used the following equation:
θ=
⎛ 2x ⎞
π
− 2arctan⎜ m ⎟
⎝ d ⎠
2
(1)
where θ is the contact angle, xm is the height, and d is the diameter of
the meniscus.
To compare with the results obtained using the above-mentioned
approach, the contact angles were also determined by drawing a
tangent to the meniscus at the point of contact of the fluids with the
solid surface, using ImageJ software. The results obtained using these
two techniques were comparable within experimental error. We also
examined the effect of gravity on the shape of the meniscus. To this
end, we calculated the Bond number for our experiments: Bo =
(ΔρgL2)/σ, in which Δρ is the difference between the density of the
liquid and that of the air, g is acceleration due to gravity, and L is the
characteristic length of the system which in this case is the radius of
the capillary tube. It gives the ratio of gravity to capillary forces. For all
the fluid systems and tube sizes we used in this study (see Table 1),
the Bond number ranged between 0.012 and 0.076 except for glycerol
experiment in 2 mm tube and Soltrol 170 test in 1.3 mm tube for
which Bond numbers were 0.194 and 0.129, respectively. We believe
gravity had negligible impact on the meniscus shape in our
experiments, but one should note that for the cases of Bond numbers
greater than 0.1 (i.e., tests with Glycerol in 2 mm diameter tubes and
with Soltrol 170 in 1.3 mm diameter tubes) the shape of the interface
might have been slightly affected by gravity.
In order to eliminate the refraction of light on the curved surface of
the tubes, which could result in a deformed meniscus image, tubes
were mounted inside a square cross section cuvette made of glass. The
cuvette was open at one end. Its closed end had a hole, made to the
size of the outer diameter of the test capillary tube. The capillary tube
was passed through the hole to make its tip available for contact with
the test fluid surface. Figure 2 shows the difference between the cases
Figure 2. Meniscus image without the glycerol bath (left) and with
glycerol bath (right).
with and without a cuvette. In the former, the space between the
capillary tube and the cuvette was filled with glycerol because it has the
same refractive index as the glass. The above-mentioned setup
eliminated any image distortion caused by the refraction of the light.
■
RESULTS AND DISCUSSION
In this section, we present and discuss all the results generated
under this study. We first compare our dynamic contact angle
data against those available in the literature. We then investigate
the ability of rise equations in predicting our measurements
with and without use of measured contact angles. Some of the
correlations available in the literature for variation of dynamic
contact angle with the velocity of moving meniscus are tested
against our experimental data. We then present a general
dynamic contact angle versus rise correlation that can be used
for a wide range of applications. These are followed by the
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Figure 3. Variation of dynamic contact angle-versus-capillary number for glycerol/air in 45° tilted and vertical glass capillary tubes and Soltrol 170/
air in vertical glass capillary tubes (top), and water/air in vertical glass capillary tubes (bottom). Hoffman’s7 experimental data are included for
comparison.
effect of thickness of water film on dynamic contact angle by
Hirasaki and Yang.30 The trend still compares relatively well
with those published by Hoffman.7 It is noteworthy that the
dynamic contact angle for a given capillary number shows,
within experimental error, weak sensitivity to the type of fluid
system and the tilt angle used. This may have important
implications for development of modeling tools used to predict
flow at the pore scale in porous media.
Dynamics of Capillary Rise. In this section, we provide an
extensive data set characterizing the dynamics of rise versus
time in capillary tubes with different internal diameters and
with different invading liquids. We compare the data with those
predicted by Washburn equation using a fixed contact angle as
well as our measured dynamic contact angles. We start with a
brief discussion of Washburn’s equation.2 It was originally
developed for a single capillary tube of uniform circular cross
section. It was assumed that the velocity of fluid penetrating the
tube would, after a short initial period, drop to a value at which
the conditions of Poiseuille flow establish and persist.
Poiseuille’s equation, neglecting any air resistance, is as follows:
study of displacements in capillary tubes with square cross
section.
Validation. The dynamic contact angle is dependent on the
velocity of the moving meniscus, or capillary number. In Figure
3 (top), we show the variation of dynamic contact angle with
capillary number for displacement of air by glycerol in circular
capillary tubes with 0.5, 1.0, and 2.0 mm ID and at 45° and 90°
tilt angles as well as those of air/Soltrol 170 in vertical tubes of
0.75, 1.0, and 1.3 mm ID. The results are compared against the
experimental data reported by Hoffman.7 The agreement is
encouraging and indicates the accuracy and reproducibility of
our measurements. Furthermore, our results are consistent
when tubes with various sizes and tilt angles are used.
Figure 3 (bottom) presents similar measurements for the air/
water system in vertical circular glass capillary tubes with 0.75,
1.0, and 1.3 mm ID. The measured values pertaining to some of
the rise tests with this fluid system show slight deviation from
those presented by Hoffman.7 This uncertainty may have been
introduced into the measured dynamic contact angle values due
to presence of a thin water film on the inner walls of the
capillary tubes, as discussed in a more detailed study on the
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Figure 4. Experimentally measured values of rise versus time for glycerol/air in 45° tilted (left column) and vertical glass capillary tubes (right
column) of different sizes. Predicted values of Washburn equation with fixed contact angle and with measured values of θd are included for
comparison.
dV =
π ΣP 4
(r + 4εr 3) dt
8ηl
(
PA + ρg (h − ls sin ψ ) +
dl
=
dt
8ηl
(2)
where dV is the volume of the liquid which flows during the
time dt through any cross section of the capillary tube, l is the
length of the column of liquid in the capillary at time t, η is the
viscosity of the liquid, ε is the coefficient of slip, r is the radius
of the capillary tube, and ΣP is the total effective pressure which
acts to force the liquid along the capillary and is the sum of
three separate pressures: the unbalanced atmospheric pressure,
PA, the hydrostatic pressure, Ph, and the capillary pressure, Ps.
The following ordinary differential equation for the
penetration velocity was derived and integrated for the rise in
single capillary tubes:2
2σ
cos
r
)
θ (r 2 + 4εr )
(3)
where g is the acceleration due to gravity, h is the height of
liquid column, ls is the linear distance between the tip of the
tube and any point along the tube, ψ is the angle that the
straight line between the tip of the tube to any point along the
tube makes with a horizontal surface, ρ is the fluid density, and
θ is the contact angle. Washburn assumed that θ was constant.
Findings of different investigations available in the literature
(e.g., Hoffman,7 Jiang et al.,9 and Bracke et al.17) as well as our
experimental results presented in the previous section show
strong sensitivity of dynamic contact angle to variations in the
velocity of the moving meniscus. Those findings along with the
fact that the velocity of the meniscus changes as the liquid rises
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Figure 5. Experimentally measured values of rise versus time for Soltrol 170 (left column) and water (right column) in vertical glass capillary tubes
of different sizes. Predicted values of Washburn equation with fixed contact angle and with measured values of θd are included for comparison.
in a capillary tube indicate that Washburn’s assumption of fixed
contact angle during the rise introduces uncertainty into the
predicted rise-versus-time values. In order to mitigate this
uncertainty, measured experimental values of dynamic contact
angle must be incorporated. We have therefore used our
measured values of this parameter in the original Washburn’s
equation and compared the predicted rise-versus-time values
against their experimental counterparts also reported in this
study. For reference, we also include predictions using a fixed
value of contact angle (i.e., zero). Two methods for
implementing the experimental values of dynamic contact
angle into Washburn’s equation were examined. The first
approach involved fitting the contact angle values with a curve
and the second approach was to implement the contact angle
measurements on a point-by-point basis. The latter was used in
this study as it led to less discrepancy between predicted results
and the measured values. The coefficient of slip was assumed to
be zero in our calculations.
The comparisons are shown in Figures 4 and 5. It is seen that
in all cases the values predicted using measured dynamic
contact angles expectedly agree much more closely with the
experimental rise counterparts than those obtained with a fixed
contact angle of zero. The agreements are encouraging,
indicating the accuracy and reproducibility of the measurements. Interestingly, when measured dynamic contact angles
are used with Washburn’s equation, the deviations between the
measured and predicted rise-versus-time values for large
capillary tubes become smaller, or comparable to, those of
smaller tubes (see Figures 4 and 5). In the case of experiments
with water, we observe a level of discrepancy that might be
attributed to the water film present in the tubes at the start of
each experiment, which could have impacted the measured
contact angle values.
Correlations for Velocity-Dependent Dynamic Contact Angle. Over the last several decades, researchers have
introduced various semiempirical correlations to describe the
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velocity-dependence of dynamic contact angle.15,20,22,42 Popescu et al.36 performed comparative analysis of the predictions
made by some of these models for dynamic contact angle
versus the velocity of the contact line for typical water and highviscosity silicone oil systems. The authors, however, did not
compare any of the predicted trends against independent
experimental counterparts mainly due to lack of such data in
the literature. In this section, we use the experimental dynamic
capillary rise data presented earlier to examine the predictive
capabilities of four correlations presented by Cox,15 Joos et
al.,20 Shikhmurzaev, 22 and Sheng.42 We discuss the details of
the models followed by comparison of the predicted trends
against our experimental measurements.
Bracke et al.17 used two experimental methods to study the
dependence of dynamic contact angle on the velocity of contact
line. In the first set of experiments, they used polyethylene/
polyethylene terephthalate solid strips drawn into pools of
different aqueous glycerol solutions, aqueous ethylenegelycol
solutions, and ordinary corn oil. In the second method, they
utilized a dry platina Wilhelmy plate. Joos et al.20 then used the
experimental data to develop a semiempirical correlation for the
dependence of dynamic contact angle to the velocity of contact
line in a circular capillary tube. They replaced the dynamic
contact angle term in Washburn’s equation2 with their
correlation for the dynamic contact angle17 and showed that
this semiempirical correlation leads to more accurate
predictions of capillary rise than that of Washburn. The
proposed empirical correlation is given by
cos(θd) = cos(θe) − 2(1 + cos(θe))Ca1/2
divided by microscopic length scales.36 The parameters used
with Cox’s model are also tabulated in Table 2.
Utilizing numerical hydrodynamic calculations, Hoffman’s7
slipping function, and eq 6, Sheng and Zhou42 linked
macroscopic flow behavior (e.g., dynamic contact angle) to
the microscopic parameters governing the contact-line region.
The authors also introduced parameters to take the effect of
surface roughness into account.
G(θ ) = G(θe) + Ca ln(K /ls)
where
G(q) =
∫0
q
dϕ[f (ϕ)]−1
and
f (ϕ) = ⎡⎣2sin ϕ{q2(ϕ2 − sin 2 ϕ) + 2q[ϕ(π − ϕ)
+ sin 2 ϕ] + (π − ϕ)2 − sin 2 ϕ}⎦⎤
÷ {q(ϕ2 − sin 2 ϕ)[(π − ϕ) + sin ϕ cos ϕ]
+ (ϕ − sin ϕ cos ϕ)[(π − ϕ)2 − sin 2 ϕ]}
In the above equations, ls is the slipping length and K is a
slipping model dependent constant.
We also considered the mathematical model proposed by
Shikhmurzaev.22 The author suggested that the surface tension
gradient caused by the flow of a liquid flowing on a solid
surface, influences the flow. This gradient, in the case of small
capillaries, determines the dynamic contact angle. The shear
stress singularity present in classical approaches is eliminated in
this model. Blake and Shikhmurzaev29 and Popescu et al.36
simplified Shikhmurzaev’s original model into eq 7:
(4)
where θd is the dynamic contact angle in radians, θe is the
equilibrium contact angle in radians, and Ca = (ηv)/σ is the
capillary number, in which η is the fluid viscosity, σ is the
surface tension, and v the velocity of contact line. Table 2 lists
all the parameters needed to use this correlation with our fluid
systems.
cos(θe) − cos(θd) =
2V (ρ2es* + ρ1es*u0)
(1 − ρ1es*)[(ρ2es* + V 2)1/2 + V ]
(7)
where
Table 2. Parameters Used with the Models Proposed by
Joos,20 Cox,15 Shikhmurzaev,22 and Sheng42
u0(θd , 0) =
sin(θd) − θd cos(θd)
sin(θd) cos(θd) − θd
fluid
surface
tension
(dyn/cm)
viscosity
(cP)
θe
(rad)
ρs1e*22
Sc
σSG
* 22
V = Sc × Ca
glycerol
Soltrol 170
water
63.47
24.83
72.8
1011.1
2.6
1.1
0
0
0
0.54
0.54
0.54
17.90
12.56
12.5
−0.07
−0.07
−0.07
*)
ρ2es* = 1 + (1 − ρ1es*)(cos(θe) − σSG
In the above equations, Ca is the capillary number, Sc is a
scaling factor that depends on the material properties, and ρs2e*
and ρs1e* are two phenomenological coefficients (see Blake and
Shikhmurzaev29 for more details). Table 2 lists values of the
parameters used with the above-mentioned model (Sc for water
is obtained from Popescu et al.36).
We compare our experimental dynamic contact angle data
for glycerol, Soltrol 170, and water with the trends predicted by
these four correlations. For each of the models, we use our
measured values of the physical parameters (i.e., viscosity and
surface tension) and correlation coefficients available in the
literature for the same fluid pair (see Table 2). Figure 6
compares our measured dynamic contact angles versus velocity
of the triple contact line against the trends predicted by the
above-mentioned correlations.
This figure shows that the model proposed by Joos generates
the best match against the Soltrol 170 data. It predicts the low
velocity trend for glycerol well, but deviates at larger velocity
In a thermodynamics-based approach, Cox15 assumed a
microscopic slip boundary condition for a moving fluid on a
solid surface. This assumption helps removing the stress
singularity at the triple contact line of the system. Cox’s analysis
resulted in the following relationship for the dynamic contact
angle versus velocity of the contact line:
ηv
G(θd) = G(θe) + χ
(5)
σ
where
G (θ ) =
∫0
θ
(6)
x − sin(x) cos(x)
dx
2sin(x)
In this correlation, v is the contact line velocity and χ ≈ 16,
which is defined as the natural logarithm of macroscopic
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Figure 6. Experimentally measured variation of dynamic contact angle versus the velocity of the contact line for glycerol (top), Soltrol 170 (middle),
and water (bottom). Predicted trends by the models of Joos,20 Cox,15 Shikhmurzaev,22 Sheng42are included for comparison.
values of 0.41, 2.9 × 10−3 and 4.1 × 10−4 for glycerol, Soltrol
170 and water respectively. Sheng’s model predicts the
experimental data for glycerol and Soltrol 170 well; however,
it slightly overpredicts the experimental data for water at almost
all velocities. In case of Shikhmurzaev’s model, one needs to use
parameters that can be found only for a very few fluid systems
in the literature; see, for instance, Popescu et al.36 and Blake
and Shikhmurzaev.29 When available, we have used parameters
from literature (i.e., for distilled water). In other cases, that is,
for Soltrol 170 and glycerol, we adjusted Sc to 12.56 and 17.90,
respectively, to obtain the best agreement with the
values. In the case of water, it overpredicts the experimental
data at both low and high velocities. Cox’s model produces a
very good fit in the case of glycerol, while it overpredicts the
Soltrol 170 experimental data particularly at higher velocities.
This model shows more significant deviation from the
experimental data in the case of water. It is important to note
that there are no fitting parameters used in the calculations
performed with these two models. This is, however, not the
case for Sheng’s and Shikhmurzaev’s models. In Sheng’s model,
the value of ls = 10−7 cm was obtained from molecular dynamic
simulations,42 while K was the fitting parameter which has
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Figure 7. General trend for variation of normalized rise, Ln, with changes in dynamic contact angle, θd, generated using the experimental data
gathered under this study. The experimental data presented by Siebold et al.28 are included for comparison.
Figure 8. Experimentally measured values of rise versus time for the AMs (top) and MTMs (bottom) for glycerol/air in 0.5, 1.0, and 2.0 mm ID
square capillary tubes. The solid line represents a slope of 1/3 for late-time AMs and 1/2 for early-time MTMs, following Ponomarenko et al.40
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versus-time for different fluid systems (i.e., water/air, Soltrol
170/air, and glycerol/air) and tube tilt angles. To the best of
our knowledge, this is the first time that an extensive
experimental data set for variation of dynamic contact angle
with capillary number during rise with different fluid systems is
reported. Measured dynamic contact angles were used to obtain
rise-versus-time trends that agree more closely with the
measured rise values (also reported in this study) than those
predicted by the Washburn equation using a fixed contact
angle. Four empirical models of velocity-dependence on
dynamic contact angle were validated by comparing their
predicted trends against our experimental data. The predictive
capabilities of the models were discussed. A new general trend
was introduced for the changes in normalized rise with
variations in dynamic contact angle. The trend was compared
successfully against the experimental data available in the
literature. This relation can be used to find the dynamic contact
angle values for a given rise, if the invading fluid density, surface
tension, tube size, and tilt angle of the system are known.
Combined with Washburn equation, one can then produce a
rise-versus-time curve for the system. Finally, we presented our
measurements of rise of glycerol in square cross section
capillary tubes with different sizes. We reported rise-versus-time
for both AMs and MTM. The AM late time scales and MTM
early time scales data showed an encouraging agreement with
the universal relationships proposed by Ponomarenko et al.40
for rise in capillary tubes with angular cross section. The
experimental data and the trends presented here can be used in
dynamic pore-scale models of flow in porous media to predict
multiphase flow functions.
experimental measurements. All the other parameters were kept
the same as the values used by Blake and Shikhmurzaev29 for
silicon oil and different glycerol/water solutions, respectively.
The model developed by Shikhmurzaev expectedly shows a
good agreement with the experimental trends at low and
moderate velocities in glycerol and water systems. The model
always overpredicts the experimental values at higher velocities.
Dynamic Contact Angle and Normalized Rise. In this
section, we present, for the systems investigated in this study, a
general trend for the changes in normalized rise (Ln = l/le,
where l is the value of rise and le is the equilibrium rise), with
variations in dynamic contact angle. We consolidated all the rise
and dynamic contact angle data (except for those of the water/
air system) generated under this study to obtain the trend
shown in Figure 7. The rise and dynamic contact angle data for
all fluid/tube-size/tilt-angle combinations follow the same
trend. Figure 7 also shows a curve fit for the data that is
given by
⎛ −(θ − b)2 ⎞1/4r
l
d
⎟
= a exp⎜
le
2c 2
⎠
⎝
(8)
where r is the radius of the capillary tube in mm, a = 1.03, b =
−2.47, and c = 27.7.
In this figure, we also include the data presented by Siebold
et al.,28 which follow the trend relatively well. There are very
limited numbers of dynamic rise-versus-contact angle data sets
available in the literature. This relation can be used to find the
dynamic contact angle value for a given rise, if the invading fluid
density, surface tension, tube size, and tilt angle of the system
are known. Combined with the Washburn equation, one can
then produce a rise-versus-time curve for the system.
Capillary Tubes with Square Cross Section. Here we
study the rise in capillary tubes with square cross section. The
cross section of a typical square capillary tube used in these
experiments is not a perfect square and has round corners. We
investigate the rise-versus-time of both AMs and MTMs in
these experiments. The data are presented in log−log plots in
Figure 8. As shown in this figure, the rise-versus-time of AMs
and MTM in 1 and 2 mm square tubes with glycerol/air fluid
system follow the universal relationship proposed by
Ponomarenko et al.40 As seen, the late time scales of AM
rise-versus-time data follow a trend proportional to t1/3, while
the early time scales of MTM rise-versus-time values follow a
t1/2 profile. As expected, the AMs rise faster ahead of MTMs.
Ransohoff and Radke16 presented a dimensionless flow
resistance parameter (β), which, among other parameters,
depends on the roundedness of the corner of the tube in which
the fluid rises. Based on the calculations presented by the
authors, the higher is the roundedness of the corner, the greater
is the value of the dimensionless flow resistance (β). And the
higher is the value of β, the lower is the average velocity of the
moving meniscus. Therefore, the large roundedness values in
our tubes makes the value of β much larger than it is in sharp
cornered channels used by Ponomarenko et al.40 This may have
contributed to the slight deviation between the slope of our AM
rise-versus-time values from the universal relationships
proposed by Ponomarenko et al.40
■
AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected].
Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS
We gratefully acknowledge the financial support of the School
of Energy Resources and the Enhanced Oil Recovery Institute
at the University of Wyoming. We extend our gratitude to
Professor George Hirasaki for his valuable comments and
Henry Plancher and Soheil Saraji of Piri Research Group for
their assistance with the capillary tube cleaning procedures and
surface tension measurements.
■
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