Simultaneous measurement of liquid surface tension and contact angle by light reflection

Daobin Luo; Lailai Qian; Liangwei Dong; Peng Shao; Zongmin Yue; Juan Wang; Bo Shi; Shengbo Wu; Yipan Qin

doi:10.1364/OE.27.016703

1. Introduction

Interfacial phenomena are ubiquitous in nature, and many of them are related to the liquid surface tension and contact angle. When a solid plate is put into a liquid, the liquid surface will deform and form curved liquid surface with a finite contact angle between liquid and solid plate. The measurements of liquid surface tension and contact angle are important in many applied science and engineering areas [1], such as, exploitation of crude oil [2], gem identification [3], metal casting [4] and the design of propellant management devices for spacecraft [5]. The traditional Wilhelmy plate and capillarity method are usually applied to measure the liquid surface tension and contact angle [6,7]. In some circumstance, one needs to measure the liquid surface shape formed due to the wetting effect [8,9]. These traditional liquid surface measurement methods cannot measure the liquid surface tension, contact angle and the liquid shape at the same time well.

Optical measurement is widely used in many research fields, including liquid surface and contact angle measurement [10,11]. Some people have measured the liquid surface tension waves and liquid surface tension by light diffraction. The low frequency liquid surface tension waves need be formed in advance and the contact angle cannot be measured proposed [12–15]. Light reflection method was applied to measure the surface shape [16]. When the liquid surface tension is known in advance, light reflection method can also be used to detect the shape of liquid surface [17,18] or the contact angle [19,20]. Optical mapping was applied to study the change in contact angle as a function of change in silver nanoparticle size controlled by thermal evaporation [21]. This method realizes nanoscale characterization and industrial monitoring as well as chemical analyses by allowing rapid contact angle measurements over large areas or large numbers of samples. Fiber optic Fabry-Perot sensor was applied to investigate the liquid surface tension, and wetting process was also shown [22]. Three-dimensional structural imaging of a droplet on a substrate is performed using optical coherence tomography which provides micrometer resolution images without contact with the sample, and the contact angle was obtained [23]. There are some situations in which the simultaneous measurement of the contact angle and surface tension is necessary. For instance, in studies concerning the effect of the adsorption of surface-active substances on solid-liquid and liquid-vapor interfacial tension or in studies concerning the measurement of the work of adhesion in molten alloy-ceramic substrate systems [24–26]. Based on the optical imaging method, some people measured the liquid surface tension and the contact angle simultaneously, this method is mainly concerned the algorithm method, for example, axisymmetric drop-shape analysis [27]. It is valuable to measure the liquid surface tension, contact angle and liquid surface shape simultaneously by optical method due to its non-contact capability and rapidity.

In this work, we put forward an effective and relatively simple light reflection method to measure the liquid surface tension, contact angle and liquid surface shape simultaneously. The corresponding relationship between the distribution of light reflection field and liquid surface shape is deduced theoretically, and the reflection distribution from the curved liquid surface due to the wetting effect was observed experimentally. A central dark field was found in the reflection pattern. The width of the dark field is related to the width of the incidence beam and the incidence spot on the curve liquid surface. The surface tension, contact angle and liquid surface shape can be measured simultaneously according to the dark field patterns, no matter if the liquid is colorless or colored.

2. Theory

2.1 Theoretical analysis of liquid surface shape

Let us consider a liquid surface of sufficiently large extent that it is flat and horizontal. We take this surface to be the x-y plane of our Cartesian coordinate frame, and the z-axis is opposite to the gravitational acceleration $g$ , $z = - g / | g | .$ When a solid plate contacts with the liquid with an angle $θ$ , the liquid surface is upward or downward depending on the interfacial tensions and terminates along a straight line on the solid surface [28]. Figure 1 shows the diagrammatic sketch of the curved liquid surface, and curve line AB is a section of the system in a vertical plane passing through the x-axis. The contact point $A (x_{A}, y_{A}, z_{A})$ between the free surface of liquid and the plate is on the top the curve AB, and point B is located on the x-y plane. We consider the liquid confined to the region bounded by the solid surface, the liquid surface S, and the x-y plane. Therefore, a function representing the free energy of the system can be defined as [29,30]:

J = \int_{B}^{A} L (z, x, {z^{'}}^{- 1}) d z + J_{A},

and

{\begin{cases} L (z, x, {z^{'}}^{- 1}) = ρ g z x + γ (\sqrt{1 + {z^{'}}^{- 2}} + {z^{'}}^{- 1}) \\ J_{A} = \frac{1}{3} ρ g z_{A}^{3} - γ z_{A} \cos θ_{c} \csc θ - γ z_{A} \tan θ \end{cases},

where

ρ

is the liquid density,

g

is the gravitational acceleration,

γ

is the liquid surface tension coefficient,

θ_{c}

is the contact angle,

z^{'} = d z / d x .

J_{A}

represents the energy related only to the location of the point A. In the integral of the Eq. (1), one considers the gravitation force, the surface tension force acting on S and the constant volume condition, respectively.

Fig. 1 Bending liquid surface on both sides of plate.

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As a matter of fact, the curved liquid surface only exists stably at the lowest free energy of the system [31]. In other words, the min{J} corresponds to the shape of the liquid surface. According to Euler-Lagrange equation like [32]

- \frac{d}{d z} \frac{\partial L}{\partial {z^{'}}^{- 1}} + \frac{\partial L}{\partial x} = 0,

the minimum of J corresponds to a value of

z_{A}

, which satisfies the equation

\frac{\partial J}{\partial z_{A}} = 0.

Based on Eqs. (3) and (4), min{J} can be found, and the relationship between $z$ and $z^{'}$ is given as:

z = α \sqrt{1 - \frac{1}{\sqrt{1 + {z^{'}}^{2}}}},

where capillary constant

α = \sqrt{2 γ / ρ g} .

According to Eq. (5), the equation of curve AB can be written as follows:

X - \frac{1}{\sqrt{2}} \tan h^{- 1} (\frac{\sqrt{2}}{\sqrt{2 - Z^{2}}}) + \sqrt{2 - Z^{2}} - C = 0,

where Z and X are dimensionless variables

z / α

and

x / α,

respectively. The constant

C

, only related to

θ_{c}

and

θ

, is then obtained as:

C = Z_{A} \tan θ - \frac{1}{\sqrt{2}} \tanh^{- 1} (\frac{\sqrt{2}}{\sqrt{2 - Z_{A}^{2}}}) + \sqrt{2 - Z_{A}^{2}},

where

Z_{A} = \sqrt{2} \cos [0.5 (θ_{c} - θ + π / 2)]

is obtained by Eq. (4). Taking Eq. (5) into Eq. (6), the slope equation of the curved liquid surface is obtained as:

X - \frac{1}{\sqrt{2}} \tan h^{- 1} (\frac{\sqrt{2 + 2 {z^{'}}^{2}}}{\sqrt{\sqrt{1 + {z^{'}}^{2}} +1}}) + \sqrt{1+ \frac{1}{\sqrt{1 + {z^{'}}^{2}}}} - C = 0.

Equation (8) show the relation between the slope of the curved surface, liquid surface tension and contact angle. It is well known that the liquid curved surface S will be determined if the slope is given at any spot on this surface.

2.2. Theoretical analysis of light reflection from the curved liquid surface

Let us consider a beam of parallel light input along opposite z-axis. A schematic diagram of the incidence beam and boundary light reflection is shown in Fig. 2. The light reflection patterns distribute on a horizontal observation screen above the liquid surface, and the distance between the horizontal liquid surface x-y plane and the screen is h $(h > > z_{A}) .$ We consider two boundary rays of the incidence beam which distribute at right and left side of the inserted plate. The distance from z-axis to the right and left boundary ray are denoted as $l_{r}$ and $l_{l},$ respectively. The width of the incidence beam is thus $l_{r} + l_{l} .$ Now, we pay attention to the boundary ray 1 and its reflection ray 1′. The distance from the point of the right boundary reflection ray arrives on the screen to the z-axis denoted as $d_{r},$ and the left one denoted as $d_{l} .$ As the reflection law on the left and right sides of the curved surface is similar, we only analysis the light reflection from the right curve liquid surface. Seeing Fig. 2, $β$ is the angle between the tangent line through the incidence point of boundary ray on the curved surface and horizontal plane, and $\tan β = - d z / d x = - z^{'} .$ Based on the geometrical optics, the angle between the right boundary ray 1 and its reflection ray 1′, $2 β,$ is expressed as:

Fig. 2 Schematic diagram of the boundary light reflection from the liquid surface.

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\tan 2 β = \frac{d_{r} - l_{r}}{h} .

A set of dimensionless quantities are defined as following:

{(\begin{matrix} D_{l} & D_{r} & L_{l} & \begin{matrix} L_{r} & \begin{matrix} L & H & X_{0} \end{matrix} \end{matrix} \end{matrix})}^{T} = \frac{1}{α} {(\begin{matrix} d_{l} & d_{r} & l_{l} & \begin{matrix} l_{r} & \begin{matrix} l & h & x_{0} \end{matrix} \end{matrix} \end{matrix})}^{T} .

When the width of the inserted plate is small enough, $l_{r}$ is equal to x value of the boundary incidence point. Based on Eqs. (9) and (10), one can get the relation:

D_{r} (X) = - H \frac{2 z^{'}}{1 - {z^{'}}^{2}} + X X > X_{1},

where

X_{1} = X (z^{'}) |_{z^{'} = - 1},

and

X (z^{'})

is given by Eq. (8).

z' = - 1

means that the reflection ray will be parallel to the horizontal plane, and this reflection ray will never arrive at the screen. So, we only consider the case of

X > X_{1} .

The position of boundary reflection light on the screen can be obtained from the first and second derivatives of Eq. (11)

{D^{'}}_{r} = \frac{d D_{r} (X)}{d X} = 1 - \frac{2 H Z {(1 + {z^{'}}^{2})}^{\frac{5}{2}}}{{(1 - {z^{'}}^{2})}^{2}},

{D^{″}}_{r} = \frac{d^{2} D_{r} (X)}{d X^{2}} = - 2 H \frac{[z^{‴} (1 + {z^{'}}^{2}) + 2 z^{'} z^{″}] {(1 - {z^{'}}^{2})}^{2} + 4 (1 - {z^{'}}^{2}) z^{'} z^{″}}{{(1 - {z^{'}}^{2})}^{4}},

where

z^{″} = d z^{'} / d x,

z^{‴} = d z^{″} / d x .

For the right part of the curved liquid surface in Fig. 2, there is

X_{0}

satisfying

{D^{'}}_{r} (X_{0}) = 0.

Based on Eqs. (12) and (13),

D_{r}

decreases with the increase of X for

X < X_{0},

but increases for

X > X_{0} .

So,

X_{0}

is an important position on the liquid surface and we marked it as P. The incidence point of boundary ray may be located left-side or right-side the critical spot P [Fig. 3]. We found that the reflection field distribution is related to both the width of the incidence beam and the critical spot P. Figure 3(a) shows that the incidence spot of boundary ray 1 is left-side the critical spot, and its reflection light arrives at point

S_{1} .

Under this condition, for any other incidence ray, e.g. the ray 2, its reflection light ray

2^{'}

is always on the right-side of point

S_{1}

when it arrives at the screen. So, there will be no reflection light left-side the point

S_{1},

and a strip-shape dark region is formed on the screen. In this case, the width of dark field, depending on the boundary ray 1, decreases with the width of incident beam.

Fig. 3 Boundary ray left-side and right-side the critical spot. (a) Boundary ray left-side the critical spot. 3(b) Boundary ray right-side the critical spot.

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Another case is that the incidence spot of the boundary light is on the right-side area of the critical spot P [Fig. 3(b)]. There must be an incidence ray 0 that is just incident on the critical spot, and its reflection light ray $0^{'}$ will arrive at point $S_{0} .$ The reflection light ray $0^{'}$ determines the width of the dark field because all reflection rays arrive at the right-side of $S_{0} .$ So, the dark field distribution keeps unchanged when the boundary ray 1 move toward the right-side of the critical spot P. That is to say, the width of the dark field depends only on the position of the critical spot P, and it is independent of the width of the incident beam when the boundary incidence ray is located on the right-side of the critical spot P.

When the inserted plate is perpendicular to the horizontal plane, i.e. $θ = 0 °,$ the curved liquid surface is symmetric about the plate [Fig. 4(a)]. If the incidence point of the left boundary reflection ray is on the left critical spot, $L_{l} = X_{0},$ we can obtain the following relation:

W = {\begin{cases} \frac{W_{0}}{2} + D_{r} \begin{matrix} When & X_{0} < L < 2 X_{0} \end{matrix} \\ W_{0} \begin{matrix} When & L > 2 X_{0} \end{matrix} \end{cases},

where

L = L_{r} + L_{l},

W

is the width of dark field,

W_{0} / 2

is the distance from z-axis to point

S_{0}

on the screen. The relation between the width of dark field and the width of incident beam is shown as Fig. 4(b).

Fig. 4 The relation between dark field distribution and boundary incident ray. (a) The left boundary ray incidence on the critical spot. (b) The dark field distribution when the left incident boundary ray keeping incidence on the critical spot and the right incident boundary ray moving right-side.

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Thus, we obtain a pattern of the reflection light distribution from the curved liquid surface. Especially, we found that the property of the dark field is related to the liquid surface tension, contact angle and the shape of liquid surface. According to the Eqs. (6)–(8), (11) and (14), the contact angle, liquid surface tension coefficient and the shape of the liquid surface can be measured simultaneously by controlling the width of incidence beam and its incidence position on the liquid surface.

3. Experiments and results

A schematic diagram of the experiment is shown in Fig. 5. Light from a He–Ne laser (0.8mW, λ = 632.8 nm) is expanded and collimated and then illustrated vertically on the surface of the liquid, and the beam diameter of the expanded beam is about 35mm. The bottom of the liquid sample cell is covered with rough black cloth, and the depth of liquid in the cell is about 5 cm. A quartz glass sheet (Sail Brand, thickness is 0.13mm) is selected as the plate inserted into liquid. A width-adjustable slit aperture is used to control the width of the incidence beam, then we can control the incidence position of the left or the right boundary incidence ray by this aperture. A CCD camera (G1UC05C, resolution: 2594 × 1944) is used to capture the reflection pattern on a viewing screen. The distance from screen to CCD and horizontal liquid surface is 12.6 cm and 17.5cm, respectively. The data detected by the CCD are input into a computer, and reflection patterns can be displayed, stored, and processed by the computer. The water that is taken from ultra-pure water system (BLH1-20L-D). The room temperature is 22°C.

Fig. 5 Schematic diagram of the experimental setup.

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In order to control the width of the incidence beam, l, we keep the left side of the slit aperture static and move its right side quantitatively. The reflection patterns are recorded when the width of the slit aperture is adjusted step by step. Figure 6(a) shows the reflection light distribution when the width of incidence beam increases. We found that the pattern contains central dark field and external bright field. The width of dark field decreases remarkably with the width of the slit aperture. This experimental finding agrees excellently with the theoretical analysis shown in Fig. 3(a), i.e., the wide of dark field depends on the incidence spot of boundary ray. Figure 6(b) displays that the dark field keeps unchanged when the width of the slit aperture increases. It agrees very well with the theoretical analysis shown in Fig. 3(b). In this case, the incidence point of boundary ray is right-side the critical spot

Fig. 6 Reflection patterns from curved liquid surface. (a) $L < 2 X_{0}$ , (b) $L > 2 X_{0}$ .

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Based on Eq. (11), we can get the slope at the incidence point on the liquid surface through the width of dark field. The surface tension coefficient and the contact angle can be measured according to Eq. (8). The shape of curved liquid surface was obtained by substituting the measurement value of surface tension coefficients and contact angle into Eqs. (6) and (7). Figure 7(a) shows the slope of different points on the curved water surface, and the full line is the fitting curve based on Eq. (8). In this experiment, the goodness-of-fit value is $R^{2} = 0.9838,$ and the critical spot is $x_{0}$ = 12.05 mm. Our experimental results show that water surface tension coefficient and the contact angle are $γ = 7.13 \times 1 0^{- 2}$ N/m and $θ_{c} = 45.8 °,$ respectively. Besides, 20% ethanol was also measured in this work. Figure 7(b) shows the slope of different points on curved ethanol surface. For 20% ethanol, our results show that the critical spot is $x_{0}$ = 10.02 mm, the goodness-of-fit value is $R^{2} = 0.9911$ and the surface tension coefficient and the contact angle are $γ = 4.21 \times 1 0^{- 2}$ N/m and $θ_{c} = 29.8 °,$ respectively.

Fig. 7 The slope of the curved liquid surface versus x-position. (a) water and (b) and 20% ethanol.

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In order to proof-test the effectiveness of our method, some comparative experiments have been done. Surface tension coefficient was measured by tearing-off method (Instrument type: FD-NST-I), and the contact angle was measured by goniometry (Instrument type: JC2000C1). The measurement results are shown in Table 1. It is found that our results are in agreement with the results measured by other means.

Table 1. The measurement results of surface tension and contact angle

View Table

We can also obtain the shape of the curve liquid surface by Eq. (6). The gravity acceleration is g = 9.8 m/s², and the densities of water and 20% ethanol are 998.23 kg/m³ and 973.50 kg/m³, respectively. Figures 8(a) and 8(b) show the measurement results of surface shape of the water and 20% ethanol, and maximum height of curved surface are 2.03 mm and 2.11 mm, respectively.

Fig. 8 The shape of the curved liquid surface. (a) Water and (b) 20% ethanol.

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4. Conclusion

This paper developed a light reflection method to measure the liquid surface tension coefficient and the contact angle simultaneously. The reflection light flied distribution from curved liquid surface is observed experimentally, and there is a dark field area in the center of the distribution. We found that there is a critical spot on the liquid surface and the width of the dark area depend on the critical spot and the width of incidence beam. The width of dark field decreases with the width of incidence beam when the boundary light is within the area of critical spot, but the dark area will keep unchanged if the boundary incidence light is located right-side of the critical spot. This phenomenon exists in the reflection characteristics of laser beams incidence on any surface like the meniscus. Combined the dark field distribution and the slope equation of curved liquid surface due to wetting effect, the surface tension and contact angle were measured, and the shape of the liquid surface was also obtained. In our experiment, measured value of surface tension and the contact angle are 71.3 mN/m and 45.8° for water, and 42.1 mN/m and 29.8° for 20% ethanol, respectively. The liquid surface shape was also presented by measuring the slope of boundary incidence spot, and maximum height of curved water and 20% ethanol surface are 2.03 mm and 2.11 mm, respectively. This work presents an effective method to measure the wetting effects of liquids, no matter if the liquid is colored or colorless.

Funding

Collaborative Innovation Program of Shaanxi (2015XT-65); Research Project of Shaanxi University of Science and Technology (2017BJ-50).

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	this work			_$γ$(N/m),tearing-off method	$θ_{c}$ (°),Goniometry
	$γ$ (N/m)	$θ_{c}$ (°)	$R^{2}$	_$γ$(N/m),tearing-off method	$θ_{c}$ (°),Goniometry
water	(7.13 ± 0.06) × 10⁻²	45.8 ± 0.9	0.9838	(7.09 ± 0.03) × 10⁻²	46.0 ± 1.1
20% ethanol	(4.21 ± 0.03) × 10⁻²	29.8 ± 0.7	0.9911	(4.16 ± 0.02) × 10⁻²	30.2 ± 1.2

Simultaneous measurement of liquid surface tension and contact angle by light reflection

Abstract

1. Introduction

2. Theory

2.1 Theoretical analysis of liquid surface shape

2.2. Theoretical analysis of light reflection from the curved liquid surface

3. Experiments and results

4. Conclusion

Funding

References

Cited By

Figures (8)

Tables (1)

Equations (14)

Optics Express