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Research on variation of static contact angle in incomplete wetting system and modeling method
Colloids and Surfaces A: Physicochem. Eng. Aspects 504 (2016) 400–406
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Colloids and Surfaces A: Physicochemical and Engineering Aspects journal homepage: www.elsevier.com/locate/colsurfa
Research on variation of static contact angle in incomplete wetting system and modeling method Jianjun Jiang a,∗ , Qiang Guo a , Bailing Wang b , Linchao Zhou a , Chumeng Xu a , Chao Deng a , Xuming Yao a , Yang Su a , Junbiao Wang a a b
Shaanxi Engineering Research Center for Digital Manufacturing Technology, Northwestern Polytechnical University, Xi’an 710072, PR China Beijing Shenzhou Aerospace Software Technology Co., Ltd., PR China
h i g h l i g h t s
g r a p h i c a l
a b s t r a c t
• The contact angle in capillary is • •
• •
generally greater than 0◦ and multivalued. The model of solid-liquid static contact angle considers the meniscus surface of liquid. The paper constructs incomplete wettability system contact angle model that could accurately predict the contact angle. The prediction error of the contact angle is less than 3% by using the prediction model of the paper. Experimental conclusions will play an important guiding role in research field of contact angle.
a r t i c l e
i n f o
Article history: Received 28 January 2016 Received in revised form 7 May 2016 Accepted 18 May 2016 Available online 2 June 2016 Keywords: Incomplete wetting Static contact angle Capillary method Meniscus shape Mathematical modeling
∗ Corresponding author. E-mail address: [email protected] (J. Jiang). http://dx.doi.org/10.1016/j.colsurfa.2016.05.051 0927-7757/© 2016 Elsevier B.V. All rights reserved.
a b s t r a c t The solid-liquid contact angle which reflects the wettability of solid-liquid system is determined by the physical nature of solid-liquid medium and environmental factors. Through experiments we discovered that the value of contact angle in capillary is generally greater than 0◦ and multivalued. In addition, the experiments show that as the tube diameter increases, the equivalent height of liquid meniscus increases; as the liquid column height increases, the equivalent height of liquid meniscus decreases. In order to build the solid-liquid static contact angle model of incomplete wetting system, we established the equivalent height function with the capillary diameter as the independent variable. Furthermore, the contact angle model was constructed based on the Jurin formula. The experiments indicated that the model was coincident with the experimental data, what is more, the forecast errors of the contact angle were less than 3%. The model had higher prediction precision compared with that of Jurin or Rayleigh formula. Meanwhile, the reasonability of the modeling method and the practicability of the model are also verified by the paper. © 2016 Elsevier B.V. All rights reserved.
J. Jiang et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 504 (2016) 400–406
1. Introduction
401
density of liquid, r is the radius of the tube bore, h is the height of liquid column.
Solid-liquid contact angle is an important physical factor to characterize the wettability performance of liquid to solid surface. The wettability performance determines the final form of the two-phase interface and affects the interface properties between the fiber and resin after cured. The solid-liquid contact angle is an important parameter to measure the wetting properties of the surface of the material, and the contact angle measurements are important technical means to detect material characteristics, many importantly interface interactional information of solidliquid and gas-liquid, gas-solid can be obtained by contact angle measurements, such as wetting ability, solid-liquid interfacial tension, surface roughness, chemical heterogeneity. Therefore, by measuring contact angle we can get the information of interface performance and so on. Therefore, the solid-liquid contact angle attracts a wide spread attention in the field of composite materials, especially in the liquid composite molding (LCM) [1–4]. For the LCM, the exact characterization of the contact angle plays an important role in the wettability performance of liquid to solid surface which determines the quality of composites [5]. The characterization of the contact angle is influenced by the structure of fabric preform, because there are a variety of heterogeneous capillaries between the fiber bundle and the fibers in the fabric preform. What is more, the heterogeneous capillaries have deep influence on the exact characterization of the contact angle and the capillary tube is similar to the structure of fabric preform precisely. Accordingly, we try to explore the model between the capillary tube diameter and solid-liquid contact angle and it is also significant for theoretical and engineering significance [especially for liquid composite molding (LCM)] to study solid-liquid contact angle by capillary method. A large number of scholars at home and abroad have paid close attention to the research of solid-liquid contact angle in capillary tube [6–10], because it is an effective method to research the solidliquid wettability. Jurin formula [11–14] and Rayleigh equation [15,16] are the fundamental formula to characterize the relationship between capillary column height and the contact angle. However, as the limitations of the study and calculation in that time, Jurin formula ignores the height of the liquid meniscus, and Rayleigh equation assumes a spherical meniscus (the static contact angle is 0◦ ). Actually, it is a general phenomenon that the static con◦ tact angle > 0 , meanwhile the experiments of Huang Renzhong [17], Li ChuanWen [18] and Li Jian [19] also confirmed the phenomenon, and our study also proved the phenomenon. Research shows that the solid-liquid contact angle is not only related to solid and liquid property, but also may be connected with the capillary diameter [17–19]. The capillary diameter is similar to the structure of fabric preform precisely. For these reasons, this study tries to explore the model about the capillary tube and solid-liquid contact angle by using different solid-liquid mediums and capillary diameters. 2. Modeling approach of solid-Liquid static contact angle
2.2. The characterization of meniscus surface 2.2.1. The actual liquid phase of capillary phenomenon The actual fluid phase of partial-wetting fluid observed by the method of capillary rise or descent as shown in Fig. 1. The phenomenon of liquid rises or declines in capillary tube is called capillarity. As can be seen in Fig. 2, after the capillary tube insert in liquid, the liquid rises along the capillary wall under the effect of concave liquid surface tension until the upward tension is equal to the gravity of the liquid column, and then the liquid inside the tube stops rising. There is a meniscus surface at the top of liquid column as shown in Fig. 1. The liquid volume of the meniscus surface is transformed into the liquid equivalent height sequentially the contact angle model that considers the impact of liquid meniscus surface is established based on the Eq. (1).
2.2.2. The characterization of meniscus surface Bashforth-Adams equation [20] (Eq. (2)) is the perfect model of meniscus appearance in gravity field.
⎧ ⎪ ⎨
d2 y dx2
3/2 ⎪ ⎩ 1 + dy 2 dx
+
x 1+
dy dx
⎫ ⎪ ⎬
dy 2 1/2 ⎪ ⎭
= gh + gy
(2)
dx
Where x is the independence variable of the meniscus curve equation, y is the meniscus curve equation. In order to accurately express the shape of liquid meniscus, Bashforth-Adams equation was derived from the Laplace equation, which could be more accurate expression of the liquid meniscus shape under the influence of gravitational field. When the liquid interface is static or equilibrium condition, namely, the equilibrium state. The shape outline of liquid obeys the Bashforth-Adams equation (Eq. (2)). Accordingly, we use the Eq. (2) to characterize the liquid meniscus surface. Furthermore, in this paper we used the different radius of capillary tubes in the experiment because the capillary tube is similar to the structure of fabric preform accurately. For this reason, we tried to explore the model between the capillary tube diameter and solid-liquid contact angle through changing the medium and the capillary tubes diameter, and in the process of contact angle measurement gravity would affect the shape of liquid. What is more, in the process of LCM the wettability of liquid to fabric preform is affected by the gravity. The influence factors of surface shape we considered are as follows: (a) Gravity; (b) The liquid surface tension ; (c) Scope (r).
2.1. Fundamental equation—Jurin formula
2.3. Modeling of solid-liquid static contact angle
Jurin formula establishes the balance relationship between the pressure of rised liquid column on the horizontal plane gh and the additional pressure of the meniscus. The equation is as follows:
The model of solid-liquid static contact angle considers the liquid of meniscus surface. What is more, the model was derived from the Jurin formula.
2cos = gh
(1)
Where is the liquid surface tension, g is the magnitude of the local gravitational accelerate, is the solid-liquid contact angle, is the
2.3.1. The correction of Rayleigh to Jurin formula ◦ The meniscus is assumed to be hemispherical ( = 0 ) in Rayleigh equation [15,21], moreover the Rayleigh equation also considers the pressure of the liquid meniscus on the horizontal
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Fig. 1. The practical liquid phase of capillary phenomenon.
Fig. 2. Feature points on semi-meniscus (a, b and c are corresponding to the experiments with capillary tubes of radius 0.15 mm, 0.5 mm and 1.4 mm,respectively).
plane. Series approximation method are used to correct Jurin formula, and the equation is given by
2
2 cos r 1r = gh 1 + − 0.1288 r 3h h Where
1 r 3h
r 3
+ 0.1312
h
(3)
is the theoretical value when the meniscus surface is
2
3
hemispherical, −0.1288 hr + 0.1312 hr is the correction value of hemisphere deviates from meniscus under the influence of gravity. ◦ In this case, if = 0 , the capillary will be completely wetted. However, the experiments show that solid-liquid static contact
◦
angle = 0 , so the calculation accuracy of Eq. (3) is affected and inaccurate. In order to obtain accurate solid-liquid static contact ◦ angle ( > 0 ), we researched the solid-liquid static contact angle ◦ modeling of incomplete wetting system ( > 0 ).
2.3.2. Solid-liquid static contact angle modeling of incomplete ◦ wetting system ( > 0 ) The laws of experiments: under the same liquid medium and same experimental method, equivalent height of the liquid meniscus (the volume of the liquid meniscus is transformed into the height of the liquid) is closely related to the capillary diameter and
J. Jiang et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 504 (2016) 400–406
the height of liquid column, respectively. As the capillary diameter increases, the equivalent height of the liquid meniscus increases. As the liquid column height rises, the equivalent height of the liquid meniscus decreases. Therefore, the state of the contact angle can be characterized by r and h. Modeling approach: Through the above analysis, we draw the following modeling methods. The pressure on the horizontal plane of liquid meniscus is reflected by the equivalent height, meanwhile, we establish the equivalent height function with the capillary radius as the independent variable. Finally, the model of contact angle is characterized by the capillary radius and liquid column height, as shown in Eqs. (4)–(6). 2 cos = g (h + A · h) r
(4)
1r ar 2 br 3 + + 3h h h
(5)
A=
A·h=
1 r + ar 2 + br 3 3
(6)
Where A is the equivalent factor of height, A · h is the equivalent height, h is the height of liquid, 13 ris the equivalent height when the meniscus surface is hemispherical, ar 2 + br 3 is the difference of the wettability of solid-liquid system and the correction value of hemisphere deviates from meniscus under the influence of gravity. The research focus: By analyzing the modeling process of the solid-liquid static contact angle, we make a detailed experimental scheme to establish the contact angle model. The specific values of a and b are obtained by numerical analysis and polynomial fitting methods, thereby establishing the solid-liquid static contact angle modeling of incomplete wetting system. 3. Experimental 3.1. Experimental program Through the analysis of modeling approach and research focus, we conducted a large number of experimental measurements and calculations. In order to obtain the theoretical value and the model of the solid-liquid static contact angle, we had performed experiments on the different solid and fluid mediums, thereby obtaining the parameters used in the modeling. 3.1.1. The measurement of liquid surface tension In the study, the liquid surface tension is calculated using the Eq. (7). In addition, the other relevant parameters of Eq. (7) are measured by Wilhelmy plate method. =
F 2l
(7)
Where F is the maximum tension that the liquid surface can bear, l is the contact line length of the hanger plate and the liquid surface. The other parameters are same as above. 3.1.2. The measurement of liquid density The buoyancy of the standard block in the liquid was measured by the method of specific gravity, and then the value was added to the Eq. (8) to calculate the volume of the block. =
Fg − Fl gVs
(8)
Where Fg is the weight of the standard block in the air, Fl is the weight of the standard block in the liquid, Vs is the volume of the standard block. The other parameters are same as above.
403
3.1.3. The measurement of liquid volume height (a) Capillary rise method: The glass capillary that was perpendicular to the liquid plane was dipped in liquid, which traveled along capillary tube wall by capillarity. Then we measured the rising height of liquid column by ruler (the measure principle is shown in Fig. 1) when the liquid was not rose and remained stable. (b) Capillary descent method: The glass capillary was placed vertically in the liquid. Then the liquid was sucked high enough (over the maximum height that glass tube can bear) using the rubber suction bulb. Afterwards, the liquid in the glass capillary would fall under the effect of gravity. Finally, the height of liquid column would be measured by ruler when liquid was no longer fall and kept stable. (c) We selected four different radius of glass capillaries as the solid medium and three kinds of the fluid as liquid medium that were combined into 12 kinds of solid-liquid systems, with which we did the experiments by the capillary rise method and the method of capillary descent respectively (each experiment is carried out three times). The arithmetic average of the three experiments as the liquid column height of the system. 3.2. Experimental materials and data 3.2.1. Experimental materials (a) Solid medium: the internal diameter of capillary tubes are 0.15 mm, 0.5 mm and 1.4 mm, respectively. Furthermore, the capillary tubes whose internal diameter is 0.25 mm are chosen for the verification experiment. (b) Fluid medium: alcohol (AR, Tianjin Tianli Chemical Reagent Co., Ltd.), distilled water (Guangzhou Watson’s Food & Beverage Co., Ltd.), Sesame oil (Xi’an xiang zheng shi pin Instrument Co., Ltd) (c) Density/Surface tension measuring instrument: specific gravity measuring instrument (type: MDY-1, Shanghai Fangrui Instrument Co., Ltd.). 3.2.2. Experimental data ◦ Experiments was operated at room temperature (14 C) Distilled water: Distill−Water = 73.29mN/m, Distill−Water = 1.000g/cm3 Sesame oil: Sesame−Oil = 36.98mN/m, Sesame−Oil = 0.919g/cm3 Alcohol: Alcohol = 22.55mN/m, Alcohol = 0.792g/cm3 4. Modeling of static contact angle in the incomplete wetting system 4.1. Morphological analysis of the actual meniscus 4.1.1. Experimental data processing and analysis We just put the column height h and radius r into the formula 2, and then applied the Matlab to obtain the theoretical numerical point set xi , yi , yi of the theoretical meniscus shape. Furthermore, the first derivative of the curve of the upper edge can be transformed to the numerical solution of solid-liquid contact angle. Fig. 2 is the numerical solution of solid–liquid contact angle and theoretical meniscus shape. The meniscus in capillary is not a complete hemispherical as shown in Fig. 2, meanwhile we can obtain some conclusions from Fig. 2. (a) Descent method reduces contact angle to a certain extent and improves the wettability, however the contact angle of descent ◦ method is not equivalent to 0 . (b) The range of wetting and contact angle are associated with the capillary diameter and the type of liquid.
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4.1.2. Characterization of the meniscus curve Numerical analysis method can only characterize the theoretical liquid shape with a series of discrete data points. In order to make a further quantification for the theoretical shape of the liquid, in this paper, we conducted polynomial fitting of discrete points, thus obtained liquid shape function. The construct of the polynomial adopted from low order to high order tried to step by step, until the fitting error reached 10−6 or 10−7 . Then we taken the last fitting result as the function of liquid shape in the experimental system. The expression of the function is the Eq. (9). Specific values of the parameters are presented in Table 2. y = a · x4 + b · x3 + c · x2 + d · x + e
(9)
We can find that the forth order term fitting parameter a is equal to 0, as shown in Table 2. Hence, it turns out that 3rd order is enough as the coefficient of forth term is zero. Meanwhile, it also demonstrate that the Eq. (9) does not have the forth order term is enough and complete. Finally, we would modify the Eq. (9) and the function of liquid shape can be derived as the following: y = b · x3 + c · x2 + d · x + e. Furthermore, the fitting parameters of Eq. (9) are listed in Table 2, and we can choose the parameter values of b, c, d, e from the table according to the certain conditions. 4.2. The calculation and fitting of equivalent height 4.2.1. Calculation of the equivalent height The solving of equivalent height used function integral method of the liquid shape, and Eq. (10) is the formula for calculating the equivalent height. The values of equivalent height are shown in Table 3.
r
A·h=
0
2xydx
(10)
r 2
Through the comparative analysis of Tables 1 and 3, we can obtain the following rules in the capillary experimental system of same liquid mediums and measuring methods.
Fig. 3. The fitting curve of equivalent height.
4.4. Model validation and comparative analysis The capillary tubes whose internal diameter is 0.25 mm are chosen for the verification experiment. And then we used Jurin Formula, Rayleigh Formula and Eq.11 to calculate solid-liquid static contact angle separately. What is more, the calculated results and the numerical solution of contact angle were compared, as shown in Table 4. As can been seen from the Table 4, the calculation results of the model are close to the numerical solution compared with those of the Jurin formula and Rayleigh formula. In this paper, the relative error of the prediction model Eq. (11) is less than 3%, which is more accurate than the previous models. Meanwhile, verification experiments are given to demonstrate the correctness and practicality of the contact angle prediction model.
(a) As the glass tube diameter increases, the equivalent height A · h of the liquid meniscus increases. (b) As the liquid column height inside the glass tube increases, the equivalent height A · h of the liquid meniscus decreases.
5. Conclusions and prospects
4.2.2. The function fitting of equivalent height We established the function of the equivalent height A · h and the radius r. Equivalent heights that were measured in the same liquid and method were fitted in accordance with Eq. (6). The fitting results are given by Table 3 and Fig. 3.
(1) Capillary descent method can improve the wettability and increase the height of capillary rise, besides, the method of capillary descent can reduce the contact angle. In addition, the effect of the method of capillary descent depends on the capillary tubes diameter and the change of the liquid surface tension, furthermore, the effect of the method will be weaken when the liquid surface tension decreases. (2) For the same liquid and measuring method, the equivalent height of liquid meniscus is related to the capillary diameter and liquid column height: With the increasing of tube diameter, the equivalent height of liquid meniscus increases. With the increasing of the liquid column height, the equivalent height of liquid decreases. (3) The model of the solid-liquid static contact angle which is constructed in the system of incomplete wetting considers the change of the wetting of the solid-liquid system. It can more reasonably reflect the actual state of the meniscus, meanwhile, the relative error is less than 3%. Compared with Jurin’s formula, the accuracy has a substantial increase.
4.3. Modeling of static contact angle According to the Eq. (4), the contact angle prediction model (Eq. (11)) can be obtained under the different materials and experimental methods.
2 cos 1 = g h + r + a × 103 r 2 + b × 106 r 3 r 3
(11)
Where a, b which are related with materials and experiment method are taken from Table 3, (1/3)r is the theoretical height of hemispherical surface of the liquid, a × 103 r 2 + b × 106 r 3 is the correction height value of meniscus under the influence of gravity. The Eq. (11) contains the liquid meniscus pressure on the horizontal plane, and it takes the influence of material, experimental method and the wettability of solid-liquid medium along with the change of radius into consideration. Furthermore, the state of the meniscus is truly reflected.
Some conclusions can be obtained by the experimental research and theoretical analysis in the paper.
Furthermore, there are several experimental phenomena that are noteworthy in Fig. 3.
J. Jiang et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 504 (2016) 400–406
405
Table 1 The liquid column height of different fluid mediums in capillary tubes (mm). r/mm
0.15 0.5 1.4
Method
↑ ↓ ↑ ↓ ↑ ↓
Distilled water
Sesame oil
Alcohol
I
II
III
Average
I
II
III
Average
I
II
III
Average
75.8 77.0 10.8 26.8 7.5 8.2
72.6 74.1 10.5 26.6 8.6 8.6
73.4 76.1 11.0 27.8 8.0 8.5
73.9 75.7 10.8 27.1 8.0 8.4
38.4 44.2 11.0 15.2 4.0 3.9
41.3 45.6 9.6 13.9 4.0 3.9
38.4 43.2 10.1 13.3 3.9 4.0
39.4 44.3 10.2 14.1 3.9 3.9
29.5 30.0 9.5 9.5 2.5 2.5
27.2 27.5 10.0 10.5 2.8 2.8
28.3 29.5 9.9 10.0 2.7 2.7
28.3 29.0 9.8 10.0 2.7 2.7
Note: “I”, “II”, “III” represent experimental group number, Average means the arithmetic average of three experiments, moreover, ↑ is rise method, ↓ is descent method in the paper.
Table 2 The fitting function of the liquid meniscus. Diameter of capillary
0.15
Measurement methods
↑
Distilled water Sesame oil Alcohol Distilled water Sesame oil Alcohol Distilled water Sesame oil Alcohol Distilled water Sesame oil Alcohol Distilled water Sesame oil Alcohol Distilled water Sesame oil Alcohol
↓
0.5
↑
↓
1.4
The type of liquid
↑
↓
y = a · x4 + b · x3 + c · x2 + d · x + e a
b
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
7.656 × 10 0 0 8.528 × 106 0 0 0 0 1.4 × 106 2.023 × 106 1.521 × 106 1.578 × 106 1.139 × 105 8.214 × 104 8.791 × 104 1.43 × 105 8.214 × 104 8.791 × 104 6
c
d
e
1622 3082 3159 1572 786 3290 382.5 743.9 277 55.4 236.6 211.4 144.7 151.6 139.1 122.9 151.6 139.1
0.0258 −0.04812 −0.05131 0.02947 −0.07821 −0.05626 −0.004896 −0.02835 0.06063 0.09274 0.06718 0.07018 0.03569 0.02461 0.027 0.04657 0.02461 0.027
−4.68 × 10−8 2.03 × 10−7 2.16 × 10−7 −5.254 × 10−8 3.284 × 10−7 2.366 × 10−7 7.682 × 10−8 4.202 × 10−7 −4.03 × 10−7 −6.293 × 10−7 −4.642 × 10−7 −4.788 × 10−7 −6.353 × 10−7 −4.35 × 10−7 −4.833 × 10−7 −8.43 × 10−7 −4.35 × 10−7 −4.833 × 10−7
Table 3 Equivalent height and fitting results. The type of liquid
Measurement methods
Distilled water
↑ ↓ ↑ ↓ ↑ ↓
Sesame oil Alcohol
A·h=
r (mm)
1 r + ar 2 + br 3 3
0.15
0.5
1.4
a
b
0.03112 0.03209 0.03006 0.03510 0.03062 0.03161
0.04626 0.1383 0.08396 0.1275 0.1245 0.1282
0.2995 0.3200 0.2613 0.2613 0.2575 0.2575
−0.7054 −0.1462 −0.4645 −0.1939 −0.2149 −0.1912
0.4429 0.051 0.2569 0.06363 0.07729 0.06038
Table 4 Validation experimental data and comparative analysis of experimental data. Distilled water Rise method Height h(mm) Average height h(mm) Numerical solution (◦ ) Equation and (◦ ) Relative error 0⁄000 Descent method Height h(mm) Average height h(mm) Numerical solution (◦ ) Equation and (◦ ) Relative error 0⁄000
Sesame oil
55.7 55.3 22.2568 Jurin 22.4358
56.1
54.2
Rayleigh 22.2261
80.4
Alcohol 28.2
27.0
Eq. (11) 22.3197
29.3 28.2 30.6432 Jurin 30.8529
19.8
20.1
Eq. (11) 30.6539
20.9 20.3 28.8132 Jurin 29.1444
Rayleigh 30.5692
Rayleigh 28.7217
Eq.11 28.7824
13.8
28.3
68.8
24.1
3.5
114.9
31.8
10.7
57.1 57.9 14.2894 Jurin 14.5873
59.0
57.5
32.5
30.4
21.5
Rayleigh 17.1785
Eq. (11) 17.2426
20.9 20.9 25.5590 Jurin 25.9456
20.2
Eq. (11) 14.2994
30.9 31.3 17.2225 Jurin 18.7783
Rayleigh 14.2675
Rayleigh 25.4741
Eq.11 25.5351
208.5
15.3
7.0
904.4
25.5
11.7
151.3
33.2
9.4
(1) For the same solid-liquid system, the fitting curve of the rise method exists median strip with that of descent method, what
is more, the width of the median strip rapid contraction in the wake of the decrease of the liquid surface tension. Meanwhile,
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the small median strip (corresponding to the small liquid surface tension) is almost always surrounded by the large median strip (corresponding to the large liquid surface tension); (2) For the different solid-liquid systems, the fitting curve that obtained by the descent method is more centralized distribution compared to that of the fitting curve generated by rise method. The above conclusions and phenomena could be used for reference in the field of solid-liquid static contact angle. Insufficiencies of this dissertation are still unable to explain the multi-valued of the contact angle and the corresponding rules. Furthermore, it is significant for theoretical and engineering significance [especially for liquid composite molding (LCM)] to study the solid-liquid contact angle of liquid mixture containing surfactant and active solid surface. Therefore, in the future work we will research the contact angle in the mixture of resin and curing agent which is the liquid of LCM, on the basis of the solid-liquid static contact angle model.Meanwhile, we will do some experiments to verify the practicability and rationality of the model. Besides, we will continue to explore the formation mechanism of contact angle and further analysis the multi-valued of the contact angle as well as the corresponding rules on the basis of the solid-liquid static contact angle model. Author contributions Jianjun Jiang, Qiang Guo and Junbiao Wang conceived and designed the experiments; Xuming Yao, and Chumeng Xu performed the experiments; Chao Deng analyzed the data; Linchao Zhou and Yang Su contributed reagents and materials; Qiang Guo and Bailing Wang wrote the paper. Conflict of interest The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgment This research is supported by “National Natural Science Foundation of China” (No. 51573148).
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Research on variation of static contact angle in incomplete wetting system and modeling method Jianjun Jiang a,∗ , Qiang Guo a , Bailing Wang b , Linchao Zhou a , Chumeng Xu a , Chao Deng a , Xuming Yao a , Yang Su a , Junbiao Wang a a b
Shaanxi Engineering Research Center for Digital Manufacturing Technology, Northwestern Polytechnical University, Xi’an 710072, PR China Beijing Shenzhou Aerospace Software Technology Co., Ltd., PR China
h i g h l i g h t s
g r a p h i c a l
a b s t r a c t
• The contact angle in capillary is • •
• •
generally greater than 0◦ and multivalued. The model of solid-liquid static contact angle considers the meniscus surface of liquid. The paper constructs incomplete wettability system contact angle model that could accurately predict the contact angle. The prediction error of the contact angle is less than 3% by using the prediction model of the paper. Experimental conclusions will play an important guiding role in research field of contact angle.
a r t i c l e
i n f o
Article history: Received 28 January 2016 Received in revised form 7 May 2016 Accepted 18 May 2016 Available online 2 June 2016 Keywords: Incomplete wetting Static contact angle Capillary method Meniscus shape Mathematical modeling
∗ Corresponding author. E-mail address: [email protected] (J. Jiang). http://dx.doi.org/10.1016/j.colsurfa.2016.05.051 0927-7757/© 2016 Elsevier B.V. All rights reserved.
a b s t r a c t The solid-liquid contact angle which reflects the wettability of solid-liquid system is determined by the physical nature of solid-liquid medium and environmental factors. Through experiments we discovered that the value of contact angle in capillary is generally greater than 0◦ and multivalued. In addition, the experiments show that as the tube diameter increases, the equivalent height of liquid meniscus increases; as the liquid column height increases, the equivalent height of liquid meniscus decreases. In order to build the solid-liquid static contact angle model of incomplete wetting system, we established the equivalent height function with the capillary diameter as the independent variable. Furthermore, the contact angle model was constructed based on the Jurin formula. The experiments indicated that the model was coincident with the experimental data, what is more, the forecast errors of the contact angle were less than 3%. The model had higher prediction precision compared with that of Jurin or Rayleigh formula. Meanwhile, the reasonability of the modeling method and the practicability of the model are also verified by the paper. © 2016 Elsevier B.V. All rights reserved.
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1. Introduction
401
density of liquid, r is the radius of the tube bore, h is the height of liquid column.
Solid-liquid contact angle is an important physical factor to characterize the wettability performance of liquid to solid surface. The wettability performance determines the final form of the two-phase interface and affects the interface properties between the fiber and resin after cured. The solid-liquid contact angle is an important parameter to measure the wetting properties of the surface of the material, and the contact angle measurements are important technical means to detect material characteristics, many importantly interface interactional information of solidliquid and gas-liquid, gas-solid can be obtained by contact angle measurements, such as wetting ability, solid-liquid interfacial tension, surface roughness, chemical heterogeneity. Therefore, by measuring contact angle we can get the information of interface performance and so on. Therefore, the solid-liquid contact angle attracts a wide spread attention in the field of composite materials, especially in the liquid composite molding (LCM) [1–4]. For the LCM, the exact characterization of the contact angle plays an important role in the wettability performance of liquid to solid surface which determines the quality of composites [5]. The characterization of the contact angle is influenced by the structure of fabric preform, because there are a variety of heterogeneous capillaries between the fiber bundle and the fibers in the fabric preform. What is more, the heterogeneous capillaries have deep influence on the exact characterization of the contact angle and the capillary tube is similar to the structure of fabric preform precisely. Accordingly, we try to explore the model between the capillary tube diameter and solid-liquid contact angle and it is also significant for theoretical and engineering significance [especially for liquid composite molding (LCM)] to study solid-liquid contact angle by capillary method. A large number of scholars at home and abroad have paid close attention to the research of solid-liquid contact angle in capillary tube [6–10], because it is an effective method to research the solidliquid wettability. Jurin formula [11–14] and Rayleigh equation [15,16] are the fundamental formula to characterize the relationship between capillary column height and the contact angle. However, as the limitations of the study and calculation in that time, Jurin formula ignores the height of the liquid meniscus, and Rayleigh equation assumes a spherical meniscus (the static contact angle is 0◦ ). Actually, it is a general phenomenon that the static con◦ tact angle > 0 , meanwhile the experiments of Huang Renzhong [17], Li ChuanWen [18] and Li Jian [19] also confirmed the phenomenon, and our study also proved the phenomenon. Research shows that the solid-liquid contact angle is not only related to solid and liquid property, but also may be connected with the capillary diameter [17–19]. The capillary diameter is similar to the structure of fabric preform precisely. For these reasons, this study tries to explore the model about the capillary tube and solid-liquid contact angle by using different solid-liquid mediums and capillary diameters. 2. Modeling approach of solid-Liquid static contact angle
2.2. The characterization of meniscus surface 2.2.1. The actual liquid phase of capillary phenomenon The actual fluid phase of partial-wetting fluid observed by the method of capillary rise or descent as shown in Fig. 1. The phenomenon of liquid rises or declines in capillary tube is called capillarity. As can be seen in Fig. 2, after the capillary tube insert in liquid, the liquid rises along the capillary wall under the effect of concave liquid surface tension until the upward tension is equal to the gravity of the liquid column, and then the liquid inside the tube stops rising. There is a meniscus surface at the top of liquid column as shown in Fig. 1. The liquid volume of the meniscus surface is transformed into the liquid equivalent height sequentially the contact angle model that considers the impact of liquid meniscus surface is established based on the Eq. (1).
2.2.2. The characterization of meniscus surface Bashforth-Adams equation [20] (Eq. (2)) is the perfect model of meniscus appearance in gravity field.
⎧ ⎪ ⎨
d2 y dx2
3/2 ⎪ ⎩ 1 + dy 2 dx
+
x 1+
dy dx
⎫ ⎪ ⎬
dy 2 1/2 ⎪ ⎭
= gh + gy
(2)
dx
Where x is the independence variable of the meniscus curve equation, y is the meniscus curve equation. In order to accurately express the shape of liquid meniscus, Bashforth-Adams equation was derived from the Laplace equation, which could be more accurate expression of the liquid meniscus shape under the influence of gravitational field. When the liquid interface is static or equilibrium condition, namely, the equilibrium state. The shape outline of liquid obeys the Bashforth-Adams equation (Eq. (2)). Accordingly, we use the Eq. (2) to characterize the liquid meniscus surface. Furthermore, in this paper we used the different radius of capillary tubes in the experiment because the capillary tube is similar to the structure of fabric preform accurately. For this reason, we tried to explore the model between the capillary tube diameter and solid-liquid contact angle through changing the medium and the capillary tubes diameter, and in the process of contact angle measurement gravity would affect the shape of liquid. What is more, in the process of LCM the wettability of liquid to fabric preform is affected by the gravity. The influence factors of surface shape we considered are as follows: (a) Gravity; (b) The liquid surface tension ; (c) Scope (r).
2.1. Fundamental equation—Jurin formula
2.3. Modeling of solid-liquid static contact angle
Jurin formula establishes the balance relationship between the pressure of rised liquid column on the horizontal plane gh and the additional pressure of the meniscus. The equation is as follows:
The model of solid-liquid static contact angle considers the liquid of meniscus surface. What is more, the model was derived from the Jurin formula.
2cos = gh
(1)
Where is the liquid surface tension, g is the magnitude of the local gravitational accelerate, is the solid-liquid contact angle, is the
2.3.1. The correction of Rayleigh to Jurin formula ◦ The meniscus is assumed to be hemispherical ( = 0 ) in Rayleigh equation [15,21], moreover the Rayleigh equation also considers the pressure of the liquid meniscus on the horizontal
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Fig. 1. The practical liquid phase of capillary phenomenon.
Fig. 2. Feature points on semi-meniscus (a, b and c are corresponding to the experiments with capillary tubes of radius 0.15 mm, 0.5 mm and 1.4 mm,respectively).
plane. Series approximation method are used to correct Jurin formula, and the equation is given by
2
2 cos r 1r = gh 1 + − 0.1288 r 3h h Where
1 r 3h
r 3
+ 0.1312
h
(3)
is the theoretical value when the meniscus surface is
2
3
hemispherical, −0.1288 hr + 0.1312 hr is the correction value of hemisphere deviates from meniscus under the influence of gravity. ◦ In this case, if = 0 , the capillary will be completely wetted. However, the experiments show that solid-liquid static contact
◦
angle = 0 , so the calculation accuracy of Eq. (3) is affected and inaccurate. In order to obtain accurate solid-liquid static contact ◦ angle ( > 0 ), we researched the solid-liquid static contact angle ◦ modeling of incomplete wetting system ( > 0 ).
2.3.2. Solid-liquid static contact angle modeling of incomplete ◦ wetting system ( > 0 ) The laws of experiments: under the same liquid medium and same experimental method, equivalent height of the liquid meniscus (the volume of the liquid meniscus is transformed into the height of the liquid) is closely related to the capillary diameter and
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the height of liquid column, respectively. As the capillary diameter increases, the equivalent height of the liquid meniscus increases. As the liquid column height rises, the equivalent height of the liquid meniscus decreases. Therefore, the state of the contact angle can be characterized by r and h. Modeling approach: Through the above analysis, we draw the following modeling methods. The pressure on the horizontal plane of liquid meniscus is reflected by the equivalent height, meanwhile, we establish the equivalent height function with the capillary radius as the independent variable. Finally, the model of contact angle is characterized by the capillary radius and liquid column height, as shown in Eqs. (4)–(6). 2 cos = g (h + A · h) r
(4)
1r ar 2 br 3 + + 3h h h
(5)
A=
A·h=
1 r + ar 2 + br 3 3
(6)
Where A is the equivalent factor of height, A · h is the equivalent height, h is the height of liquid, 13 ris the equivalent height when the meniscus surface is hemispherical, ar 2 + br 3 is the difference of the wettability of solid-liquid system and the correction value of hemisphere deviates from meniscus under the influence of gravity. The research focus: By analyzing the modeling process of the solid-liquid static contact angle, we make a detailed experimental scheme to establish the contact angle model. The specific values of a and b are obtained by numerical analysis and polynomial fitting methods, thereby establishing the solid-liquid static contact angle modeling of incomplete wetting system. 3. Experimental 3.1. Experimental program Through the analysis of modeling approach and research focus, we conducted a large number of experimental measurements and calculations. In order to obtain the theoretical value and the model of the solid-liquid static contact angle, we had performed experiments on the different solid and fluid mediums, thereby obtaining the parameters used in the modeling. 3.1.1. The measurement of liquid surface tension In the study, the liquid surface tension is calculated using the Eq. (7). In addition, the other relevant parameters of Eq. (7) are measured by Wilhelmy plate method. =
F 2l
(7)
Where F is the maximum tension that the liquid surface can bear, l is the contact line length of the hanger plate and the liquid surface. The other parameters are same as above. 3.1.2. The measurement of liquid density The buoyancy of the standard block in the liquid was measured by the method of specific gravity, and then the value was added to the Eq. (8) to calculate the volume of the block. =
Fg − Fl gVs
(8)
Where Fg is the weight of the standard block in the air, Fl is the weight of the standard block in the liquid, Vs is the volume of the standard block. The other parameters are same as above.
403
3.1.3. The measurement of liquid volume height (a) Capillary rise method: The glass capillary that was perpendicular to the liquid plane was dipped in liquid, which traveled along capillary tube wall by capillarity. Then we measured the rising height of liquid column by ruler (the measure principle is shown in Fig. 1) when the liquid was not rose and remained stable. (b) Capillary descent method: The glass capillary was placed vertically in the liquid. Then the liquid was sucked high enough (over the maximum height that glass tube can bear) using the rubber suction bulb. Afterwards, the liquid in the glass capillary would fall under the effect of gravity. Finally, the height of liquid column would be measured by ruler when liquid was no longer fall and kept stable. (c) We selected four different radius of glass capillaries as the solid medium and three kinds of the fluid as liquid medium that were combined into 12 kinds of solid-liquid systems, with which we did the experiments by the capillary rise method and the method of capillary descent respectively (each experiment is carried out three times). The arithmetic average of the three experiments as the liquid column height of the system. 3.2. Experimental materials and data 3.2.1. Experimental materials (a) Solid medium: the internal diameter of capillary tubes are 0.15 mm, 0.5 mm and 1.4 mm, respectively. Furthermore, the capillary tubes whose internal diameter is 0.25 mm are chosen for the verification experiment. (b) Fluid medium: alcohol (AR, Tianjin Tianli Chemical Reagent Co., Ltd.), distilled water (Guangzhou Watson’s Food & Beverage Co., Ltd.), Sesame oil (Xi’an xiang zheng shi pin Instrument Co., Ltd) (c) Density/Surface tension measuring instrument: specific gravity measuring instrument (type: MDY-1, Shanghai Fangrui Instrument Co., Ltd.). 3.2.2. Experimental data ◦ Experiments was operated at room temperature (14 C) Distilled water: Distill−Water = 73.29mN/m, Distill−Water = 1.000g/cm3 Sesame oil: Sesame−Oil = 36.98mN/m, Sesame−Oil = 0.919g/cm3 Alcohol: Alcohol = 22.55mN/m, Alcohol = 0.792g/cm3 4. Modeling of static contact angle in the incomplete wetting system 4.1. Morphological analysis of the actual meniscus 4.1.1. Experimental data processing and analysis We just put the column height h and radius r into the formula 2, and then applied the Matlab to obtain the theoretical numerical point set xi , yi , yi of the theoretical meniscus shape. Furthermore, the first derivative of the curve of the upper edge can be transformed to the numerical solution of solid-liquid contact angle. Fig. 2 is the numerical solution of solid–liquid contact angle and theoretical meniscus shape. The meniscus in capillary is not a complete hemispherical as shown in Fig. 2, meanwhile we can obtain some conclusions from Fig. 2. (a) Descent method reduces contact angle to a certain extent and improves the wettability, however the contact angle of descent ◦ method is not equivalent to 0 . (b) The range of wetting and contact angle are associated with the capillary diameter and the type of liquid.
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4.1.2. Characterization of the meniscus curve Numerical analysis method can only characterize the theoretical liquid shape with a series of discrete data points. In order to make a further quantification for the theoretical shape of the liquid, in this paper, we conducted polynomial fitting of discrete points, thus obtained liquid shape function. The construct of the polynomial adopted from low order to high order tried to step by step, until the fitting error reached 10−6 or 10−7 . Then we taken the last fitting result as the function of liquid shape in the experimental system. The expression of the function is the Eq. (9). Specific values of the parameters are presented in Table 2. y = a · x4 + b · x3 + c · x2 + d · x + e
(9)
We can find that the forth order term fitting parameter a is equal to 0, as shown in Table 2. Hence, it turns out that 3rd order is enough as the coefficient of forth term is zero. Meanwhile, it also demonstrate that the Eq. (9) does not have the forth order term is enough and complete. Finally, we would modify the Eq. (9) and the function of liquid shape can be derived as the following: y = b · x3 + c · x2 + d · x + e. Furthermore, the fitting parameters of Eq. (9) are listed in Table 2, and we can choose the parameter values of b, c, d, e from the table according to the certain conditions. 4.2. The calculation and fitting of equivalent height 4.2.1. Calculation of the equivalent height The solving of equivalent height used function integral method of the liquid shape, and Eq. (10) is the formula for calculating the equivalent height. The values of equivalent height are shown in Table 3.
r
A·h=
0
2xydx
(10)
r 2
Through the comparative analysis of Tables 1 and 3, we can obtain the following rules in the capillary experimental system of same liquid mediums and measuring methods.
Fig. 3. The fitting curve of equivalent height.
4.4. Model validation and comparative analysis The capillary tubes whose internal diameter is 0.25 mm are chosen for the verification experiment. And then we used Jurin Formula, Rayleigh Formula and Eq.11 to calculate solid-liquid static contact angle separately. What is more, the calculated results and the numerical solution of contact angle were compared, as shown in Table 4. As can been seen from the Table 4, the calculation results of the model are close to the numerical solution compared with those of the Jurin formula and Rayleigh formula. In this paper, the relative error of the prediction model Eq. (11) is less than 3%, which is more accurate than the previous models. Meanwhile, verification experiments are given to demonstrate the correctness and practicality of the contact angle prediction model.
(a) As the glass tube diameter increases, the equivalent height A · h of the liquid meniscus increases. (b) As the liquid column height inside the glass tube increases, the equivalent height A · h of the liquid meniscus decreases.
5. Conclusions and prospects
4.2.2. The function fitting of equivalent height We established the function of the equivalent height A · h and the radius r. Equivalent heights that were measured in the same liquid and method were fitted in accordance with Eq. (6). The fitting results are given by Table 3 and Fig. 3.
(1) Capillary descent method can improve the wettability and increase the height of capillary rise, besides, the method of capillary descent can reduce the contact angle. In addition, the effect of the method of capillary descent depends on the capillary tubes diameter and the change of the liquid surface tension, furthermore, the effect of the method will be weaken when the liquid surface tension decreases. (2) For the same liquid and measuring method, the equivalent height of liquid meniscus is related to the capillary diameter and liquid column height: With the increasing of tube diameter, the equivalent height of liquid meniscus increases. With the increasing of the liquid column height, the equivalent height of liquid decreases. (3) The model of the solid-liquid static contact angle which is constructed in the system of incomplete wetting considers the change of the wetting of the solid-liquid system. It can more reasonably reflect the actual state of the meniscus, meanwhile, the relative error is less than 3%. Compared with Jurin’s formula, the accuracy has a substantial increase.
4.3. Modeling of static contact angle According to the Eq. (4), the contact angle prediction model (Eq. (11)) can be obtained under the different materials and experimental methods.
2 cos 1 = g h + r + a × 103 r 2 + b × 106 r 3 r 3
(11)
Where a, b which are related with materials and experiment method are taken from Table 3, (1/3)r is the theoretical height of hemispherical surface of the liquid, a × 103 r 2 + b × 106 r 3 is the correction height value of meniscus under the influence of gravity. The Eq. (11) contains the liquid meniscus pressure on the horizontal plane, and it takes the influence of material, experimental method and the wettability of solid-liquid medium along with the change of radius into consideration. Furthermore, the state of the meniscus is truly reflected.
Some conclusions can be obtained by the experimental research and theoretical analysis in the paper.
Furthermore, there are several experimental phenomena that are noteworthy in Fig. 3.
J. Jiang et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 504 (2016) 400–406
405
Table 1 The liquid column height of different fluid mediums in capillary tubes (mm). r/mm
0.15 0.5 1.4
Method
↑ ↓ ↑ ↓ ↑ ↓
Distilled water
Sesame oil
Alcohol
I
II
III
Average
I
II
III
Average
I
II
III
Average
75.8 77.0 10.8 26.8 7.5 8.2
72.6 74.1 10.5 26.6 8.6 8.6
73.4 76.1 11.0 27.8 8.0 8.5
73.9 75.7 10.8 27.1 8.0 8.4
38.4 44.2 11.0 15.2 4.0 3.9
41.3 45.6 9.6 13.9 4.0 3.9
38.4 43.2 10.1 13.3 3.9 4.0
39.4 44.3 10.2 14.1 3.9 3.9
29.5 30.0 9.5 9.5 2.5 2.5
27.2 27.5 10.0 10.5 2.8 2.8
28.3 29.5 9.9 10.0 2.7 2.7
28.3 29.0 9.8 10.0 2.7 2.7
Note: “I”, “II”, “III” represent experimental group number, Average means the arithmetic average of three experiments, moreover, ↑ is rise method, ↓ is descent method in the paper.
Table 2 The fitting function of the liquid meniscus. Diameter of capillary
0.15
Measurement methods
↑
Distilled water Sesame oil Alcohol Distilled water Sesame oil Alcohol Distilled water Sesame oil Alcohol Distilled water Sesame oil Alcohol Distilled water Sesame oil Alcohol Distilled water Sesame oil Alcohol
↓
0.5
↑
↓
1.4
The type of liquid
↑
↓
y = a · x4 + b · x3 + c · x2 + d · x + e a
b
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
7.656 × 10 0 0 8.528 × 106 0 0 0 0 1.4 × 106 2.023 × 106 1.521 × 106 1.578 × 106 1.139 × 105 8.214 × 104 8.791 × 104 1.43 × 105 8.214 × 104 8.791 × 104 6
c
d
e
1622 3082 3159 1572 786 3290 382.5 743.9 277 55.4 236.6 211.4 144.7 151.6 139.1 122.9 151.6 139.1
0.0258 −0.04812 −0.05131 0.02947 −0.07821 −0.05626 −0.004896 −0.02835 0.06063 0.09274 0.06718 0.07018 0.03569 0.02461 0.027 0.04657 0.02461 0.027
−4.68 × 10−8 2.03 × 10−7 2.16 × 10−7 −5.254 × 10−8 3.284 × 10−7 2.366 × 10−7 7.682 × 10−8 4.202 × 10−7 −4.03 × 10−7 −6.293 × 10−7 −4.642 × 10−7 −4.788 × 10−7 −6.353 × 10−7 −4.35 × 10−7 −4.833 × 10−7 −8.43 × 10−7 −4.35 × 10−7 −4.833 × 10−7
Table 3 Equivalent height and fitting results. The type of liquid
Measurement methods
Distilled water
↑ ↓ ↑ ↓ ↑ ↓
Sesame oil Alcohol
A·h=
r (mm)
1 r + ar 2 + br 3 3
0.15
0.5
1.4
a
b
0.03112 0.03209 0.03006 0.03510 0.03062 0.03161
0.04626 0.1383 0.08396 0.1275 0.1245 0.1282
0.2995 0.3200 0.2613 0.2613 0.2575 0.2575
−0.7054 −0.1462 −0.4645 −0.1939 −0.2149 −0.1912
0.4429 0.051 0.2569 0.06363 0.07729 0.06038
Table 4 Validation experimental data and comparative analysis of experimental data. Distilled water Rise method Height h(mm) Average height h(mm) Numerical solution (◦ ) Equation and (◦ ) Relative error 0⁄000 Descent method Height h(mm) Average height h(mm) Numerical solution (◦ ) Equation and (◦ ) Relative error 0⁄000
Sesame oil
55.7 55.3 22.2568 Jurin 22.4358
56.1
54.2
Rayleigh 22.2261
80.4
Alcohol 28.2
27.0
Eq. (11) 22.3197
29.3 28.2 30.6432 Jurin 30.8529
19.8
20.1
Eq. (11) 30.6539
20.9 20.3 28.8132 Jurin 29.1444
Rayleigh 30.5692
Rayleigh 28.7217
Eq.11 28.7824
13.8
28.3
68.8
24.1
3.5
114.9
31.8
10.7
57.1 57.9 14.2894 Jurin 14.5873
59.0
57.5
32.5
30.4
21.5
Rayleigh 17.1785
Eq. (11) 17.2426
20.9 20.9 25.5590 Jurin 25.9456
20.2
Eq. (11) 14.2994
30.9 31.3 17.2225 Jurin 18.7783
Rayleigh 14.2675
Rayleigh 25.4741
Eq.11 25.5351
208.5
15.3
7.0
904.4
25.5
11.7
151.3
33.2
9.4
(1) For the same solid-liquid system, the fitting curve of the rise method exists median strip with that of descent method, what
is more, the width of the median strip rapid contraction in the wake of the decrease of the liquid surface tension. Meanwhile,
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the small median strip (corresponding to the small liquid surface tension) is almost always surrounded by the large median strip (corresponding to the large liquid surface tension); (2) For the different solid-liquid systems, the fitting curve that obtained by the descent method is more centralized distribution compared to that of the fitting curve generated by rise method. The above conclusions and phenomena could be used for reference in the field of solid-liquid static contact angle. Insufficiencies of this dissertation are still unable to explain the multi-valued of the contact angle and the corresponding rules. Furthermore, it is significant for theoretical and engineering significance [especially for liquid composite molding (LCM)] to study the solid-liquid contact angle of liquid mixture containing surfactant and active solid surface. Therefore, in the future work we will research the contact angle in the mixture of resin and curing agent which is the liquid of LCM, on the basis of the solid-liquid static contact angle model.Meanwhile, we will do some experiments to verify the practicability and rationality of the model. Besides, we will continue to explore the formation mechanism of contact angle and further analysis the multi-valued of the contact angle as well as the corresponding rules on the basis of the solid-liquid static contact angle model. Author contributions Jianjun Jiang, Qiang Guo and Junbiao Wang conceived and designed the experiments; Xuming Yao, and Chumeng Xu performed the experiments; Chao Deng analyzed the data; Linchao Zhou and Yang Su contributed reagents and materials; Qiang Guo and Bailing Wang wrote the paper. Conflict of interest The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgment This research is supported by “National Natural Science Foundation of China” (No. 51573148).
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