Xiongheng Bian,
Haibo Huang‡
* and
Liguo Chen‡*
Robotics & Microsystem Center, Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215123, China. E-mail: hbhuang@suda.edu.cn; chenliguo@suda.edu.cn
First published on 27th May 2020
The formation of a liquid bridge in non-parallel plates is very common and the stability (whether or not it can move spontaneously) of such liquid bridges has been studied a lot for industry, e.g. in printing applications. It is generally considered that the liquid bridge stability is determined by Contact Angle (CA), Contact Angle Hysteresis (CAH), the position of the liquid bridge (represented as P) and the dihedral angle (θ) between non-parallel plates. The stability equation is θ = f(CA, CAH, P). Since P is a process quantity, which is difficult to determine, so it is also difficult to obtain the critical equation for the stability of the liquid bridge. In the previous study (J. Colloid Interface Sci., 2017, 492, 207–217), based on the fitting simulation results, the critical equation about CA, CAH and θ is obtained, as θ = f(CA, CAH). However, in some special cases, the results are still biased (e.g. the weak hydrophilic situation). In this paper, unlike simulation, we get the critical equation θ = f(CA, CAH) from a theoretical point of view. For the first time, by in-depth analysis of the process of liquid bridge formation, the theoretical calculation equation of P is obtained as P = f(CA, CAH, θ). And then, combining the equations θ = f(CA, CAH, P) and P = f(CA, CAH, θ), the theoretical equation is obtained. A lot of simulations and experiments were performed to verify our theoretical equation. Furthermore, comparing our equation with the previous equation, it was found that our equation is more consistent with the experimental results (error less than 0.2°). Finally, the importance of considering the liquid bridging process (the function of P) for stability analysis is illustrated by comparing the results with those not considered (the difference is more than 20% in some cases). The outputs of this paper provide in-depth theoretical support for the analysis and application of liquid bridges.
However, there are still three shortcomings in the current research. Firstly, since the critical equation θ = f(CA, CAH)16 is obtained by fitting the experimental and simulation results, and the influence of P is ignored, the equation works well in most cases, but there are still deviations in some special cases (for example, the weak hydrophilic situation). Secondly, since P is decided by the liquid bridge formation process, then, how to calculate the value of P by analyzing the process of liquid bridge formation is still unknown. Third, the importance of considering the process of liquid bridge formation is still unknown.
In this paper, based on equation θ = f(CA, CAH, P), we try to find a theoretical method to calculate P, so as to obtain a theoretical equation about θ = f(CA, CAH). First, from a geometric perspective, the critical equation about θ = f(CA, CAH, P) was given out. Then, through theoretical analysis, the influence of bridge formation process on its post-bridge position is studied, and the theoretical calculation equation P = f(CA, CAH, θ) is obtained, and verified by simulations. Furthermore, the theoretical curves, as θ = f(CA, CAH), for judging the stability of the liquid bridge are obtained for the first time. After that, the theoretical curves are compared with the existing fitting equation,16 and found it more consistent with the experimental results in the weak hydrophilic situation. Finally, the situation where the formation process needs to be considered are identified. The outputs of this paper provide a new theoretical support for the analysis and application of the liquid bridge.
(1) |
(2) |
• For the case of inward motion: eqn (1) and (2) can be transformed into the equations about R, as and . According to Laplace equation, |R1| < |R2| is necessary to make the spontaneous inward motion. Besides that, CAs also have to satisfy α1 = αa and α2 = αr (subscripts a and r represent the advancing and receding angles of the droplets on the surface, respectively). Therefore, it can be found that when and θ > αCAH, the requirements can be met. More precisely, the conditions of inward motion can be written as:
(3) |
• For the case of outward motion: to achieve the spontaneous outward motion, CAs have to satisfy α1 = αr and α2 = αa. Therefore, the conditions of θ is:
(4) |
(1) Only on the hydrophobic surface can the liquid bridge move outwards spontaneously.
(2) θ will be smaller and approaching to zero with the decreasing of LD/S. However, with the spontaneous outward motion, LD/S has to be smaller, so the liquid bridge cannot exit nonparallel plates completely.
In conclusion, the critical condition for the liquid bridge to move inward is , and the critical condition for the liquid bridge to move outward is . It shows that the conditions for stable liquid bridges should be associated with the coupling of CA, CAH, θ and LD/S. Among them, CA, CAH and θ are determined by the surface properties and structural properties of nonparallel plates. LD/S is determined by the formation process of the liquid bridge. Then, in next sections, the influence of liquid bridge formation on LD/S is discussed.
(5) |
Fig. 3 The model of liquid bridge forming process (a) before bridge forming process (b) at the beginning of bridge forming process (c) after bridge forming process (which decide the later stability). |
During the bridge forming process, with the droplet wetting along the up-plate and receding on the bottom plate, as shown in Fig. 3b and c, LDI goes down to LDBT, and SI goes up to SBT (here, subscripts I, BT and UP represent the initial state, bottom plate and upper plate respectively). Since the exist of CAH, LDBT is larger than LDUP and SBT is smaller than SUP, but these difference are very small and negligible when calculating the value of LD/S (the details is demonstrated in Section 4.1.2). Therefore, it can be found that . LD/S after bridge forming can be calculated by:
(6) |
In fact, the process of liquid bridge formation is also the process that liquid droplets wetting along the upper plate. Since α1 is always greater than α2 when in the hydrophilic nonparallel plates (according to eqn (1) and (2)), the inner contact line stays still while the outer contact line recedes, that is, SI ≈ SUP. In addition, in order to simplify the calculation, the liquid bridge can be seen as cylindrical in shape after its formation, and its average height is HB (as shown in Fig. 3c). Such height is similar to the height of the contact point (marked as HC, as shown in Fig. 3b) when the upper plate contacts the droplets and is also the height of the droplets in the initial state (marked as HD, as shown in Fig. 3a). Then, the average height and volume after the formation of the liquid bridge can be written as:
HB ≈ (SUP + LDUP/2)tanθ | (7) |
(8) |
Meanwhile, for the initial droplet, the average height and volume before the formation of the liquid bridge can be written as (the calculation process is offered in the ESI 3†):
HD = LDI/(2sinαa)(1 − cosαa) | (9) |
(10) |
According eqn (7)–(10), HB ≈ HD and SI ≈ SUP, eliminate the variables (HB, HD, V) in them and substitute into the equation , the expression of without considering CAH can be written as:
(11) |
To verify the relationship between Δ(LD/S), θ and αa in eqn (11), verification simulations by surface evolver are carried out. In these simulations, CAHs are ignored, αa has different values (varying from 60° to 80°) and θ also has different values (varying from 1° to 6°). According to the simulation results shown in Fig. 4b, the curves are approximately linear. The approximate lines with different αa are , and respectively. Compared with the theoretical results obtained from eqn (11), as shown in Fig. 4a (the approximate lines are , and respectively), the theoretical curves and the simulation curves are basically the same.
Fig. 4 Verify the relationship between Δ(LD/S), θ and αa in eqn (11). (a) The theoretical relationship between Δ(LD/S) and θ with different αa. (b) The simulation results between Δ(LD/S) and θ with different αa. |
In addition, more simulations were carried out to verify the correctness of the eqn (11) for a larger range of αa (varying from 30° to 170°). Simulation results are shown in Fig. 5. The solid line is the theoretical curve obtained by the eqn (11) and the points are the simulations results. The theory and simulation results are consistent. When αa is greater than 130°, Δ(LD/S) is approximately 0, so in this case, the bridge formation process does not need to be considered.
ΔLCAH ≈ Htan(αCAH/2) | (12) |
Fig. 6 Analysis about the CAH effect based on simulation (a) simulation result without CAH (b) simulation result with αCAH = 30° (c) theoretical analysis model about the effect of αCAH. |
According to the structural equation (H ≈ Sθ), it can be found . Combining the eqn (11), the equation of Δ(LD/S) considering CAH can be obtained as:
(13) |
The relationship between Δ(LD/S) and αCAH is further analyzed by eqn (13) and simulations (obtained by surface evolver). The theoretical curves in Fig. 7 are obtained by eqn (13) and the points in Fig. 7 are obtained through simulations. In these simulations, θ is set to 1° and αCAH has three different values (varying from 10° to 30°). The following two points can be obtained:
Fig. 7 Verify the relationship between Δ(LD/S) and αCAH in eqn (13) and demonstrate that the effect of αCAH on Δ(LD/S) is negligible. |
(1) The curves almost coincide with each other at different αCAH, indicating that the influence of αCAH on Δ(LD/S) is almost negligible.
(2) Considering the effect of αCAH, as shown in the local enlarged image, the greater αCAH, the greater Δ(LD/S). The simulation results are also consistent with the prediction of eqn (13) (the maximum error is less than 0.002). The correctness of the equation is verified.
In conclusion, the influence of CAH on Δ(LD/S) is so small and can be ignored, which explains why the theoretical model in the third section and eqn (6) and (7) can be used in this paper.
Fig. 8 Verify the theoretical curves with the simulations, experiment results and compare with previous study. |
In order to verify these theoretical curves, simulations and experiments were carried out. Based on the simulation model of the third section, the simulation results are also obtained in the Fig. 8, which are found to be basically consistent with the theoretical curves (error less than 0.5°). In the experimental part, polystyrene (PS, αa ≈ 89.5°, CAH ≈ 10.2°), poly(ethyl methacrylate) (PEMA, αa ≈ 79.4°, CAH ≈ 10.5°), silicon (αa ≈ 44.5°, CAH ≈ 21.2°) and glass (αa ≈ 75.6°, CAH ≈ 29.2°) surfaces are adopted. The CAs and CAHs were measured using the sessile drop method and obtained by programming based on OpenCV. Measurements was repeated for five times for each surface. The experiments were carried out on the platform built in our previous study.20 The bottom plate was placed horizontally, and then the downward movement of the upper plate was controlled by a micro motor. As soon as the upper plate came into contact with droplets at a low speed (0.01 mm s−1), it stopped moving immediately. The critical angle of the stable liquid bridge was obtained by repeatedly locating θ when the liquid bridge was stable and unstable, and all the experiments were repeated five times and averaged for each case. The experimental results are shown in Fig. 8 (the asterisk marked), which also in line with the theoretical curves (the error was less than 0.5°). Comparing with the existing simulation fitting curve obtained by Alidad (the dotted line in Fig. 8),16 the two have a good fit (the difference is less than 0.2°) in most situations, but slightly different when αa is larger than 60°. It can be found that the critical angle of PS surface is about 9.2° by experiment (the asterisk marked), 8.8° by our theoretical curve (the solid line) and 7.5° by the previous study (the dotted line). It means that when αa was greater than 60°, our curves are more consistent with the experimental results and has a better performance.
In conclusion, comparing our method with the previous methods, Table 1 can be obtained:
As shown in the Fig. 9, the theoretical curves (consider the process of liquid bridge formation) are obtained with different CAHs (the dotted line) by using the eqn (4), (6) and (13). Compared with the case that the formation process of liquid bridge is not considered (without consider the eqn (13)), it can be found that the difference between these two curves increases first and then decreases as αa increases. If αa is greater than 110°, the influence of the formation process of the liquid bridge on the stability can be ignored. But according to the research in Section 2, to achieve the inward motion, αa also should smaller than (at the right part the red dotted line in Fig. 9). In this region, the results with and without considering the process of bridge formation are quite different (in some cases, the difference is more than 20%). Therefore, the process of bridge formation should be considered for inwards motion.
Fig. 9 Comparisons of results between considering and without considering bridge forming by theoretical analysis. |
Compared with previous studies,10,15,21–23 the paper provided the following new research progress.
(a) This research conclusion points out that the formation process of liquid bridge has influence on its later stability for the first time.
(b) The theoretical equation about the influence of formation process on bridge position is presented as P = f(CA, CAH, θ), and then, the theoretical curves for the stability determination of the liquid bridge are obtained as. The theoretical curves are verified by simulations and experiments.
(c) For inward motion, the process of bridge formation should be considered in all the cases.
Footnotes |
† Electronic supplementary information (ESI) available. See DOI: 10.1039/d0ra03438j |
‡ Present addresses: Robotics & Microsystem Center & Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University Suzhou 215123, China. |
This journal is © The Royal Society of Chemistry 2020 |