# A Predictive Model of Capillary Forces and Contact Diameters between Two Plates Based on Artificial Neural Network

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}) were employed to evaluate the prediction accuracy of the GA-ANN model, theoretical solution method of the Young–Laplace equation and simulation approach based on the minimum energy method. The results showed that the values of MSE of capillary force and contact diameter using GA-ANN were 10.3 and 0.0001, respectively. The values of R

^{2}were 0.9989 and 0.9977 for capillary force and contact diameter in regression analysis, respectively, demonstrating the accuracy of the proposed predictive model. The sensitivity analysis was conducted to investigate the influence of input parameters, including liquid volume and separation distance, on the capillary force and contact diameter. The liquid volume and separation distance played dominant roles in affecting the capillary force and contact diameter.

## 1. Introduction

## 2. ANN Model and GA Optimization

#### 2.1. ANN Model

_{1}and the output layer f

_{2}were defined as follows:

_{i}is the number of neurons in last layer and N

_{o}is that in next layer. In the present ANN structure: ${w}^{1}~U(-\sqrt{6/17},\text{}\sqrt{6/17})$ and ${w}^{2}~U(-\sqrt{6/15},\text{}\sqrt{6/15})$. The biases b

^{1}and b

^{2}were initially set to zero.

^{2}was used to check the performance of ANN [36]. The accuracy of the predictive outputs is measured by R

^{2}, which is written as

^{−7}, preventing the divisor from being 0. The value of α is set by the GA. If α is too large, the changes in updated parameters will be too great, resulting in the oscillation of loss. If α is too small, the learning process will be slow.

^{2}). ANN is trained by a complete dataset once it is termed an epoch. The number of epochs is defined as ep. If ep is too small, the ANN will not be well trained, leading to a large difference between predictive values and actual values. If ep is too large, the network will be overfitted with poor generalization performance, which means the network has good performance on the training dataset while bad performance on the testing dataset.

#### 2.2. Optimization of ANN Using GA

_{1}to U

_{2}, the corresponding decoding equation is

_{i}is the i-th value of the binary string. For n, ep and bs are integers; the final results of them need to round.

_{k}that the k-th individual is selected is expressed as

_{max}is the maximum in S, pop is the population size and ε is set to 10

^{−4}, preventing p

_{k}from being 0. When the MSE of one individual gets smaller, p

_{k}is larger. The number of selections is equal to pop to maintain the population, which is set to 30.

- Step 1
- Generating initial population. The initial population consisting of 20 chromosomes is randomly generated. Each chromosome represents a set of values of n, α, ep and bs. The present generation i is initially set to 0. The total generation number Gen is set to 100.
- Step 2
- Training ANN. i adds one. ANN is trained and MSE is calculated under the condition that each chromosome is represented.
- Step 3
- Optimization. The processes of selection, crossover and mutation are conducted as i is not equal to Gen+1.
- Step 4
- Generation of the best ANN parameters. Step 2 and step 3 are repeated until i is equal to Gen+1. The chromosome with minimum MSE is selected as the optimal set of values of n, α, ep and bs.

## 3. Experiments

#### 3.1. Experimental Setup

^{−5}g. The contact diameter was calculated by processing the liquid bridge images captured by the microscopes. The top silicon wafer and the bottom wafer were attached to the PMMA substrates separately. The top substrate was glued to a single-probe microgripper controlled by a four-axis precision stage with a resolution of 0.125 μm. The bottom substrate was placed on the analytical balance with a cubic foam pad. The instruments above were installed on a vibration isolation table to reduce the vibration transmission.

_{s}and θ

_{r}is the viscosity, surface tension, density, static angle and receding angle, respectively. θ

_{s}and θ

_{r}were calculated using a contact angle goniometer (JC2000D1, POWEREACH). The stretching speed U was 10 μm/s. All experiments were conducted at an ambient temperature of 20 ± 2°.

_{c}(defined as ${\lambda}_{\mathrm{C}}=\sqrt{\gamma /\rho g}$), the capillary number Ca (defined as $Ca=\mu U/\gamma $), and the Weber number We (defined as $We=\rho {U}^{2}L/\gamma $, where L is the characteristic length which was considered to be 1 mm) were calculated, as shown in Table 2. λ

_{c}was larger than the radius of the droplet used, indicating the surface tension was dominant, and the influence of gravity was negligible. Ca and We were much less than 1, indicating that the inertial force and viscous force could be neglected [39].

#### 3.2. Experimental Data

## 4. Results and Discussions

#### 4.1. ANN Training

_{std}is the standard deviation of datasets.

^{2}of capillary force and contact diameter for GA-ANN were 0.9993 and 0.9988, respectively. The R

^{2}value of capillary force for gANN was 0.9796, and that of contact diameter was 0.9824. Therefore, the GA-ANN model predictions are closer to the real values of the training dataset than the gANN model.

#### 4.2. Modeling

#### 4.2.1. Theoretical Model

_{1}, θ

_{2}are the contact angles on the top plate and bottom plate, respectively, H is the separation distance between plates and R

_{1}is the contact radius of the liquid bridge on the top plate, R

_{2}on the bottom plate. The symmetric axis of the liquid bridge is defined as Z-axis. The shape of the liquid bridge profile is meniscus due to the pressure difference between the inside liquid pressure (P

_{i}) and the outside air pressure (P

_{o}). The meniscus profile is axisymmetric and expressed by the coordinates (X, Z). A (X

_{A}, Z

_{A}) and B (X

_{B}, 0) are the coordinates of nodes where the profile terminates on the top and bottom plates.

_{L}and surface tension force F

_{S}. F

_{L}derives from the pressure difference, and the vertical component of surface tension force consists of F

_{S}. Thus, the capillary force is given as

_{A1}, ΔP

_{1}and ΔP

_{2}based on the boundary condition θ

_{1}. Two candidate θ

_{2}could be obtained. If the target θ

_{2}is within the range of the two candidates θ

_{2}, ΔP

_{1}and ΔP

_{2}will be adjusted to a fixed value based on a dichotomy search method, which will result in a candidate V

_{1}. Similarly, for a given X

_{A2}, a candidate V

_{2}can be obtained. If the target V is within the range of two candidates V, the profile of the liquid bridge would be further adjusted to reach the target V. X

_{B}is obtained in the solution process and compared with X

_{A}. If X

_{B}is not equal to X

_{A}, it indicates that the solution is non-stable; otherwise, the solution is the stable and correct solution.

#### 4.2.2. Simulation Model

_{sl}γ

_{sl}), the solid–gas (A

_{sg}γ

_{sg}) and the liquid–gas (A

_{lg}γ

_{lg}) interfacial energies, where A and γ are the area and surface tension of the interface, respectively. Thus, the total interfacial energy E of the liquid bridge system is expressed as

^{−6}. Figure 8c shows that a stable liquid bridge with minimal energy is established.

_{b}are the pressure difference, contact line length and the contact area of the liquid bridge on the bottom plate, respectively. Their values can be obtained from the evolved SE model, as well as the contact diameter.

#### 4.3. Comparison of GA-ANN, gANN, Simulation and Theoretical Solutions

_{r}because of contact angle hysteresis (CAH) [20]. In general, the predicted capillary force by the four methods exhibited good agreement with the experimental values on the 27 testing samples, as shown in Figure 9a. All four methods can be used to predict the capillary force with good accuracy.

^{2}gets closer to 1, the prediction gets more accurate. R

^{2}of capillary force for GA-ANN, SE and theoretical solutions are 0.9989, 0.9109 and 0.9114, respectively. In terms of contact diameter, R

^{2}of SE and theoretical solutions are 0.4389 and 0.4468, respectively, indicating SE and theoretical solution methods are not suitable for predicting contact diameter. Inversely, R

^{2}of contact diameter for GA-ANN is 0.9977, which shows the strong prediction ability of ANN.

#### 4.4. Effects of Input Parameters on Capillary Force and Contact Diameter

## 5. Conclusions

^{2}). In terms of GA-ANN, the MSE of the capillary force and contact diameter was 10.3 and 0.0001, respectively. The regression analysis showed that for GA-ANN, R

^{2}of the capillary force was 0.9989, and that of the contact diameter was 0.9977. The sensitivity analysis showed that the capillary force was subject to liquid volume while the contact diameter was subject to the separation distance. The developed ANN model enabled the precise prediction of the capillary force and contact diameter, providing a powerful tool for studying liquid bridges.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Symbol | Name |

n | Number of neurons in the hidden layer |

f | Activation function |

w | Weight |

b | Bias |

h | Output of hidden layer |

t | Predicted value of ANN |

y | Actual value of dataset |

R_{2} | Correlation coefficient |

α | Initial learning rate |

ep | Number of epochs |

bs | Number of samples passed to ANN at once |

p | Selected probability in the GA optimization process |

pop | Population size in GA |

Gen | Total generation number |

μ | Viscosity |

γ | Surface tension |

ρ | Density |

θ_{s} | Static contact angle |

θ_{r} | Receding contact angle |

U | Stretching speed |

λ_{c} | Capillary length |

L | Characteristic length of system |

V | Liquid volume |

H | Separation distance |

F | Capillary force |

D | Contact diameter |

GA-ANN | ANN optimized by GA |

gANN | ANN employing general parameters |

Abbreviation | Name |

CAH | Contact angle hysteresis |

ANN | Artificial neural network |

GA | Genetic algorithm |

MSE | Mean square error |

EG | Ethylene Glycol |

Ca | Capillary number |

We | Weber number |

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**Figure 5.**Experimental data: capillary force and contact diameter versus separation distance of different liquids: (

**a**) 50 wt% EG, (

**b**) EG and (

**c**) glycerol.

**Figure 8.**Surface evolution of the liquid bridge system: (

**a**) initial definition, (

**b**) deformation of evolution and (

**c**) finished evolution with a stable liquid bridge.

**Figure 11.**Capillary force and contact diameter versus input parameters: (

**a**) changing V with various H (θ = 36°, γ = 50 mN/m), (

**b**) changing H with various V (θ = 36°, γ = 50 mN/m).

Liquids | μ (Pa s) | γ (mN/m) | ρ (g/cm^{3}) | θ_{s} (°) | θ_{r} (°) |
---|---|---|---|---|---|

Ethylene glycol | 0.021 | 48.4 | 1.11 | 41.7 | 34.4 |

50 wt% ethylene glycol | 0.004 | 57 | 1.07 | 50.3 | 40.4 |

Glycerol | 0.243 | 63.4 | 1.26 | 42.6 | 33.3 |

Liquids | λ_{c} (mm) | Ca | We |
---|---|---|---|

Ethylene glycol | 2.109 | 4.34 × 10^{−6} | 5.28 × 10^{−4} |

50 wt% ethylene glycol | 2.331 | 7.02 × 10^{−7} | 2.67 × 10^{−3} |

Glycerol | 2.266 | 3.83 × 10^{−5} | 5.18 × 10^{−5} |

Models | MSE of Capillary Force | MSE of Contact Diameter | R^{2} of Capillary Force | R^{2} of Contact Diameter |
---|---|---|---|---|

GA-ANN | 10.3 | 0.0001 | 0.9989 | 0.9977 |

gANN | 244.706 | 0.0011 | 0.9748 | 0.9764 |

SE solutions | 865.883 | 0.0268 | 0.9109 | 0.4389 |

Theoretical solutions | 860.581 | 0.0265 | 0.9114 | 0.4468 |

Neuron Number | Weight | |||||
---|---|---|---|---|---|---|

Liquid Volume (x_{1}) | Separation Distance (x_{2}) | Contact Angle (x_{3}) | Surface Tension (x_{4}) | Capillary Force (t_{1}) | Contact Diameter (t_{2}) | |

1 | 0.272 | 2.657 | 3.642 | −1.608 | −0.34 | 1.218 |

2 | −0.234 | −3.530 | −0.280 | 1.944 | 0.073 | −1.300 |

3 | 0.702 | −1.779 | 1.869 | −1.682 | 0.450 | −0.802 |

4 | 1.818 | −1.430 | 0.163 | −1.043 | −0.801 | 0.219 |

5 | 1.769 | −1.570 | −0.290 | −0.597 | −0.060 | 0.425 |

6 | −3.487 | 1.240 | −0.307 | 0.159 | 2.647 | 0.817 |

7 | −2.176 | 1.248 | 0.141 | 0.214 | 2.013 | 1.052 |

8 | 2.566 | −4.556 | 0.490 | 1.668 | 0.039 | −0.464 |

9 | 1.481 | −1.650 | 0.520 | 1.775 | −0.811 | 0.865 |

10 | 0.593 | 2.880 | −0.978 | −1.805 | −0.639 | −0.288 |

11 | 0.978 | −4.842 | −0.817 | −0.700 | −0.909 | −0.123 |

12 | −1.150 | 2.577 | 0.790 | −0.118 | 0.489 | 0.409 |

13 | 0.614 | −0.858 | 3.061 | −1.881 | −0.120 | −0.636 |

Sum of products for capillary force | −17.970 | 10.159 | −0.076 | 2.239 | - | - |

Importance of capillary force | 1 | 2 | 4 | 3 | - | - |

Percentage | 59.0% | 33.3% | 2.4% | 7.3% | - | - |

Sum of products for contact diameter | −4.975 | 12.646 | 2.093 | −0.747 | - | - |

Importance of contact diameter | 2 | 1 | 3 | 4 | - | - |

Percentage | 24.3% | 61.8% | 10.2% | 3.7% | - | - |

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## Share and Cite

**MDPI and ACS Style**

Huang, C.; Fan, Z.; Fan, M.; Xu, Z.; Gao, J. A Predictive Model of Capillary Forces and Contact Diameters between Two Plates Based on Artificial Neural Network. *Micromachines* **2023**, *14*, 754.
https://doi.org/10.3390/mi14040754

**AMA Style**

Huang C, Fan Z, Fan M, Xu Z, Gao J. A Predictive Model of Capillary Forces and Contact Diameters between Two Plates Based on Artificial Neural Network. *Micromachines*. 2023; 14(4):754.
https://doi.org/10.3390/mi14040754

**Chicago/Turabian Style**

Huang, Congcong, Zenghua Fan, Ming Fan, Zhi Xu, and Jun Gao. 2023. "A Predictive Model of Capillary Forces and Contact Diameters between Two Plates Based on Artificial Neural Network" *Micromachines* 14, no. 4: 754.
https://doi.org/10.3390/mi14040754