Permeability Model of Liquid Microcapsule Based on Multiple Linear Regression Method

Abstract

The release rate of liquid core material from microcapsules is crucial for the surface properties of self-protective metal/liquid microcapsule composite plating coating. However, there is no method to accurately predict the release rate of microcapsules. In this paper, the permeability experiments of different shell membranes and core materials were carried out using the weight loss method, and the permeability model of liquid microcapsules was studied based on a multiple linear regression method. The results show that three-variable mathematical model C, including membrane porosity, the viscosity of core material, and membrane thickness is suitable to describe permeability, and the membrane thickness is the most significant influence factor. Additionally, the accuracy of model C was experimentally verified, and the error of the permeation rate is about 2.06% between predictive and experimental values.

1. Introduction

Nowadays, composite materials containing microcapsules have attracted more and more attention due to their magical self-repairing properties [1,2,3,4,5,6]. The self-healing microcapsule as a new material has the characteristics of easy dispersion in the matrix and intelligent self-healing. The repair principle is that repaired material is stored inside microcapsules and enclosed with the capsule shell membrane. The internal core material is released gradually or quickly to modify the properties of materials in service [7]. In our previous research, the functional Cu-based and Ni-based composite coatings containing liquid microcapsules were prepared by electrolytic co-deposition [8,9,10,11]. The process of core material on the coating surface is divided into two steps, as seen in Figure 1: (i) the core material of microcapsules diffuses into the matrix through the shell material; (ii) the released core material from the microcapsules diffuses to the coating surface. The diffusion rates of core material in the shell and matrix are crucial to realize controllable release. If the release rule is mastered, the controlled release of microcapsule will become possible. Unfortunately, there is no method to accurately predict the release properties of microcapsules. Herein, the first step was mainly discussed in this research.

As is known, the release of microcapsules is closely related to many factors, such as the porosity of shells, viscosity of core material, and shell thickness. Yuan [12] studied the chloride ion transport model in a chloride-activated self-healing concrete system. Zhu [13] synthesized the self-healing microcapsule using in-situ polymerization with dicyclopentadiene.

DCPD served as the capsule core, and urea-formaldehyde resin was the shell material. Moreover, the author built the quantitative structure–property relationship (QSPR) model using the SPSS statistical analysis tool software. In our previous research, hydrophobic agent, organosilicon resin, and lubricant were usually used as the liquid core materials, and polyvinyl alcohol (PVA), gelatin, or methylcellulose (MC) as the shell materials of the microcapsule [14].

In this paper, the shell and core materials above are used as research objects, and a series of permeability experiments were carried out under ambient temperatures and moistures. We attempt to seek a statistical rule about permeability without considering the change in environmental factors. The multiple linear regression (MLR) method is an effective tool for quick predictions [15,16,17]. Here, it is first introduced to constitute a suitable permeability model based on the permeability data obtained in experiments. This paper provides a new method to analyze experimental data and is favorable to the establishment of mathematical models.

2. Materials and Experiments

2.1. Preparation of Shell Membranes of Microcapsule

We prepared 2 wt.% methylcellulose (MC), polyvinyl alcohol (PVA), and gelatin aqueous solutions, respectively, which were purchased from YiLi fine chemicals Co., Ltd. of Beijing, China. After deformation, the calculated solutions were poured on the plastic mold with a certain area and dried at room temperature for 24 h in a vacuum drying oven. Hence, the films with 5 μm, 15 μm, and 30 μm thickness were obtained.

2.2. Characterization of Membranes

The cross-section photographs of shell membranes were characterized using scanning electron microscopy (SEM, HITACHI S-530, 20 kV, Tokyo, Japan). Moreover, the cross-section fractal dimensions of membranes were used to characterize their porosity.

2.3. Permeability of Shell Membranes

The core materials permeability of the films was measured by weighing. BH-102 hydrophobic agent and L-MH46 lubricating oil were adopted as the core materials of the microcapsules, and their measured viscosities were 15 m²/s and 32 m²/s, respectively. The orifices (Φ12 mm) of the cylindrical tube were closed by different shell membranes, which were filled with 3 mL core materials. These tubes were placed downward in air at 25 °C and weighed every 2 h. The residual core material around the orifice was cleaned using a hexane solution before weighing. Then the weight loss of the total tube was calculated, and each test was repeated at least three times to ensure reproducibility. The permeability data were analyzed using the multiple linear regression method with the Statistical Package for Social Science 16.0.2 (SPSS) software.

3. Results and Discussion

3.1. Cross-Section Images and Fractal Dimensions D_f of Membranes

The SEM graphs of the cross-section of the shell membranes are shown in Figure 2. As observed, the cross-sectional samples have different porosity characterized by the cross-section fractal dimensions D_f. In this study, the area dimension D_f can be determined by the box-counting method [18,19] which is based on the membrane image analysis of a sufficiently large section. SEM images of the shell membranes are in the JEPG format with true color and should be transferred to the gray-scale formation with Adobe Photo Version 24.1 software. Before analyzing an image, a threshold has to be determined in order to distinguish pores from the background obtaining a binary image. Then the cross-section fractal dimensions D_f are obtained using the Matlab 9.12 program according to the box-counting method. Simultaneously, the pore areas of films are also calculated.

Table 1. Calculated cross-section fractal dimension D_f and pore area of membranes.

Shell Materials	Pore Area (%)	Cross-Section Fractal Dimension Df
PVA film	0.0019	2.00
Gelatin film	5.3120	2.29
MC film	16.7062	2.51

displays the fractal dimensions D_f and pore areas of these three membranes.

3.2. Permeability Analysis of Shell Membranes

Figure 3 depicts the permeability properties of three shell membranes with different thicknesses of the membranes under different systems. As seen, the mass loss all increases linearly with time. The slope of the fitting line represents the permeation rate of the film, which is almost a fixed value. These values of permeation rates are listed in

Table 2. Permeation rates of different thickness membranes under different conditions.

Fractal Dimension (Df)	Viscosity of Core (m2/s)	Membrane Thickness (μm)	Permeation Rate (mg/cm2·h)
MC (Df = 2.51)	15	5	40
	15	15	8
	15	30	0.75
	32	5	23
	32	15	3.2
	32	30	0.14
Gelatin (Df = 2.29)	15	5	32
	15	15	6.4
	15	30	0.36
	32	5	16
	32	15	2.4
	32	30	0.06
PVA (Df = 2.00)	15	5	26
	15	15	2.53
	15	30	0.0053
	32	5	5
	32	15	0.2
	32	30	0.0014

.

3.3. Multiple Linear Regression Theory

Multiple linear regression analysis is an effective tool to confirm the relationship between the dependent variable and several independent variables. The attained expression is called a regression equation or mathematical model, which can predict the value of the dependent variable. The MLR method requires a certain linear relationship between the dependent variable and explanatory variables.

Usually, the MLR model is typically expressed as [20]

$$Y_i={\beta }_0+{\displaystyle \sum _{i=1}^p{\beta }_i{x}_i}$$

(1)

where $Y_i$ represents the observed value of the dependent variable, $x_i$ and ${\beta }_i$ denote the explanatory variables and the regression coefficients for the constant and the variables, respectively. $p$ is the number of explanatory variables.

3.4. Model Selection and Assessment

We try to present the permeability mathematical model according to the 18 sets of experimental data in Table 2. The three measured explanatory variables are used in this work, including membrane porosity (fractal dimension D_f), viscosity of core material (VCM), and membrane thickness (MT). The variables should be properly transformed to satisfy the requirement of the MLR method when it is necessary. After transformations, three variables are finally identified to constitute the models: D_f, 1/VCM, 1/(sqrt (MT)).

Three models are obtained after MRL, and the statistical results of the models for the experimental data are listed in

Table 3. The statistic results of the models for the complete data.

Model	Variables	R	Unstandardized Coefficients B	Standardized Coefficients	F	Sig.	Durbin-Watson
A	Constant 1/(sqrt (MT))	0.839	0.360 584.476	0.839	38.184	0.000
B	Constant 1/VCM 1/(sqrt (MT))	0.891	−9.784 207.197 584.476	0.299 0.839	28.951	0.000
C	Constant Df 1/VCM 1/(sqrt (MT))	0.920	−40.414 13.513 207.197 584.476	0.230 0.299 0.839	25.879	0.000	2.180

. The correlation coefficient R shows the serial correlations between the dependent variable and explanatory variables. As observed, a significant serial correlation is obtained for the models based on the whole data set (n = 18). Comparing these three models, the correlation coefficient R₃ = 0.920 in model C shows the optimal serial correlation. Moreover, the Durbin–Watson value (DW) is 2.180, which is a statistic to test the serial correlation with a first-order autoregression. It illustrates the linear independence among the explanatory variables. Consequently, D_f, 1/VCM, and 1/(sqrt (MT)) in this study are linearly independent.

Table 3 shows the ANOVA analysis results of models after MLR. It can be seen that the F values are 38.184 (Sig. (p) = 0.000 < 0.05) for model A, 28.951 (p = 0.000000 < 0.05) for model B and 25.879 (p = 0.000000 < 0.05) for model C. All three models show a significant linear relationship. Based on the discussion above, it can be considered that model C is more suitable for describing the relationship between the dependent variable and independent variables. It is also concluded from standardized coefficients that the influence of 1/(sqrt (MT)) on the dependent variable is most significant. Hence, the optimal permeability mathematical model is obtained as follows

$$Y=-40.4+13.5·D_f+\frac{207.2}{VCM}+\frac{584.5}{{MT}^2}$$

(2)

Three models are investigated to validate the rationality of the model we built. The measured values Y and the prediction values calculated using models A, B, and C are shown in Figure 4. The results show certain differences among the prediction values of the three models. Especially, the predictions made by model C are more consistent with the actually measured values. This implies that model C predicts the permeation rate Y more accurately than models A and B.

A further model evaluation based on indices is implemented, but only part of the results are presented in

Table 4. Comparison of the predictive capacity of model A, model B, and model C (n = 18).

Model	R2	$\overline{\mathit{O}}$	$\overline{\mathit{P}}$	SP	MAE	RMSE
A	0.705	9.22	9.25	7.0674	3.96	6.66
B	0.794	9.22	9.24	6.0929	4.19	5.56
C	0.847	9.22	9.21	5.4344	1.45	2.72

for brevity. The second block of Table 4 represents the goodness of fit of the three models, which are the average of the observed and predicted values, respectively. S_P denotes the standard errors of the predicted values. MAE (mean absolute error) and RMSE (root mean squared error) are respectively defined as [21]

$$MAE={\displaystyle \sum _{i=1}^N\left|P_i-O_i\right|}/N$$

(3)

$$RMSE={\left[{\displaystyle \sum _{i=1}^N{\left(P_i-O_i\right)}^2/N}\right]}^{1/2}$$

(4)

where $P_i$ and $O_i$($i=1,2,\dots \dots ,N$) are predictions and observations, respectively. The results indicate that model C fits better than models A and B with a higher R² and lower standard error S_P, which might be attributed to the slight change in the actual permeation rate data. On the other hand, model C has a more satisfactory predictive capacity with lower levels of MAE and RMSE. These results are in accordance with that of Figure 4. Therefore, it is reasonable to deduce that model C (Equation (2)) is more suitable for predicting the permeation rate of shell membranes of microcapsules under ambient temperature and moisture.

In the course of linear regression, we hypothesize the residual follows a normal distribution. Figure 5 depicts the normal P-P plot of the regression standardized residual. All the scattered points show a linear relationship which illustrates the dependent variable indeed coming from a normal population.

A new verification test was carried out at ambient temperatures and moistures to validate the good predictive capacity of model C. The MC membrane with 21 μm thickness and organosilicon resin is adopted as the experimental membrane and permeability media, respectively. The measured viscosity of organosilicon resin under room temperature is about 36 m²/s. The true permeability curve and fitted line of the MC membrane-organosilicon resin system are shown in Figure 6. As observed, the permeation quantity increases linearly with time. The fitted line depicts the high correlation coefficient of 0.997, and its slope about 0.559 is calculated.

Meanwhile, the predictive permeation rate of 0.566 is also calculated by Equation (2). Comparing the true and predictive values, it can be found that the error is about 2.06%. This result illuminates the predictive capability and the reliability of model C.

Further study will focus on the permeability process of released core material through the metal coatings. It would also be interesting to build another complex mathematical model based on the coating thickness and structure data. It will be beneficial to the implementation of controlled release.

4. Conclusions

For studying the permeability rule of liquid microcapsules within composite plating coatings, the permeability experimental data of different core materials and shell membranes were obtained by the weight loss method. The permeability models of liquid microcapsules are established and verified. The results could be summarized as follows:

(1)

Cross-section fractal dimensions of D_f of PVA, gelatin, and MC membrane are 2.00, 2.29, and 2.51, respectively, according to the box-counting method.

(2)

Multiple linear regression is first used to constitute the permeability model of liquid microcapsules used for electrolytic co-deposition.

(3)

Three-variable mathematical models, including membrane porosity, the viscosity of core material, and membrane thickness, are suitable to describe permeability, and the membrane thickness is the most significant influencing factor.

(4)

The accuracy of model C was experimentally verified, and the error of permeation rate is about 2.06% between predictive and experimental values.