Application of Thermodynamic Methods in Enhancing the Antibacterial Performance of Textile Materials

The antibacterial performance of textile materials refers to the ability of materials to effectively inhibit or kill harmful microorganisms, such as bacteria and fungi, when in contact with them. This performance not only reduces the growth of bacteria on the surface of textiles and prevents odor, but also lowers the risk of skin diseases and allergic reactions caused by bacterial infections. In the application of antibacterial textiles, the strength of the antibacterial performance is directly related to the health and comfort of the wearer. Therefore, improving the antibacterial ability of textiles has become an important topic in textile material research. The evaluation of antibacterial performance is usually based on comprehensive measurements of antibacterial rate, antibacterial durability, and inhibition effects against different microorganisms.

The main approaches to enhancing the antibacterial performance of textile materials include both physical and chemical methods. Physical methods mainly involve using nanotechnology to embed nanoparticles into textile materials. These nanoparticles have strong antibacterial effects and can effectively inhibit bacterial growth through mechanisms such as releasing reactive oxygen species or damaging the cell membrane. Chemical methods involve chemically treating textile materials, such as coating antibacterial agents or copolymerizing antibacterial functional groups into the fibers, to enhance the antibacterial performance of the textiles. These antibacterial agents can react with the bacterial cell wall, preventing its reproduction or directly killing the bacteria. Moreover, with the increasing awareness of environmental protection and health, the use of organic and bio-based antibacterial agents has gradually increased in recent years. These natural or biodegradable antibacterial substances are not only safe and environmentally friendly, but also reduce the negative impact on the environment. Figure 1 shows a schematic diagram of the microstructure of textile materials with different antibacterial agents introduced.

Temperature rise has a significant impact on the antibacterial performance of textile materials, especially in sports, outdoor, or other high-temperature environments. The textile materials may undergo changes in their surface structure or functional groups due to temperature rise, which in turn affects the release of antibacterial agents and their antibacterial effects. Therefore, by incorporating the temperature rise factor and analyzing the antibacterial behavior of textile materials under different temperature conditions using thermodynamic methods, the antibacterial effects in actual usage scenarios can be more accurately simulated. In this study, we first establish a multivariable linear regression model, combining temperature rise with other factors to quantitatively analyze the changing trends of antibacterial performance under different conditions. Relevant research can reveal the intrinsic relationship between temperature and material antibacterial performance, providing a theoretical basis for optimizing the antibacterial performance of textile materials.

Specifically, we take the maximum temperature rise and average temperature rise as the dependent variables. The relationship between these two variables and the content of antibacterial agents and the structure of the textile is quantitatively described through the multivariable linear regression model. Through the regression model, we can clearly identify how temperature changes interact with variations in antibacterial agent content and material structure, leading to the enhancement or decline of antibacterial performance. In thermodynamic analysis, temperature changes affect the molecular motion state and reaction activity. Therefore, the maximum temperature rise and average temperature rise in the model are key factors for describing the impact of temperature rise on antibacterial performance. The expression of the multivariable linear regression model constructed is as follows:

$b={{\varepsilon }_{0}}+{{\alpha }_{1}}{{a}_{1}}+{{\alpha }_{2}}{{a}_{2}}+\cdots +{{\alpha }_{o}}{{a}_{o}}+\gamma $         (1)

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Through the construction of the multivariable linear regression model, the changes in the maximum temperature rise and the average temperature rise can be explained in two parts. The relationship between these two variables and the antibacterial agent content and material structure is quantitatively described through the multivariable linear regression model. First, the temperature rise change caused by antibacterial agent content and structural changes is expressed as b=α₀+α₁a₁+α₂a₂+...+α_oa_o, which reflects the linear response of antibacterial performance during temperature rise, illustrating the thermodynamic interaction between the antibacterial agent and the textile material. For example, the release of the antibacterial agent may be positively correlated with temperature, where the temperature rise leads to a rapid release of the antibacterial agent, thereby enhancing antibacterial performance. Second, the error term γ in the regression model represents fluctuations in antibacterial performance caused by other unknown factors or random environmental changes. These factors may include experimental errors, material inhomogeneities, etc. By estimating the regression coefficients α₀, α₁, α₂,…, we can quantify the contributions of temperature rise, antibacterial agent content, and material structure to antibacterial performance, further revealing the complex relationship between temperature rise and antibacterial performance in the thermodynamic process. Taking the conditional expectation of both sides, we get the multivariable linear regression equation:

$R\left( b \right)={{\alpha }_{0}}+{{\alpha }_{1}}{{a}_{1}}+{{\alpha }_{2}}{{a}_{2}}+\cdots +{{\alpha }_{o}}{{a}_{o}}$    (2)

In order to accurately estimate the unknown parameters in the multivariable linear regression equation, statistical tests of the model are necessary. The goodness of fit is typically evaluated using the coefficient of determination R², which indicates the proportion of variation explained by the regression model relative to the total variation. In the constructed model, the relationship between temperature rise and antibacterial performance is influenced by multiple factors, including the temperature rise being related not only to the release rate of antibacterial agents but also to the material structure changes induced by temperature. Therefore, the goodness-of-fit test of the regression model can help determine the explanatory power of temperature rise on antibacterial performance, ensuring that the constructed model accurately reflects the thermodynamic dynamics during the temperature rise process. A high R² value indicates that the impact of temperature rise on antibacterial performance is well represented in the regression model. In contrast, a low R² value may suggest that the model is not yet refined and has failed to capture key thermodynamic factors effectively. Suppose the degrees of freedom for SSE and SST are denoted by v-o-1 and v-1, respectively. The mean square error is expressed as SSE/v-o-1, with the specific definition given by:

$\overline{{{R}^{2}}}=1-\frac{\frac{SSE}{v-o-1}}{\frac{SST}{v-1}}$ (3)

The significance test of the regression equation is an important step in verifying whether the explanatory variables have a significant effect on the dependent variable, i.e., temperature rise. We conduct the significance tests using the F-test and t-test. The F-test is used to determine whether the entire regression equation is significant, i.e., whether variables such as antibacterial agent content and material structure jointly influence the temperature rise. If the p-value of the F-test is smaller than the significance level, the regression equation is significant, indicating that there is a significant linear relationship between antibacterial agent content, material structure changes, and temperature rise. The t-test checks whether each regression coefficient is significantly different from zero, i.e., whether each independent variable significantly contributes to the temperature rise. In the context of thermodynamics, the significance tests help researchers identify which factors, such as the concentration of the antibacterial agent and the type of material structure, significantly influence antibacterial performance under different temperature rise conditions, thereby further optimizing the design of textile materials. Let the number of explanatory variables in the multivariable linear regression equation be denoted by O, and the mathematical definition of the F-test is:

$F=\frac{{\sum\limits_{u=1}^{v}{{{\left( {{b}_{u}}-\bar{b} \right)}^{2}}}}/{O}\;}{{\sum\limits_{u=1}^{v}{{{\left( {{b}_{u}}-\bar{b} \right)}^{2}}}}/{\left( v-O-1 \right)}\;}$    (4)

The relationship between the F-statistic and R² is given by:

$F=\frac{{{{R}^{2}}}/{O}\;}{{\left( 1-{{R}^{2}} \right)}/{\left( v-O-1 \right)}\;}$     (5)

Residual analysis examines the errors in the regression model to ensure that the assumptions of the regression model are met. In thermodynamic regression models, residual analysis is particularly important because the impact of temperature on antibacterial performance may be influenced by multiple factors such as experimental errors and material inhomogeneities, which are reflected in the residuals. The residuals should satisfy the assumptions of being independently and identically distributed, with a mean of zero and constant variance. If the residuals exhibit autocorrelation or heteroscedasticity, it indicates that the regression model may be biased and requires further improvement. For instance, if the residuals exhibit time-series correlation, it may suggest that there are some unconsidered temperature effects or thermal reaction processes during temperature rise, which causes the relationship between temperature rise and antibacterial performance to be incompletely captured. In this study, we use the DW test method to infer whether autocorrelation exists in the sample sequence, with the formula given by:

$FQ=\frac{\sum\limits_{s=2}^{v}{{{\left( {{r}_{s}}-{{r}_{s-1}} \right)}^{2}}}}{\sum\limits_{s=2}^{v}{{{r}_{s}}^{2}}}$     (6)

Multicollinearity analysis is a key step in testing whether there is high correlation between the explanatory variables in the regression equation. If antibacterial agent content and material structure are highly correlated under certain conditions, it may lead to multicollinearity, that is, an excessively strong linear dependence between the explanatory variables, which could affect the estimation and explanatory power of the regression model parameters. This is because material temperature rise is influenced by multiple factors, including the concentration of antibacterial agents, material structure, and temperature changes. If there is a strong correlation between these factors, such as a high concentration of antibacterial agent possibly causing changes in material structure, certain regression coefficients in the regression equation may become unstable, affecting the reliability of the model. In this study, tolerance and variance inflation factors (VIF) are used for specific analysis, as shown in the following formulas:

$To{{l}_{u}}=1-R_{u}^{2}$ (7)

$VI{{F}_{u}}=\frac{1}{1-R_{u}^{2}}$    (8)

From a thermodynamic perspective, the heating process is not just a simple temperature change, but involves a series of complex thermal effects, such as heat conduction inside and outside the material, molecular dynamics, and chemical reactions. High temperatures can increase the rate of molecular movement, which accelerates the diffusion of antibacterial agent molecules to the surface of the textile material, thereby improving its antibacterial activity. At the same time, the increase in temperature can also strengthen the interaction between the antibacterial agent and the textile material, potentially promoting the dissociation, release, or binding of the antibacterial agent to bacterial cells, thereby enhancing antibacterial effectiveness. However, these changes are usually nonlinear and are affected by factors such as the thermal stability of the material, the type of antibacterial agent, and its distribution within the textile material. Therefore, the relationship between temperature rise and antibacterial performance needs to be revealed through quantitative analysis, and correlation analysis provides an effective method for this.

In the specific analysis process, the first step is to calculate the correlation coefficients of the samples to evaluate the linear relationship between temperature rise and antibacterial performance. Under high-temperature conditions, temperature rise changes can have a direct or indirect impact on antibacterial performance, especially since high temperatures may accelerate the release of antibacterial agents or intensify antibacterial reactions. Therefore, the influence of temperature on antibacterial agent behavior must be considered when constructing the correlation model. By performing statistical analysis on temperature rise values and antibacterial performance data from multiple samples, the Pearson correlation coefficient obtained can quantify the strength of the linear relationship between temperature rise and antibacterial performance. A high correlation coefficient indicates a significant positive correlation between temperature rise and antibacterial effect, meaning that increased temperature can effectively enhance antibacterial performance. Conversely, a low correlation coefficient may suggest that the impact of temperature is small or that antibacterial performance is dominated by other factors such as antibacterial agent concentration or material structure. The formula for calculating the correlation coefficient is as follows:

$e=\frac{\sum\limits_{u=1}^{v}{\left( {{a}_{u}}-\bar{a} \right)\left( {{b}_{u}}-\bar{b} \right)}}{\sqrt{\sum\limits_{u=1}^{v}{{{\left( {{a}_{u}}-\bar{a} \right)}^{2}}}\sum\limits_{u=1}^{v}{{{\left( {{b}_{u}}-\bar{b} \right)}^{2}}}}}$    (9)

$e=1-\frac{6\sum\limits_{u=1}^{v}{F_{u}^{2}}}{v\left( {{v}^{2}}-1 \right)}$       (10)

$\pi =\left( I-N \right)\frac{2}{v\left( v-1 \right)}$  (11)

In addition to calculating the correlation coefficient, another critical step is determining whether the correlation is significant. Under high-temperature conditions, the antibacterial performance of textile materials may be affected by more complex thermal effects, such as pyrolysis reactions, molecular structural changes, and material thermal stability. Thus, the relationship between temperature rise and antibacterial performance may not be purely linear. To test whether the correlation is significant, researchers need to perform hypothesis testing, typically using the t-test to determine whether the correlation coefficient significantly differs from zero. If the test results show that the p-value is smaller than the significance level (typically set at 0.05), it can be concluded that there is a significant linear relationship between temperature rise and antibacterial performance. Under high-temperature conditions, the effect of temperature on antibacterial performance becomes more pronounced. If the significance test passes, it indicates that temperature rise is indeed an important factor influencing antibacterial performance. On the contrary, if the test does not pass, it may suggest that the temperature rise effect is insufficient to cause significant changes in antibacterial performance, or that improvements in antibacterial performance depend more on other thermodynamic factors, such as the initial concentration of the antibacterial agent or the thermal response of the material surface.

Furthermore, the correlation analysis conducted under high-temperature conditions should also take into account the interactions between various factors. In practical studies of antibacterial performance of textile materials, temperature changes, antibacterial agent concentration, material structural types, and environmental factors may interact in complex ways, with these factors collectively determining the antibacterial performance. For example, under high-temperature conditions, the antibacterial agent may release rapidly within a short time, whereas at lower temperatures, it may release more slowly. This behavior is closely related to the material's thermal diffusion characteristics and the thermal stability of the antibacterial agent. To fully understand the relationship between temperature rise and antibacterial performance, in addition to single-factor correlation analysis, multi-factor analysis or multivariable regression analysis is required. This approach can simultaneously consider the interactions between temperature rise and other factors, revealing more complex thermodynamic effects. By analyzing the contributions of different factors to antibacterial performance, we can better understand the release mechanism of the antibacterial agent and its behavior in different materials under high-temperature conditions, thereby providing strong data support for the design and optimization of antibacterial textiles.

From the basic performance data in Table 4, we can observe the differences in antibacterial agent content, weight, air permeability, water vapor permeability, and maximum temperature rise values across different samples. Figure 3 shows the linear correlation between maximum temperature rise and antibacterial agent content (a), weight (b), air permeability (c), and water vapor permeability (d). From the figure, it is clear that the antibacterial agent content ranges from 24% to 84%. Among the samples, the one with a polyester-cotton ratio of 24/74 has the lowest antibacterial agent content (24%), while the sample with a ratio of 84/14 has the highest content (84%). There are also variations in weight and air permeability. For example, the polyester-cotton ratio of 24/74 has a weight of 223.5 g/m² and an air permeability of 612.56 mm/s, while the sample with an 84/14 ratio has a weight of 238.6 g/m² and an air permeability of 1325.23 mm/s. This suggests that higher antibacterial agent content may be associated with higher weight and better air permeability. The water vapor permeability varies widely, from a minimum of 135.26 g/(m²·h) to a maximum of 178.25 g/(m²·h), which may be closely related to the fiber structure, moisture absorption performance, and fabric type. In terms of the maximum temperature rise, the values range from 2.19℃ to 5.61℃. Among the samples, those with higher antibacterial agent content, such as nonwoven fabrics (41% and 84%), exhibit higher temperature rises (5.61℃ and 2.7℃, respectively), while the woven and knitted fabric samples with lower antibacterial agent content show lower maximum temperature rise values.

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Figure 4. Bar chart of antibacterial test results for textile materials

From the experimental results, it can be concluded that the antibacterial agent content has a clear impact on the antibacterial performance of textile materials. The antibacterial effect against Staphylococcus aureus and Escherichia coli significantly improves as the antibacterial agent content increases, particularly when the content reaches 85%, where the antibacterial effect approaches saturation, with inhibition rates of 88% and 94%, respectively. This indicates that appropriately increasing the antibacterial agent content can effectively improve the antibacterial performance of textile materials. However, it is important to note that although increasing the antibacterial agent content improves the antibacterial performance, the improvement in antibacterial effect becomes less significant with further increases in content. Therefore, in actual production, it is essential to choose an appropriate antibacterial agent content to balance cost and performance. The results of this study provide experimental evidence for optimizing antibacterial functional textiles and suggest that when developing antibacterial materials, attention should be paid to the relationship between antibacterial agent content and performance to achieve the best antibacterial effect.