## 4 Mathematical Model
The selection of appropriate mathematical functions for modeling the intricate relationship between the dependent and independent variables holds paramount significance in data analysis. This becomes especially crucial when confronted with a scatter plot manifesting a nonlinear form: notably, characterized by a discernible parabolic shape in the data points. In delving further into the rationale for this choice, the decision to test both the quadratic polynomial function and the exponential function is based on their distinct abilities to capture and articulate the underlying dynamics between the property of interest and the volume fraction. Quadratic polynomial function:  
\[f_{1}(x)=a_{0}+a_{1}x+a_{2}x^{2} \tag{36}\]  
where \(a_{0}\), \(a_{1}\) and \(a_{2}\) are adjustable coefficients. The initial function selected for our analysis is the quadratic polynomial function. This type of function is frequently utilized to characterize physical phenomena demonstrating nonlinear responses. It typically displays an inverted U-shaped curve featuring a vertex at a specific location along the independent variable axis. The quadratic polynomial function is particularly advantageous in modeling scenarios wherein the phenomenon exhibits a peak response at a certain point, followed by a subsequent decrease. Its capability to accurately represent such characteristics makes it an ideal choice for our study's requirements. Exponential function:  
\[f_{2}(x)=\alpha\mathrm{e}^{\beta x} \tag{37}\]  
where \(\alpha\) and \(\beta\) are adjustable coefficients and x is the independent variable. The second function selected for our analysis is the exponential function, which is frequently employed to represent nonlinear relationships. Depending on the value of the coefficient \(b\), the exponential function can depict either exponential growth or decay. It proves especially effective in modeling scenarios wherein phenomena demonstrate rapid growth or decline and eventually reach a state of long-term saturation. The choice between polynomial and exponential fitting methods depends on the data's nature. Polynomial fitting is typically utilized for data exhibiting an overall trend with random fluctuations around it, whereas exponential fitting is more suited for data that show exponential growth or reduction. Both the exponential and polynomial functions follow a similar operational principle that incorporates three common methods. The first is the least squares method, which aims to find the coefficients \(a_{0}\), \(a_{1}\), and \(a_{2}\) that minimize the sum of the squared differences between the data points and the respective points on the exponential curve or polynomial function. This statistical method is widely used to identify the best-fit line or curve that most accurately represents the data. The second method is the maximum likelihood method, which seeks to determine the parameter values that maximize the probability of the observed data under the exponential function. This approach is extensively used in probability theory and statistics for parameter estimation for various distributions based on observed data. The third method employed in our analysis is the nonlinear least squares method: a versatile approach applicable to various nonlinear functions, including both exponential and polynomial types. This method focuses on minimizing the cumulative squared differences between the data points and their corresponding values as predicted by the nonlinear function. Widely utilized in scientific research and engineering, the nonlinear least squares method is integral for optimizing and determining the best-fit model for experimental data. Through the application of these methods--least squares, maximum likelihood, and nonlinear least squares--we are equipped to accurately identify the most suitable model that aligns with our data, be it polynomial or exponential in nature.