## 3 Materials and Methods
### Design of Experiments  
A design of experiments (DoE) consists of a series of runs that help to investigate the effects of input variables (factors) on an output variable (response). The factors that can influence the response are selected and evaluated. The main goal can be to identify significant factors that affect the response, identify factor interactions, to find optimal factor settings, match a target value or to build a predictive model, using the minimum number of experiments possible [19]. DoE is usually divided in three stages, starting with a screening design (identify the most important factors and reduce the number of factors that will be optimized), followed by a response surface design (construct a design that models a quadratic function of continuous factors, identifying factor settings that optimize the response) and finally the method validation. Several models can be used for screening (e.g., full factorial and fractional factorial). Factorial designs are used to study the effects of two or more variables on the response. Since the size of full factorial designs increases exponentially with the number of input variables, fractional factorial designs are preferred when dealing with high-dimensional problems. The performed \(2^{7-3}\) fractional factorial screening design (16 experiments, 1/8 of the full factorial design with 128 experiments) corresponds to a resolution IV design, meaning that the main effects are not confounded with other two-factor interactions. Another option would be to apply a \(2^{7-4}\) fractional factorial, that would result in only 8 experiments. However, this design corresponds to a resolution III design, which means it is possible to estimate the main effects, but these may be confounded with two-factor interactions. So, the \(2^{7-3}\) fractional factorial was preferred, even though it meant to conduct double the experiments. After performing the screening design and identifying the significant factors, a response surface methodology (RSM) is applied in order to model a quadratic function of continuous factors, identifying factor settings that optimize the response. Some examples include central composite design (CCD), Plackett-Burman design (PBD), Box-Behnken design (BBD), and Doehlert matrix design. The performed central composite designs consist of two-level full or fractional factorial design, coupled with two-star points (\(-\alpha\), +\(\alpha\)) that represent new extreme values (low and high) for each factor in the design that allow the estimation of curvature. It also includes center points in order to measure stability and inherent variability of the model. Each factor value (\(X_{i}\)) must be transformed into coordinates within a scale with dimensionless values (\(x_{i}\)), which must be proportional to its location in the experimental space. It enables the investigation of variables of different orders of magnitude without influencing the evaluation of the smallest. This means that the models operate on coded dimensionless input values \(x_{i}\) (\(-1\), +1) instead of actual factor values (\(X_{i}\)), that are converted by the following equation [20] (Equation (1)):  
\[x_{i}=\frac{X_{i}-X_{0}}{\Delta X} \tag{1}\]  
where \(X_{0}\) refers to the value of variable \(i\) in the center of the domain (\(x_{i}\) = 0) and \(\Delta X\) refers to the difference of that variable between \(x_{i}\) = +1 and \(x_{i}\) = 0. After conducting the experiments, the experimental responses must be adjusted to a quadratic model [20; 21] (Equation (2)):  
\[Y=b_{0}+\sum_{i=1}^{k}b_{i}x_{i}+\sum_{i=1}^{k}b_{ii}x_{i}^{2}+\sum_{j>1}^{k} \sum_{i=1}^{k}b_{ij}x_{i}x_{j} \tag{2}\]  
where \(Y\) corresponds to the process response, \(x_{i}\) to the codified independent variable, \(b_{0}\) is the interception term, \(b_{i}\) is the influence of the variable \(i\) in the response, \(b_{ii}\) is the parameter that determines the shape of the curve, and \(b_{ij}\) corresponds to the effect of the interaction among variables \(i\) and \(j\).