## 3 Solubility Parameter  
Research concerning PCR materials utilizes the SP as a relative measure of \"good\" or \"bad\" solvents, according to the solvating ability of a diluent. There are three types of SP values that are commonly used, _i.e._, the one-component SP (the Hildebrand solubility parameter), a two-component solubility (physical-chemical) parameter, and [PERSON]'s three-component SP value [30, 31, 32, 33, 34, 35]. The Hildebrand SP (\(\delta\)) of the porogen and the polymer [36], respectively, are defined as shown in Equation (1) below:\[\delta=\sqrt{C}=\sqrt{\frac{\Delta H-RT}{V_{m}}} \tag{1}\]  
where \(C\) is the cohesive energy density, \(\Delta H\) is the heat of vaporization, \(R\) is the gas constant, \(T\) is the temperature, and \(V_{m}\) is the molar volume of the solvent. The SI unit for \(\delta\) is MPa\({}^{1/2}\) while the conventional unit is (J/cm\({}^{3}\))\({}^{1/2}\). The closer the match between the porogen and the polymer implies a \"good\" solvent since solvents with similar SP values are generally miscible. In other words, if (\(|\Delta\delta|\)) is near zero (where \(|\Delta\delta|=|\delta_{\text{resin}}\cdot\delta_{\text{solvent}}|\)), then miscibility is favored. In cases where \(|\Delta\delta|>3\) MPa\({}^{1/2}\), the solvent is considered \"bad\" for values between 1-3 MPa\({}^{1/2}\), the solvent possesses intermediate solubility and is considered \"good\" and \"bad\". However, this criterion has some exceptions when the resin and diluent are both polar or have specific polar-directional group interactions. Therefore, an extension to [PERSON]'s theory was proposed by [PERSON] [32] where the SP was divided into three contributions, _i.e._, dispersion forces (\(\delta_{d}\)), dipolar interactions (\(\delta_{p}\)) and H-bond capacity (\(\delta_{h}\)), as shown in Equation (2). \[\delta_{T}=\sqrt{\left(\delta_{d}^{~{}2}+\delta_{p}^{~{}2}+\delta_{h}^{~{}2} \right)} \tag{2}\]  
The above three-dimensional SP (\(\delta_{T}\)) is assumed to be a vector sum where the three components are treated as solubility coordinates. [PERSON] reported resin solubility values using a three-dimensional model and concluded that doubling of the dispersion forces (\(\delta_{d}\)), a spherical volume of solubility would be formed for each resin. The sphere is described in Figure 2 where the SP value of a given compound can be visualized as a fixed point and referred to as a tri-point in a three-dimensional space. The coordinates of the tri-point are located by means of three SP values (\(\delta_{d}\), \(\delta_{p}\) and \(\delta_{h}\)). When the resin is mixed with a single solvent or multiple solvents with SP values lying within the sphere, the compatibility is considered \"good\" while those solvents lying outside the sphere are considered \"bad\" (_cf._ Figure 3 in [45]). **Figure 2.** A three-dimensional plot of the three solubility parameter (SP) contributions _i.e._, dispersion forces (\(\delta_{d}\)), dipolar interactions (\(\delta_{p}\)), and H-bond capacity (\(\delta_{h}\)), to the total SP (\(\delta_{T}\)). The tri-point represents a hypothetical polymer. Interaction radius is the shortest distance between the tri-point and the solvent. To aid the SP criterion of [PERSON]'s methodology for the selection of \"good\" versus \"bad\" solvents, the use of relative energy difference (_RED_) was proposed, as shown in Equation (3). In this approach, values of \(RED<1\) imply that the solvent is \"good\" or the solvent is \"bad\" when \(RED>1\). \[RED=\frac{R_{a}}{R_{o}} \tag{3}\]  
where \(R_{a}\) is the distance between centre of the polymer solubility sphere for the solvent-solvent system and \(R_{o}\) is the radius of the sphere; where \(\rm{R_{a}}\) is calculated according to Equation (4). \[R_{a}=\sqrt{4(\delta_{ak}-\delta_{as})^{2}+(\delta_{ak}-\delta_{as})^{2}+( \delta_{ak}-\delta_{as})^{2}} \tag{4}\]  
where \(\delta_{ak}\) and \(\delta_{asS}\) are the [PERSON] solubility parameters for the resin and solvent, respectively.