# A Systematic Experimental and Computational Analysis of Commercially Available Aliphatic Polyesters
## 2 Materials and Methods
### Mathematical Modeling  
The starting point for the computational analysis is the modeling framework proposed by [PERSON] et al. [17] and [PERSON] et al. [18; 19], chosen for its validated results for the degradation of bulk-eroding polymers [20]. Model development has been discussed in detail in previous works [19], and is model equations are summarized here for the sake of completeness.  
Since the materials under investigation undergo bulk hydrolysis, the system is modeled as an isothermal semibatch reactor with a constant volume. The adopted modeling approach is based on population balances solved through the method of the moments and it has been chosen for the comprehensive description (both diffusion phenomena and autocatalysis are accounted for) and its validated results. Polymer degradation can be described according to this kinetic scheme:  
\[P_{n+m}+H_{2}O+H^{+}\to P_{n}+P_{m}+H^{+} \tag{3}\]  
where \(P_{n+m}\), \(P_{n}\), and \(P_{m}\) are polymer chains containing \(n+m\), \(n\), and \(m\) repeating units, respectively, \(H_{2}O\) is water and \(H^{+}\) is employed to indicate that the reaction is catalyzed in acidic environments.  
According to the adopted modeling framework, polymer degradation can be characterized by a system of partial differential equations that account for time and spatial evolution of monomer (Equation (4)), oligomers up to nonamers (Equation (5)), water (Equation (6)) and statistical moments of the first three orders of chain length distribution [21] (Equations (7)-(9)):  
\[\frac{\partial C_{M}}{\partial t}=\frac{\partial}{\partial z}\bigg{(}D_{n} \frac{\partial C_{M}}{\partial z}\bigg{)}+2k_{d}C_{w}(\mu_{0}-C_{M})\mu_{0} \tag{4}\]  
\[\frac{\partial C_{n}}{\partial t}=\frac{\partial}{\partial z}\bigg{(}D_{n} \frac{\partial C_  
\[\frac{\partial\mu_{2}}{\partial t}=\sum_{j=1}^{9}j^{2}\frac{\partial}{\partial z} \left(D_{j}\frac{\partial C_{j}}{\partial z}\right)+\frac{k_{d}C_{w}\mu_{0}}{3} \left(\mu_{1}-2\frac{\mu_{2}^{2}}{\mu_{1}}+\frac{\mu_{2}\mu_{1}}{\mu_{0}}\right) \tag{9}\]  
where \(C_{M}\), \(C_{n}\), and \(C_{w}\) are molar concentration of monomer units, polymer chains with n repeating units and water, respectively, \(\mu_{j}\) is \(j\)-th order moment of chain length distribution, \(D_{M}\), \(D_{n}\), and \(D_{w}\) are the diffusion coefficients for monomer, oligomers, and water, respectively, \(t\) is time, \(z\) is the cylinder axial coordinate, and \(k_{d}\) is the degradation kinetic constant. Equations (4)-(9) are written under the assumption that only oligomers up to nonamers can appreciably diffuse through the matrix and adopt axial coordinate as the characteristic diffusion length.  
Because of the application of the method of the moments, the details about the chain length distribution over time in space is lost. However, the average properties of interest can be easily computed starting from the moments  
\[MW_{n}=\frac{\mu_{1}}{\mu_{0}}MW_{mon} \tag{10}\]  
\[MW_{w}=\frac{\mu_{2}}{\mu_{1}}MW_{mon} \tag{11}\]  
\[PD=\frac{\mu_{2}\mu_{1}}{\mu_{1}^{2}} \tag{12}\]  
where \(MW_{n}\) is number average molecular weight, \(MW_{w}\) is weight average molecular weight, \(PD\) is polydispersity, and \(MW_{mon}\) is the molecular weight of the repeating unit. The diffusion coefficient varies in time and space, because it dynamically increases according to degradation extent, since chain scissions open new and wider diffusion paths. Diffusion coefficients are functions of molecular weight, according to the following expression:  
\[D_{i}(t,z)=D_{i}^{0}exp\left[2.5\left(1-\frac{MW_{n}(t,z)}{MW_{n}(t=0)}\right) ^{0.5}\right] \tag{13}\]  
where \(D_{i}\) is the diffusion coefficient of the \(i\)-th species (monomer, oligomer, water) and \(D_{i}^{0}\) is the diffusion coefficient in the undegraded matrix. Details concerning the boundary conditions, initial conditions, and numerical solution are reported in the Supporting Information.