Structural characterization was carried out by analyzing torsional potentials involving C-O and C-C rotations: the analysis was focused on computing energies with respect to the dihedral angles considered, C₁-O₂-C₃-C₄ and O₂-C₃-C₄-O₅, as also shown in Figure 1.

PGA structure was first optimized in vacuo in order to obtain the global minimum energy structure, which was used for subsequent computations (Figure 2). C₁-O₂-C₃-C₄ and O₂-C₃-C₄-O₅ angle values of this minimum energy geometry were estimated equal to −87° and −164°, respectively.

Then, energy scan was done through quantum chemistry computations, changing dihedral angles with a step of 10° and optimizing the structure at every step.

Analyzing torsional potentials involving C₁-O₂-C₃-C₄ angle, energy profile shows one global minimum at −80°, which corresponds to the global optimized geometry (Figure 2), and a global maximum at 0° (Figure 3A), that offers an energy barrier of 6.87 Kcal/mol with respect to minimum. There is, moreover, a local minimum at 70° (Figure 3B), with a barrier with respect to maximum of 5.97 Kcal/mol.

The energy peak corresponds to a geometry in which the repulsion between oxygen atoms is the strongest one, since they are close and almost aligned; in the local minimum, repulsions between oxygen atoms are minimized except for the oxygen of the esters bond, which are aligned.

Referring to O₂-C₃-C₄-O₅ angle, energy profile shows a global minimum at −180° (Figure 3C) and a global maximum at −90° (Figure 3D); there are also a local minimum at −30° (Figure 3E) and a local maximum at 120° (Figure 3F).

The energy barrier between global minimum and maximum is equal to 4.27 Kcal/mol, while the barrier between the global minimum and the local maximum is equal to 4.05 Kcal/mol. The two maxima have a similar energy value; they present a difference in energy of 0.22 Kcal/mol. The energy barrier between the local minimum and the global and local maxima is equal to 2.42 Kcal/mol and to 2.20 Kcal/mol, respectively.

In the global minimum structure, oxygen repulsions are minimized, while in the global maximum, oxygen atoms are closer and in an unfavorable structure. In contrast, the local minimum structure presents a favorable conformation for oxygen atoms, except for the ones involved in esteric bonds, which are close each other. The local maximum exhibits a structure similar to the global maximum one, where oxygen interactions are not preferred.

The same torsional barriers were computed by means of molecular mechanics, using GAFF force field as implemented in AMBER8^® suite of programs. The comparison between the energy profiles computed through quantum chemistry and those obtained via molecular mechanics (Figure 4) shows a deep disagreement between the results. It can be deduced that parameterization of the dihedral term of GAFF force field cannot characterize this torsional behavior.

P_LLA structure was first optimized in vacuo in order to obtain a good guess for further computations. C₁-O₂-C₃-C₄ and O₂-C₃-C₄-O₅ angle of values of the minimum energy structure are equal to 70° and −164°, respectively.

Energy profile of C₁-O₂-C₃-C₄ angle shows a global minimum at 70°, which corresponds to the optimized geometry shown in Figure 5, and a local minimum at −70°, represented in Figure 6D. There is also a global maximum at −10° (Figure 6A) and a local maximum at −120° (Figure 6C), which have a barrier of 6.96 Kcal/mol and 6.18 Kcal/mol, respectively. There is, in addition, one local maximum and one local minimum near 120°, but they were not taken into account due to the small difference in energy (less than 0.2 Kcal/mol).

The global minimum corresponds to the overall optimized geometry, where a helix conformation minimizes the interactions between oxygen atoms: in the maximum energy structure, oxygen atom positions are less favorable, increasing repulsion interactions. The same situation can be seen in the local maximum structure.

The local minimum energy geometry exhibits a helix structure similar to the overall minimum structure, but the energy is higher because of the repulsion between carbonyl oxygen atoms.

O₂-C₃-C₄-O₅ angle, energy profile shows a global minimum at −160°, which corresponds to the overall optimized geometry, and a global maximum at 110° (Figure 6B). Moreover, there are a local minimum at −40° and a local maximum at −110° (Figure 6E and 6F).

In the global maximum structure, oxygen atoms of oligomer backbone are close each other, and thus are subjected to unfavorable repulsion interaction. The global minimum energy geometry is the same as before, where the helix conformation optimizes distances between oxygen atoms.

The local minimum structure has an optimized helix structure except for two close oxygen atoms of the ester bond, while the local maximum structure does not optimize oxygen atom positions.

Also, in this case, GAFF force field is not able to reproduce the torsional behavior of PLLA, as shown in Figure 7. As regards C₁-O₂-C₃-C₄ angle, the global minimum is well characterized, but the other configurations are not in agreement with quantum chemistry computations. Energy profile concerning O₂-C₃-C₄-O₅ angle shows a bad agreement with quantum chemistry torsional barriers. In order to understand the poor agreement of energy profiles, structures obtained through molecular mechanics can be compared with the corresponding ones characterized by means of quantum chemistry. It can be seen, indeed, that molecular mechanics geometries are very different from quantum chemistry ones. Even if a great number of minimization steps are performed, in order to obtain minimum energy structures using atoms coordinates as variables, the system is not able to relax properly. This can be related to a parameterization that is not suitable for such systems; GAFF force field, indeed, is intended to be used only with small organic molecules, and not with polymers. For the sake of completeness, an example referring to the PGA global maximum with respect to O₂-C₃-C₄-O₅ angle is here reported (Figure 8): a deep disagreement is evident since this structure represents a global minimum for molecular mechanics, while it is a global maximum for quantum chemistry.

Diffusion coefficients were computed by means of Einstein equation [28], starting from molecule trajectories:

Where D is the diffusion coefficient, t is the time, and r(t) is the position of the molecules at time t. The term between brackets in equation (1) is also known as the mean square displacement (MSD), that is, the quadratic distance covered by the molecule with respect to its initial position, i.e., r(0). In particular, the motion of the center of mass of the molecules was here considered, while angle brackets indicate that mean square displacement used in the computation of diffusion coefficient is averaged on all solute molecules. The limit operator has a physical meaning: the simulation must be long enough in order to attain Brownian motion regime, since the equation is valid only under this hypothesis. Basically, mean square displacement plotted against time gives a straight line with a slope equal to 6D. In order to verify the Brownian motion regime, and thus the validity of its linear relationship, it is possible to plot log(MSD) against log(t): if the slope is equal to one, then MSD is a linear function of time and Einstein equation is valid [28]. Figure 9B shows log(t) vs. log(MSD) estimated for the lactic acid monomer. Computed diffusion coefficient for lactic acid is equal to (1.006 ± 0.032) × 10⁻⁵ cm²/s, and it is in good agreement with the experimental value of 0.993 × 10⁻⁵ cm²/s reported by Ribeiro et al. [29]. The slope of the log(t) vs. log(MSD), as shown in Figure 9D, is close to one and the value obtained is surely representative of the behavior of the system. Figure 9A shows t vs. MSD plot for glycolic acid, whose computed self-diffusion coefficient is equal to (1.194 ± 0.062) × 10⁻⁵ cm²/s.

Up to our best knowledge, an experimental self diffusion coefficient for glycolic acid is not available in literature, but the computed result can be considered reasonable, since it is very similar to the one of lactic acid. The slope of log(t) vs. log(MSD) is very close to one and thus diffusion regime is reached.

Starting structures were P_LLA and PGA oligomers composed by three monomeric units. First, structure geometries were optimized in vacuo by means of density functional theory (DFT) using a Becke 3 parameters [30] and Lee-Yang-Parr [31] (B3LYP) functional for exchange and correlation energy, and a 6–31G(d,p) basis set [32], implemented in Gaussian03 suite of programs [33]. In this way, it was possible to find an overall optimized structure, which represents an energy minimum used as an initial estimate. Dihedral angles were changed stepwise from −180° to 180° at 10° intervals. Structure optimization was carried out at each step. Indeed, while the dihedral angle of interest was restrained to a certain value, the molecule geometry was optimized in order to obtain the minimum energy structure related to that dihedral angle value. In this way, it was possible to obtain the most probable structure at every step. In order to investigate torsional barriers by means of molecular mechanics, GAFF force field [27] implemented in AMBER8^® suite of programs [26] was chosen. Optimized P_LLA and PGA structures were placed in vacuo, changing dihedral angles values from −180° to 180° with a step of 5°. At every step, a minimization procedure of 10⁶ cycles was carried out, applying a restrain to the investigated dihedral angles. As in the DFT approach, while the dihedral angle of interest was restrained, the molecule geometry was optimized through energy minimization in order to obtain the most probable structure with respect to that dihedral angle value. In particular, 100,000 minimization cycles were run with the steepest descent algorithm, while the remaining ones were accomplished through conjugate gradient algorithm, which is more efficient when the system is close to an energy minimum. Atomic charges were computed starting from electrostatic potentials (ESP) calculated through quantum chemistry at B3LYP/6–31G(d,p) level; then, charges were fitted following RESP (Restrained Electrostatic Potentials) [34] formalism. As mentioned, this analysis aims to assess whether GAFF force field is able to reproduce torsional barriers, and thus if it can provide reliable predictions about oligomers conformations. Analysis was carried out in vacuo in order to neglect the solvent effect [35].

In order to provide further proof as to the suitability of the adopted method, i.e. the chosen force field, molecular dynamics simulations were used to compute monomers’ self diffusion coefficients. Lactic acid and glycolic acid structures were first optimized by DFT calculations in implicit water. Becke 3 parameters [30] and Lee-Yang-Parr [31] (B3LYP) functions for exchange and correlation energy, and 6–31G(d,p) basis set [32] were adopted for the geometry optimization, as implemented in Gaussian03 suite of programs [33]. Water was modeled with the integral equation formalism polarizable continuum model [36] (IEFPCM) at a temperature of 300 K. In order to compute atomic charges, electrostatic potentials (ESPs) were calculated at the B3LYP/6–31G(d,p) level; charges were then fitted through RESP [34] formalism. Ten molecules of each compound were solvated in a cubic box with a lateral size of 60 Å, containing about 9000 explicit TIP4P water molecules [37]; cutoff for non-bonded interactions was set to 15 Å, and simulations were carried out using periodic boundary conditions. Long-range electrostatic interactions were evaluated through the Particle Mesh Ewald method. As mentioned, simulations were performed using GAFF force field as implemented in AMBER8^® suite of programs: in particular, the following simulation protocol was adopted. First of all, a 2000-cycle minimization, in which the solute is restrained with harmonic potential k(Δx)² (where Δx is the displacement and k is the force constant, equal to 500 Kcal mol⁻¹ Å⁻²), was performed in order to cut out bad contacts which derive by the random placing of solvent molecules. 1000 minimization steps were realized through the steepest descent algorithm, while the remaining 1000 were accomplished through the conjugate gradient algorithm, which is more efficient when the system is close to an energy minimum. Then a 3500-cycle minimization was performed in order to minimize the energy of the whole system, without restraints. In this case, 2000 minimization steps were done through the steepest descent algorithm, and the remaining ones were realized with conjugate gradient algorithm. The temperature was raised from 0 K to 300 K by a simulated annealing of 20 ps at constant volume, imposing a weak restraint (k = 10 Kcal mol⁻¹ Å⁻²) on the solute with the purpose of avoiding wild fluctuations, and afterwards the system was relaxed with a 100 ps run at constant pressure in order to reach the correct density of the solution. Finally, molecular dynamics simulations were carried out, investigating a time span of 5 ns. SHAKE algorithm was used for all covalent bonds involving hydrogen atoms, thus allowing a time step of 2 fs. Simulations were run under steady conditions (300 K and 1 atm). Temperature and pressure were controlled using Langevin dynamics (with a collision frequency equal to 1.0 ps⁻¹) and isotropic position scaling, respectively.