PLGA–PEG–PLGA self-aggregation study via fragment dissipative particle dynamics and quantum determined interaction parameters

In this work, we used Conductor-like Screening Model for Real Solvents (COSMO-RS) to calculate the parameters that characterize the interactions between molecular segments in a coarse-grained representation of the PLGA–PEG–PLGA mesomolecule. The computed activity coefficients at infinite dilution were then used to obtain the thermodynamic Flory–Huggins interaction parameters, which were subsequently transferred to Dissipative Particle Dynamics simulations. In these simulations, beads interact through repulsive conservative parameters to investigate the self-aggregation of the PLGA–PEG–PLGA triblock copolymer. The parameters were then applied in Dissipative Particle Dynamics (DPD) simulations at varying copolymer concentrations. Self assembling at different concentrations was studied. Transitions from core-shell spherical micelles to onion-like, columnar and lamellar structures were obtained in terms of copolymer concentration, setting the optimal concentration range for different drug loaded vehicles.

1 Introduction

Polymeric micelles formed from amphiphilic block copolymers in aqueous solution have been extensively studied in recent decades as drug delivery systems Cabral et al. [1], Hossen et al. [2]. They offer high in vivo stability, efficient drug loading, good biocompatibility, and effective targeted drug release. These properties contribute to enhanced chemotherapeutic efficacy and reduced drug toxicity, Fukushima [3], Liao et al. [4], Zhang et al. [5]. Although polymeric micelles have attracted considerable interest as drug carriers, a comprehensive understanding of their structures and morphologies upon drug loading remains limited.

Various polymeric micelles can serve as drug delivery vehicles; among them, PLGA-b-PEG-b-PLGA micelles—composed of poly (lactic acid-co-glycolic acid)-b-poly (ethylene glycol)-b-poly (lactic acid-co-glycolic acid)—have garnered significant attention, Zhang et al. [6], Yan et al. [7]. The PLGA-b-PEG-b-PLGA copolymers materials have advantages such as biocompatibility, degradability, thermosensitivity and controlled release, Yu et al. [8]. The capacity of PLGA-b-PEG-b-PLGA micelles to solubilize hydrophobic drugs stems from their core–shell structure, Chen et al. [9]. Extensive studies have been conducted on their drug-loading capabilities, for example, it has been showed that PLGA-b-PEG-b-PLGA can carry doxorubicin (DOX) drugs, Wang et al. [10], in this study they found that due to the molecular interactions between hydrophobic blocks, PLGA-b-PEG-b-PLGA have exhibited long lasting maintenance after subcutaneous injection in vivo, Chang et al. [11]. Khorshid et al. [12] explored the influence of hydrophylic PEG block length keeping the hydrophobic PLGA blocks constant. They observed that, as the temperature varied, the aggregate structure transitioned from spherical core–shell micelles to cylindrical micelles and eventually to packed cylindrical arrangements.

Since experimental techniques alone have limitations in revealing the detailed molecular distribution and dynamic behavior of drug-loaded systems, computational simulations have been widely employed to study drug encapsulation and release mechanisms. Among computational approaches, dissipative particle dynamics (DPD) has shown to be an efficient mesoscopic simulation method, well-suited for studying complex multiphase systems. It has been successfully used to investigate the formation, drug distribution, and release processes in drug-loaded micelles, Hoogerbrugge and Koelman [13]. DPD is a coarse-grained simulation method used to study surfactant aggregation in solution over extended timescales. In this mesoscopic approach, molecules are represented as beads, allowing for efficient modeling of self-assembly processes. Beyond surfactants, DPD has also been applied to explore more complex liquid systems, Groot and Rabone [14]. Yang et al. [15] used DPD simulations to investigate comicellization behavior, drug distribution patterns, and dual pH/reduction-responsive drug release in mixed micelles. Similarly, Kuru et al. employed coarse-grained DPD simulations to study the morphology, drug encapsulation, and release characteristics of PEG–PLA–PEG amphiphilic block copolymer systems. In more recent years Wang et al. [10] used DPD to study the morphologies and structures of the PLGA-b-PEG-b-PLGA influenced by the copolymer concentration and composition. The structures reported were micelles observed as spherical, onionlike, columnar, and lamellar structures.

Several coarse-grained computational approaches have been reported for simulating these types of dynamics, Cooke et al. [16], Cooke and Deserno [17]. Typically, these models are parameterized using inverse Monte Carlo techniques based on atomistic simulations, Elezgaray and Laguerre [18], Shelley et al. [19]. However, such parameterizations present limitations particularly regarding temperature, which is constrained to the range used during calibration. For example, the widely used MARTINI force field is valid only between 270 and 330 K, Marrink et al. [20]; simulations beyond this range may yield inaccurate results. Another significant limitation lies in modeling interactions between polar compounds, where the forces are often underestimated.

Nivón-Ramírez et al. [21] used a methodology to estimate coarse-grained repulsion parameters $a_{ij}$ using quantum chemistry calculations. For the rapid generation of phase equilibrium data—relevant in applications such as process simulations—simplified numerical models like equations of state (EOS) or excess Gibbs free energy $(g^E)$ formulations are typically employed, Shimoyama and Iwai [22]. Among them, COSMO-type $g^E$ models and the Peng–Robinson (PR) EOS are widely used to describe the phase behavior of multicomponent mixtures.Grensemann and Gmehling [23] demonstrated that COSMO-based models can predict fluid-phase coexistence with high reliability, even for systems lacking experimental parameters, Merker et al. [24]. In this context, comparing the predictive performance of these approaches is essential to validate coarse-grained interaction parameters obtained from quantum-chemical methods.

Solvent effects are incorporated through the Conductor-like Screening Model (COSMO) and the COSMO-Real Solvent (COSMO-RS) model, which allow for the computation of chemical potentials $({\mu }_{ij})$ for species i and j in a pure solvent or mixture. From these values, activity coefficients $({\gamma }_{ij})$ are derived and used to calculate the Flory-Huggins interaction parameter $({\chi }_{ij})$.According to Groot and Warren, a linear relationship connects ${\chi }_{ij}$ to the mesoscopic parameter $a_{ij}$ via $a_{ij}=25+3.5{\chi }_{ij}^{\infty }$ Since ${\chi }_{ij}^{\infty }$ is temperature-dependent, so is $a_{ij}$. This approach enables the modeling of molecular interactions without the need for empirical force field parameterization, and it is applicable to any chemical species.

COSMO-type models do not rely on tabulated parameters, as they only require quantum mechanical solvation calculations as input to predict phase equilibria. Consequently, they are not limited by missing parameter values and can be applied to a broader range of chemical species. This is particularly valuable in industrial settings, where new or poorly characterized compounds are often of interest [25].

In this study, the interaction parameters of the PLGA-b-PEG-b-PLGA copolymer were validated through a modified COSMO-RS methodology at 298.15 K. These parameters were subsequently employed in Dissipative Particle Dynamics (DPD) simulations to examine the influence of copolymer concentration on the self-assembly of unloaded (blank) structures. The article is organized as follows: in Section 2 we present the DPD methodology with the COSMO-RS method, in Section 3 we display our findings using first principle and mesoscopic techniques and finally in Section 4 we provide an insight from our work.

2 Materials and methods

2.1 Dissipative particle dynamics

DPD simulations are based on Newton’s equations of motion and aim to represent fluid behavior and interactions as simply as possible at the mesoscopic scale, minimizing computational cost. The core idea is to model interactions using a pairwise repulsion parameter $a_{ij}$, while preserving the essential physico-chemical characteristics of the system. The DPD model represents point-like particles that interact through defined force sets. Physically, each mesomolecule is composed of a group of beads that move coherently and are connected by harmonic springs Groot and Warren [26], Groot and Rabone [14].

DPD is a coarse-grained molecular dynamics method in which the system’s time evolution is governed by the equations of motion, that is $\frac{d{\mathbf{p}}_i(t)}{dt}={\mathbf{f}}_i(t)$, where ${\mathbf{p}}_i$ is the momentum of the bead $i$ and ${\mathbf{f}}_i(t)$ is the total force acting on it.

The value of the force $\mathbf{F}$ applied to the DPD particles $i$ and $j$ separated a distance $r_{ij}$, has three contributions as defined in Equation 1, a conservative force $({\mathit{F}}_{\mathit{i}\mathit{j}}^{\mathit{C}})$ which is related to the chemical nature of the DPD particles (as shown in Equation 2), a dissipative force $({\mathit{F}}_{\mathit{i}\mathit{j}}^{\mathit{D}})$ that models the friction between particles related to viscosity as described in Equation 3 and a random force ${\mathit{F}}_{\mathit{i}\mathit{j}}^{\mathit{R}}$ that imitates accounts for thermal fluctuations:

${\mathit{F}}_{\mathit{i}}={\displaystyle \sum _{j\ne i}}{\mathit{F}}^{\mathit{C}}+{\mathit{F}}^{\mathit{D}}+{\mathit{F}}^{\mathit{R}},(1)$

Where,

${\mathit{F}}_{\mathit{i}\mathit{j}}^{\mathit{C}}=-a_{ij}{w}^c\left(r_{ij}\right)\widehat{{\mathit{r}}_{\mathit{i}\mathit{j}}},(2)$

${\mathit{F}}_{ij}^D=-\gamma w^D\left(r_{ij}\right)\left({\widehat{\mathit{r}}}_{\mathit{i}\mathit{j}}\cdot {\mathit{v}}_{\mathit{i}\mathit{j}}\right){\widehat{\mathit{r}}}_{\mathit{i}\mathit{j}},(3)$

${\mathit{F}}_{ij}^R=\sigma w^R\left(r_{ij}\right){\widehat{r}}_{ij}{\zeta }_{ij}/\sqrt{\delta t}.(4)$

All forces act within a cutoff distance $r_c$; if the separation between particles $i$ and $j$ exceeds $r_c$ $(r_{ij}>r_c)$, the force becomes zero. All beads are assumed to have an effective diameter equal to $r_c$, which is set as the unit length. The term $\zeta$ in Equation 4 is a Gaussian random variable with zero mean and unit variance. The functions $w^c(r_{ij}),w^D(r_{ij})$ and $w^R(r_{ij})$ are weight functions corresponding to the conservative, dissipative, and random forces, respectively. The parameter $a_{ij}$ defines the strength of the conservative force ${\mathit{F}}_{\mathit{i}\mathit{j}}^{\mathit{C}}$ and governs the thermodynamic behavior of the DPD fluid. Meanwhile,$\gamma$ and $\sigma$ denote the amplitudes of the dissipative and random (noise) forces, respectively.

To determine the interaction parameters $(a_{ij})$, we adopted the approach proposed by Groot [27] as a foundation, refining it through quantum mechanical calculations based on the Conductor-like Screening Model (COSMO) and Conductor-like Screening Model Real Solvents (COSMO-RS), with additional corrections to account for temperature effects, Oviedo-Roa et al. [28]; Nivón-Ramírez et al. [29]; Alasiri and Chapman [30]; Saathoff [31].

2.2 Scaling from atomistic models to mesoscopic representations

In the coarse-graining process employed for DPD modeling, all beads are assumed to occupy the same volume $(V_{\text{bead}})$, irrespective of their chemical identity Maiti and McGrother [32]; Rajkamal and Vedantam [33]. The PLGA-b-PEG-b-PLGA copolymer was modeled according to the scheme proposed by Wang et al. [34], as illustrated in Figure 1. In this representation, the polymer is segmented into three components: lactic acid (L) and glycolic acid (G), which form the hydrophobic PLGA blocks, and polyethylene glycol (PEG), which constitutes the central hydrophilic segment. This amphiphilic configuration promotes micelle or nanoparticle formation in aqueous environments Lee et al. [35].

Figure 1

Figure 1. (A) Schematic representation of the PLGA-b-PEG-b-PLGA copolymer. The lactic acid (L) block is shown in yellow, the glycolic acid (G) block in flesh color, and the polyethylene glycol (E) segment in dark. (B) Coarse-grained DPD representation of the copolymer. (C) Atomistic snapshot of three water molecules (H₂O) corresponding to one coarse-grained water bead. (D) Coarse-grained DPD representation of a single water bead.

To determine the volume of each bead $(V_{\text{bead}})$, we used Equation 5:

$V_{\text{bead}}=\frac{V_{\text{molecule}}}{\lambda },(5)$

where $\lambda$ denotes the number of fragments into which the solute molecule is divided $(\lambda =5)$, and $V_{\text{molecule}}$ is the molecular volume obtained from quantum-mechanical geometry optimizations. The choice of $\lambda =5$ was selected to ensure that the bead volume closely approximated the molecular volume, minimizing deviations, while also preserving the distinctive characteristics of the hydrophilic and hydrophobic blocks in the copolymer.

The number of water molecules in one bead is calculated by Equation 6:

$N_{\text{water}}=\frac{V_{\text{bead}}}{V_{\text{water}}},(6)$

where $V_{\text{water}}$ is the volume of a single water molecule. In these simulations, all beads are assumed to have the same mass, equivalent to the combined mass of the $N_{\text{water}}$ solvent molecules contained in a water bead. For this work, $N_{\text{water}}=2.75$. In practice, it is rounded to three to simplify the model, as illustrated in Figure 1C.

Molecular volume calculations for the solute the molecule and incorporate a continuous approximation of the solvent environment via the conductor-like screening model (COSMO), Klamt and Eckert [36]; Klamt [37]. In order to maintain consistency with the atomistic-quantum calculations of the Flory–Huggins parameter, ${\chi }_{ij}$, COSMO was used throughout.

The infinite-dilution activity coefficient, ${\gamma }_{ij}^{\infty }$, was obtained in natural logarithmic form at room temperature. With ${\gamma }_{ij}^{\infty }$ in hand, the Flory–Huggins thermodynamic interaction parameter, ${\chi }_{ij}^{\infty }$, is derived using the liquid lattice model:

${\chi }_{ij}^{\infty }=\underset{{\varphi }_j\to 1}{\lim }{\chi }_{ij}=\ln \left({\gamma }_{ij}^{\infty }\right)+\ln \left(v_{ij}\right)-\left(1-\frac{1}{v_{ij}}\right),(7)$

where $v_{ij}=\frac{v_j}{v_i}$ is the ratio of molecular volumes between the solvent and solute fragments. To evaluate the infinite-dilution activity coefficients required in Equation 7, we employed COSMO with a statistical thermodynamic treatment founded on the surface charge distribution of the molecules, as shown in Equation 8:

$\ln \left({\gamma }_{ij}^{\infty }\right)=\frac{{\mu }_{ij}^{\text{PSEUDO}}-{\mu }_i^P}{RT}.(8)$

Next, the interaction parameters were determined through a geometry optimization of the PLGA-b-PEG-b-PLGA polymer using density functional theory (DFT) within the TURBOMOLE software suite, accessed through the TMoleX 2025 interface. The DFT calculations were conducted under the COSMO continuous solvation model, Klamt [38], with the def-TZVP (triple-zeta valence polarized) atomic basis set, Schäfer et al. [39,40] and the Becke-Perdew (BP) functional, specifically B88-VWN-P86, Ahlrichs et al. [41]. Both geometric optimization and self-consistent field (SCF) procedures were implemented with an energy convergence criterion of $10^{-9}$ Ha. Following optimization, energy calculations were performed on each neutral, segmented molecule using the same level of theory.

The liquid mixture was then modeled with COSMOthermX, Klamt and Schüürmann [42] under the COSMO-RS framework, effectively treating the molecules as if immersed in an ideal conductor. The parameter file BP_TZVP_C21_0111.ctd, Diedenhofen and Klamt [43], was selected to match the chosen basis set and level of theory.

In homogeneous systems, the interaction parameters for identical species simplify to $v_{ii}=1$, $\ln ({\gamma }_{ii}^{\infty })=0$, and ${\chi }_{ii}^{\infty }=0$, Nivón-Ramírez et al. [29], reflecting a uniform interaction within a single species. Consequently, the activity coefficient equals 1, and the corresponding repulsion coefficient is constant.

For identical molecules, the following DPD repulsion parameters follow, Xu et al. [44]:

$a_{ii}=k_{B}T\frac{k^{-1}{N}_{\text{water}}-1}{2\alpha {\rho }_{\text{DPD}}},(9)$

where $k_{B}T$ represents the thermal energy (reduced to one at 298.15 K), and $k^{-1}=15.9835$ is the dimensionless compressibility at this temperature. The term $N_{\text{water}}$ is the number of water molecules contained in one coarse-grained bead, $\alpha$ is a proportionality constant (0.101 $\pm 0.001$) adjusting the relationship between the Flory–Huggins parameter and the DPD repulsion parameters, Groot and Warren [45], and ${\rho }_{\text{DPD}}=3$ (in DPD units) is the chosen numerical density. It is important to note that the mapping scheme of three water molecules per bead $(N_{\text{water}}=3)$ establishes both the fragmentation and the bead volumes; adopting a different scheme would require a complete re-parameterization, including fragments, bead volumes, sigma profiles, activity coefficients, $v_{ij}$, and $a_{ij}$.

According to Groot and Rabone (Equation 10) [46], for a system at ${\rho }_{\text{DPD}}=3$:

$\chi =0.231\pm 0.001\mathrm{\Delta }a,(10)$

where $\mathrm{\Delta }a=a_{ij}-a_{ii}$ is the excess repulsion between dissimilar particle types. Moreover, adopting a coarse-graining factor of $N_{\text{water}}\approx 3$ makes $\chi$ directly proportional to the bead size. Hence, when multiple elementary molecules are clustered into one bead, $\chi$ is scaled following:

$\chi =N_{\text{water}}{\chi }_{ij}^{\infty }.(11)$

Because atomistic simulations work at the molecular scale, it becomes necessary to adjust $\chi$ to reflect the coarse-grained representation. Thus, if a water bead comprises roughly three water molecules, Maiti and McGrother [47], the solute–solvent interaction parameter is modified by:

$a_{ij}=a_{ii}+\frac{N_{\text{water}}{\chi }_{ij}^{\infty }}{0.231}.(12)$

In Equation 12, $a_{ij}$ is the repulsion parameter between beads, ${\chi }_{ij}$ is the Flory–Huggins parameter for components $i$ and $j$, and $k_{B}T$ is taken as the energy unit in DPD. A reduced cutoff radius $r_c=1$ is used, determined by $r_c=\sqrt{{\rho }_{\text{DPD}}{V}_{\text{bead}}}$. In this work, the cutoff radius $(r_c)$ in real units is 6.28 Å, and each bead’s effective diameter is approximately $0.86r_c$. With this convention, both $r_c$ and $k_{B}T$ are set to one within the DPD framework, Khedr and Striolo [48].

3 Results and discussion

3.1 Interaction parameters

Table 1 presents the molecular volumes at 298.15 K used for temperature correction. Based on these volumes, a bead volume of 82.61 ${\text{Å}}^3$ is obtained, corresponding to $N_{\text{water}}=2.75$.

Table 1

Table 1. Molecular properties of polyethylene glycol (E), glycolic acid (G), lactic acid (L), and a reference system of three water molecules (W) at 298.15 K.

The ratios of molecular volumes employed to compute the solute–solvent interaction parameters are shown in Table 2, along with the corresponding DPD interaction parameters $a_{ij}$. The diagonal terms $a_{ii}$ were taken as 78, in accordance with Equation 9.

Table 2

Table 2. DPD interaction parameters between different structural units.

Using Equation 11 and the volumetric coefficients $v_{ij}$, $\ln (Y_{ij}^{\inf })$, and $X_{ij}^{\inf }$, the values of $a_{ij}$ were determined. Table 2 shows that the largest $a_{ij}$ arises for the water (W)–lactic acid (L) combination, signifying a lower affinity between those two components. By contrast, ethylene glycol (E)–glycolic acid (G) exhibits a moderate $a_{ij}$ (91.13), suggesting more compatibility. These distinctions in $a_{ij}$ values are vital for explaining the solubility and mixture behavior in fluid dynamics simulations. They align with the observations by Merve et al. and Yildiz et al. Kuru et al. [49]; Yildiz and Kacar [50], though they differ from the values reported by Wang et al. Wang et al. [34]. Supplementary Figure S3 in the Supplementary Material illustrates the workflow of our methodology.

Through the COSMO method, the polarization charge density (SCD) is also obtained. Positive SCD values designate partially negative charge regions, whereas negative values correspond to regions of partial positive charge. The DFT-COSMO calculations generate a SCD distribution, which is transformed into sigma profiles ($\sigma$-profiles), Scheffczyk et al. [51]. Then, COSMO-RS uses these profiles to compute local contact energies and chemical potentials of mixture components, Klamt et al. [52].

Figure 2 displays the SCD profiles for water (W; magenta), lactic acid (L; blue), glycolic acid (G; green), and ethylene glycol (E; red). The $x$-axis corresponds to $\sigma$ values, whereas the $y$-axis shows the distribution function $\rho (\sigma )$. Water presents two prominent peaks around 0.018 e/${\text{Å}}^2$ and $-0.016$ e/${\text{Å}}^2$, consistent with its well-known hydrogen-bonding capability Mullins et al. [53]. Lactic acid displays peaks at about $-0.007$ e/${\text{Å}}^2$ and 0.010 e/${\text{Å}}^2$, correlated with polar regions in its carboxyl group and less negative sites elsewhere. Glycolic acid has a strong negative peak at $-0.008$ e/${\text{Å}}^2$, indicating a high capacity to accept hydrogen bonds, and a positive peak near 0.010 e/${\text{Å}}^2$. Ethylene glycol shows a negative region around $-0.002$ e/${\text{Å}}^2$, due to the oxygen atom in its structure and a maximum positive peak of about 0.014 e/${\text{Å}}^2$ in its carbon backbone.

Figure 2

Figure 2. SCD profiles of each molecular fragment: (W) water (magenta line), (L) lactic acid (blue line), (G) glycolic acid (green line), and (E) ethylene glycol (red line).

Based on these profiles, good miscibility is expected between water (W) and glycolic acid (G), as both exhibit pronounced negative $\sigma$ peaks. Lactic acid (L) and ethylene glycol (E) can also interact favorably, albeit less strongly than W–G. Because all fragments are predominantly polar and capable of hydrogen bonding, they are generally miscible, with W–G showing the highest affinity. This conclusion is supported by the interaction parameters in Table 2 showed by the repulsion parameters.

3.2 Dissipative particle dynamics results

The DPD simulations are performed in a $20\times 20$ $\times$ 20 cubic box with periodic boundary conditions, these conditions were guided by standard practices in dissipative particle dynamics (DPD) simulations, where the simulation domain should be sufficiently large to accommodate the characteristic length scales of the self-assembled structures and minimize finite-size effects Groot and Warren [26], Al-Jabri and Rodgers [54]. We used DL_MESO, Seaton et al. [55]. It is worth mention that previous simulations have showned that this box is sufficient to avoid the finite-size effects, Guo et al. [56]. The simulations were carried out in the canonic ensamble (NVT). The simulation temperature used was $k_{B}T=1$, we set the bead density to $\rho =3$ with a fricction coefficient $\gamma =4.5$ and a cutoff radius $r_c=1$. These conditions ensured the system relaxation. The initial configuration of the components was randomly placed in the simulation box. We run the simulation for $4\times 10^5$ steps with a time step $\mathrm{\Delta }t=0.05$ to balance numerical stability with computational efficiency until thermodynamic equilibrium was achived.

We work with the PLGA-b-PEG-b-PLGA copolymer and water, the PLGA is the hydrophobic block, PEG is the hydrophilic block. Since we propose a more complete coarse grained model based on quantum calculations, we refer to the model in Figure 1 to represent the beads of the PLGA-b-PEG-b-PLGA copolymer and for water. As shown, several atoms or functional groups are represented as a DPD bead as we highlighted by color. We fragment the PLGA-b-PEG-b-PLGAThe in 5 parts: lactic acid (L) block shown in yellow, the glycolic acid (G) block in flesh color, and the polyethylene glycol (E) segment in dark. We represent three water molecules by one blue bead.

3.3 Morphologies and structures of the PLGA-b-PEG-b-PLGA micelles

The ${\mathrm{P}\mathrm{L}\mathrm{G}\mathrm{A}}_3$-$\mathrm{b}$-${\mathrm{P}\mathrm{E}\mathrm{G}}_7$-$\mathrm{b}$-${\mathrm{P}\mathrm{L}\mathrm{G}\mathrm{A}}_3$ copolymer was selected to represent the real ${\mathrm{P}\mathrm{L}\mathrm{G}\mathrm{A}}_{1200}$-$\mathrm{b}$-${\mathrm{P}\mathrm{E}\mathrm{G}}_{1450}$-$\mathrm{b}$-${\mathrm{P}\mathrm{L}\mathrm{G}\mathrm{A}}_{1200}$ copolymer, Wang et al. [10]. The $\mathrm{P}\mathrm{L}\mathrm{G}\mathrm{A}$-$\mathrm{b}$-$\mathrm{P}\mathrm{E}\mathrm{G}$-$\mathrm{b}$-$\mathrm{P}\mathrm{L}\mathrm{G}\mathrm{A}$ copolymers are theoretically predicted to form core–shell spherical micelles in aqueous environments, Yu et al. [57], Wang et al. [10]. To investigate the influence of $\mathrm{P}\mathrm{L}\mathrm{G}\mathrm{A}$-$\mathrm{b}$-$\mathrm{P}\mathrm{E}\mathrm{G}$-$\mathrm{b}$-$\mathrm{P}\mathrm{L}\mathrm{G}\mathrm{A}$ concentration $c_{cp}$ on micelle morphologies and structures, DPD simulations were performed using ${\mathrm{P}\mathrm{L}\mathrm{G}\mathrm{A}}_3$-$\mathrm{b}$-${\mathrm{P}\mathrm{E}\mathrm{G}}_7$-$\mathrm{b}$-${\mathrm{P}\mathrm{L}\mathrm{G}\mathrm{A}}_3$ concentrations ranging from 1.0% to 40%. It can be observed in Figure 3A that ${\mathrm{P}\mathrm{L}\mathrm{G}\mathrm{A}}_3$-$\mathrm{b}$-${\mathrm{P}\mathrm{E}\mathrm{G}}_7$-$\mathrm{b}$-${\mathrm{P}\mathrm{L}\mathrm{G}\mathrm{A}}_3$ copolymers are capable of self-assembling into micelles in aqueous solution at a low copolymer concentration of $c_{cp}=1.0\%$. The micellar microstructure exhibits significant variations as the concentration of ${\mathrm{P}\mathrm{L}\mathrm{G}\mathrm{A}}_3$-$\mathrm{b}$-${\mathrm{P}\mathrm{E}\mathrm{G}}_7$-$\mathrm{b}$-${\mathrm{P}\mathrm{L}\mathrm{G}\mathrm{A}}_3$ copolymer increases from $c_{cp}=1.0\%$ to $5\%$, the hydrophobic beads L, E form the hidrophobyc core of the micelle and the hydrophilic bead G sorrounds the block arrange. As the concentration of ${\mathrm{P}\mathrm{L}\mathrm{G}\mathrm{A}}_3$-$\mathrm{b}$-${\mathrm{P}\mathrm{E}\mathrm{G}}_7$-$\mathrm{b}$-${\mathrm{P}\mathrm{L}\mathrm{G}\mathrm{A}}_3$ copolymer increases from $c_{cp}=7.5\%$ to $15\%$, the structures packed and indicate onionlike structures, Guo et al. [58]. However, as the concentration of ${\mathrm{P}\mathrm{L}\mathrm{G}\mathrm{A}}_3$-$\mathrm{b}$-${\mathrm{P}\mathrm{E}\mathrm{G}}_7$-$\mathrm{b}$-${\mathrm{P}\mathrm{L}\mathrm{G}\mathrm{A}}_3$ increases to $c_{cp}=20\%$, the E, L, and G beads exhibit the widest spatial distribution, corresponding to a columnar structure. When $c_{cp}$ further increases from 30% to 40%, the distribution of the E, L, and G beads becomes narrower again, with the E beads positioned on both sides of the L and G beads, which is characteristic of a lamellar structure, Bänsch et al. [59].

Figure 3

Figure 3. Representative morphology snapshots of the ${\mathrm{P}\mathrm{L}\mathrm{G}\mathrm{A}}_3$-$\mathrm{b}$-${\mathrm{P}\mathrm{E}\mathrm{G}}_7$-$\mathrm{b}$-${\mathrm{P}\mathrm{L}\mathrm{G}\mathrm{A}}_3$ system in their final state: (a) C = 1%, (b) C = 1.5%, (c) C = 2.5%, (d) C = 5%, (e) C = 7.5%, (f) C = 10%, (g) C = 15%, (h) C = 20%, (i) C = 25%, (j) C = 30%, (k) C = 40%.

Figure 3 demonstrates that the micellization process of ${\mathrm{P}\mathrm{L}\mathrm{G}\mathrm{A}}_3$-$\mathrm{b}$-${\mathrm{P}\mathrm{E}\mathrm{G}}_7$-$\mathrm{b}$-${\mathrm{P}\mathrm{L}\mathrm{G}\mathrm{A}}_3$ in aqueous solution follows a mechanism similar to that observed for most other polymeric micelles, Yang et al. [15], Wang et al. [10]. Having confirmed that the PLGA-b-PEG-b-PLGA copolymers form micelles in aqueous solution, we next investigate the effects of the copolymer concentration and composition on the morphologies structures of the micelle and to obtain a clearer perspective, we made transversal cuts of each structure, and these can be found in the Supplementary Material (Supplementary Figure S1). To assess whether the system reaches equilibrium, the time evolution of the radial distribution functions for the PLGA-b-PEG-b-PLGA copolymer beads was analyzed. Figure 4 shows that at the initial state $t=0$, the $\mathrm{P}\mathrm{L}\mathrm{G}\mathrm{A}$-$\mathrm{b}$-$\mathrm{P}\mathrm{E}\mathrm{G}$-$\mathrm{b}$-$\mathrm{P}\mathrm{L}\mathrm{G}\mathrm{A}$ copolymers and water are randomly mixed. The corresponding curve exhibits a low peak at $r<1$, indicating weak interactions between the ${\mathrm{P}\mathrm{L}\mathrm{G}\mathrm{A}}_3$-$\mathrm{b}$-${\mathrm{P}\mathrm{E}\mathrm{G}}_7$-$\mathrm{b}$-${\mathrm{P}\mathrm{L}\mathrm{G}\mathrm{A}}_3$ beads. As time evolution increases due to the significantly aggregate, the width and height of the peak exhibit an increase. And at the final state $(t=4\times 10^5)$, the width and height of the peak exhibit almost no change, which indicates that the system has reached equilibrium. To obtain more quantitative insights, we plotted the radius of gyration as a function of concentration, as shown in Supplementary Figure S2 of the Supplementary Material. As shown in Supplementary Figure S2, the radius of gyration increases with concentration, confirming that aggregation is favorable, promoting micelle fusion and structural deformation when the concentrations increases from 15%.

Figure 4

Figure 4. Radial distribution functions between beads of the $PLG{A}_3-b-PE{G}_7-b-PLG{A}_3$ micelles at different simulation times.

Our findings align with the results reported by Wang Wang et al. [10], who reported similar effects of copolymer concentration on micelle morphology. Specifically, at concentrations below 10%, the micelles formed core-shell spherical structures, while at higher concentrations, they transitioned into onionlike, columnar, and lamellar structures. These findings are consistent with experimental reports of similar micelle size trends with increasing copolymer concentration Khorshid et al. [12], and also with the report by Shen Shen et al. [60], that demonstrates comparable structural transitions in PEG-PLGA micelles as the copolymer concentration increased.

Our results show that the G bead behaves like de hidrophilic core sorrounded by the E bead and finally the hidrophobic core L. With the findings of our study we show that this given difference is due to the theory used to calculate the interaction parameters, Wang et al. [10] used the Hildebrand theory, Hildebrand [61], while we used the COSMO methodology, Mullins et al. [53]. Table 3 highlights the differences between our parameters and those reported by Wang. Our model offers a more detailed representation by calculating energy interactions at the molecular level, whereas Hildebrand theory can introduce errors when estimating cohesive energy Ovejero et al. [62]. Moreover, a key limitation of the Hildebrand solubility parameter is its applicability only to non-polar or weakly polar systems. Since it primarily accounts for dispersion forces, it neglects important dipole–dipole and hydrogen bonding interactions, which are essential in polar solvents and polymer systems. As a result, it may lead to inaccurate solubility predictions in systems with strong intermolecular forces, Barton [63], Venkatram et al. [64].

Table 3

Table 3. Interaction parameters for the conservative force.

4 Discussion

In this study, we demonstrated that the coarse-graining method significantly influences the calculation of interaction parameters, which in turn affect the chemical behavior of the system. Methods such as COSMO and COSMO-RS have proven to be superior approaches for performing bottom-up parameterization. Polymeric micelles formed by $\mathrm{P}\mathrm{L}\mathrm{G}\mathrm{A}$-$\mathrm{b}$-$\mathrm{P}\mathrm{E}\mathrm{G}$-$\mathrm{b}$-$\mathrm{P}\mathrm{L}\mathrm{G}\mathrm{A}$ copolymers were investigated using Dissipative Particle Dynamics (DPD) simulations. The resulting morphologies and structures were found to depend on molecular parameters, particularly the copolymer concentration and composition.

As the $\mathrm{P}\mathrm{L}\mathrm{G}\mathrm{A}$-$\mathrm{b}$-$\mathrm{P}\mathrm{E}\mathrm{G}$-$\mathrm{b}$-$\mathrm{P}\mathrm{L}\mathrm{G}\mathrm{A}$ concentration increases from $c_{cp}=2.5\%$ to $40\%$, the micellar structures undergo a morphological transition from spherical (core–shell) to onion-like (core–middle layer–shell), and eventually to columnar and lamellar structures. The onion-like structures consist of a hydrophilic PEG core, a hydrophobic PLGA middle layer, and a hydrophilic PEG shell. Our results reveal that the micelle structures and morphologies are highly dependent on the concentration of the $\mathrm{P}\mathrm{L}\mathrm{G}\mathrm{A}$-$\mathrm{b}$-$\mathrm{P}\mathrm{E}\mathrm{G}$-$\mathrm{b}$-$\mathrm{P}\mathrm{L}\mathrm{G}\mathrm{A}$ copolymer making it an excellent candidate for drug loaded micelles. From a practical point of view, a design rule can be established as follows: the relevant regime for in vivo applications corresponds to the copolymer concentration window where spherical or short wormlike micelles are stable, from C = 1.5% to C = 15%.

Data availability statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Author contributions

AC-C: Conceptualization, Investigation, Validation, Visualization, Writing – original draft, Writing – review and editing. MC-C: Investigation, Visualization, Writing – review and editing. IS-H: Conceptualization, Funding acquisition, Supervision, Writing – review and editing.

Funding

The author(s) declare that financial support was received for the research and/or publication of this article. A.AC-C. acknowledges CONAHCYT for supporting the scholarship CVU: 1184262. M.A. C-C acknowledges CONAHCYT for supporting the scholarship CVU: 1183460. A.AC-C. appreciates the technical support of Alejandro de León Cuevas, Alejandro Ávalos and Luis Alberto Aguilar Bautista from Laboratorio Nacional de Visualización Científica Avanzada (LAVIS-UNAM). A. A. Coreno-Cortés thanks Dr. Rodolfo Gómez Balderas and Dra. Roxana Mitzaye del Castillo Vazquez for their technical support. IS-H. acknowledges UMDI-J-FC UNAM for partial financial support under grant number 115377. IS-H. is grateful to LANCAD-UNAM-DGTIC-276.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Generative AI statement

The author(s) declare that no Generative AI was used in the creation of this manuscript.

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Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphy.2025.1694078/full#supplementary-material

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