- UMDI-Facultad de Ciencias, Universidad Nacional Autónoma de México Campus Juriquilla, Querétaro, México
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
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
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
DPD is a coarse-grained molecular dynamics method in which the system’s time evolution is governed by the equations of motion, that is
The value of the force
Where,
All forces act within a cutoff distance
To determine the interaction parameters
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
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
where
The number of water molecules in one bead is calculated by Equation 6:
where
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,
The infinite-dilution activity coefficient,
where
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
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
For identical molecules, the following DPD repulsion parameters follow, Xu et al. [44]:
where
According to Groot and Rabone (Equation 10) [46], for a system at
where
Because atomistic simulations work at the molecular scale, it becomes necessary to adjust
In Equation 12,
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
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
Using Equation 11 and the volumetric coefficients
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 (
Figure 2 displays the SCD profiles for water (W; magenta), lactic acid (L; blue), glycolic acid (G; green), and ethylene glycol (E; red). The
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
3.2 Dissipative particle dynamics results
The DPD simulations are performed in a
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
Figure 3. Representative morphology snapshots of the
Figure 3 demonstrates that the micellization process of
Figure 4. Radial distribution functions between beads of the
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].
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
As the
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.
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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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Keywords: self-assembly, activity coefficients, flory-huggins, first principles, dissipative particle dynamics
Citation: Coreno-Cortés AA, Cortés-Cuán MA and Santamaría-Holek I (2025) PLGA–PEG–PLGA self-aggregation study via fragment dissipative particle dynamics and quantum determined interaction parameters. Front. Phys. 13:1694078. doi: 10.3389/fphy.2025.1694078
Received: 27 August 2025; Accepted: 06 October 2025;
Published: 20 October 2025.
Edited by:
Edward John Sambriski, Delaware Valley University, United StatesReviewed by:
Saravana Prakash Thirumuruganandham, SIT Health, Quito, EcuadorSegun Oke, Alabama A & M University, United States
Copyright © 2025 Coreno-Cortés, Cortés-Cuán and Santamaría-Holek. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
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