# Implicit-Solvent Coarse-Grained Simulations of Linear–Dendritic Block Copolymer Micelles

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Simulation Model

## 3. Theory

## 4. Results

#### 4.1. Model Parameters

#### 4.2. Linear Block Copolymers, $g=0$

#### 4.3. Linear–Dendritic Block Copolymers

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Lazzari, M.; Liu, G.; Lecommandoux, S. Block Copolymers in Nanoscience; Wiley-VCH Verlag GmbH & Co. KGaA: Weinheim, Germany, 2006. [Google Scholar]
- Mane, S.R.; Sathyan, A.; Shunmugam, R. Biomedical Applications of pH-Responsive Amphiphilic Polymer Nanoassemblies. ACS Appl. Nano Mater.
**2020**, 3, 2104–2117. [Google Scholar] [CrossRef] - Schacher, F.H.; Rupar, P.A.; Manners, I. Functional block copolymers: Nanostructured materials with emerging applications. Angew. Chem. Int. Ed.
**2012**, 51, 7898. [Google Scholar] [CrossRef] - Tritschler, U.; Pearce, S.; Gwyther, J.; Whittell, G.R.; Manners, I. 50th Anniversary Perspective: Functional Nanoparticles from the Solution Self-Assembly of Block Copolymers. Macromolecules
**2017**, 50, 3439. [Google Scholar] [CrossRef] - Rösler, A.; Vandermeulen, G.W.; Klok, H.A. Advanced drug delivery devices via self-assembly of amphiphilic block copolymers. Adv. Drug Deliv. Rev.
**2012**, 64, 270–279. [Google Scholar] [CrossRef] - Elsabahy, M.; Wooley, K.L. Design of polymeric nanoparticles for biomedical delivery applications. Chem. Soc. Rev.
**2012**, 41, 2545–2561. [Google Scholar] [CrossRef] - Lyu, Z.; Ding, L.; Tintaru, A.; Peng, L. Self-Assembling Supramolecular Dendrimers for Biomedical Applications: Lessons Learned from Poly(amidoamine) Dendrimers. Accounts Chem. Res.
**2020**, 53, 2936–2949. [Google Scholar] [CrossRef] - Chen, J.; Zhu, D.; Liu, X.; Peng, L. Amphiphilic Dendrimer Vectors for RNA Delivery: State-of-the-Art and Future Perspective. Accounts Mater. Res.
**2022**, 3, 484–497. [Google Scholar] [CrossRef] - Israelachvili, J.N. Intermolecular and Surface Forces; Academic Press: Cambridge, MA, USA, 2011. [Google Scholar]
- Israelachvili, J.N.; Mitchell, D.J.; Ninham, B.W. Theory of self-assembly of hydrocarbon amphiphiles into micelles and bilayers. J. Chem. Soc. Faraday Trans. 2
**1976**, 72, 1525–1568. [Google Scholar] [CrossRef] - Liggins, R.; Burt, H. Polyether–polyester diblock copolymers for the preparation of paclitaxel loaded polymeric micelle formulations. Adv. Drug Deliv. Rev.
**2002**, 54, 191–202. [Google Scholar] [CrossRef] - Volkmar Weissig, T.E. Pharmaceutical Nanotechnology; Humana: New York, NY, USA, 2019. [Google Scholar]
- Zheng, X.; Xie, J.; Zhang, X.; Sun, W.; Zhao, H.; Li, Y.; Wang, C. An overview of polymeric nanomicelles in clinical trials and on the market. Chin. Chem. Lett.
**2021**, 32, 243–257. [Google Scholar] [CrossRef] - Lee, C.C.; MacKay, J.A.; Fréchet, J.M.; Szoka, F.C. Designing dendrimers for biological applications. Nat. Biotechnol.
**2005**, 23, 1517. [Google Scholar] [CrossRef] [PubMed] - Mane, S.R.; Sarkar, S.; Rao, V.N.; Sathyan, A.; Shunmugam, R. An Efficient Method to Prepare a New Class of Regioregular Graft Copolymer via a Click Chemistry Approach. RSC Adv.
**2015**, 5, 74159. [Google Scholar] [CrossRef] - Vinciguerra, D.; Degrassi, A.; Mancini, L.; Mura, S.; Mougin, J.; Couvreur, P.; Nicolas, J. Drug-Initiated Synthesis of Heterotelechelic Polymer Prodrug Nanoparticles for in Vivo Imaging and Cancer Targeting. Biomacromolecules
**2019**, 20, 2464. [Google Scholar] [CrossRef] [PubMed] - Matyjaszewski, K.; Spanswick, J. Controlled/living radical polymerization. Mater. Today
**2005**, 8, 26–33. [Google Scholar] [CrossRef] - Hawker, C.J.; Wooley, K.L. The Convergence of Synthetic Organic and Polymer Chemistries. Science
**2005**, 309, 1200–1205. [Google Scholar] [CrossRef] - Laurini, E.; Aulic, S.; Marson, D.; Fermeglia, M.; Pricl, S. Cationic Dendrimers for siRNA Delivery: An Overview of Methods for In Vitro/In Vivo Characterization. In Design and Delivery of SiRNA Therapeutics; Ditzel, H.J., Tuttolomondo, M., Kauppinen, S., Eds.; Springer: New York, NY, USA, 2021; pp. 209–244. [Google Scholar] [CrossRef]
- Tomalia, D.A.; Baker, H.; Dewald, J.; Hall, M.; Kallos, G.; Martin, S.; Roeck, J.; Ryder, J.; Smith, P. A New Class of Polymers: Starburst-Dendritic Macromolecules. Polym. J.
**1985**, 17, 117–132. [Google Scholar] [CrossRef] - Aharoni, S.; Murthy, N. SPHERICAL NON-DRAINING BOC-POLY (alpha, epsilon-L-LYSINE) MACROMOLECULES: SAXS AND VISCOSITY STUDIES. Polymer
**1983**, 24, 132–136. [Google Scholar] - Neelov, I.; Falkovich, S.; Markelov, D.; Paci, E.; Darinskii, A.; Tenhu, H. Molecular Dynamics of Lysine Dendrimers. Computer Simulation and NMR. In Dendrimers in Biomedical Applications; The Royal Society of Chemistry: London, UK, 2013; pp. 99–114. [Google Scholar] [CrossRef]
- Moorefield, C.N.; Newkome, G.R. Unimolecular micelles: Supramolecular use of dendritic constructs to create versatile molecular containers. C. R. Chim.
**2003**, 6, 715–724. [Google Scholar] [CrossRef] - Cao, W.; Zhu, L. Synthesis and Unimolecular Micelles of Amphiphilic Dendrimer-like Star Polymer with Various Functional Surface Groups. Macromolecules
**2011**, 44, 1500–1512. [Google Scholar] [CrossRef] - Boris, D.; Rubinstein, M. A Self-Consistent Mean Field Model of a Starburst Dendrimer: Dense Core vs Dense Shell. Macromolecules
**1996**, 29, 7251–7260. [Google Scholar] [CrossRef] - Murat, M.; Grest, G.S. Molecular Dynamics Study of Dendrimer Molecules in Solvents of Varying Quality. Macromolecules
**1996**, 29, 1278–1285. [Google Scholar] [CrossRef] - Kłos, J.S.; Sommer, J.U. Properties of Dendrimers with Flexible Spacer-Chains: A Monte Carlo Study. Macromolecules
**2009**, 42, 4878–4886. [Google Scholar] [CrossRef] - Okrugin, B.; Neelov, I.; Leermakers, F.; Borisov, O. Structure of asymmetrical peptide dendrimers: Insights given by self-consistent field theory. Polymer
**2017**, 125, 292–302. [Google Scholar] [CrossRef] - Shavykin, O.; Mikhailov, I.; Darinskii, A.; Neelov, I.; Leermakers, F. Effect of an asymmetry of branching on structural characteristics of dendrimers revealed by Brownian dynamics simulations. Polymer
**2018**, 146, 256–266. [Google Scholar] [CrossRef] - Gorzkiewicz, M.; Konopka, M.; Janaszewska, A.; Tarasenko, I.I.; Sheveleva, N.N.; Gajek, A.; Neelov, I.M.; Klajnert-Maculewicz, B. Application of new lysine-based peptide dendrimers D3K2 and D3G2 for gene delivery: Specific cytotoxicity to cancer cells and transfection in vitro. Bioorganic Chem.
**2020**, 95, 103504. [Google Scholar] [CrossRef] - Gorzkiewicz, M.; Kopeć, O.; Janaszewska, A.; Konopka, M.; Pȩdziwiatr-Werbicka, E.; Tarasenko, I.I.; Bezrodnyi, V.V.; Neelov, I.M.; Klajnert-Maculewicz, B. Poly(lysine) Dendrimers Form Complexes with siRNA and Provide Its Efficient Uptake by Myeloid Cells: Model Studies for Therapeutic Nucleic Acid Delivery. Int. J. Mol. Sci.
**2020**, 21, 3138. [Google Scholar] [CrossRef] - Shi, X.; Lesniak, W.; Islam, M.T.; MuÑiz, M.C.; Balogh, L.P.; Baker, J.R. Comprehensive characterization of surface-functionalized poly(amidoamine) dendrimers with acetamide, hydroxyl, and carboxyl groups. Colloids Surfaces A Physicochem. Eng. Asp.
**2006**, 272, 139–150. [Google Scholar] [CrossRef] - Trinchi, A.; Muster, T.H. A Review of Surface Functionalized Amine Terminated Dendrimers for Application in Biological and Molecular Sensing. Supramol. Chem.
**2007**, 19, 431–445. [Google Scholar] [CrossRef] - Caminade, A.M.; Turrin, C.O. Dendrimers for drug delivery. J. Mater. Chem. B
**2014**, 2, 4055–4066. [Google Scholar] [CrossRef] - Sheveleva, N.N.; Markelov, D.A.; Vovk, M.A.; Mikhailova, M.E.; Tarasenko, I.I.; Neelov, I.M.; Lähderanta, E. NMR studies of excluded volume interactions in peptide dendrimers. Sci. Rep.
**2018**, 8, 8916. [Google Scholar] [CrossRef] - Sheveleva, N.N.; Markelov, D.A.; Vovk, M.A.; Tarasenko, I.I.; Mikhailova, M.E.; Ilyash, M.Y.; Neelov, I.M.; Lahderanta, E. Stable Deuterium Labeling of Histidine-Rich Lysine-Based Dendrimers. Molecules
**2019**, 24, 2481. [Google Scholar] [CrossRef] [PubMed] - Sheveleva, N.N.; Markelov, D.A.; Vovk, M.A.; Mikhailova, M.E.; Tarasenko, I.I.; Tolstoy, P.M.; Neelov, I.M.; Lähderanta, E. Lysine-based dendrimer with double arginine residues. RSC Adv.
**2019**, 9, 18018–18026. [Google Scholar] [CrossRef] - Yang, H.; Lopina, S.T. Penicillin V-conjugated PEG-PAMAM star polymers. J. Biomater. Sci. Polym. Ed.
**2003**, 14, 1043–1056. [Google Scholar] [CrossRef] - Luong, D.; Kesharwani, P.; Deshmukh, R.; Mohd Amin, M.C.I.; Gupta, U.; Greish, K.; Iyer, A.K. PEGylated PAMAM dendrimers: Enhancing efficacy and mitigating toxicity for effective anticancer drug and gene delivery. Acta Biomater.
**2016**, 43, 14–29. [Google Scholar] [CrossRef] - Liu, X.; Liu, C.; Zhou, J.; Chen, C.; Qu, F.; Rossi, J.J.; Rocchi, P.; Peng, L. Promoting siRNA delivery via enhanced cellular uptake using an arginine-decorated amphiphilic dendrimer. Nanoscale
**2015**, 7, 3867–3875. [Google Scholar] [CrossRef] - Xiong, Y.; Ke, R.; Zhang, Q.; Lan, W.; Yuan, W.; Chan, K.N.I.; Roussel, T.; Jiang, Y.; Wu, J.; Liu, S.; et al. Small Activating RNA Modulation of the G Protein-Coupled Receptor for Cancer Treatment. Adv. Sci.
**2022**, 9, 2200562. [Google Scholar] [CrossRef] - Borisov, O.V.; Zhulina, E.B.; Leermakers, F.A.; Müller, A.H. Self-assembled structures of amphiphilic ionic block copolymers: Theory, self-consistent field modeling and experiment. In Self Organized Nanostructures of Amphiphilic Block Copolymers I; Muller, A.H., Borisov, O.V., Eds.; Springer: Berlin/Heidelberg, Germany, 2011; pp. 57–129. [Google Scholar] [CrossRef]
- Zhang, Q.; Lin, J.; Wang, L.; Xu, Z. Theoretical modeling and simulations of self-assembly of copolymers in solution. Prog. Polym. Sci.
**2017**, 75, 1–30. [Google Scholar] [CrossRef] - Diaz, J.; Pinna, M.; Zvelindovsky, A.V.; Pagonabarraga, I. Hybrid Time-Dependent Ginzburg-Landau Simulations of Block Copolymer Nanocomposites: Nanoparticle Anisotropy. Polymers
**2022**, 14, 1910. [Google Scholar] [CrossRef] - Tan, H.; Wang, W.; Yu, C.; Zhou, Y.; Lu, Z.; Yan, D. Dissipative particle dynamics simulation study on self-assembly of amphiphilic hyperbranched multiarm copolymers with different degrees of branching. Soft Matter
**2015**, 11, 8460–8470. [Google Scholar] [CrossRef] - Tan, H.; Yu, C.; Lu, Z.; Zhou, Y.; Yan, D. A dissipative particle dynamics simulation study on phase diagrams for the self-assembly of amphiphilic hyperbranched multiarm copolymers in various solvents. Soft Matter
**2017**, 13, 6178–6188. [Google Scholar] [CrossRef] - Lebedeva, I.O.; Zhulina, E.B.; Borisov, O.V. Theory of linear–dendritic block copolymer micelles. ACS Macro Lett.
**2018**, 7, 42–46. [Google Scholar] [CrossRef] [PubMed] - Lebedeva, I.O.; Zhulina, E.B.; Borisov, O.V. Self-assembly of linear-dendritic and double dendritic block copolymers: From dendromicelles to dendrimersomes. Macromolecules
**2019**, 52, 3655–3667. [Google Scholar] [CrossRef] - Suek, N.W.; Lamm, M.H. Computer Simulation of Architectural and Molecular Weight Effects on the Assembly of Amphiphilic Linear-Dendritic Block Copolymers in Solution. Langmuir
**2008**, 24, 3030–3036. [Google Scholar] [CrossRef] [PubMed] - Lin, Y.L.; Chang, H.Y.; Sheng, Y.J.; Tsao, H.K. Photoresponsive Polymersomes Formed by Amphiphilic Linear–Dendritic Block Copolymers: Generation-Dependent Aggregation Behavior. Macromolecules
**2012**, 45, 7143–7156. [Google Scholar] [CrossRef] - Lin, C.M.; Li, C.S.; Sheng, Y.J.; Wu, D.T.; Tsao, H.K. Size-Dependent Properties of Small Unilamellar Vesicles Formed by Model Lipids. Langmuir
**2012**, 28, 689–700. [Google Scholar] [CrossRef] - Márquez-Miranda, V.; Araya-Durán, I.; Camarada, M.B.; Comer, J.; Valencia-Gallegos, J.A.; González-Nilo, F.D. Self-Assembly of Amphiphilic Dendrimers: The Role of Generation and Alkyl Chain Length in siRNA Interaction. Sci. Rep.
**2016**, 6, 29436. [Google Scholar] [CrossRef] - Milchev, A.; Bhattacharya, A.; Binder, K. Formation of Block Copolymer Micelles in Solution: A Monte Carlo Study of Chain Length Dependence. Macromolecules
**2001**, 34, 1881–1893. [Google Scholar] [CrossRef] - Cooke, I.R.; Deserno, M. Solvent-free model for self-assembling fluid bilayer membranes: Stabilization of the fluid phase based on broad attractive tail potentials. J. Chem. Phys.
**2005**, 123, 224710. [Google Scholar] [CrossRef] [Green Version] - Cooke, I.R.; Kremer, K.; Deserno, M. Tunable generic model for fluid bilayer membranes. Phys. Rev. E
**2005**, 72, 011506. [Google Scholar] [CrossRef] - Kremer, K.; Grest, G.S. Dynamics of entangled linear polymer melts: A molecular-dynamics simulation. J. Chem. Phys.
**1990**, 92, 5057–5086. [Google Scholar] [CrossRef] - Weik, F.; Weeber, R.; Szuttor, K.; Breitsprecher, K.; de Graaf, J.; Kuron, M.; Landsgesell, J.; Menke, H.; Sean, D.; Holm, C. ESPResSo 4.0—An extensible software package for simulating soft matter systems. Eur. Phys. J. Spec. Top.
**2019**, 227, 1789–1816. [Google Scholar] [CrossRef] - Available online: www.espressomd.org (accessed on 29 January 2023).
- Zhulina, E.B.; Borisov, O.V. Theory of Block Polymer Micelles: Recent Advances and Current Challenges. Macromolecules
**2012**, 45, 4429–4440. [Google Scholar] [CrossRef] - de Gennes, P.J. Solid State Physics; Academic Press: New York, NY, USA, 1978; p. 1. [Google Scholar]
- Zhulina, Y.B.; Birshtein, T.M. Conformations of block-copolymer molecules in selective solvents (micellar structures). Polym. Sci. USSR
**1985**, 27, 570. [Google Scholar] [CrossRef] - Halperin, A. Polymeric micelles: A star model. Macromolecules
**1987**, 20, 2943. [Google Scholar] [CrossRef] - Birshtein, T.M.; Zhulina, E.B. Scaling theory of supermolecular structures in block copolymer-solvent systems: 1. Model of micellar structures. Polymer
**1989**, 30, 170. [Google Scholar] [CrossRef] - Evans, D.F.; Wennerström, H. The Colloidal Domain: Where Physics, Chemistry, Biology, and Technology Meet; Wiley: Hoboken, NJ, USA, 1999. [Google Scholar]
- Shavykin, O.V.; Leermakers, F.A.M.; Neelov, I.M.; Darinskii, A.A. Self-Assembly of Lysine-Based Dendritic Surfactants Modeled by the Self-Consistent Field Approach. Langmuir
**2018**, 34, 1613–1626. [Google Scholar] [CrossRef] - LaRue, I.; Adam, M.; Zhulina, E.B.; Rubinstein, M.; Pitsikalis, M.; Hadjichristidis, N.; Ivanov, D.A.; Gearba, R.I.; Anokhin, D.V.; Sheiko, S.S. Effect of the Soluble Block Size on Spherical Diblock Copolymer Micelles. Macromolecules
**2008**, 41, 6555–6563. [Google Scholar] [CrossRef] - Pickett, G.T. Classical Path Analysis of end-Grafted Dendrimers: Dendrimer Forest. Macromolecules
**2001**, 34, 8784. [Google Scholar] [CrossRef] - Zook, T.C.; Pickett, G.T. Hollow-Core Dendrimers Revised. Phys. Rev. Lett.
**2003**, 90, 015502. [Google Scholar] [CrossRef] - Zhulina, E.B.; Leermakers, F.A.M.; Borisov, O.V. Ideal mixing in multicomponent brushes of branched macromolecules. Macromolecules
**2015**, 48, 8025. [Google Scholar] [CrossRef]

**Figure 1.**WCA potential (red) and hydrophobic–hydrophobic potential, ${V}_{\mathrm{BB}}$ (blue), as a function of the interparticle separation. The arrow indicates the width ${w}_{c}$ of the attractive part of ${V}_{\mathrm{BB}}$.

**Figure 2.**(

**a**) Sketch of a linear–dendritic block copolymer. (

**b**) Exemplary initial configuration. (

**c**) Exemplary micelle at equilibrium.

**Figure 3.**Total energy per macromolecule (left) and asphericity (right) versus aggregation number. The arrows indicate the equilibrium aggregation number. Other system parameters: ${N}_{\mathrm{A}}=25$, ${N}_{\mathrm{B}}=5$ ($g=0$), ${w}_{c}=1.6$, ${k}_{\mathrm{B}}T=1.0$, ${\u03f5}_{\mathrm{attr}}=1.0$.

**Figure 4.**Equilibrium size versus (

**a**) ${\u03f5}_{\mathrm{attr}}$ and versus (

**b**) ${w}_{c}$. Other system parameters: (

**a**) ${N}_{\mathrm{A}}=25$, ${N}_{\mathrm{B}}=5$ ($g=0$), ${w}_{c}=1.6$ in (

**a**), ${k}_{\mathrm{B}}T=1.0$, ${\u03f5}_{\mathrm{attr}}=1.0$ in (

**b**).

**Figure 5.**(

**a**) Equilibrium aggregation number and (

**b**) equilibrium radius of gyration versus degree of polymerization of the hydrophilic block ${N}_{\mathrm{A}}$. (

**c**) Normalized equilibrium aggregation number versus normalized degree of polymerization of the hydrophilic block. We consider linear neutral block copolymers ($g=0$) with ${N}_{\mathrm{B}}=5$. Continuous lines correspond to power law behavior predictions from mean-field theory and dash-dotted lines correspond to predictions from scaling theory [42]. In (

**b**), circles and triangles are results for micelle and core radii of gyration, respectively, from simulations, while blue and green lines correspond to corona and core size scaling predictions from theory. In (

**c**), filled circles correspond to simulation results and empty symbols to experimental data from Ref. [66] for linear block copolymers formed from polystyrene-b-polyisoprene with different lengths of polystyrene block, namely 39 kDa and 19 kDa. Other system parameters: ${w}_{c}=1.8$, ${\u03f5}_{\mathrm{attr}}=0.6$ and ${k}_{\mathrm{B}}T=1.0$.

**Figure 6.**(

**a**) Equilibrium aggregation number and (

**b**) equilibrium radius of gyration versus degree of polymerization of the hydrophobic block ${N}_{\mathrm{B}}$. We consider linear neutral block copolymers with ${N}_{\mathrm{A}}=30$. Continuous lines correspond to power law behavior predictions from mean-field theory [42], and dash-dotted lines correspond to predictions from scaling theory [59]. In (

**b**), circles and triangles are results for micelle and core radii of gyration, respectively, from simulations, while blue and green lines correspond to corona and core size scaling predictions from theory. Other system parameters: ${w}_{c}=1.8$, ${\u03f5}_{\mathrm{attr}}=1.0$ and ${k}_{\mathrm{B}}T=1.0$.

**Figure 7.**(

**a**) Equilibrium aggregation number and (

**b**) equilibrium radius of gyration versus degree of polymerization of the hydrophilic block ${N}_{\mathrm{A}}$. We consider linear charged block copolymers ($g=0$) with ${N}_{\mathrm{B}}=5$. Lines correspond to scaling behavior predictions from scaling theory [42]. Blue lines correspond to the limit of small aggregation number, where $p\ll {\alpha}^{-1/2}{\lambda}_{\mathrm{B}}^{-1}\sigma $; meanwhile, the green line corresponds to the limit of large aggregation number, where $p\gg {\alpha}^{-1/2}{\lambda}_{\mathrm{B}}^{-1}\sigma $. Other system parameters: ${w}_{c}=2.5$, ${\u03f5}_{\mathrm{attr}}=1.0$, ${k}_{\mathrm{B}}T=1.0$, ${Z}_{\mathrm{A}}={N}_{\mathrm{A}}$ and ${Z}_{\mathrm{B}}=0$.

**Figure 8.**(

**a**) Equilibrium aggregation number and (

**b**) equilibrium radius of gyration versus degree of polymerization of the hydrophobic block ${N}_{\mathrm{B}}$. We consider linear–dendritic neutral block copolymer with ${N}_{\mathrm{A}}=120$ ($g=3$, $q=3$, ${N}_{\mathrm{s}}=3$). Lines correspond to scaling behavior predictions from mean-field SCF theory [47]. In (

**b**), the blue line corresponds to the scaling of the corona size $\propto {N}_{\mathrm{B}}^{2/11}$ and the green line to the scaling of the core $\propto {N}_{\mathrm{B}}^{7/11}$. Other system parameters: ${w}_{c}=1.8;2.5$, ${\u03f5}_{\mathrm{attr}}=1.0$ and ${k}_{\mathrm{B}}T=1.0$.

**Figure 9.**(

**a**) Equilibrium aggregation number and (

**b**) equilibrium radius of gyration versus degree of polymerization of the hydrophilic block ${N}_{\mathrm{A}}$. We consider linear–dendritic neutral block copolymer with ${N}_{\mathrm{B}}=15$. Different type of dots correspond to simulation results with variations in ${N}_{A}$ by changing the indicated parameter, see Equation (5): red circle indicates variation of $g=0,1,2,3$, with $q=3$ and ${N}_{s}=15$; orange square for variation of $q=1,2,3$ with $g=3$ and ${N}_{s}=9$; and green triangle variation of ${N}_{s}=2,3,4,5,6$ with $g=3$ and ${N}_{s}=3$. Lines correspond to scaling behavior predictions from mean-field SCF theory [47]. In (

**b**), line corresponds to the scaling of the corona size $\propto {N}_{\mathrm{A}}^{6/11}$. Other system parameters: ${w}_{c}=2.5$, ${\u03f5}_{\mathrm{attr}}=1.0$ and ${k}_{\mathrm{B}}T=1.0$. con.

**Figure 10.**(

**a**) Equilibrium aggregation number and (

**b**) equilibrium radius of gyration versus degree of polymerization of the hydrophobic block ${N}_{\mathrm{B}}$. We consider linear–dendritic charged block copolymers with ${N}_{\mathrm{A}}=14$ ($g=2$, $q=2$, ${N}_{\mathrm{s}}=2$). In (

**b**), circle corresponds to the radius of gyration of the whole micelle and triangles to the radius of gyration of the core. Dashed lines correspond to least-squared-method fittings, while dotted lines correspond to scaling predictions from SCF theory for linear block copolymers with ionic hydrophilic block: (

**a**) equilibrium aggregation number ${p}_{\mathrm{eq}}\propto {N}_{\mathrm{B}}^{2/3}$ for small aggregation number, with $p\ll {\alpha}^{-1/2}{\lambda}_{\mathrm{B}}^{-1}\sigma $, and ${p}_{\mathrm{eq}}\propto {N}_{\mathrm{B}}^{2}$ for large aggregation number, with $p\gg {\alpha}^{-1/2}{\lambda}_{\mathrm{B}}^{-1}\sigma $; (

**b**) core $R\propto {N}_{\mathrm{B}}$ and corona independent of ${N}_{\mathrm{B}}$ [42]. Other system parameters: ${w}_{c}=2.5$, ${\u03f5}_{\mathrm{attr}}=1.0$ and ${k}_{\mathrm{B}}T=1.0$.

**Figure 11.**(

**a**) Equilibrium aggregation number and (

**b**) equilibrium radius of gyration versus degree of polymerization of the hydrophilic block ${N}_{\mathrm{A}}$. We consider linear–dendritic charged block copolymers. Different type of dots correspond to simulation results with variation in ${N}_{A}$ by changing the indicated parameter, see Equation (5): red circle indicates variation of $g=0,1$, with $q=2$, ${N}_{s}=5$ and ${N}_{B}=5$; orange square for variation of $q=1,2$ with $g=2$, ${N}_{s}=5$ and ${N}_{B}=5$; and green triangle for variation of ${N}_{s}=3,4,5,6,7$ with $g=2$, ${N}_{s}=2$ and ${N}_{B}=10$. In (

**a**), gray circles correspond to simulation results from Figure 7 for linear charged block copolymers. Dotted lines correspond to scaling predictions from SCF theory for linear block copolymers with ionic hydrophilic blocks [42]: Blue lines correspond to the limit of small aggregation number, where $p\ll {\alpha}^{-1/2}{\lambda}_{\mathrm{B}}^{-1}\sigma $; meanwhile, the green line corresponds to the limit of large aggregation number, where $p\gg {\alpha}^{-1/2}{\lambda}_{\mathrm{B}}^{-1}\sigma $. Other system parameters: ${w}_{c}=2.5$, ${\u03f5}_{\mathrm{attr}}=1.0$ and ${k}_{\mathrm{B}}T=1.0$.

**Figure 12.**Equilibrium aggregation number and equilibrium radius of gyration for different block copolymers architectures. For insets (

**a**,

**b**), case 1 is plotted, with ${N}_{\mathrm{A}}=21$ and ${N}_{\mathrm{B}}=10$ and a topology of the hydrophilic block according to the following: system 1—$g=0$ and ${N}_{\mathrm{s}}=21$, corresponding to a linear block copolymer; system 2—$g=1$, $q=2$ and ${N}_{\mathrm{s}}=7$; and system 3—$g=2$, $q=2$ and ${N}_{\mathrm{s}}=3$. For insets (

**c**,

**d**), case 2 is plotted, with ${N}_{\mathrm{A}}=60$ and ${N}_{\mathrm{B}}=20$ and topology of the hydrophilic block according to the following: system 1—$g=0$ and ${N}_{\mathrm{s}}=60$, corresponding to a linear block copolymer; system 2—$g=1$, $q=2$ and ${N}_{\mathrm{s}}=20$; system 3—$g=1$, $q=3$ and ${N}_{\mathrm{s}}=15$; and system 4—$g=3$, $q=2$ and ${N}_{\mathrm{s}}=4$. In (

**b**,

**d**), full bar represents the radius of gyration of the whole micelle, while light blue and light red bars correspond to the radius of gyration of only the core. Other system parameters: ${w}_{c}=1.8$, ${\u03f5}_{\mathrm{attr}}=1$, ${Z}_{\mathrm{A}}=21$ and ${Z}_{\mathrm{B}}=0$, for charged case.

**Figure 13.**Density profiles for different particle/bead types. In (

**a**,

**b**), results for case 1 are plotted for neutral and charged hydrophilic blocks, respectively. In (

**c**,

**d**), results for case 2 are plotted for neutral and charged hydrophilic blocks, respectively. Hydrophobic beads are represented by lightly colored continuous lines, hydrophilic beads by colored continuous lines and counterions are represented by colored dashed lines. Other system parameters as in Figure 12.

**Table 1.**Sketch representation of the simulated linear–dendritic block copolymers of case 1 and case 2.

System 1 | System 2 | System 3 | System 4 | |
---|---|---|---|---|

Case 1 | ||||

${N}_{\mathrm{A}}=21$, ${N}_{\mathrm{B}}=10$ | $g=0$, ${N}_{\mathrm{s}}=21$ | $g=1$, $q=2$, ${N}_{\mathrm{s}}=7$ | $g=2$, $q=2$, ${N}_{\mathrm{s}}=3$ | |

Case 2 | ||||

${N}_{\mathrm{A}}=60$, ${N}_{\mathrm{B}}=20$ | $g=0$, ${N}_{\mathrm{s}}=60$ | $g=1$, $q=2$, ${N}_{\mathrm{s}}=20$ | $g=1$, $q=3$, ${N}_{\mathrm{s}}=15$ | $g=3$, $q=2$, ${N}_{\mathrm{s}}=4$ |

**Table 2.**Ratio of counterions inside the corona ${Q}_{\mathrm{in}}$ and micelle net charge ${Z}_{\mathrm{net}}$ for micelles from Figure 13b.

${\mathit{Q}}_{\mathbf{in}}$ | ${\mathit{Z}}_{\mathbf{net}}$ [1/e] | ||
---|---|---|---|

Case 1 | system 1 | $0.856\pm 0.005$ | $39\pm 1$ |

system 2 | $0.82\pm 0.01$ | $31\pm 1$ | |

system 3 | $0.77\pm 0.01$ | $24\pm 1$ | |

Case 2 | system 1 | $0.852\pm 0.002$ | $87\pm 1$ |

system 2 | $0.834\pm 0.005$ | $69\pm 2$ | |

system 3 | $0.81\pm 0.01$ | $58\pm 3$ | |

system 4 | $0.80\pm 0.01$ | $37\pm 2$ |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Brito, M.E.; Mikhtaniuk, S.E.; Neelov, I.M.; Borisov, O.V.; Holm, C.
Implicit-Solvent Coarse-Grained Simulations of Linear–Dendritic Block Copolymer Micelles. *Int. J. Mol. Sci.* **2023**, *24*, 2763.
https://doi.org/10.3390/ijms24032763

**AMA Style**

Brito ME, Mikhtaniuk SE, Neelov IM, Borisov OV, Holm C.
Implicit-Solvent Coarse-Grained Simulations of Linear–Dendritic Block Copolymer Micelles. *International Journal of Molecular Sciences*. 2023; 24(3):2763.
https://doi.org/10.3390/ijms24032763

**Chicago/Turabian Style**

Brito, Mariano E., Sofia E. Mikhtaniuk, Igor M. Neelov, Oleg V. Borisov, and Christian Holm.
2023. "Implicit-Solvent Coarse-Grained Simulations of Linear–Dendritic Block Copolymer Micelles" *International Journal of Molecular Sciences* 24, no. 3: 2763.
https://doi.org/10.3390/ijms24032763