Dendrimers vs. Hyperbranched Polymers: Studies of the Polymerization Process Based on Monte Carlo Simulations
Polanowski Piotr 1, Hałagan Krzysztof 1, Sikorski Andrzej 2*
1 Lodz University of Technology
Department of Molecular Physics
ul. Żeromskiego 116, 90-924 Łódź, Poland2 University of Warsaw
Faculty of Chemistry
ul. Pasteura 1, 02-093 Warsaw, Poland
∗E-mail: sikorski@chem.uw.edu.pl
Received:
Received: 9 September 2022; accepted: 15 September 2022; published online: 28 September 2022
DOI: 10.12921/cmst.2022.0000020
Abstract:
A simple model was developed for studies of the polymerization process of branched polymers. Monte Carlo simulations were carried out by means of the Dynamic Lattice Liquid algorithm. A living polymerization in bulk of dendrimers and hyperbranched polymers was studied. The mass and structure of both types of macromolecules were investigated. The influence of the functionality of well-defined cores on the structure of the system was also examined. The differences in the kinetics in the formation of both architectures and changes to their structures were discussed. It was shown that both architecture exhibit low dispersity which was found to be higher for hyperbranched macromolecules.
Key words:
dendrimers, Dynamic Lattice Liquid model, hyperbranched polymers, lattice models, Monte Carlo method
References:
[1] D.A. Tomalia, J.M.J. Frechet, Discovery of dendrimers and dendritic polymers: A brief historical perspective, J. Polym. Sci. Part A Polym. Chem. 40, 2719–2728 (2002).
[2] D. Astruc, E. Boisselier, C. Ornelas, Dendrimers designed for functions: From physical, photophysical, and supramolecular properties to applications in sensing, catalysis, molecular electronics, photonics, and nanomedicine, Chem. Rev. 110, 1857–1959 (2010).
[3] C. Gao, D. Yan, Hyperbranched polymers: from synthesis to applications, Progress in Polymer Science 29, 183–275 (2004).
[4] S.E. Seo, C.J. Hawker, The beauty of branching in polymer science, Macromolecules 53, 3257–3261 (2020).
[5] D.A. Tomalia, Birth of a new macromolecular architecture: Dendrimers as quantized building blocks for nanoscale synthetic polymer chemistry, Prog. Polym. Sci. 30, 294–324 (2005).
[6] D.A. Tomalia, Dendrons/Dendrimers: Quantized, nanoelement like building blocks for soft-soft and soft-hard nanocompound synthesis, Soft Matter 6, 456–474 (2009).
[7] D.A. Tomalia, J.B. Christensen, U. Boas, Dendrimers, Dendrons and Dendritic Polymers: Discovery, Applications, and the Future, Cambridge University Press: Cambridge, UK (2012).
[8] C.J. Hawker, J.M.J. Frechet, Preparation of polymers with controlled molecular architecture: A new convergent approach to dendritic macromolecules, J. Am. Chem. Soc. 112, 7638–7647 (1990).
[9] A.K. Patri, I.J. Majoros, J.R. Baker, Dendritic polymer macromolecular carriers for drug delivery, Curr. Opin. Chem. Biol. 6, 466–471 (2002).
[10] R. Hourani, A. Kakkar, Advances in the elegance of chemistry in designing dendrimers, Macromol. Rapid Commun. 31, 947–974 (2010).
[11] K. Inoue, Functional dendrimers, hyperbranched and stars polymers, Prog. Polym. Sci. 25, 453–471 (2000).
[12] P. Kesharwani, K. Jain, N.K. Jain, Dendrimer as nanocarrier for drug delivery, Prog. Polym. Sci. 39, 268–307 (2014).
[13] S. Mignani, S. El Kazzouli, M. Bousmina, J.P. Majoral, Dendrimer space concept for innovative nanomedicine: A futuristic vision for medicinal chemistry, Prog. Polym. Sci., 38, 993–1008 (2013).
[14] P.M. Maiti, T. Çagın, G. Wang, W.A Goddard, Structure of PAMAM dendrimers: Generations 1 through 11, Macromolecules 37, 6236–6254 (2004).
[15] P. Polanowski, K. Hałagan, A. Sikorski, Star Polymers vs. Dendrimers – Studies on the synthesis based on computer simulations, Polymers 14, 2522 (2022).
[16] G.A. Pilkington, J.S. Pedersen, W.H. Briscoe, Dendrimer nanofluids in the concentrated regime: From polymer melts to soft spheres, Langmuir 31, 3333–3342 (2015).
[17] J.M.J. Frechet, Functional polymers and sendrimers: Reactivity, molecular architecture, and interfacial energy, Science 263, 1710–1715 (1994).
[18] T.H. Mourey, S.R. Turner, M. Rubinstein, J.M.J. Frechet,
C.J. Hawker, K.L. Wooley, Unique behavior of dendritic macromolecules: Intrinsic viscosity of polyether Dendrimers, Macromolecules 25, 2401–2406 (1992).
[19] A.W. Bosman, H.M. Janssen, E.W. Meijer, About dendrimers: Structure, physical properties, and applications, Chem. Rev. 99, 1665–1688 (1999).
[20] R. La Ferla, Conformations and dynamics of dendrimers and cascade macromolecules, J. Chem. Phys. 106, 688–700 (1997).
[21] Y.J. Sheng, S.Y. Jiang, H.K. Tsao, Radial size of a starburst dendrimer in solvents of varying quality, Macromolecules 35, 7865–7868 (2002).
[22] C.N. Likos, M. Ballauff, Equilibrium structure of dendrimer: Results and open questions, Top. Curr. Chem. 245, 239–252 (2005).
[23] M. Ballauff, C.N. Likos, Dendrimers in solution: Insight from theory and simulation, Angew. Chem. Int. Ed. 43, 2998–3020 (2004).
[24] G. Giupponi, D.M.A. Buzza, Monte Carlo simulation of dendrimers in variable solvent quality, J. Chem. Phys. 120, 10290–10298 (2004).
[25] J.S. Kłos, J.U. Sommer, Properties of dendrimers with flexible spacer-chains: A Monte Carlo study, Macromolecules 42, 4878–4886 (2009).
[26] E. Wawrzynska, S. Eisenhaber, P. Parzuchowski, A. Sikorski, G. Zifferer, Simulation of hyperbranched polymers. 1. Properties of regular three generation dendrimers, Macromol. Theory Simul. 23, 288–299 (2014).
[27] E. Wawrzynska, A. Sikorski, G. Zifferer, Monte Carlo simulation studies of regular and irregular dendritic polymers, Macromol. Theory Simul. 24, 477–489 (2015).
[28] W.D. Tian, Y.Q. Ma, Coarse-grained molecular simulation of interacting dendrimers, Soft Matter 7, 500–505 (2011).
[29] S. Kanchi, S. Suresh, U.D. Priyakumar, K.G Ayappa, P.K. Maiti, Molecular Dynamics study of the structure, flexibility, and hydrophilicity of PETIM dendrimers: A comparison with PAMAM dendrimers, J. Phys. Chem. B 119, 12990–13001 (2015).
[30] K. Karatasos, D.B. Adolf, G.R. Davies, Statics and dynamics of model dendrimers as studied by Molecular Dynamics simulations, J. Chem. Phys. 115, 5310–5318 (2001).
[31] A.O. Kurbatov, N.K. Balabaev, M.A. Mazo, E.Y. Kramarenko, Molecular Dynamics simulations of single siloxane dendrimers: Molecular structure and intramolecular mobility of terminal groups, J. Chem. Phys. 148, 014902 (2018).
[32] D.A. Markelov, A.N. Shishkin, V.V. Matveev, A.V. Penkova, E. Lähderanta, V.I. Chizhik, Orientational mobility in dendrimer melts: Molecular Dynamics simulations, Macromolecules 49, 9247–9257 (2018).
[33] F. Khabaz, R. Khare, Effect of chain architecture on the size, shape, and intrinsic viscosity of chains in polymer solutions: A molecular simulation study, J. Chem. Phys. 141, 214904 (2014).
[34] C. Yu, L. Ma, K. Li, S. Li, Y. Liu, Y. Zhou, D. Yan, Molecular dynamics simulation studies of hyperbranched polyglycerols and their encapsulation behaviors of small drug molecules, Phys. Chem. Chem. Phys. 32, 22446–22457 (2016).
[35] M.L. Mansfield, L.I. Klushin, Monte Carlo studies of dendrimer macromolecules, Macromolecules 26, 4262–4268 (1993).
[36] E.G. Timoshenko, Y.A. Kuznetsov, R. Connolly, Conformations of dendrimers in dilute solution, J. Chem. Phys. 117, 9050–9062 (2002).
[37] C.R. Yates, W. Hayes, Synthesis and applications of hyperbranched polymers, Eur. Polym. J. 40, 1257–1281 (2000).
[38] M. Seiler, Hyperbranched polymers: Phase behavior and new applications in the field of chemical engineering, Fluid Phase Equil. 241, 155–174 (2006).
[39] B.I. Voit, Hyperbranched polymers: a chance and a challenge, C. R. Chimie 4, 821–832 (2003).
[40] I.-Y. Jeon, H.-J. Noh, J.-B. Baek, Hyperbranched macromolecules: From synthesis to applications, Molecules 23, 657 (2018).
[41] T. Higashihara, Y. Segawa, W. Sinananwanich, M. Ueda, Synthesis of hyperbranched polymers with controlled degree of branching, Polym. J. 44, 14–29 (2012).
[42] B.I. Voit, A. Lederer, Hyperbranched and Highly Branched Polymer Architectures-Synthetic Strategies and Major Characterization Aspects, Chem. Rev. 109, 5924–5973 (2009).
[43] A. Kumar, G.J. Rai, P. Biswas, Conformation and intramolecular relaxation dynamics of semiflexible randomly hyperbranched polymers, J. Chem. Phys. 138, 104902 (2013).
[44] L. Li, Y. Lu, L. An, C. Wu, Experimental and theoretical studies of scaling of sizes and intrinsic viscosity of hyperbranched chains in good solvents, J. Chem. Phys. 138, 114908 (2013).
[45] A. Jurjiu, R. Dockhorn, O. Mironova, J.-U. Sommer, Two universality classes for random hyperbranched polymers, Soft Matter 10, 4935–4946 (2014).
[46] D. Konkolewicz, R.G. Gilbert, A. Gray-Weale, Randomly hyperbranched polymers, Phys. Rev. Lett. 98, 238301 (2007).
[47] D. Konkolewicz, O. Thorn-Seshold, A. Gray-Weale, Models for randomly hyperbranched polymers: Theory and simulation, J. Chem. Phys. 129, 054901 (2008).
[48] D. Konkolewicz, A. Gray-Weale, S. Perrier, Describing the structure of a randomly hyperbranched polymer, Macromol. Theory Simul. 19, 219–227 (2010).
[49] D.M.A. Buzza, Power law polydispersity and fractal structure of hyperbranched polymers, Eur. Phys J. E 13, 79–86 (2004).
[50] E.L. Richards, D. Martin, A. Buzza, G.R. Davies, Monte Carlo simulation of random branching in hyperbranched polymers, Macromolecules 40, 2210–2218 (2007).
[51] P.F. Sheridan, D.B. Adolf, A.V. Lyulin, I. Neelov, G.R. Davis, Computer simulations of hyperbranched polymers: The influence of the Wiener index on the intrinsic viscosity and radius of gyration, J. Chem. Phys. 117, 7802–7812 (2002).
[52] S.V. Lyulin, K. Karatasos, A.A. Darinskii, S. Larin, A. Lyulin, Structural effects in overcharging in complexes of hyperbranched polymers with linear polyelectrolytes, Soft Matter 4, 453–457 (2008).
[53] S.V. Lyulin, E.V. Reshetnikov, A.A. Darinskii, A.V. Lyulin, Structural behavior of hyperbranched polymers in solvents of various qualities: Brownian Dynamics simulation, Polym. Sci. A 53, 827–845 (2011).
[54] T.C. Le, B.D. Todd, P.J. Davis, A. Uhlherr, The effect of interbranch spacing on structural and rheological properties of hyperbranched polymer melts, J. Chem. Phys. 131, 164901 (2009).
[55] I.M. Neelov, D.B. Adolf, Brownian Dynamics simulation of hyperbranched polymers under elongational flow, J. Phys. Chem. B 108, 7627–7636 (2004).
[56] A.V. Lyulin, D.B. Adolf, G.R. Davies, Computer simulations of hyperbranched polymers in shear flows, Macromolecules 34, 3783–3789 (2001).
[57] T. Pakula, Simulation on the completely occupied lattices, [In:] Simulation Methods for Polymers, Eds. M. Kotelyanskii, D.N. Theodorou, Marcel Dekker, New York, NY, USA, Basel, Switzerland (2004).
[58] M. von Smoluchowski, Versuch einer Mathematischen Theorie der Koagulations Kinetic Kolloider Lousungen, Phys. Z. Chem. 92, 129–168 (1917).
[59] R. Jullien, R. Botet, Aggregation and Fractal Aggregate, World Scientific, Singapore (1987).
[60] M. Tirado-Miranda, A. Schmitt, J. Callejas-Fernández, A. Frenández-Barbero, Dynamic scaling and fractal structure of small colloidal clusters, Colloid. Surface A 162, 67–73 (2000).
[61] P. Van Dongen, M.H. Ernst, Dynamic scaling in the kinetics of clustering, Phys. Rev. Lett. 54, 1396–1399 (1985).
[62] M. Rubinstein, R.H. Colby, Polymer Physics, Oxford University Press, Oxford, UK (2003).
A simple model was developed for studies of the polymerization process of branched polymers. Monte Carlo simulations were carried out by means of the Dynamic Lattice Liquid algorithm. A living polymerization in bulk of dendrimers and hyperbranched polymers was studied. The mass and structure of both types of macromolecules were investigated. The influence of the functionality of well-defined cores on the structure of the system was also examined. The differences in the kinetics in the formation of both architectures and changes to their structures were discussed. It was shown that both architecture exhibit low dispersity which was found to be higher for hyperbranched macromolecules.
Key words:
dendrimers, Dynamic Lattice Liquid model, hyperbranched polymers, lattice models, Monte Carlo method
References:
[1] D.A. Tomalia, J.M.J. Frechet, Discovery of dendrimers and dendritic polymers: A brief historical perspective, J. Polym. Sci. Part A Polym. Chem. 40, 2719–2728 (2002).
[2] D. Astruc, E. Boisselier, C. Ornelas, Dendrimers designed for functions: From physical, photophysical, and supramolecular properties to applications in sensing, catalysis, molecular electronics, photonics, and nanomedicine, Chem. Rev. 110, 1857–1959 (2010).
[3] C. Gao, D. Yan, Hyperbranched polymers: from synthesis to applications, Progress in Polymer Science 29, 183–275 (2004).
[4] S.E. Seo, C.J. Hawker, The beauty of branching in polymer science, Macromolecules 53, 3257–3261 (2020).
[5] D.A. Tomalia, Birth of a new macromolecular architecture: Dendrimers as quantized building blocks for nanoscale synthetic polymer chemistry, Prog. Polym. Sci. 30, 294–324 (2005).
[6] D.A. Tomalia, Dendrons/Dendrimers: Quantized, nanoelement like building blocks for soft-soft and soft-hard nanocompound synthesis, Soft Matter 6, 456–474 (2009).
[7] D.A. Tomalia, J.B. Christensen, U. Boas, Dendrimers, Dendrons and Dendritic Polymers: Discovery, Applications, and the Future, Cambridge University Press: Cambridge, UK (2012).
[8] C.J. Hawker, J.M.J. Frechet, Preparation of polymers with controlled molecular architecture: A new convergent approach to dendritic macromolecules, J. Am. Chem. Soc. 112, 7638–7647 (1990).
[9] A.K. Patri, I.J. Majoros, J.R. Baker, Dendritic polymer macromolecular carriers for drug delivery, Curr. Opin. Chem. Biol. 6, 466–471 (2002).
[10] R. Hourani, A. Kakkar, Advances in the elegance of chemistry in designing dendrimers, Macromol. Rapid Commun. 31, 947–974 (2010).
[11] K. Inoue, Functional dendrimers, hyperbranched and stars polymers, Prog. Polym. Sci. 25, 453–471 (2000).
[12] P. Kesharwani, K. Jain, N.K. Jain, Dendrimer as nanocarrier for drug delivery, Prog. Polym. Sci. 39, 268–307 (2014).
[13] S. Mignani, S. El Kazzouli, M. Bousmina, J.P. Majoral, Dendrimer space concept for innovative nanomedicine: A futuristic vision for medicinal chemistry, Prog. Polym. Sci., 38, 993–1008 (2013).
[14] P.M. Maiti, T. Çagın, G. Wang, W.A Goddard, Structure of PAMAM dendrimers: Generations 1 through 11, Macromolecules 37, 6236–6254 (2004).
[15] P. Polanowski, K. Hałagan, A. Sikorski, Star Polymers vs. Dendrimers – Studies on the synthesis based on computer simulations, Polymers 14, 2522 (2022).
[16] G.A. Pilkington, J.S. Pedersen, W.H. Briscoe, Dendrimer nanofluids in the concentrated regime: From polymer melts to soft spheres, Langmuir 31, 3333–3342 (2015).
[17] J.M.J. Frechet, Functional polymers and sendrimers: Reactivity, molecular architecture, and interfacial energy, Science 263, 1710–1715 (1994).
[18] T.H. Mourey, S.R. Turner, M. Rubinstein, J.M.J. Frechet,
C.J. Hawker, K.L. Wooley, Unique behavior of dendritic macromolecules: Intrinsic viscosity of polyether Dendrimers, Macromolecules 25, 2401–2406 (1992).
[19] A.W. Bosman, H.M. Janssen, E.W. Meijer, About dendrimers: Structure, physical properties, and applications, Chem. Rev. 99, 1665–1688 (1999).
[20] R. La Ferla, Conformations and dynamics of dendrimers and cascade macromolecules, J. Chem. Phys. 106, 688–700 (1997).
[21] Y.J. Sheng, S.Y. Jiang, H.K. Tsao, Radial size of a starburst dendrimer in solvents of varying quality, Macromolecules 35, 7865–7868 (2002).
[22] C.N. Likos, M. Ballauff, Equilibrium structure of dendrimer: Results and open questions, Top. Curr. Chem. 245, 239–252 (2005).
[23] M. Ballauff, C.N. Likos, Dendrimers in solution: Insight from theory and simulation, Angew. Chem. Int. Ed. 43, 2998–3020 (2004).
[24] G. Giupponi, D.M.A. Buzza, Monte Carlo simulation of dendrimers in variable solvent quality, J. Chem. Phys. 120, 10290–10298 (2004).
[25] J.S. Kłos, J.U. Sommer, Properties of dendrimers with flexible spacer-chains: A Monte Carlo study, Macromolecules 42, 4878–4886 (2009).
[26] E. Wawrzynska, S. Eisenhaber, P. Parzuchowski, A. Sikorski, G. Zifferer, Simulation of hyperbranched polymers. 1. Properties of regular three generation dendrimers, Macromol. Theory Simul. 23, 288–299 (2014).
[27] E. Wawrzynska, A. Sikorski, G. Zifferer, Monte Carlo simulation studies of regular and irregular dendritic polymers, Macromol. Theory Simul. 24, 477–489 (2015).
[28] W.D. Tian, Y.Q. Ma, Coarse-grained molecular simulation of interacting dendrimers, Soft Matter 7, 500–505 (2011).
[29] S. Kanchi, S. Suresh, U.D. Priyakumar, K.G Ayappa, P.K. Maiti, Molecular Dynamics study of the structure, flexibility, and hydrophilicity of PETIM dendrimers: A comparison with PAMAM dendrimers, J. Phys. Chem. B 119, 12990–13001 (2015).
[30] K. Karatasos, D.B. Adolf, G.R. Davies, Statics and dynamics of model dendrimers as studied by Molecular Dynamics simulations, J. Chem. Phys. 115, 5310–5318 (2001).
[31] A.O. Kurbatov, N.K. Balabaev, M.A. Mazo, E.Y. Kramarenko, Molecular Dynamics simulations of single siloxane dendrimers: Molecular structure and intramolecular mobility of terminal groups, J. Chem. Phys. 148, 014902 (2018).
[32] D.A. Markelov, A.N. Shishkin, V.V. Matveev, A.V. Penkova, E. Lähderanta, V.I. Chizhik, Orientational mobility in dendrimer melts: Molecular Dynamics simulations, Macromolecules 49, 9247–9257 (2018).
[33] F. Khabaz, R. Khare, Effect of chain architecture on the size, shape, and intrinsic viscosity of chains in polymer solutions: A molecular simulation study, J. Chem. Phys. 141, 214904 (2014).
[34] C. Yu, L. Ma, K. Li, S. Li, Y. Liu, Y. Zhou, D. Yan, Molecular dynamics simulation studies of hyperbranched polyglycerols and their encapsulation behaviors of small drug molecules, Phys. Chem. Chem. Phys. 32, 22446–22457 (2016).
[35] M.L. Mansfield, L.I. Klushin, Monte Carlo studies of dendrimer macromolecules, Macromolecules 26, 4262–4268 (1993).
[36] E.G. Timoshenko, Y.A. Kuznetsov, R. Connolly, Conformations of dendrimers in dilute solution, J. Chem. Phys. 117, 9050–9062 (2002).
[37] C.R. Yates, W. Hayes, Synthesis and applications of hyperbranched polymers, Eur. Polym. J. 40, 1257–1281 (2000).
[38] M. Seiler, Hyperbranched polymers: Phase behavior and new applications in the field of chemical engineering, Fluid Phase Equil. 241, 155–174 (2006).
[39] B.I. Voit, Hyperbranched polymers: a chance and a challenge, C. R. Chimie 4, 821–832 (2003).
[40] I.-Y. Jeon, H.-J. Noh, J.-B. Baek, Hyperbranched macromolecules: From synthesis to applications, Molecules 23, 657 (2018).
[41] T. Higashihara, Y. Segawa, W. Sinananwanich, M. Ueda, Synthesis of hyperbranched polymers with controlled degree of branching, Polym. J. 44, 14–29 (2012).
[42] B.I. Voit, A. Lederer, Hyperbranched and Highly Branched Polymer Architectures-Synthetic Strategies and Major Characterization Aspects, Chem. Rev. 109, 5924–5973 (2009).
[43] A. Kumar, G.J. Rai, P. Biswas, Conformation and intramolecular relaxation dynamics of semiflexible randomly hyperbranched polymers, J. Chem. Phys. 138, 104902 (2013).
[44] L. Li, Y. Lu, L. An, C. Wu, Experimental and theoretical studies of scaling of sizes and intrinsic viscosity of hyperbranched chains in good solvents, J. Chem. Phys. 138, 114908 (2013).
[45] A. Jurjiu, R. Dockhorn, O. Mironova, J.-U. Sommer, Two universality classes for random hyperbranched polymers, Soft Matter 10, 4935–4946 (2014).
[46] D. Konkolewicz, R.G. Gilbert, A. Gray-Weale, Randomly hyperbranched polymers, Phys. Rev. Lett. 98, 238301 (2007).
[47] D. Konkolewicz, O. Thorn-Seshold, A. Gray-Weale, Models for randomly hyperbranched polymers: Theory and simulation, J. Chem. Phys. 129, 054901 (2008).
[48] D. Konkolewicz, A. Gray-Weale, S. Perrier, Describing the structure of a randomly hyperbranched polymer, Macromol. Theory Simul. 19, 219–227 (2010).
[49] D.M.A. Buzza, Power law polydispersity and fractal structure of hyperbranched polymers, Eur. Phys J. E 13, 79–86 (2004).
[50] E.L. Richards, D. Martin, A. Buzza, G.R. Davies, Monte Carlo simulation of random branching in hyperbranched polymers, Macromolecules 40, 2210–2218 (2007).
[51] P.F. Sheridan, D.B. Adolf, A.V. Lyulin, I. Neelov, G.R. Davis, Computer simulations of hyperbranched polymers: The influence of the Wiener index on the intrinsic viscosity and radius of gyration, J. Chem. Phys. 117, 7802–7812 (2002).
[52] S.V. Lyulin, K. Karatasos, A.A. Darinskii, S. Larin, A. Lyulin, Structural effects in overcharging in complexes of hyperbranched polymers with linear polyelectrolytes, Soft Matter 4, 453–457 (2008).
[53] S.V. Lyulin, E.V. Reshetnikov, A.A. Darinskii, A.V. Lyulin, Structural behavior of hyperbranched polymers in solvents of various qualities: Brownian Dynamics simulation, Polym. Sci. A 53, 827–845 (2011).
[54] T.C. Le, B.D. Todd, P.J. Davis, A. Uhlherr, The effect of interbranch spacing on structural and rheological properties of hyperbranched polymer melts, J. Chem. Phys. 131, 164901 (2009).
[55] I.M. Neelov, D.B. Adolf, Brownian Dynamics simulation of hyperbranched polymers under elongational flow, J. Phys. Chem. B 108, 7627–7636 (2004).
[56] A.V. Lyulin, D.B. Adolf, G.R. Davies, Computer simulations of hyperbranched polymers in shear flows, Macromolecules 34, 3783–3789 (2001).
[57] T. Pakula, Simulation on the completely occupied lattices, [In:] Simulation Methods for Polymers, Eds. M. Kotelyanskii, D.N. Theodorou, Marcel Dekker, New York, NY, USA, Basel, Switzerland (2004).
[58] M. von Smoluchowski, Versuch einer Mathematischen Theorie der Koagulations Kinetic Kolloider Lousungen, Phys. Z. Chem. 92, 129–168 (1917).
[59] R. Jullien, R. Botet, Aggregation and Fractal Aggregate, World Scientific, Singapore (1987).
[60] M. Tirado-Miranda, A. Schmitt, J. Callejas-Fernández, A. Frenández-Barbero, Dynamic scaling and fractal structure of small colloidal clusters, Colloid. Surface A 162, 67–73 (2000).
[61] P. Van Dongen, M.H. Ernst, Dynamic scaling in the kinetics of clustering, Phys. Rev. Lett. 54, 1396–1399 (1985).
[62] M. Rubinstein, R.H. Colby, Polymer Physics, Oxford University Press, Oxford, UK (2003).