ВЫСОКОМОЛЕКУЛЯРНЫЕ СОЕДИНЕНИЯ, Серия C, 2013, том 55, № 7, с. 933-962
COARSE GRAINED SIMULATIONS OF NEUTRAL AND CHARGED DENDRIMERS
© 2013 t. J. S. K+os" c and J.-U. Sommer" b
a Leibniz Institute of Polymer Research Dresden e. V., 01069 Dresden, Germany b Institute for Theoretical Physics, Technische Universitat Dresden, 01069 Dresden, Germany c Faculty of Physics, A. Mickiewicz University, Umultowska 85, 61-614 Poznan, Poland
Abstract — Dendrimers are macromolecules with a regular-treelike, branched architecture of their skeleton. In terms of the branching number and the number of terminal groups they represent an extreme case among branched polymers. Dendrimers can occur in neutral and various charged states. Due to their highly branched architecture excluded volume effects are of great importance and conformational properties and monomer distribution profiles of dendrimers differ considerably from those of linear polymers. We give an overview of the state-of-the-art knowledge of physical properties of dendrimers as seen from coarse-grained computer simulations. Our main focus is on isolated dendrimers with flexible spacers both in the neutral and in the charged state, as well as complexation of dendrimers with oppositely charged linear polyelectrolytes. We briefly address problems of adsorption and concentration effects in dendrimer solutions and outline recent progress and open questions in this field.
Starburst dendrimers are macromolecules consisting of linear chains (called spacers) arranged in a hierarchical, treelike structure [8, 147]. The most widespread methods of making such highly branched molecules by chemical synthesis are referred to as divergent and convergent, respectively. In the former approach, starting from the core, new branched units of monomers are attached to the outermost units so as to form the next generation. In the latter, first the branched arms (dendrons) are prepared, and finally attached to the multifunctional core [35, 36]. The dendritic architecture is characterized by three parameters: The total number of generations G, spacer length S, functionality of the branching units f, and degree of polymerization N (S, G, f). Dendrimers exhibit the highest fraction of end groups per monomer units among all possible polymer architectures. The latter can be functionalized which makes dendrimers particularly interesting for applications. Due to the highly branched structure of dendrimers their molecular weigth N grows exponentially with G and linearly
with S: N = ASeaG, where the constants A and a are given by the dendrimer's architecture.
Apart from purely scientific value dendrimers prove useful in industry, biomedicine, pharmacy and materials engineering. To name but a few, lithographic materials, nanoscale catalysts, drug delivery systems, rheol-
ogy modifiers, bioadhesives, and MRI contrast agents are examples of their potential applications [53, 62, 93, 136]. Dendrimers were used to deliver oligonucleotides to the cell [16, 156], they enhance cytosolic and nuclear availability as indicated by confocal microscopy as well as cell uptake and transfection efficiency of plasmid DNA . Guest-host nanodevices such as gold/PAMAM (polyamidoamine) nanocomposites are potentially very useful agents for improving the imaging and radiation treatment of cancer .
Theoretical studies on dendrimers make use of mean-field models, self-consistent methods, renor-malization group techniques and Flory-type approaches. On- and off-lattice simulations, on the other hand, involve the kinetic self-avoiding-walk, Brownian/molecular dynamics and Monte Carlo algorithms. These methods enable a direct insight into conformational properties of dendrimers, spatial distribution of monomers and terminal groups, and phase properties [4, 6, 13, 15, 23, 26, 29, 46, 51, 66, 67, 76, 81, 91, 100, 101, 104, 105, 122, 126, 134, 145, 152, 158, 159]. Simulations allow an inspection of dynamical behaviour of dendrimers as well. Actually, dendrimer translational self-diffusion, the size and shape fluctuations, rotational mobility, and elastic motions were considered [51, 77, 80].
Experiments, on the other hand, employ photochemical and spectroscopic probe methods, mass spectrometry, translational diffusion and viscometry [41, 113, 153]. Furthermore, transmission electron
(TEM) and atomic force microscopy (AFM) as well as small-angle neutron (SANS) and X-ray (SAXS) scattering methods are used to elucidate the shape and internal structure of dendrimers [40, 68, 69, 115, 116, 118, 122, 128].
The research on dendrimers is not limited to neutral molecules since their properties can also be tuned by electrostatic effects. With this respect, of increasing interest are weak dendritic polyelectrolytes whose charge in solution can be modulated by changing the solution pH. For instance, PAMAM and poly(propy-leneimine) dendrimers acquire positive charge because they have primary amine groups at the terminal units and tertiary amine groups at the branching points which become protonated as the solution pH-value is lowered from around 7 down to 4 [43, 90, 92, 108, 112, 150]. In other words, under physiological conditions only the terminal groups bear positive charges, whereas in more acidic environments both the terminal groups and branching groups are charged.
Whether or not the above charge modulation has a pronounced effect on conformational and structural properties of charged dendrimers has been the subject of scientific deliberation for over a decade. For instance, simulations indicate that the dendrimer size increases as pH decreases from neutral to low, although the actual value of the swelling parameter depends on the applied model. Within the the Debye-Huckel approximation which treats free ions implicitly the radius of gyration was shown to increase by as much as 70% , whereas other approaches with explicit ions lead to a much weaker dependence of dendrimer conformations on ionic strength [63, 90, 92, 110]. Molecular dynamics simulations taking counterions into account explicitly as well as mean field theory [25, 74] predict much lower swelling in agreement with small-angle neutron scattering (SANS) measurements on PAMAMs [11, 108].
It is therefore apparent that for dendritic polyelectrolytes the degrees of freedom of counterions have to be taken into account explicitly due to the importance of ion valence, ion trapping in the dendrimer volume, ion condensation and its effect on conformational changes of dendrimers. Actually, as the strength of electrostatic interactions increases both under neutral and low pH conditions counterions penetrate not only the molecule periphery but, in the first place, its interior [2, 5, 22, 25, 34, 55, 73, 94-96, 141]. This leads to the reduction of the dendrimer effective charge, screening of electrostatic repulsion between charged monomers and non-monotonous behavior of den-drimer size with the strength of electrostatic interactions. Last but not least ions also play a crucial role in the formation of complexes comprised of charged dendrimers and linear polyanions and in self-organization in solutions of charged dendrimers [47-49, 88, 89, 138, 142], though many important aspects of com-
plexation such as dendrimer overcharging by linear polyelectrolytes were revealed within the Debye-Huckel approach as well [61, 78, 84, 87]. Obviously, it is dendrimer-DNA binding [33, 39, 44, 45, 64, 103, 124, 127, 131, 133, 137], which being of immense interest in biosciences inspires theoretical investigations.
Physical properties of charged dendrimers were the subject of intensive experimental studies [11, 109, 120, 157]. For instance, small-angle X-ray scattering (SAXS) and conductivity measurements were made for dilute solutions of PAMAM molecules with univalent and divalent counterions . Among others the latter showed that divalent counterions are more strongly condensed on the dendrimers and thus more effective in reducing their charge. As indicated by small-angle neutron scattering (SANS), SAXS and transmission electron microscopy PAMAM dendrim-ers are useful for forming gold nano-clusters within their interior [31, 32]. A number of EPR and UV-vis spectroscopy measurements demonstarated that at various temperatures and pH divalent metal ions can be distributed both inside and outside the molecule [111, 151]. Furthermore, apart from dendrimers' ability to encapsulate smaller molecules, with the use of optical reflectometry, atomic force microscopy (AFM), light scattering, SANS and electrophoretic mobility measurements a number of recent works were devoted to their adsorption properties on silica surfaces and latex particles as well as to formations of defined supramolecular assemblies based on ionic interaction in aqueous solution [9, 72, 114, 125].
In this review we focus on the application of coarse grained simulation models to explore universal properties of dendrimers. In the next section we discuss the properties of isolated and neutral dendrimers.
ISOLATED NEUTRAL DENDRIMERS
In this section we give an overview of the confor-mational properties of neutral, isolated dendrimers with excluded volume as seen from theory and coarsed grained computer simulations. The first theoretical work on self-avoiding dendrimers in an athermal solvent was done by de Gennes and Hervet . Using a modified version of the Edwards self consistent field method the authors considered dendrimers composed of very long spacers with trifunctional branching monomers. Based on the assumption that all the segments from a given generation are localized in a shell around the molecule's center they arrived at radial monomer density profiles which increase strictly monotonously from the center towards the molecules' periphery. Additionally, they found that the radius of gyration, Rg, of dendrimers scales with the degree of poly
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