Simulations of Neutral and Charged Dendrimers in Solvents of Varying Quality

Analytical Model to Deduce the Conformational and Dynamical Behavior in Dendrimers: A Review

This review utilizes an optimized Rouse-Zimm discrete hydrodynamic model and the preaveraged Oseen tensor, which accurately consider hydrodynamic interactions to study model dendrimers. We report the analytical theories that have been previously developed for the creation of generalized analytical models for dendrimers. These generalized theories were used to assess the conformational and dynamical behavior of the dendrimers. By including stiffness in the bonds, the neglect of excluded volume interactions may be somewhat offset. This is true at least in the case of short spacers. While the topological limitations on the directions and orientations of the individual bond vectors in dendrimers implement semiflexibility, the intensity of these contacts was determined by the potential geometric orientations of the bonds, and later on the excluded volume interactions in dendrimers, which were described in terms of the effective co-volume between nearest non-bonded monomers and modeled using the delta function pseudopotential. With the aid of the models developed, the authors condensed various conformational and dynamic properties of dendrimers that depend on their degree of semiflexibility and the strength of the excluded volume. These analyses came to the conclusion that the flexible dendrimer in one limit and the earlier described freely rotating model of dendrimers in the other constitute a highly generalized way of capturing a wide range of conformations in the developed mathematical model in dendrimers.

Simulation of Nondilute Dendrimer Systems with the Bond Fluctuation Model

Using the bond fluctuation model, we performed Monte Carlo simulations for solutions of generation 4 and 5 dendrimers with only an intermediate unit between the branching points at different concentrations, ranging from moderately dilute solutions to concentrated systems close to the melt behavior. This model may represent different real types of dendrimer families. We obtained the mean sizes, asphericities, displacement of units, scattering functions, radial distribution functions and structure factors. We compared the results obtained for the last two properties with much faster Monte Carlo simulations of point-like dendrimers using global potentials obtained through the study of binary interactions. The latter procedure provided good reproductions of these properties but failed in the reproduction of the scattering functions in the range of higher concentrations. In this range, the scattering function cannot be described as the product of the structure function and the form factor, because the intensity decreases when the density of the dendrimer units becomes more homogenous.

Binding mechanisms in dendrimer-surfactant complexes

Molecular dynamics simulations were employed to investigate the impact of interactions between dendritic polyeclectrolytes and amphiphilic surfactants on the supramolecular complex formation. We recognize two crucial parameters that govern association of surfactants within dendrimers: surfactant hydrophobicity, ε^{*}, and dendrimer generation, G. We find that depending on the values of ε^{*} and G encapsulation of surfactants by dendrimers is either noncooperative or cooperative. The noncooperative binding is characterized by absorption of surfactants as unimers, whereas in cooperative binding absorption of unimers is followed by aggregate formation through hydrophobic attractions between the surfactant tails. Our results provide guidelines for controlled encapsulation of guest molecules in dendrimer-based guest-host complexes.

The self-diffusion of polymethylsilsesquioxane (PMSSO) dendrimers in diluted solutions and melts

Recently developed non-functional derivatives of polymethylsilsesquioxane (PMSSO) dendrimers of the first to fifth generation were characterized by 1H, 13C and 29Si NMR spectroscopy. The self-diffusion and NMR relaxation of PMSSO dendrimers in dilute solutions of toluene and melts were investigated in a wide temperature range (-50-80 °C). The hydrodynamic radii of dendrimers were determined from the self-diffusion coefficients measured in diluted solutions according to the Stokes-Einstein equation. The hydrodynamic radius of PMSSO dendrimers as a function of molecular mass follows a power law with the scaling exponent of 0.32 ± 0.02 in the investigated temperature range. The temperature dependences of the self-diffusion coefficients of dendrimers were described by the Arrhenius-type equation. The activation energies of self-diffusion of dendrimers in diluted toluene solutions are identical for different generations while the dependence of activation energy for dendrimers in melts shows a maximum for the third generation (G3) dendrimer. Taking into account the absence of specific interactions in PMSSO dendrimer melts the observed behavior was ascribed to the manifestation of interpenetration of dendrimer molecules. For low generations (G1 and G2) the short length of the branches does not considerably affect the translational diffusion while for higher generations (G4 and G5) the densification of the structure prevents significant interpenetration.

Understanding the Thermodynamics of the Binding of PAMAM Dendrimers to Graphene: A Combined Analytical and Simulation Study

We investigate the thermodynamics of the binding of a poly(amidoamine) dendrimer to an uncharged graphene sheet as a function of the pH level using umbrella sampling simulations and a mean-field theory for generations three and four. We find that the dendrimer strongly binds to the graphene sheet ( O (100) kcal/mol) from our potential of mean force (PMF) calculations. In specific, we find that the dendrimer binds the most at neutral pH (∼7) and the least at low pH (∼4). We explain this nonmonotonic nature of the dendrimer's adsorption by studying the interactions contributing to the PMF, i.e., the dendrimer-graphene, dendrimer-water, and dendrimer-ion interactions. We also corroborate our PMF calculations with molecular mechanics generalized Born surface area analysis and free energies obtained from a mean-field theory of Flory-Huggins-Debye-Hückel type [ Muthukumar , M. , J. Chem. Phys. 2010 , 132 , 084901 ], including electrostatic interactions. We find that the van der Waals interactions between the dendrimer and the graphene alone cannot capture the accurate trends in the binding free energies (BEs) as a function of pH. The solvent and the counterions present in the system are also found to have a major influence on these trends. We demonstrate that the dendrimer-graphene and dendrimer-water interactions become favorable, whereas the dendrimer-ion interaction becomes unfavorable, as the dendrimer binds to graphene. These opposing effects lead to the observed nonmonotonicity in the BE trends. Our theoretical model also reproduces these trends in the subinteractions contributing to the PMF. To the best of our knowledge, this is a novel attempt where an equivalence between theory and simulations is made in the context of the dendrimer's adsorption.

Dynamics of a Polymer Network Modeled by a Fractal Cactus

In this paper, we focus on the relaxation dynamics of a polymer network modeled by a fractal cactus. We perform our study in the framework of the generalized Gaussian structure model using both Rouse and Zimm approaches. By performing real-space renormalization transformations, we determine analytically the whole eigenvalue spectrum of the connectivity matrix, thereby rendering possible the analysis of the Rouse-dynamics at very large generations of the structure. The evaluation of the structural and dynamical properties of the fractal network in the Rouse type-approach reveals that they obey scaling and the dynamics is governed by the value of spectral dimension. In the Zimm-type approach, the relaxation quantities show a strong dependence on the strength of the hydrodynamic interaction. For low and medium hydrodynamic interactions, the relaxation quantities do not obey power law behavior, while for slightly larger interactions they do. Under strong hydrodynamic interactions, the storage modulus does not follow power law behavior and the average displacement of the monomer is very low. Remarkably, the theoretical findings with respect to scaling in the intermediate domain of the relaxation quantities are well supported by experimental results from the literature.

Charge and hydration structure of dendritic polyelectrolytes: molecular simulations of polyglycerol sulphate

Macromolecules based on dendritic or hyperbranched polyelectrolytes have been emerging as high potential candidates for biomedical applications. Here we study the charge and solvation structure of dendritic polyglycerol sulphate (dPGS) of generations 0 to 3 in aqueous sodium chloride solution by explicit-solvent molecular dynamics computer simulations. We characterize dPGS by calculating several important properties such as relevant dPGS radii, molecular distributions, the solvent accessible surface area, and the partial molecular volume. In particular, as the dPGS exhibits high charge renormalization effects, we address the challenges of how to obtain a well-defined effective charge and surface potential of dPGS for practical applications. We compare implicit- and explicit-solvent approaches in our all-atom simulations with the coarse-grained simulations from our previous work. We find consistent values for the effective electrostatic size (i.e., the location of the effective charge of a Debye-Hückel sphere) within all the approaches, deviating at most by the size of a water molecule. Finally, the excess chemical potential of water insertion into dPGS and its thermodynamic signature are presented and rationalized.

Dynamics of a Complex Multilayer Polymer Network: Mechanical Relaxation and Energy Transfer

In this paper, we focus on the mechanical relaxation of a multilayer polymer network built by connecting identical layers that have, as underlying topologies, the dual Sierpinski gasket and the regular dendrimer. Additionally, we analyze the dynamics of dipolar energy transfer over a system of chromophores arranged in the form of a multilayer network. Both dynamical processes are studied in the framework of the generalized Gaussian structure (GSS) model. We develop a method whereby the whole eigenvalue spectrum of the connectivity matrix of the multilayer network can be determined iteratively, thereby rendering possible the analysis of the dynamics of networks consisting of a large number of layers. This fact allows us to study in detail the crossover from layer-like behavior to chain-like behavior. Remarkably, we highlight the existence of two bulk-like behaviors. The theoretical findings with respect to the decomposition of the intermediate domain of the relaxation quantities, as well as the chain-like behavior, are well supported by experimental results.

Dendritic polyelectrolytes revisited through the Poisson-Boltzmann-Flory theory and the Debye-Hückel approximation

The properties of a dendritic polyelectrolyte in equilibrium with a reservoir of monovalent salts are investigated using the cell model and the Poisson-Boltzmann-Flory theory. Within this approach we use the Debye-Hückel approximation to solve the Poisson-Boltzmann equation and minimize the semi-grand potential of the system with respect to the size of the molecule which enables us to inspect its conformations as well as the electric field, the ionic density profile, the overall charge density, the effective charge of the dendrimer and the osmotic pressure based on their response to the salt concentration and the dendrimer charge. The model predicts pronounced trapping of salt ions, a local charge neutrality and a zero electric field in the volume of the molecule as well as oscillations of the density profiles and the electric field in the vicinity of the dendrimer-bulk interface. As a result of ion trapping and screening of Coulomb interactions monovalent salts are found to have a minor effect on the size of the dendrimer. Specifically, the dendrimer exists in slightly swollen states as compared to the neutral molecule which indicates that the conformational properties of the polyelectrolyte depend weakly on monovalent salts. These observations harmonise with the equilibrium behavior of the dendrimer pressure, the internal pressure and the bulk pressure, respectively.

Molecular dynamics simulations of single siloxane dendrimers: Molecular structure and intramolecular mobility of terminal groups

Molecular dynamics simulations of two types of isolated siloxane dendrimers of various generations (from the 2nd to the 8th) have been performed for temperatures ranging from 150 K to 600 K. The first type of dendrimer molecules has short spacers consisting of a single oxygen atom. In the dendrimers of the second type, spacers are longer and comprised of two oxygen atoms separated by a single silicon atom. A comparative analysis of molecular macroscopic parameters such as the gyration radius and the shape factor as well as atom distributions within dendrimer interior has been performed for varying generation number, temperature, and spacer length. It has been found that the short-spacer dendrimers of the 7th and 8th generations have a stressed central part with elongated bonds and deformed valence angles. Investigation of the time evolution of radial displacements of the terminal Si atoms has shown that a fraction of the Si groups have a reduced mobility. Therefore, rather long time trajectories (of the order of tens of nanoseconds) are required to study dendrimer intramolecular dynamics.

Dendritic polyelectrolytes as seen by the Poisson–Boltzmann–Flory theory

The conformational and electrostatic properties of dendritic polyelectrolytes accompanied by counterions are investigated using the Poisson–Boltzmann–Flory theory.

Dynamics of a Polymer Network Based on Dual Sierpinski Gasket and Dendrimer: A Theoretical Approach

In this paper we focus on the relaxation dynamics of a multihierarchical polymer network built through the replication of the dual Sierpinski gasket in the form of a regular dendrimer. The relaxation dynamics of this multihierarchical structure is investigated in the framework of the generalized Gaussian structure model using both Rouse and Zimm approaches. In the Rouse-type approach, we show a method whereby the whole eigenvalue spectrum of the connectivity matrix of the multihierarchical structure can be determined iteratively, thereby rendering possible the analysis of the Rouse-dynamics at very large generations. Remarkably, the general picture that emerges from both approaches, even though we have a mixed growth algorithm and the monomers interactions are taken into account specifically to the adopted approach, is that the multihierarchical structure preserves the individual relaxation behaviors of its constituent components. The theoretical findings with respect to the splitting of the intermediate domain of the relaxation quantities are well supported by experimental results.

Relaxation dynamics of a multihierarchical polymer network

In this work, we study the relaxation dynamics of a multihierarchical polymer network built by replicating the Vicsek fractal in dendrimer shape. The relaxation dynamics is investigated in the framework of the generalized Gaussian structure model by employing both Rouse and Zimm approaches. In the Rouse-type approach, we show the iterative procedure whereby the whole eigenvalue spectrum of the connectivity matrix of the multihierarchical structure can be obtained. Remarkably, the general picture that emerges from both approaches, even though we have a mixed growth algorithm, is that the obtained multihierarchical structure preserves the individual relaxation behaviors of its components. The theoretical findings with respect to the splitting of the intermediate domain of the relaxation quantities are well supported by experimental results.

Relaxation dynamics of Sierpinski hexagon fractal polymer: Exact analytical results in the Rouse-type approach and numerical results in the Zimm-type approach

In this paper, we focus on the relaxation dynamics of Sierpinski hexagon fractal polymer. The relaxation dynamics of this fractal polymer is investigated in the framework of the generalized Gaussian structure model using both Rouse and Zimm approaches. In the Rouse-type approach, by performing real-space renormalization transformations, we determine analytically the complete eigenvalue spectrum of the connectivity matrix. Based on the eigenvalues obtained through iterative algebraic relations we calculate the averaged monomer displacement and the mechanical relaxation moduli (storage modulus and loss modulus). The evaluation of the dynamical properties in the Rouse-type approach reveals that they obey scaling in the intermediate time/frequency domain. In the Zimm-type approach, which includes the hydrodynamic interactions, the relaxation quantities do not show scaling. The theoretical findings with respect to scaling in the intermediate domain of the relaxation quantities are well supported by experimental results.

Dendrimer solutions: a Monte Carlo study

We study the conformational properties of dendrimers with flexible spacers in solutions over a wide range of concentrations from dilute solutions to melts. By combining large scale computer simulations using the bond fluctuation model with scaling arguments we identify the semi-dilute regime of dendrimers which is controlled by the concentration behavior of the linear spacers. Associated with this observation we find that the decrease in the size of flexible dendrimers is accompanied by increasing interpenetration between the molecules with increasing concentration of the solution. In the melt state we show that the size of individual dendrimers follows the scaling prediction for isolated dendrimers at the θ-point rather than that of collapsed dendrimers. The pair correlation functions between the centers of dendrimers indicate that for short spacers dendrimer solutions retain the morphological characteristics of simple liquids. For long spacers the functions reveal high penetration of neighboring dendrimers in the melt state. Our studies show that flexible dendrimers in solution can be understood with arguments similar to those of linear polymers. The role of generation is to influence the particular form of the crossover-function.

Hydrodynamic behaviors of amphiphilic dendritic polymers with different degrees of amidation

This work highlights the structure evolution and the response to solvent quality of ADPs.

Molecular dynamics simulation studies of hyperbranched polyglycerols and their encapsulation behaviors of small drug molecules

Computer simulation could disclose more details about the conformations of HPGs and their encapsulation behaviors of guest molecules.

Charged dendrimers under the action of AC electric fields: breathing characteristics of molecular size, polarizations, and ion distributions

Langevin dynamics simulations are performed to study the response of charged dendrimers in alternating current electric fields in 3:1 salt solutions. Time evolutions of molecular size show breathing characteristics which take saw-tooth-like patterns in square-wave electric fields and undulated sine-function ones in sine-wave fields. Detailed study reveals how the dendrimer and condensed ions oscillate in the electric fields, which result in polarization of the molecule. To effect a significant deformation of the dendrimer, the applied field amplitude must be larger than some critical strength Ecrit and the field frequency smaller than a threshold fcrit. The response behavior is characterized by two relaxation times in square-wave fields, both of which decrease linearly with the strong field strength larger than Ecrit. In sine-wave fields, the molecular size exhibits interesting hysteretic behavior in plotting the curves with the field variation. A Maxwell-Wagner type polarization theory is derived and proved by simulations, which connects fcrit with the strength of the applied electric field.

Adsorption of branched and dendritic polymers onto flat surfaces: a Monte Carlo study

Using Monte Carlo simulations based on the bond fluctuation model we study the adsorption of starburst dendrimers with flexible spacers onto a flat surface. The calculations are performed for various generation number G and spacer length S in a wide range of the reduced temperature τ as the measure of the interaction strength between the monomers and the surface. Our simulations indicate a two-step adsorption scenario. Below the critical point of adsorption, τc, a weakly adsorbed state of the dendrimer is found. Here, the dendrimer retains its shape but sticks to the surface by adsorbed spacers. By lowering the temperature below a spacer-length dependent value, τ*(S) < τc, a step-like transition into a strongly adsorbed state takes place. In the flatly adsorbed state the shape of the dendrimer is well described by a mean field model of a dendrimer in two dimensions. We also performed simulations of star-polymers which display a simple crossover-behavior in full analogy to linear chains. By analyzing the order parameter of the adsorption transition, we determine the critical point of adsorption of the dendrimers which is located close to the critical point of adsorption for star-polymers. While the order parameter for the adsorbed spacers displays a critical crossover scaling, the overall order parameter, which combines both critical and discontinuous transition effects, does not display simple scaling. The step-like transition from the weak into the strong adsorbed regime is confirmed by analyzing the shape-anisotropy of the dendrimers. We present a mean-field model based on the concept of spacer adsorption which predicts a discontinuous transition of dendrimers due to an excluded volume barrier. The latter results from an increased density of the dendrimer in the flatly adsorbed state which has to be overcome before this state is thermodynamically stable.

Charged dendrimers in trivalent salt solutions under the action of DC electric fields

The structural properties and electrophoretic mobility of charged dendrimers in 3:1 electrolyte solutions subjected to direct current electric fields are studied using molecular dynamics simulations. The simulated dendrimer size is studied in zero fields and found to scale as R(g) ∼ N(0.29). The dendrimers exhibit shape distortions when the applied electric field is larger than some critical value, which scales with the number of dendrimer monomers as E(z,crit) ∼ N(0.39(6)). Families of curves, such as the curves of the square of radius of gyration, the asphericity, the degree of prolateness, and the electrophoretic mobility of dendrimers, are shown to collapse to single, master curves in electric fields through appropriate scaling. This reflects the fractal characteristics of these systems. The density profile of the surface monomers and salt cations reveals two pronounced combination effects between the polarization of dendrimer complexes and stripping-off of the condensed salt cations from the dendrimer surface.