^{1}, Oliver Thorn-Seshold

^{1}and Angus Gray-Weale

^{1,a)}

### Abstract

We derive theoretical models for the structures of randomly hyperbranched polymers in solution, and test them against computer simulations. The models are based on the same basic approach: Building a structure by the random assembly of “simple units,” which may be monomers, linear chains, or larger branched species. Comparisons to simulation reported here show that the conformations of hyperbranched species, i.e., their radii of gyration and full density profiles, are accurately described by this approach. These stringent tests complement previous tests against experiment. We include the effects of solvent quality at the mean-field level. Our model works best for hyperbranched structures, but also reproduces very well the simulated density profiles of dendrimers. The models reported here provide a simple, but realistic, picture of the physical influences that affect the conformations of hyperbranched species.

A.G.-W. would like to thank the Australian Research Council for a grant, and the Australian Centre for Advanced Computing and Communications for computer time. We are all very grateful to Professor R. G. Gilbert and Dr. Patrice Castignolles for many essential discussions and for their time and advice.

I. INTRODUCTION

II. THEORY

A. Neglect of repulsive forces

B. Simple units

1. Oligomeric simple units

2. -particles as simple units

3. Other possible simple units

4. A limiting case

C. Mean-field treatment of interactions

1. Entropic contribution

2. Enthalpic contribution

D. Random selection of simple units

E. Summary of equations to be solved

III. SIMULATIONS

IV. RESULTS

A. Testing the probability of addition

B. Radii of gyration

C. Density profiles

1. Monte Carlo simulations

2. Molecular dynamics simulations of dendrimers and hyperbranched polymers

3. Molecular dynamics simulations of the effect of solvent quality on polymer conformation

V. CONCLUSIONS

### Key Topics

- Polymers
- 121.0
- Mean field theory
- 66.0
- Polymer structure
- 21.0
- Solvents
- 17.0
- Probability theory
- 13.0

## Figures

Examples of simple units that might be used to build up a hyperbranched structure. Top: The structure of a linear chain of glucose units with links. The link at a branch point is also shown. Bottom left: Electron microscope image showing the rosette structure of particles of rat liver glycogen, and their component particles (image kindly provided by Stapleton, see Refs. 21 and 22 for similar images). These particles are roughly spherical and have diameters distributed between about 20 and . Bottom right: The distribution of the lengths of these linear glucose chains obtained for glycogen from a debranching experiment (reproduced from Ref. 20).

Examples of simple units that might be used to build up a hyperbranched structure. Top: The structure of a linear chain of glucose units with links. The link at a branch point is also shown. Bottom left: Electron microscope image showing the rosette structure of particles of rat liver glycogen, and their component particles (image kindly provided by Stapleton, see Refs. 21 and 22 for similar images). These particles are roughly spherical and have diameters distributed between about 20 and . Bottom right: The distribution of the lengths of these linear glucose chains obtained for glycogen from a debranching experiment (reproduced from Ref. 20).

The process of adding new simple units to a hyperbranched structure. At the top, the actual formation of a link is illustrated. In case of glucose monomers illustrated in Fig. 1, it is an link that is formed. At the bottom, we show the evolution of the density profile that our theory predicts. The predicted density profile is an ensemble-average quantity for structures like that shown at the top. The left-hand pane of the density profile shows the density of the existing polymer ensemble, with simple units (dotted line), and the density of the new simple unit to be added (dashed line, distribution centered at , the point of addition). The center pane shows the new density profile for the ensemble of polymers with simple units, with the new unit added at (solid line). The right-hand pane shows the density for the new ensemble (again, solid line), a weighted average of the solid curve in the center over the addition point . This weighted average is described in the text in a simple case in Sec. II A and for the full theory in Sec. II C. The original density is shown for comparison in the center and right-hand panes (dots). The difference between the curves in the right-hand pane is of Eq. (3).

The process of adding new simple units to a hyperbranched structure. At the top, the actual formation of a link is illustrated. In case of glucose monomers illustrated in Fig. 1, it is an link that is formed. At the bottom, we show the evolution of the density profile that our theory predicts. The predicted density profile is an ensemble-average quantity for structures like that shown at the top. The left-hand pane of the density profile shows the density of the existing polymer ensemble, with simple units (dotted line), and the density of the new simple unit to be added (dashed line, distribution centered at , the point of addition). The center pane shows the new density profile for the ensemble of polymers with simple units, with the new unit added at (solid line). The right-hand pane shows the density for the new ensemble (again, solid line), a weighted average of the solid curve in the center over the addition point . This weighted average is described in the text in a simple case in Sec. II A and for the full theory in Sec. II C. The original density is shown for comparison in the center and right-hand panes (dots). The difference between the curves in the right-hand pane is of Eq. (3).

The radial density profile (solid curve, right axis) of a randomly hyperbranched structure of monomers, with . is the length of a step in the random walk used to model the oligomers that are attached to create the hyperbranched structure. The fit probability (dashes, left axis) is the chance that if we attempt to add a new monomer to the polymer at radius , there will be enough room. The addition probability is the probability that if we attempt to add at , we will find a potential branch point to join to and also enough room to accommodate the new monomer.

The radial density profile (solid curve, right axis) of a randomly hyperbranched structure of monomers, with . is the length of a step in the random walk used to model the oligomers that are attached to create the hyperbranched structure. The fit probability (dashes, left axis) is the chance that if we attempt to add a new monomer to the polymer at radius , there will be enough room. The addition probability is the probability that if we attempt to add at , we will find a potential branch point to join to and also enough room to accommodate the new monomer.

Testing the fit and addition probabilities against simulation. The simulated data come from the Monte Carlo simulations described in Sec. III. The curves are the functional forms used in the mean-field models for [in this case Eq. (25)]. Top: The probability that on adding at a point , enough free volume will be found to accommodate the new unit. The four curves are for adding new oligomeric chains of lengths 1, 2, 4, and 10. These are adding to randomly hyperbranched structures with and of about 1000 monomers. Bottom: The filled symbols are for the same systems as in the upper plot, but now the probability that a free branch point will be found has been included. The empty symbols are of the same quantity, but for addition to a hyperbranched polymer with .

Testing the fit and addition probabilities against simulation. The simulated data come from the Monte Carlo simulations described in Sec. III. The curves are the functional forms used in the mean-field models for [in this case Eq. (25)]. Top: The probability that on adding at a point , enough free volume will be found to accommodate the new unit. The four curves are for adding new oligomeric chains of lengths 1, 2, 4, and 10. These are adding to randomly hyperbranched structures with and of about 1000 monomers. Bottom: The filled symbols are for the same systems as in the upper plot, but now the probability that a free branch point will be found has been included. The empty symbols are of the same quantity, but for addition to a hyperbranched polymer with .

Here we show the probabilities for successful addition plotted not against the density experienced by the first monomer added, but against the average density experienced by the added chain. This approach is more appropriate to the mean-field theory and consequently gives similar values of for all added chain lengths (see Table III).

Here we show the probabilities for successful addition plotted not against the density experienced by the first monomer added, but against the average density experienced by the added chain. This approach is more appropriate to the mean-field theory and consequently gives similar values of for all added chain lengths (see Table III).

Radii of gyration obtained from the Monte Carlo simulations described in Sec. III, compared with the model predictions. Both the oligomer- and sphere-SAR models agree well for the range simulated.

Radii of gyration obtained from the Monte Carlo simulations described in Sec. III, compared with the model predictions. Both the oligomer- and sphere-SAR models agree well for the range simulated.

The simulated density profiles compared with the model’s predictions. The same parameters were used to fit the profiles at three degrees of polymerization, 800, 4000, and 10 000. We show the agreement for the smallest (top) and largest (bottom) of these three. The result is similar. We use here the oligomer-SAR.

The simulated density profiles compared with the model’s predictions. The same parameters were used to fit the profiles at three degrees of polymerization, 800, 4000, and 10 000. We show the agreement for the smallest (top) and largest (bottom) of these three. The result is similar. We use here the oligomer-SAR.

Here we fit the model to density profiles reported by Neelov and Adolf (Ref. 32) for molecular dynamics simulations of HBP and dendrimer models. Our model agrees very well in the hyperbranched case, but fails to capture the shell-like structure of the dendrimer, as expected.

Here we fit the model to density profiles reported by Neelov and Adolf (Ref. 32) for molecular dynamics simulations of HBP and dendrimer models. Our model agrees very well in the hyperbranched case, but fails to capture the shell-like structure of the dendrimer, as expected.

Here we test the use of the Flory–Huggins parameter in describing the density profiles. The four sets of data are for a model dendrimer of the eighth generation studied by molecular dynamics and reported by Murat and Grest in Ref. 33. The parameters emerging from our fit of the model to these simulation data give the correct dependence of on temperature (see Table VI).

Here we test the use of the Flory–Huggins parameter in describing the density profiles. The four sets of data are for a model dendrimer of the eighth generation studied by molecular dynamics and reported by Murat and Grest in Ref. 33. The parameters emerging from our fit of the model to these simulation data give the correct dependence of on temperature (see Table VI).

## Tables

A table summarizing the various versions of the models studied in this paper. The three models listed here are a representative selection of those that might be constructed from the elements described in this paper. Each is tested against a particular set of simulations described in Sec. III.

A table summarizing the various versions of the models studied in this paper. The three models listed here are a representative selection of those that might be constructed from the elements described in this paper. Each is tested against a particular set of simulations described in Sec. III.

Parameters obtained by fitting the probabilities obtained from simulation to the oligomer-SAR addition probability [Eq. (25)]. These are the parameters corresponding to the fits in Fig. 4.

Parameters obtained by fitting the probabilities obtained from simulation to the oligomer-SAR addition probability [Eq. (25)]. These are the parameters corresponding to the fits in Fig. 4.

Parameters obtained by fitting the probabilities obtained from simulation to the oligomer-SAR addition probability [Eq. (25)]. These are the parameters corresponding to the fits in Fig. 5.

Parameters obtained by fitting the probabilities obtained from simulation to the oligomer-SAR addition probability [Eq. (25)]. These are the parameters corresponding to the fits in Fig. 5.

Parameters obtained from fitting the oligomer-SAR model to the simulated radii of gyration.

Parameters obtained from fitting the oligomer-SAR model to the simulated radii of gyration.

Parameters obtained from fitting to the HBP and dendrimer models reported by Neelov and Adolf (Ref. 32).

Parameters obtained from fitting to the HBP and dendrimer models reported by Neelov and Adolf (Ref. 32).

Parameters obtained from fitting to the dendrimer models reported by Murat and Great. These differ in the variation of the solvent quality by varying the temperature.

Parameters obtained from fitting to the dendrimer models reported by Murat and Great. These differ in the variation of the solvent quality by varying the temperature.

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