What is a dendrimer?

A dendrimer is a highly branched polymer, as shown in the schematic below, and consists of a core where a monomer unit is attached.

A dendrimer has a core, branching units and end groups

 

The first monomer unit, in this example, has a functionality of three with one reactive site attached to the core or focal point and the other two making up a branching unit. This is considered the first generation or G1. The branching unit is then reacted with further monomer to produce G2 and a molecule with four end groups.

For a given functionality (f ) the number of end groups scales as

NE = [f-1]g

where g is the generation number; 0 (the core only), 1, 2, … Likewise, the dendrimer molecular mass (M) scales as

M ~ [f-1]g

An example of a poly(benzyl ether) dendrimer is given below. This is a G4 dendrimer with an -OH core.

A fourth generation poly(benzyl ether) dendrimer

 

A reference to this type of dendrimer is

C. J. Hawker and J. M. J. Fréchet,  “Preparation of polymers with controlled molecular architecture.  A new convergent approach to dendritic macromolecules,” J. Am. Chem. Soc., 112 (1990) 7638-7647.

General references to dendrimers are

J. M. J. Fréchet,  “Functional Polymers and Dendrimers: Reactivity, Molecular Architecture, and Interfacial Energy,” Science, 263 (1994) 1710-1715.

D. A. Tomalia,  “Dendrimer Molecules,” Scientific American, May (1995) 62-66.

B. I. Voit,  “Dendritic polymers:  from aesthetic macromolecules to commercially interesting materials,” Acta Polymer, 46 (1995) 87-99.

Dendrimers show some unique properties, one is the intrinsic viscosity going through a maximum at the fourth generation (see e.g. T. H. Mourey, S. R. Turner, M. Rubenstein, J. M. J. Fréchet, C. J. Hawker and K. L. Wooley,  “Unqiue Behaviour of Dendritic Macromolecules: Intrinsic Viscosity of polyether Dendrimers,” Macromolecules, 25 (1992) 2401-2406). This is sometimes denoted as a globular transition which may be more apparent below.

Through Einstein's relation for suspension viscosity the above can be rationalized. Einstein showed the intrinsic viscosity ([h]) for hard spheres is given by

[h] = 5/2 f/c

where f is the volume fraction of polymer in a solvent (ratio of polymer volume to total volume) and c the polymer concentration (ratio of polymer mass to total volume). Thus, [h] is merely the ratio of polymer volume (V) to polymer mass (M).

The dendrimer volume is taken to a first approximation as

V ~ g3

or the linear dimension grows steadily with generation number thereby making the volume increase accordingly. Using the above molecular mass scaling one can find

[h] ~ V/M ~ g3/[f-1]g

This equation has a maximum (amazingly) at G4 which is where Mourey et al. found their maximum for poly(benzyl ether) dendrimers. In general, the intrinsic viscosity maximum (gmax) is given by

gmax = 3/ln(f-1)

for any functionality.

This is a crude calculation, yet, does serve to show the reason why the peculiar intrinsic viscosity maximum is seen. The molecular volume can at most scale with g3 while the mass is set by the functionality and scales with a power law in generation number. Eventually the power law become the dominant factor and the intrinsic viscosity decreases, in other words, the volume does not suddenly "collapse," rather the mass increases faster than the volume. A more detailed analysis is given in an appendix below, yet, it does not reveal new insight.

Note the molecular density (r ~ 1/[h]) will asymptotically scale as

r ~ ln(f-1) [f-1]g

and will thus tend to infinity. This has led to the speculation that dendrimers will not be made above a certain generation as the crowding will become too great. This seems to be the case since dendrimers of generation greater than 7-10 are not reported. (n.b. The range of maximum generation number is due to the various branching units used. Some are more flexible than others and can be made to a larger maximum generation. The work of Mansfield and Klushin (Macromolecules, 26 (1993) 4262-4268) is interesting since they have shown the end groups saturate the entire molecule and are not localized at the molecular periphery. This prediction certainly depends on the branching unit flexibility, yet, is provocative. Further work by Mansfield shows at high generation that the molecular density is approximately constant (Macromolecules, 33 (2000) 8043-8049).)

Appendix - Generation number for a maximum in the intrinsic viscosity

The volume is written as

V ~ gn+1

where n is an exponent assumed equal to 2 in the above simple analysis. For linear polymers, this exponent is 1 (Rouse model), 1/2 (Zimm model) and 4/5 (Good solvent) as demonstrated by Doi and Edwards (M. Doi and S.F. Edwards, The Theory of Polymer Dynamics, Oxford (1989)). Of course, a rigid rod polymer will have n equal to 2 also.

A dendrimer's molecular mass can be more accurately written as scaling

M ~ {[f-1]g - 1}/[f-2] + r [f-1]g

where r is the ratio of the terminal to branching unit's molecular mass (0.752 for poly(benzyl ether) dendrimers). This equation has the advantage of reducing to the proper limit when f tends to 2 for linear polymers, M ~ g and is easily proven when l-Hôpital's rule is used.

The intrinsic viscosity is merely V/M (plus unknown scaling factors) and can be shown to have a maximum at the following generation number

0 = gmax - {[n+1]/ln(f-1)} ´ {1-1/[{1+r[f-2]}[f-1]gmax]}

Assuming n equal to 2, gmax is found to be 4.19 while the above approximate treatment yielded 4.33. Taking n as a measure of monomer flexibility one finds gmax equal to 2.62 for n equal to 1. The intrinsic viscosity maximum appears to be between generation 4 and 5 suggesting poly(benzyl ether) dendrimers are fairly stiff and each generation merely adds onto the molecular dimension in a linear manner.

Continuing this line of thought could lead to the conclusion that intrinsic viscosity maxima at generation numbers greater than G5 represents imperfect dendrimers as demonstrated now. Fractional powers for f may be used to represent imperfect dendrimers based on the expected chemical reaction. In fact, I will take fractional powers as a model for hyperbranched polymers. Assuming

f = 2.5, r = 1

one finds the following results  

n

gmax

0.5

2.85

0.8

3.72

1.0

4.31

2.0

7.10

So, even if the intrinsic viscosity maximum is at generation 4 it is not entirely clear if a perfect dendrimer is present. Other conclusions can be made based on this model such as a hyperbranched polymer will have a gmax at larger values for a given flexibility compared to a dendrimer. This technique could be used to determine the functionality should a dendrimer and a hyperbranched polymer be made with a given monomer unit. However, we have noted hyperbranched polymers are polydisperse in nature and this may make this technique useless unless the hyperbranched polymer is fractionated.

Back to the M.E. Mackay web page

© 2000-2006 All rights reserved. M.E. Mackay