# Dendrimers

Dendrimers (from the aincient greek δένδρον, meaning tree [1]). Dendrimers can be characterised by three parameters: functionality (${\displaystyle f}$), spacer length (${\displaystyle P}$) and number of generations (${\displaystyle G}$). The number of monomers (${\displaystyle N}$) in a dendrimer is given by

${\displaystyle N=1+fP{\frac {(f-1)^{G+1}-1}{f-2}}}$

## Density profile

#### Dense shell model

de Gennes and Hervet [2] calculated that for self-avoiding dendrimers in a good solvent, the density profile increases from a minimum at the centre of the dendrimer to a maximum at its outer surface, i.e. a dense outer shell with a hollow centre. Note this leads to a limit of

${\displaystyle G_{\mathrm {max} }\approx 2.88(\ln P+1.5)}$

However, recent work by Zook and Pickett [3] has shown that the de Gennes and Hervet model was flawed.

#### Dense core model

Most studies support the dense core model of Lescanec and Muthukumar[4] despite early uptake of the dense shell model. Boris and Rubinstein pointed out that the structure of the dendrimer is a result of the competition between the entropy and excluded volume [5], neither of which terms favouring a hollow centre.

It has been suggested that the radius of gyration (${\displaystyle R_{G}}$) scales as [6]

${\displaystyle R_{G}\propto N^{1/3}}$

where ${\displaystyle N}$ is the number of monomers. This implies a compact structure.

#### Ideal dendrimer

For an ideal dendrimer, consisting of non-interacting monomers, ${\displaystyle R_{G}}$ is given by [7]

${\displaystyle R_{G\mathrm {ideal} }\propto {\sqrt {PG}}}$

#### Chen-Cui scaling law

The Chen-Cui scaling law is given by [8]:

${\displaystyle R_{G}\propto (PG)^{1-\nu }N^{2\nu -1}}$

where ${\displaystyle \nu }$ is the Flory exponent.

## Specific dendrimers

• Star polymers (${\displaystyle G=0}$)