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[[Image:single_dendrimer.png|thumb|right|A single dendrimer molecule (G4 [[PAMAM (dendrimer) | PAMAM]], solvent not shown)]]
'''Dendrimers'''. Dendrimers can be characterised by three parameters: functionality (<math>f</math>), spacer length (<math>P</math>) and number of generations (<math>G</math>). The number of monomers (<math>N</math>) in a dendrimer is given by
'''Dendrimers''' (from the aincient greek δένδρον, meaning tree <ref>[http://dx.doi.org/10.1002/anie.199001381 Donald A. Tomalia, Adel M. Naylor and William A. Goddard III "Starburst Dendrimers: Molecular-Level Control of Size, Shape, Surface Chemistry, Topology, and Flexibility from Atoms to Macroscopic Matter", Angewandte Chemie International Edition in English '''29''' pp. 138-175 (1990)]</ref>). Dendrimers can be characterised by three parameters: functionality (<math>f</math>), spacer length (<math>P</math>) and number of generations (<math>G</math>). The number of monomers (<math>N</math>) in a dendrimer is given by


:<math>N= 1 +fP \frac{(f-1)^{G+1}-1}{f-2}</math>
:<math>N= 1 +fP \frac{(f-1)^{G+1}-1}{f-2}</math>
==Density profile==
==Density profile==
====Dense shell model====
====Dense shell model====
de Gennes and Hervet <ref>[http://dx.doi.org/10.1051/jphyslet:01983004409035100 P. G. de Gennes and H. Hervet "Statistics of «starburst» polymers", Journal de Physique Lettres '''44''' pp. 351-360 (1983)]</ref> 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
<ref>[http://dx.doi.org/10.1051/jphyslet:01983004409035100 P. G. de Gennes and H. Hervet "Statistics of «starburst» polymers", Journal de Physique Lettres '''44''' pp. 351-360 (1983)]</ref>
 
:<math>G_{\mathrm{max}} \approx 2.88 (\ln P + 1.5) </math>
 
However, recent work by Zook and Pickett <ref>[http://dx.doi.org/10.1103/PhysRevLett.90.015502 Thomas C. Zook and Galen T. Pickett "Hollow-Core Dendrimers Revisited", Physical Review Letters '''90''' 015502 (2003)]</ref> has shown that the de Gennes and Hervet model was flawed.
 
====Dense core model====
====Dense core model====
Most studies support the dense core model of Lescanec and Muthukumar<ref>[http://dx.doi.org/10.1021/ma00210a026 Robert L. Lescanec and M. Muthukumar "Configurational characteristics and scaling behavior of starburst molecules: a computational study", Macromolecules '''23''' pp. 2280-2288 (1990)]</ref>
Most studies support the dense core model of Lescanec and Muthukumar<ref>[http://dx.doi.org/10.1021/ma00210a026 Robert L. Lescanec and M. Muthukumar "Configurational characteristics and scaling behavior of starburst molecules: a computational study", Macromolecules '''23''' pp. 2280-2288 (1990)]</ref>
despite early uptake of the dense shell model.
<ref>[http://dx.doi.org/10.1021/ma960397k David Boris and Michael Rubinstein "A Self-Consistent Mean Field Model of a Starburst Dendrimer:  Dense Core vs Dense Shell", Macromolecules '''29''' pp. 7251-7260 (1996)]</ref> 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]]
<ref>[http://dx.doi.org/10.1021/ma960397k David Boris and Michael Rubinstein "A Self-Consistent Mean Field Model of a Starburst Dendrimer:  Dense Core vs Dense Shell", Macromolecules '''29''' pp. 7251-7260 (1996)]</ref>, neither of which terms favouring a hollow centre.


==Radius of gyration==
==Radius of gyration==
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:<math>R_G \propto N^{1/3}</math>
:<math>R_G \propto N^{1/3}</math>


where <math>N</math> is the number of monomers. This implies a compact structure.
where <math>N</math> is the number of monomers.
====Ideal dendrimer====
====Ideal dendrimer====
For an ''ideal'' dendrimer, consisting of non-interacting monomers, <math>R_G</math> is given by <ref>[http://dx.doi.org/10.1039/FT9969204151 Wilfried Carl "A Monte Carlo study of model dendrimers", Journal of the Chemical Society, Faraday Transactions '''92''' pp. 4151-4154 (1996)]</ref>
For an ''ideal'' dendrimer, consisting of non-interacting monomers, is given by <ref>[http://dx.doi.org/10.1039/FT9969204151 Wilfried Carl "A Monte Carlo study of model dendrimers", Journal of the Chemical Society, Faraday Transactions '''92''' pp. 4151-4154 (1996)]</ref>
 
 
:<math>R_{G \mathrm{ideal} } \propto \sqrt{PG}</math>
 
====Chen-Cui scaling law====
The Chen-Cui scaling law is given by <ref>[http://dx.doi.org/10.1021/ma9514636 Zheng Yu Chen and Shi-Min Cui "Monte Carlo Simulations of Star-Burst Dendrimers", Macromolecules '''29''' pp. 7943-7952 (1996)]</ref>:
 


:<math>R_G \propto  (PG)^{1-\nu}N^{2\nu-1}</math>


where <math>\nu</math> is the [[Flory exponent]].
:<math>R_{G{\mathrm{ideal}} \propto \sqrt{PG}</math>


==Specific dendrimers==
==Specific dendrimers==
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