ISSN 1811�2382, Polymer Science, Ser. C, 2013, Vol. 55, No. 1, pp. 125–153. © Pleiades Publishing, Ltd., 2013.

125

1 INTRODUCTION

Starburst dendrimers are macromolecules consist�ing of linear chains (called spacers) arranged in a hier�archical, treelike structure [8, 147]. The most wide�spread methods of making such highly branched mol�ecules by chemical synthesis are referred to asdivergent and convergent, respectively. In the formerapproach, starting from the core, new branched unitsof monomers are attached to the outermost units so asto form the next generation. In the latter, first thebranched arms (dendrons) are prepared, and finallyattached to the multifunctional core [35, 36]. Thedendritic architecture is characterized by threeparameters: The total number of generations G, spacerlength S, functionality of the branching units f, anddegree of polymerization N(S, G, f). Dendrimersexhibit the highest fraction of end groups per mono�mer units among all possible polymer architectures.The latter can be functionalized which makes den�drimers particularly interesting for applications. Dueto the highly branched structure of dendrimers theirmolecular weigth N grows exponentially with G andlinearly with S: , where the constants A andα are given by the dendrimer’s architecture.

Apart from purely scientific value dendrimers proveuseful in industry, biomedicine, pharmacy and materi�als engineering. To name but a few, lithographic mate�rials, nanoscale catalysts, drug delivery systems, rheol�ogy modifiers, bioadhesives, and MRI contrast agentsare examples of their potential applications [53, 62,93, 136]. Dendrimers were used to deliver oligonucle�

1 The article is published in the original.

α

=

GN ASe

otides to the cell [16, 156], they enhance cytosolic andnuclear availability as indicated by confocal micros�copy as well as cell uptake and transfection efficiencyof plasmid DNA [60]. Guest�host nanodevices such asgold/PAMAM (polyamidoamine) nanocompos�itesare potentially very useful agents for improving theimaging and radiation treatment of cancer [52].

Theoretical studies on dendrimers make use ofmean�field models, self�consistent methods, renor�malization group techniques and Flory�typeapproaches. On� and off�lattice simulations, on theother hand, involve the kinetic self�avoiding�walk,Brownian/molecular dynamics and Monte Carloalgorithms. These methods enable a direct insight intoconfor�mational properties of dendrimers, spatial dis�tribution of monomers and terminal groups, and phaseproperties [4, 6, 13, 15, 23, 26, 29, 46, 51, 66, 67, 76,81, 91, 100, 101, 104, 105, 122, 126, 134, 145, 152,158, 159]. Simulations allow an inspection of dynam�ical behavior of dendrimers as well. Actually, den�drimer translational self�diffusion, the size and shapefluctuations, rotational mobility, and elastic motionswere considered [51, 77, 80].

Experiments, on the other hand, employ photo�chemical and spectroscopic probe methods, massspectrometry, translational diffusion and viscometry[41, 113, 153]. Furthermore, transmission electron(TEM) and atomic force microscopy (AFM) as well assmall�angle neutron (SANS) and X�ray (SAXS) scat�tering methods are used to elucidate the shape andinternal structure of dendrimers [40, 68, 69, 115, 116,118, 122, 128].

Coarse Grained Simulations of Neutral and Charged Dendrimers1

J. S. K osa,c and J.�U. Sommera,b

aLeibniz Institute of Polymer Research Dresden e. V., Dresden, 01069 GermanybInstitute for Theoretical Physics, Technische Universitat Dresden, Dresden, 01069 Germany

cFaculty of Physics, A. Mickiewicz University, Umultowska 85, 61�614 Poznan, Polande�mail: klos@ipfdd.de

Abstract—Dendrimers are macromolecules with a regular�treelike, branched architecture of their skeleton.In terms of the branching number and the number of terminal groups they represent an extreme case amongbranched polymers. Dendrimers can occur in neutral and various charged states. Due to their highlybranched architecture excluded volume effects are of great importance and conformational properties andmonomer distribution profiles of dendrimers differ considerably from those of linear polymers. We give anoverview of the state–of–the–art knowledge of physical properties of dendrimers as seen from coarse�grained computer simulations. Our main focus is on isolated dendrimers with flexible spacers both in the neu�tral and in the charged state, as well as complexation of dendrimers with oppositely charged linear polyelec�trolytes. We briefly address problems of adsorption and concentration effects in dendrimer solutions and out�line recent progress and open questions in this field.

DOI: 10.1134/S1811238213070023

l

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The research on dendrimers is not limited to neu�tral molecules since their properties can also be tunedby electrostatic effects. With this respect, of increasinginterest are weak dendritic polyelectrolytes whosecharge in solution can be modulated by changing thesolution pH. For instance, PAMAM and poly(propy�leneimine) dendrimers acquire positive chargebecause they have primary amine groups at the termi�nal units and tertiary amine groups at the branchingpoints which become protonated as the solution pH�value is lowered from around 7 down to 4 [43, 90, 92,108, 112, 150]. In other words, under physiologicalconditions only the terminal groups bear positivecharges, whereas in more acidic environments boththe terminal groups and branching groups are charged.

Whether or not the above charge modulation has apronounced effect on conformational and structuralproperties of charged dendrimers has been the subjectof scientific deliberation for over a decade. Forinstance, simulations indicate that the dendrimer sizeincreases as pH decreases from neutral to low,although the actual value of the swelling parameterdepends on the applied model. Within the Debye�Hückel approximation which treats free ions implic�itly the radius of gyration was shown to increase by asmuch as 70% [154], whereas other approaches withexplicit ions lead to a much weaker dependence ofdendrimer conformations on ionic strength [63, 90,92, 110]. Molecular dynamics simulations takingcounterions into account explicitly as well as meanfield theory [25, 74] predict much lower swelling inagreement with small�angle neutron scattering(SANS) measurements on PAMAMs [11, 108].

It is therefore apparent that for dendritic polyelec�trolytes the degrees of freedom of counterions have tobe taken into account explicitly due to the importanceof ion valence, ion trapping in the dendrimer volume,ion condensation and its effect on conformationalchanges of dendrimers. Actually, as the strength ofelectrostatic interactions increases both under neutraland low pH conditions counterions penetrate not onlythe molecule periphery but, in the first place, its inte�rior [2, 5, 22, 25, 34, 55, 73, 94–96, 141]. This leads tothe reduction of the dendrimer effective charge,screening of electrostatic repulsion between chargedmonomers and non�monotonous behavior of den�drimer size with the strength of electrostatic interac�tions. Last but not least ions also play a crucial role inthe formation of complexes comprised of chargeddendrimers and linear polyanions and in self�organi�zation in solutions of charged dendrimers [47–49, 88,89, 138, 142], though many important aspects of com�plexation such as dendrimer overcharging by linearpolyelectrolytes were revealed within the Debye�Hückel approach as well [61, 78, 84, 87]. Obviously, itis dendrimer�DNA binding [33, 39, 44, 45, 64, 103,124, 127, 131, 133, 137], which being of immenseinterest in biosciences inspires theoretical investiga�tions.

Physical properties of charged dendrimers were thesubject of intensive experimental studies [11, 109, 120,157]. For instance, small�angle X�ray scattering(SAXS) and conductivity measurements were madefor dilute solutions of PAMAM molecules with univa�lent and divalent counterions [109]. Among others thelatter showed that divalent counterions are morestrongly condensed on the dendrimers and thus moreeffective in reducing their charge. As indicated bysmall�angle neutron scattering (SANS), SAXS andtransmission electron microscopy PAMAM dendrim�ers are useful for forming gold nano�clusters withintheir interior [31, 32]. A number of EPR and UV�visspectroscopy measurements demonstrated that at var�ious temperatures and pH divalent metal ions can bedistributed both inside and outside the molecule [111,151]. Furthermore, apart from dendrimers’ ability toencapsulate smaller molecules, with the use of opticalreflectometry, atomic force microscopy (AFM), lightscattering, SANS and electrophoretic mobility mea�surements a number of recent works were devoted totheir adsorption properties on silica surfaces and latexparticles as well as to formations of defined supramo�lecular assemblies based on ionic interaction in aque�ous solution [9, 72, 114, 125].

In this review we focus on the application of coarsegrained simulation models to explore universal prop�erties of dendrimers. In the next section we discuss theproperties of isolated and neutral dendrimers.

ISOLATED NEUTRAL DENDRIMERS

In this section we give an overview of the confor�mational properties of neutral, isolated dendrimerswith excluded volume as seen from theory and coarsegrained computer simulations. The first theoreticalwork on self�avoiding dendrimers in an athermal sol�vent was done by de Gennes and Hervet [15]. Using amodified version of the Edwards self consistent fieldmethod the authors considered dendrimers composedof very long spacers with trifunctional branchingmonomers. Based on the assumption that all the seg�ments from a given generation are localized in a shellaround the molecule’s center they arrived at radialmonomer density profiles which increase strictlymonotonously from the center towards the molecules’periphery. Additionally, they found that the radius ofgyration, , of dendrimers scales with the degree of

polymerization as . Due to the formerfinding the notion of the hollow�center or the dense�shell picture of dendrimers has been used in the litera�ture.

As we present below coarse grained computer sim�ulations contradict the dense�shell model of dendrim�ers and provide evidence for a completely differentidea instead. Actually, computer simulations lead tothe so called dense�core description of dendrimerswhere radial monomer density profiles decrease with

gR

N 1/5gR N∼

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the distance from the molecules’ center, and the ter�minal monomers are delocalized throughout thewhole molecules’ interior. The latter effect is known asbackfolding. With respect to scaling of the dendrimers’size a few different formulas were proposed, most ofwhich however, are of rather empirical nature, see thetable. The dendrimer’s size is controlled by excludedvolume effects due to the exponential growth of thenumber of monomers at higher generations. As laterpointed out and proved by a number of authors, thecorrect scaling law for the radius of gyration of den�drimers, Rg, in various solvents can be derived analyti�cally through minimization of the Flory�type, mean�field free energy per thread (a linear branch of GSmonomers reaching from the center of the dendrimerto the end�monomer)

(0.1)

with respect to , where T is the absolute tempera�ture, kB the Boltzmann constant, denotes theexcluded volume parameter describing the strength ofthe two�body interactions between monomers, and is the three�body interaction parameter [6, 13, 24, 59,130]. The first term in Eq. (0.1) corresponds tostretching of the thread using the ideal extension

, the second represents the two and three�body mean field interaction energy of the thread withall the other monomers. Eq. (0.1) leads to the scaling

⎛ ⎞= + + ,⎜ ⎟

⎝ ⎠B

2 2

3 6g

g g

RF N NGS wk T GS R R

v

gRv

w

20gR GS∼

relationship for for dendrimers in various solvents[Eq. (0.2)]

(0.2)

Interestingly, the scaling relation Eq. (0.2) (a) isequivalent with the prediction in Ref. [15], but noassumption about an inhomogeneous density distribu�tion is made. The question of the monomer distribu�tion within an isolated dendrimer can be answered bydirect computer simulations. According to Eq. (0.2)scaling of the size of dendrimers is predicted to dependon the quality of solvent in which the molecules areimmersed. For this reason it is convenient to overviewcoarse grained simulations of isolated dendrimers withrespect to various solvent conditions the calculationswere performed for.

Dendrimers in a Good Solvent

Various simulation studies addressed the confor�mational properties of neutral, isolated dendrimersunder good solvent conditions. One of the first simu�lation study was conducted by Chen and Cui usingbead�rod model in which dendrimers are made up ofidentical, freely�rotating rigid bonds with hard�spherebeads attached to their ends and arranged into a star�burst architecture [13]. By performing Monte Carlo

gR

( )

( )

2/5 1/5

1/4 1/4

1/3

good solvent a

solvent b

poor solvent c

( )

( )

( )

g

NSG

R SG N

N

⎧ ,⎪⎪

, θ⎨⎪

, .⎪⎩

∼

Scaling relations for the dendrimers’ radius of gyration obtained from coarse�grained simulations. MC, MD and BD stand forMonte Carlo, molecular dynamics and Brownian dynamics methods, respectively

Ref. Good solvent θ�solvent Poor solvent Density profile Method

[122] N1/5(GS)2/5 dense�core lattice MC

[13] N1/5(GS)2/5 dense�core off�lattice MC

[73] N1/5(GS)2/5 dense�core MD

[54] N1/5(GS)2/5 dense�core lattice MC

[130] N1/5(GS)2/5 N1/3 off�lattice MC

[24] N1/5(GS)2/5 N1/4(GS)1/4 N1/3 dense�core lattice MC

[26] N0.24�large G dense�core off�lattice MC

N1/3�small G dense�core

[66] N0.22S0.5 dense�core off�lattice kinetic growth

[51] N0.35 dense�core MD

[23] N1/3 dense�core lattice MC

[77] N0.31 dense�core BD

[145, 146] N7/20S1/4 N1/3S1/6 N1/3 dense�core off�lattice MC

[104] N1/3 N1/3 N1/3 dense�core MD

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simulations based on a generalized off�lattice pivotmethod on dendrimers of various generations andspacers in an athermal, implicit solvent, they showedthat the size of dendrimers follows the scaling law Eq.(0.2) (a). Additionally, since the monomer distribu�tions around the molecules’s center obtained by theauthors display a maximum at the center and the end�monomer distributions indicate penetration of all themolecules’ volume by the end�monomers, the simula�tions display a dense�core picture of dendrimers.

Following the generalized Flory�type theory Shenget al. derived Eq. (0.2) (a), (c) for the radius of gyrationof dendrimers under good and poor solvent condi�tions, and performed off�lattice Monte Carlo simula�tions [130]. In their approach dendrimers were mod�eled as freely jointed, square�well dendritic chains inan implicit solvent whose quality was controlled by ashort range, attractive potential between nonbondedbeads. Using the data the authors did confirm thatwhen the generation number and spacer length arevaried independently the relation Eq. (0.2) (a) isobeyed for dendrimers in a good solvent.

Brownian dynamics simulations along with amean�field theory with higher�order excluded volumeinteractions included were used by Lyulin et al. whoinvestigated conformational properties of neutral den�drimers with short spacers in dilute solutions of differ�ent quality [81]. In the simulations all the nonbondedbeads interacted via a Lennard�Jones potential inwhich the attractive term was modified by a screeningfactor whose value regulated the quality of the solvent.In terms of the behavior of the linear�expansion factorthe calculations confirmed the proposed mean�fieldapproximation for solvents ranging from good to poor.In particular, the scaling Eq. (0.2) (a) was recoveredand confirmed numerically in the case of a good sol�vent. Furhermore, for a good solvent the authorsfound that the effective fractal dimension of dendrim�ers varies with the generation number. Actually, theyargued it is less than 3 in the region of small generationnumbers and goes up to 3.5 for large generations. Theradial monomer density profiles calculated withrespect to the molecules’ center possess maximum atthe core subsequently followed by a monotonicdecrease to a local minimum, a plateau toward theedge and a monotonic decay again. The terminalbeads were shown to be dispersed all over the den�drimer, and the simulations confirmed the dense�corepicture of neutral dendrimers. The same conclusionabout spatial distributions of monomers and terminalgroups for dendrimers in good solvents was confirmedby Lyulin et al. in Ref. [76].

The scaling formula for the radius of gyration wasalso tested by Lin et al. by means of moleculardynamics simulations, see Ref. [73]. Although, as welater discuss in more detail, the work is primarilydevoted to dendritic polyelectrolytes, the authorsalso performed simulations of neutral dendrimers of

various generations and spacers, and were able toconfirm Eq. (0.2) (a).

Rathgeber et al. carried out Monte Carlo simula�tions of dendrimers in a good solvent using the coop�erative motion algorithm. This method is based onrepresenting macromolecules on a fcc lattice asassemblies of beads connected by nonbreakable bondsin a way corresponding to the molecule skeleton.Mobility of beads is provided by the concept of coop�erative molecular rearrangements. The authors founda good agreement between their athermal solvent dataand Eq. (0.2) (a) [122]. With respect to the spatial dis�tribution of terminal groups around the dendrimers’center their results confirm the phenomenon of back�folding. The obtained radial monomer density profilesfor high generation molecules show pronounced pla�teaus in the molecules' domain and decay monotoni�cally to zero in a narrow peripherial region. For lowgeneration molecules the plateaus disappear and theprofiles decay to zero starting from a small distancefrom the center. As reported by the authors the simu�lations show a very narrow density hole at the mole�cules’ center for dendrimers with spacer S = 1,whereas for their results support the dense�coreidea (plots not shown). Furthermore, it is also demon�strated that all features of the measured SAXS spectracorrespond with the form factors determined for thesimulated systems.

Using a configurational�bias Monte Carlo schemeon a cubic lattice [23] and Flory�type theory the prop�erties of dendrimers with a fixed spacer length in vari�able solvents were studied by Giupponi and Buzza[24]. Using the Flory�type free energy function theyderived Eq. (0.2) for the scaling of dendrimers’ radiusof gyration for any solvent conditions, and proved itcomputationally. The authors point out that althoughin agreement with Murat and Grest [104] the radius of

gyration follows the law reasonably in allsolvents, a more accurate description of the behaviorof dendrimers in a good solvent is provided by Eq.(0.2) (a). In particular, it is shown that Eq. (0.2) (a)provides the best fit to the athermal simulation data ascompared with the other existing formulas found byLescanek and Muthukumar, de Gennes and Hervet,and Timoshenko et al. [15, 66, 145]. Furthermore, thesimulations indicated that the intramolecular radialdensity profiles are dense�corelike and terminalgroups are delocalized throughout the dendrimer.

By systematic variation of the generation numberand spacer length dendrimers in an athermal solvent

were simulated by K os and Sommer with the use ofthe Monte Carlo algorithm based on the bond fluctu�ation model [54]. In this lattice method, a bead is asimple cube represented by eight lattice sites on acubic lattice. Beads are connected with bonds so as toform the required molecular architecture. In themodel different beads must not occupy even one samelattice site due to the excluded volume condition, and

2S ≥

1/3gR N∼

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COARSE GRAINED SIMULATIONS OF NEUTRAL AND CHARGED DENDRIMERS 129

the bond vectors have to belong to a certain set ofallowed vectors only. The allowed vectors are chosen issuch a way that the excluded volume condition pre�vents them from crossing each other. In the frame�work of the Monte Carlo approach trial moves areperformed by displacing a bead by one lattice site andby testing the new conformation and bonds againstthe above mentioned restrictions on the beads andbonds [18].

The authors noted that within the mean�fieldmodel spacers can be seen as nearly unperturbedchains in a good solvent. In order to see this, the solu�tion of the mean�field model for athermal solvent con�ditions Eq. (0.2) (a) can be rewritten in the followingform:

(0.3)

where corresponds to the mean�field model,and denotes the number of spacer chains.This indicates that all length scales and densities canbe rescaled by the size and self�density of spacers. Inparticular, as demonstrated in Fig. 1a the calculatedand rescaled radius of gyration obeys the predictedbehavior Eq. (0.3). A stringent proof of the above con�jecture is also given in Fig. 1b where we display thedensity profiles applying spacer scaling. In this plot the

self�density of spacers given by is used to renor�malize the total density, and the distance is rescaled bythe unperturbed size of the spacers. The collapse of thedata for various spacer length into a single mastercurve can be observed. As a consequence dendrimerconformations in a good solvent are due to rearrange�ments of spacers rather than to their stretching, anddendrimers under good solvent conditions display uni�

1/5/ 2( )R S nGν

,∼

/3 5ν =

/n N S=

1 3S − ν

versal properties with respect to the length of spacersup to a high generation number.

Based on the analysis of a variety of density profilesthe calculations also confirmed the dense�core pictureof dendrimers and backfolding of terminal groups. Forlow generation dendrimers their radial mass densityprofile is shown to decay strictly monotonously withthe distance from the molecule’s center, while a sharpdecrease in the density near the center and a broadplateau in the interior occur for higher generationmolecules, see also Fig. 1b. For dendrimers of highgenerations the backfolding phenomenon is best pro�nounced for the terminal groups and the outerbranches, whereas the inner branches are localized.With respect to the instantaneous shape of dendrimersthe authors found that there is a smooth transitionfrom oblate to spherical shapes in going from low tohigh generations.

It should be stressed that, like off�lattice simula�tions in which solvent quality is modeled by varyingthe attraction term in the Lennard�Jones potential, itis also possible to consider the effect of implicit solventwithin the bond fluctuation model. This can be simplyachieved by introducing a short�range attractivepotential between beads [Eq. (0.4)]

(0.4)

where r denotes the distance between a pair of beadsand rc the interaction cutoff radius. In the simulationsthe quality of solvent is controlled by the reduced sol�vent temperature τs > 0. Due to high efficiency of thebond fluctuation model, by lowering the reduced tem�perature τs all the solvent regions can be localized and

−⎧−τ , ≤= ⎨

, ,⎩B

if

otherwise

1

0s cr rU

k T

4

101 102

Rg/Sν

(N/S)G2

2

8

16

(a)

103

12345678910

0.8

0 8

ρmS3ν −1

r/Sν

0.4

(b)

12

123456

4 16

Fig. 1. (a) Scaling plot of vs. for dendrimers (1) G1, (2) G2, (3) G3, (4) G4, (5) G5, (6) G6, (7) G7, (8) G8, (9) G9,and (10) G10 in an athermal solvent. The dashed line indicates the slope of 1/5 as predicted by the mean�field model. (b) Rescaled

radial monomer density of the G6 dendrimer in an athermal solvent vs. the rescaled distance from the dendrimer’s center

of mass for variable spacer length S = (1) 1, (2) 2, (3) 4, (4) 8, (5) 16, and (6) 32.

/gR Sν / 2( )N S G

3 1mS ν−

ρ

/r Sν

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examined for dendrimers in a wide range of genera�tions and spacers. The choice of rc = 4u, where u is thelattice constant, secures that well below the θ�pointwhere the poor solvent conditions are reached, freez�ing effects connected with the Monte Carlo samplingare avoided. In particular, for dendrimers in a goodsolvent at high τs�values the bond fluctuation methodprovides another confirmation of Eq. (0.2) (a) and thedense�core picture of dendrimers. As we demonstratein the following section for the �point for den�drimers is achieved at .

A couple of works deviate from the interpretationof the dendrimer conformation in terms of the simplemean�field concept. Lescanec and Muthukumarapplied an off�lattice kinetic growth algorithm of self�avoiding walks to construct dendrimer molecules [66].They found that the dendrimer size scales with thenumber of monomers and spacer length as ~

, with and .They also found backfolding of the terminal units andradial monomer distributions that decay strictlymonotonously with the distance from the molecules’center, which in turn, contradicted the dense�shellmodel. Relaxation was not performed on the grownstructures, and it is not clear if the obtained resultsrefer to dendrimers at equilibrium.

Brownian dynamics simulations were carried outby Murat and Grest who studied dendrimers in sol�vents ranging from athermal to poor [104]. In theapplied model the chemical bonds were represented bythe FENE potential (the finite extensible nonlinearelastic potential) and the interactions between thenonbonded monomers by a Lennard�Jones potentialwith a varying cutoff distance, which enabled manipu�lating solvent quality. The simulation results are in

favour of the common scaling behavior fordendrimers under any solvent conditions. For this rea�son the authors concluded that unlike Lescanec andMuthukumar [66], and Mansfield [97] the fractaldimension of dendrimers is equal to three. The simu�lations also resulted in radial densities for all mono�mers and monomers belonging to subsequent internalgenerations which confirm the phenomenon of back�folding and dense�core structure of dendrimers. Actu�ally, the monomer density profiles were shown toexhibit a high value near the molecules’ core and a pla�teau region separated by a local minimum. The calcu�lations also indicated segregation of the dendronswhich decreases with increasing dendrimers’ genera�tion at a given solvent quality.

Similar to Murat and Grest [104] the scaling for�

mula was found to be roughly valid by Giup�poni and Buzza who applied a configurational�biasedlattice Monte Carlo scheme obeying the detailed bal�ance condition for self�avoiding dendrimers ina good solvent [23]. This, in turn, again suggested that

4cr u= θ

8 5sτ ≈ .

gR

N Sν β 0 22 0 02ν = . ± . 0 50 0 02β = . ± .

1/3gR N∼

1/3gR N∼

1 8G G−

dendrimers behave like compact spacefilling objectswith constant density. However, as argued by theauthors, a more careful examination of the data leadsto a conclusion that there ia a small but systematicdeviation from the constant density scaling, which isconsistent with some intrinsic viscosity results on den�drimers. Note, that as we have already pointed out theauthors later showed [24] that Eq. (0.2) (a) should beused instead. The calculations also showed that theradial density profiles are dense�corelike and decaymonotonously with the distance from the molecules’center for low generation dendrimers. For high gener�ation dendrimers the calculated profiles are alsodense�corelike and possess a shallow minimum nearthe center followed by a broad plateau. In agreementwith the dense�core description of dendrimer mole�cules, the terminal groups were found to penetrate themolecules’ interior. Furthermore, the scattering formfactors from the simulation data are reported to be ingood agreement with the SAXS measurements.

Timoshenko et al. studied dendrimers withvarious spacer length under good and poor solventconditions as well as across the coil–to–globule tran�sition by means of the Gaussian self�consistentmethod and off�lattice Monte Carlo simulations [145,146]. The latter model incorporated harmonic springsfor connected monomers and the Lennard�Jones pair�wise potential with a hard core and with a dimension�less parameter controlling the solvent quality. Based

on the simulation data the scaling law for the radius of gyration of dendrimers in a good sol�vent was claimed.

Using the “bead�thread” model with variablethread�to�bead size ratio the size and conformationsof flexible dendrimers with a fixed spacerlength S = 1 in an athermal solvent were examinedwithin the off�lattice Monte Carlo scheme by Götzeand Likos [26]. On the grounds of their data theauthors concluded that the dependence of the radiusof gyration on the molecular weight only slightly devi�

ates from the law for small molecules,whereas for high generation numbers the relation

is obeyed instead. The latter stronglyresembles Eq. (0.2) (a) for S = 1 and for molecularweigth corresponding to dendrimers. Also thesimulations provided evidence of the distribution ofend monomers throughout the dendrimers’ volume.The obtained radial density profiles are dense�corelikewith a broad plateau, and the form factors are in verygood agreement with the experimental SANS mea�surements.

A number of computational works focus on thebehavior of dendrimers’ size and conformations ingood solvents, but the precise form of scaling of theradius of gyration is not explicitly considered in them.

Lyulin et al. carried out Brownian dynamics simu�lations based on the bead�rod model on den�

1 7G G−

7/20 1/4gR N S∼

4 9G G−

1/3gR N∼

0 24gR N .

∼

7 9G G−

1 5G G−

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drimers with spacers S = 1 in an athermal solvent [82].The excluded�volume interactions were described bythe Lennard�Jones potential, and the solvent wastreated as a viscous medium acting on the dendrimerbeads via frictional and random forces. The simula�tions confirmed that the size of neutral dendrimers canbe adequately described within the framework of theFlory�type mean�field theory. The fractal dimensioncalculated in the publication for dendrimers is

. The obtained results also confirm the con�cept of the dense�core picture of dendrimers. Actually,the density profiles obtained from the calculationshave a maximum at the center and decrease with thedistance from the center. Furthermore, it was alsodemonstrated that the terminal groups are distributedover the entire dendrimer volume with a feebly markedmaximum.

In a series of works Mansfield et al. studied den�drimer conformations using “wiggle based” MonteCarlo simulations [100, 101]. In Ref. [97] Mansfieldcarried on his studies on dendrimers and consideredthe question of dendron segregation as well as den�drimers’ fractal self�similarity under athermal solventconditions. His Monte Carlo calculations indicatedthat separate dendrons of single, isolated dendrimersspontaneously segregate from each other and that thesame effect is observed for subdendrons. According tothe author the segregation is due to the moleculararchitecture of dendrimers. Individual segments of thedendrimer are significantly crowded by “close rela�tives” and there is no room for “distant cousins”. It isalso conjectured that dendrons are likely to mix whenunder high density conditions, for instance, in poorsolvents. Furthermore, the simulation data indicatedthat over a narrow length scale dendrimers are self�similar objects, with the fractal dimension between 2.4and 2.8.

For the same diamond�lattice model, using analo�gies between the physics of transport and of randomwalks Mansfield estimated the hydrodynamic radiusand the intrinsic viscosity of the model dendrimers[99]. He demonstrated that the ratio between thehydrodynamic radius and the radius of gyration variesbetween values typical of linear chains for low genera�tion dendrimers and ones typical of spheres for largerdendrimers. In agreement with experiments, heshowed that the intrinsic viscosity of the dendrimersexhibits a maximum at G = 6. Also, his calculationsconfirmed the dense�corelike radial monomer densi�ties and distribution of the end groups all over the mol�ecules’ interior. In agreement with the previous stud�ies, the behavior of pair distribution functions hefound indicate that the examined dendrimers fill spacein a fractal way with effective fractal dimension closeto three. However, it is concluded that the range overwhich this behavior is observed is too small to applythe idea of self�similarity. Furthermore, structure fac�tors are reported to be in qualitative agreement withthe results of X�ray scattering studies and in disagree�

2 77fd = .

ment with neutron scattering measurements on end�labeled PAMAM dendrimers. However, the authorexplains that these discrepancies may be due to the factthat the end groups of PAMAM’s are charged andtherefore repel each other.

We note that according to Wallace et al. part of thework by Mansfield does not obey the detailed balanceconditions [152].

Dendrimers in a θ�Solvent

In the literature different definitions of the �pointfor dendrimers were proposed. Lyulin et al. defined theθ�point as the characteristic energy of the excludedvolume interactions when the dendrimer linear expan�sion factor is equal to unity [81]. In this case the �temperature depends on the generation number andspacer length. Another definition of the �point wassuggested by Sheng et al., see Ref. [130]. In theirapproach the �point is identified as the temperaturebelow which the scaling relation Eq. (0.2) (a) for thegood solvent case no longer holds, i.e., when dataobtained for dendrimers of various generations andspacers start to diverge. Such a definition, in turn,implies that the �point is independent of the genera�tion number and spacer length. Subsequently, in theirsimulations of dendrimers Murat and Grest assumedthat the dendrimer �point is the same as for linearchains [104]. Similarly, Neelov and Adolf performedBrownian dynamics simulations of perfectly brancheddendrimers at the �point for chains and obtained ananalogous scaling relation for the dendrimer size

[106]. Timoshenko et al. identified theapproximated dendrimer �point as one where thechange in the specific energy slope is maximal [145,146].

Based on the Flory�type approach Giupponi andBuzza gave the precise definition of the �temperaturefor dendrimers [24]. According to them the true �point for dendritic molecules is localized when theappropriately rescaled computational data obey Eq.(0.2) (b). By definition within this approach the den�drimer �temperature is not a property of a single den�drimer. The authors proved computationally that den�drimers shrink as the solvent becomes poorer and thatEq. (0.2) (b) agrees better with the data as compared

both with the relation of Murat and Grest[104] and with that of Timoshenko et al. [145]. Fur�thermore, the simulations showed that, like the goodsolvent case, the radial density profiles measured withrespect to the center of the molecules are dense�core�like and terminal groups are delocalized throughoutthe dendrimer. However, in comparison with den�drimers under athermal solvent conditions, the calcu�lated density in the plateau region is higher, the widthof the density plateau is narrower, and the local density

θ

θ

θ

θ

θ

θ

θ

0 3gR N .

∼

θ

θ

θ

θ

1/3gR N∼

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K OS, SOMMERL/

minimum near the core observed in the athermal casetends to disappear for the poorer solvent.

As we have already pointed out in the previous sec�tion, with the use of the bond fluctuation model thescaling behavior of dendrimers in various solvents canbe examined for a fair range of the generation numberand spacer length. By applying the method it is possi�ble to observe that dendrimers reduce their size andthat their conformations become more and morecompact as the reduced solvent temperature is grad�ually decreased, see Fig. 2. Freezing effects can belargely avoided by extending the range of the attractive

sτ

potential, , over several lattice units, see Eq. 0.4. Inparticular, the �point can be detected as temperatureat which Eq. (0.2) (b) holds. Actually, as it is displayedin Fig. 3a for the obtained simulation dataagrees with the scaling law very well at fordendrimers of various generations and spacers. Notethat by analogy with the athermal solvent case all thedata displayed were rescaled using spacer scaling at the

�point. This is achieved by rewriting Eq. (0.2) (b)according to:

(0.5)

crθ

4cr u=

8 5sτ = .

θ

1/2 1/4 1/4/ .R S n G∼

10

0 0.15

Rg

τs−1

20

30 (a)

0.30

0.6

0 20

ρm

r

0.3

(b)123456

40

7

Fig. 2. (a) Radius of gyration, , vs. , the �point is indicated with the vertical line, (b) radial monomer density, , vs. the

distance from the dendrimer’s center of mass at varying = (1) 5, (2) 6, (3) 7, (4) 8, (5) 8.5, (6) 9, and (7) 100 for the G5 den�drimer with spacer S = 8.

gR1

s−

τ θ mρ

sτ

102

Rg/S1/2

(N/S)G

4

8

16 (a)

103

12345

0.6

10

ρmS1/2

r/S1/2

0.4

(b) 123456

0 20

0.2

0

Fig. 3. (a) Scaling plot of vs. (N/S)G for dendrimers (1) G3, (2) G4, (3) G5, (4) G6, and (5) G7 at the �point for

. The dashed line indicates the slope of 1/4 as predicted by the mean�field model. (b) Rescaled radial monomer density

of the G6 dendrimer at the �point vs. the rescaled distance from the dendrimer’s center of mass for variable spacerlength S = (1) 1, (2) 2, (3) 4, (4) 8, (5) 16, and (6) 32.

1/2/gR S θ 8 5sτ = .

4cr u=

1/2mSρ θ

1/2/r S

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COARSE GRAINED SIMULATIONS OF NEUTRAL AND CHARGED DENDRIMERS 133

In this case spacers behave like ideal chains, and

their extension, , defines a characteristic lengthscale for dendrimers under theta solvent conditions. InFig. 3b we display the radial monomer density profilesfor various spacer length calculated with respect to thecenter of the molecules. According to the mean�fieldpicture the profiles are rescaled by the extension of

spacers and by their self�density, , respectively. Itis seen that except for the shortest spacers the profilesobey the scaling prediction reasonably well.

Further discussion of the θ�point based on simula�tion results can be found in Ref. [104, 145, 146].

Dendrimers in a Poor Solvent

Independently of the applied model and simulationmethod, all the computational studies clearly indicatethat dendrimers collapse considerably in going fromgood to poor solvents, and the general relation Eq.(0.2) (c) was obtained. Actually, apart from the goodsolvent case, based on the generalized Flory�type the�ory Sheng et al. derived Eq. (0.2) (c) for the dendrim�ers’ size in a poor solvent and confirmed them numer�ically [130]. In their approach poor solvent conditionswere achieved through lowering the reduced tempera�ture below the �region so that all their data started tocollapse onto a single curve given by Eq. (0.2) (c).

The same conclusion about the behavior of den�drimers in poor solvents, and the scaling relation forthe radius of gyration in particular, was reached byMurat and Grest [104]. Their computational dataobtained from Brownian dynamics simulations car�ried out at the temperature at which linear chains arestrongly collapsed were satisfactorily fitted with theformula Eq. (0.2) (c). The authors also found that inpoor solvents the dense�core picture of dendrimers isvalid as well, but the density at the plateau increasesconsiderably. Furthermore, the overlap between den�drons was reported to become the most pronounced asthe solvent becomes poor.

Also, as demonstrated by off�lattice Monte Carlosimulations performed by Timoshenko et al. dendrim�ers contract significantly as the solvent changes fromgood to poor, and in the latter case Eq. (0.2) (c) isobeyed [145, 146]. The radial monomer densitiesmeasured with respect to the molecules' center werefound to be dense�corelike. However, in comparisonwith good solvent conditions, for poor solvents theglobular conformation of a dendrimer is compact anduniform without any hollow domains near the center,and the degree of packing is larger. Moreover, the sim�ulations showed that there occurs bond stretchingwhich is the highest near the core and decreases withincreasing inner generation number. The stretchingwas found to decrease with solvent quality.

Giupponi and Buzza found that under poor solventconditions radial density profiles are dense�corelikeand terminal groups fold back into the molecule’s

1/2S

1/2S −

θ

interior [24]. However, the actual value of density inthe plateau region is higher and the width of the pla�teau narrower as compared with molecules in athermalsolvents. It was also observed that in poor solvents evenlow generation molecules possess a density plateauwhich indicates that the dendrimers collapse into acompact, space filling state. Furthermore, for solventconditions in between and the completely collapsed

state the authors suggest a relation deviat�ing from Eq. (0.2) (c). Similar conclusions about theprofile and distribution of monomers and terminalgroups were obtained in Refs. [22, 81].

Like good and �solvents, the poor solvent case andthe corresponding scaling prediction Eq. (0.2) (c) canbe examined effectively using the bond fluctuationmodel. Within this approach, the poor solvent condi�tions can be attained by lowering the reduced solventtemperature far below the �point. Although, dueto high monomer density inside the molecules, spacerscaling does not hold for dendrimers in poor solvents,it is convenient to rewrite Eq. (0.2) (c) in the followingform:

(0.6)and to rescale all the data accordingly. In Fig. 4a

is plotted versus for all the visited sys�tems at and . It is seen that at fixed Gthere is still some dispersion of the data connectedwith different S�values, which however, becomes lesspronounced as G is increased. The best fit to the calcu�lated radius of gyration of the examined dendrimersroughly agrees with the predicted behavior Eq. (0.6),though the actual value of the exponent is 0.28 instead1/3. This behavior might be explained with the contri�bution of a diffusive surface layer as predicted for lin�ear chains in the collapsed state and which contributesto the extension and to the profile up to rather highmolar masses. In Fig. 4b we display the radial mono�mer profiles calculated relative to the center of mass ofdendrimers and rescaled in accordance with Eq. (0.6).This approach is quit reasonable since the plots over�lap fairly well. The shape of the plots confirms that thedendrimers take very dense, compact conformations.Up to some distance from the center the monomerdensity is nearly constant and beyond that it goes downto zero rapidly. The monomer distributions are inqualitative agreement with those reported in Ref. [24],and thus are in favour of the dense�core picture ofdendrimers under poor solvent conditions.

Dynamical Aspects

Some aspects of dynamical behavior of isolateddendrimers such as dendrimer translational self�diffu�sion, the size and shape fluctuations, rotational mobil�ity, and elastic motions were examined in the litera�ture. Karatasos et al. carried out molecular dynamicssimulations of dendrimers with spacers

θ

0 26gR N .

∼

θ

sτ θ

1/3 1/3/R S n ,∼

1/3/gR S /N S4cr u= 5sτ =

3 6G G− 2AB

134

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S = 2 in explicit solvent solutions at constant temper�ature corresponding to the condition of linear poly�mers [51]. In their calculations the bonds between thebeads were represented by a harmonic potential andthe nonbonded interactions by a Lennard�Jonespotential. The calculations showed that the center�of�mass diffusion coefficient scales with molecular

weight as in accordance with the theoretical pre�diction for the diffusion of dendrimers under condi�tions within the Zimm approximation [12]. Further�more, based on the data the authors proposed the“dynamic layering” visualization of local, dynamicbehavior of dendrimers. According to them, close tothe dendrimer’s periphery the principal relaxationmechanism consists in the fast, molecular weightindependent, local bond reorientation. On the otherhand, close to the core the local motion is slower anddoes depend on the molecular size. The intermediatetime scales between the two extremes are likely todepend on the topological features of dendrimers suchas the branching functionality and spacer length.

Based on the bead�rod freely jointed model Lyulinet al. performed Brownian dynamics simulations ofneutral dendrimers to examine some of their dynamicproperties [80, 83]. In the calculations the solvent wastreated implicitly as a continuous viscous medium andthe excluded volume interactions were accounted forby the Lennard�Jones repulsive potential correspond�ing to an athermal solvent. The hydrodynamic inter�actions were introduced by means of the Rotne–Prager–Yamakawa tensor, see also Ref [12]. In thesimulations the authors considered three types ofmotion in dendrimers which included the motion ofthe whole molecule, the size and shape fluctuations,

θ

0 5N − .

θ

and the local reorientations of the individual mono�mers. The calculations demonstrated that the molecu�lar�mass dependence of the self�diffusion coefficientis in agreement with theoretical predictions for themodel with preaveraged hydrodynamic interactionsand in quantitative agreement with NMR results fordilute solutions of poly(propylene imine) dendrimers.The hydrodynamic radius was found to be smaller thanthe radius of gyration. The simulations also indicatedthat dynamics of the size fluctuations for a dendrimerwith rigid spacers differs from the theoretical predic�tion for one with flexible spacers. The relaxation ofthese fluctuations is practically insensitive to the pres�ence of hydrodynamic interactions and turns out to befaster. The behavior of the molecule reminds that of anelastic body in a viscous medium. Moreover, it wasfound that the orientational mobility of bonds fromthe different inner generations of dendrimers increasesfrom the core to the periphery, and mobility of the ter�minal groups is independent of the generation num�ber.

By employing the above model Lyulin et al. alsoexamined dendritic polymers up to six generationsunder the influence of shear [77]. Their simulationsindicated that the onset of shear�thinning occurs atlower shear rates for larger generation dendrimers. Inthe dependence of zero shear rate intrinsic viscosity onthe generation number the maximum occurs which islocated at G ~ 4. The maximum is seen only when thehydrodynamic interactions are invoked, and becomesmore pronounced when the strength of these interac�tions is increased. Furthermore, the authors con�cluded that the hydrodynamic radius obtained fromtranslational diffusion coefficient is close both to thatcalculated from the zero�shear viscosity and to the

102

Rg/S1/3

N/S

8

4

(a)

103

12345

0.6

10

ρm

r/S1/3

0.4

(b)

123456

0 15

0.2

05

Fig. 4. (a) Scaling plot of vs. for dendrimers (1) G3, (2) G4, (3) G5, (4) G6, and (5) G7 in a poor solvent at

for . The dashed line indicates the slope of 1/3, and the solid line the slope of 0.28. (b) Radial monomer density of the

G6 dendrimer vs. the rescaled distance from the dendrimer’s center of mass for variable spacer length S = (1) 1, (2) 2, (3)4, (4) 8, (5) 16, and (6) 32.

1/3/gR S /N S 5sτ =

4cr u= mρ

1/3/r S

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COARSE GRAINED SIMULATIONS OF NEUTRAL AND CHARGED DENDRIMERS 135

average radius of gyration of dendrimers. On the otherhand, the preaveraged approach results in a largerhydrodynamic radius. In contrast to linear chainsshear flow leads to a monotonous increase in theradius of gyration of dendrimers.

Mato et al. applied molecular dynamics simula�tions to study carbosilane dendrimers of the 5th gener�ation in a good solvent. Their calculations were carriedout using the AMBER force field in the united atomapproximation with Lennard�Jones particles as thesolvent molecules. The main focus of this investigationwas on the internal structure of the dendrimer, dynam�ics of trans�gauche transitions of single bonds and fluc�tuations of the branching points. Among other find�ings this study demonstrated that the one barriermechanism of conformational transitions observed inlinear chains is also valid for conformational rear�rangements inside dendrimers with hindered rotationaround single bonds. The confor�mational transitionsalso determine the kinetics of fluctuations of the dis�tances between the core and the branching points aswell as of the terminal groups [102].

ISOLATED CHARGED DENDRIMERS

As we shall present in this section, coarse grainedcomputer simulation studies clearly indicate that suchparameters as the solution pH, counterion/saltvalence and concentration as well as the effectivestrength of electrostatic interactions all have a tremen�dous influence on the conformational properties ofisolated, charged dendrimers. Due to high computa�tional costs of simulating dendritic polyelectrolytes anumber of studies were carried out using the Debye�Hückel approximation. Within this approach counte�rions are treated implicitly, and the Coulombic inter�actions between a pair of beads carrying charges ofvalence z and z' at a distance r from each other is givenby

(0.7)

where T is the absolute temperature, kB the Boltzmannconstant,

(0.8)

the Bjerrum length and

(0.9)

denotes the inverse Debye screening length, respec�tively. In the definition of the Bjerrum length e standsfor the elementary charge and � for the dielectric con�stant of the solvent. In the formula for the inverseDebye screening length and denote the concen�tration and valence of the ith ion in the solution.

−κ

= λ ,BB

'rU ezz

k T r

λ = ,BB

2ek T�

⎡ ⎤κ = πλ⎢ ⎥

⎢ ⎥⎣ ⎦∑

1/2

B ,24 i i

i

c z

ic iz

Using the Debye�Hückel approximation and theMonte Carlo off�lattice scheme Welch and Muthuku�mar employed a bead�spring model to examine thedilute solution behavior of dendritic polyelectrolytesin solvents of various ionic strengths [154]. In theircalculations the bonded monomers interacted via theFENE potential and the nonbonded ones via theMorse potential, respectively. The free parameters ofthe latter were set to such values that good solvent con�ditions were simulated. With respect to the size ofcharged dendrimers the authors predicted that athigher salt concentrations the radius of gyration scales

with the molecular weight according to ∝

[(v + , where is the effectiveexcluded volume interaction magnitude. This corre�sponds to the mean�field scaling behavior derivedfrom Eq. (0.1), where the excluded volume isenhanced by screened electrostatic interactions. Thesimulations indicated “smart behavior” of dendrim�ers: As the ionic strength changes from high to lowdendrimers take compact (“dense�core”) and moreopen (“dense�shell”) conformations in a reversibleway and their size changes by about 65 percent. Wenote that the term “dense�shell” is used in a slightlydifferent meaning as presented in the original work byde Gennes and Hervet [15], and the results rather dis�play a swelling of the dendrimers with decreasing saltconcentration. This effect is also displayed by theradial density profiles. At high salt concentrations theradial monomer density profiles were shown to decaystrictly monotonously with the distance from the mol�ecules’ center, whereas in the other extreme a mini�mum occurs near the core which is followed by a sec�ondary peak near the periphery. However, in bothcases, unlike the prediction by de Gennes and Hervet[15], the calculated monomer density is very high inthe vicinity of the center and reduces considerablywith the distance from it. No matter the value of theinverse Debye screening length, the terminal groupswere found to be dispersed throughout the molecules.However, the location of maximum terminal groupdensity shifts toward the molecules’ periphery withdecreasing inverse Debye screening length. Theauthors also report a qualitative agreement between theform factors they calculated for various salt concentra�tions and experimental observations for PAMAMsstudied via SANS.

The same approximation was applied by Lyulin etal. who used Brownian dynamics simulations to studythe equilibrium, static properties of terminallycharged dendrimers up to the sixth generation withspacers S = 1 in a good solvent [81, 82]. The authorsemployed the bead�rod model of dendrimers withexcluded volume, and considered the solvent animplicit viscous medium affecting the beads via fric�tional and random forces. In the simulations the ratiobetween the Bjerrum length and the rod length was set to one so as to refer to room temperature for a

1/4/gR N

πλ κ5/4 1/5

B/ 24 ) ]N v

λB/l

136

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K OS, SOMMERL/

typical flexible polymer. The calculations showed that,as compared with neutral dendrimers, the size ofcharged dendrimers increases weakly with increasingDebye radius. The fractal dimension calculated forcharged dendrimers was found to be smaller than inthe neutral case and to decrease with increasing Debyeradius. According to the authors, this means that anincrease in Debye radius leads to an increase in theamount of “vacant” space inside the dendrimer whichbecomes much more open in accordance with thefindings in Ref. [154]. This is also displayed by theradial monomer density profiles, which in comparisonwith neutral molecules, flatten considerably and oscil�late in the plateau region. Due to a higher internalstress in the system the oscillations become more pro�nounced at larger values of the Debye length. It mustbe stressed that although the monomer density profilesreduce their value in the dendrimers’ domain, theystill peak near the center which contradicts the earliesttheoretical prediction [15]. The terminal groups werefound to be distributed all over the entire dendrimervolume with the maximum that shifts toward theperiphery due to electrostatic repulsion.

Like for neutral dendrimers, some aspects ofdynamical behavior of dendritic polyelectrolytes wastaken up by Lyulin et al. as well [80, 83]. By employingthe Debye�Hückel potential and Brownian dynamicssimulations they studied dynamics of dilute solutionsof rigid, terminally charged dendrimers with explicitexcluded�volume, electrostatic, and hydrodynamicinteractions in an athermal solvent. The hydrody�namic interactions were taken into account via theRotne–Prager–Yamakawa tensor. The focus of thestudy was on the motion of a dendrimer as a whole, thesize and shape fluctuations and the local reorienta�tions of the individual monomers. The authors foundthat due to the rigid character of the dendrimer modelan increase in the Debye length does not result in a sig�nificant change of the self�diffusion coefficient ascompared to the neutral case. They also found that thehydrodynamic radius of dendrimers is smaller than theradius of gyration. The relaxation time connected withrotation of a dendrimer as a whole is reported to beslightly larger than that for neutral dendrimers, andthe relaxation time of a monomer�to�core vector wasfound to increase slightly with the increase in theDebye length. With respect to the change of the latterrelaxation time with molecular weight of a dendrimer,

its proportion to was confirmed. Furthermore, forthe autocorelation function of the radius of gyrationthe simulations indicated that an increase in theDebye length leads to a decrease in the correspondingrelaxation time. According to the authors this is due toelectrostatic repulsion which increases the effectiverigidity of a dendrimer, which in turn, results in adecrease in the corresponding characteristic times.Moreover, the dynamics of internal “pulsations” in adendrimer were described with the correlation func�tion for the squared distance between the core and the

1 3N .

monomers of a given generation shell inside the den�drimer. The corresponding relaxation times werefound to be independent of the number of exteriorgenerations, except for large values of the Debyelength where the relaxation time decreases. Theauthors argued that the latter effect is due to additionalstretching of a dendrimer caused by the electrostaticrepulsion between the charged monomers, whichleads to the decrease of the fluctuations of internalmotions and to the faster decrease in the relaxationtime. Finally, the behavior of the calculated orienta�tional autocorrelation function shows that the Debyelength affects mainly the orientational behavior of thebonds attached to the core, while the bonds of all theother inner generations reveal only a weak sensitivityto the Debye length.

As we already pointed out, in the case of explicitions simulations become computationally much moredemanding. However, due to significant computerdevelopments during the last few years, explicit treat�ment of ions became possible. In return, these simula�tions revealed the importance of the phenomena of iontrapping in the dendrimer’s volume and ion condensa�tion. In particular, it was possible to explain swelling ofcharged dendrimers observed in simulations using theconcept of osmotic pressure exerted by counterions onthe dendrimer they are trapped in. The calculationsalso provided microscopic insight into the behavior ofdendrimers in various ionic environments, includingmultivalent salt.

The total Coulomb energy of N ions with valences

and positions ri in the cubic box of size with peri�odic boundaries is given by [1, 121]

(0.10)

The first sum in Eq. (0.10) is over all integer vectorsk, and the prime indicates that for k = 0 terms with

must be skipped. At present there exist a few effi�cient methods of calculating Eq. (0.10) which includethe Standard 3D Ewald method, Particle�Particle–

Particle–Mesh method, Particle Mesh Ewald(PME) method and Smooth Particle Mesh Ewald(SPME) method. The other ways of computingEq. (0.10) are referred to as the convergence factormethod and the fast multipole method (FMM). Foran overview of these methods see Ref. [37].

Galperin et al. examined the effect of explicitcounterions on the spatial structure of the G5 termi�nally charged dendrimer with spacers S = 2 in dilutesalt�free solutions by the off�lattice Monte Carlomethod using the bead�spring model of dendrimers ina good solvent [22]. In the calculations the bonds wererepresented by a harmonic potential and the excludedvolume interactions by the Morse potential. The den�drimer was simulated in a cubic cell with repulsivewalls and a size that considerably exceeded the charac�

iz 3L

= =

= λ .

+∑ ∑∑B

B 1 1

1 '2

N Ni j

iji j

z zUk T L

kr k

i j=

3( )P M

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COARSE GRAINED SIMULATIONS OF NEUTRAL AND CHARGED DENDRIMERS 137

teristic size of the dendrimer. For this reason, unlikemost of the recent simulation studies, the calculationof the Ewald sum was not performed and the totalelectrostatic energy of the system was just the sumtaken over all the pairs of charges interacting via theCoulomb potential without screening. The authorsfound that as the Bjerrum length is increased Coulombrepulsion between the end groups bearing like chargescauses the molecule to swell. However, at the Bjerrumlength greater than unity the behavior is inverted anddendrimer shrinks with further increase in the Bjerrumlength. According to the authors, this change is causedby counterion condensation which leads to effectivecompensation for repulsion between the chargedgroups of the dendrimer. In the limit of large Bjerrumlength the size of the molecule turns out to be smallerthan that of the neutral dendrimer, which as pointedout by the authors, is due to multipole attractionbetween the charges present in the system. The calcu�lated radial monomer density profiles correspond to adense�core and backfolding of the terminal groups wasdetected. Furthemore, with increasing the Bjerrumlength, counterions were found to be more stronglytrapped inside the dendrimer by electrostatic interac�tion.

The importance of explicit treatment of counteri�ons and their effect on the properties of dendritic poly�electrolytes was remarkably demonstrated by Gur�tovenko et al. [34]. Based on the freely jointed “bead�spring” model, they performed molecular dynamicssimulations of terminally charged G4 dendrimers in agood solvent without salt. Solvent molecules as well ascounterions were explicitly included as interactingbeads. The bonds were modeled as harmonic springsand the nonbonded interactions were described by aLennard�Jones potential. The long�range electrostaticinteractions between the charged units were handledusing the particle�mesh Ewald (PME) method. Themain goal of the study was to inspect the effect ofcounterions along with the strength of electrostaticinteractions, as determined by the values of the Bjer�rum length, on the structural properties of chargeddendrimers.

The simulations showed that the size of dendrimerschanges non�monotonously with the Bjerrum length:An increase in the strength of electrostatic interactionsfrom zero results in a pronounced swelling of the den�drimer up to some maximum, followed by shrinking ofthe molecule in the limit of high values of the interac�tion parameter. As argued by the authors such a behav�ior is due to a subtle interplay between two oppositeeffects that occur as the Bjerrum length is increased.The first is the electrostatic repulsion between thecharged terminal beads which promotes swelling of thedendrimer. The second is a strengthening of counte�rion condensation due to the attractive interactionsbetween counterions and the charged dendrimerbeads, which causes screening of the dendrimer chargeand leads to shrinking of the molecule. The calcula�

tions also demonstrated that condensation of counte�rions on the dendrimer is accompanied by a decreasein the molecule’s hydration. Actually, an increase inthe Bjerrum length promotes ion condensation, whichin turn, leads to a loss of solvent molecules from theion’s first hydration shell. The calculated radial num�ber densities of the dendrimer beads, terminal groups,counterions and the solvent molecules correspond andcomplement the behavior of the dendrimer’s size. Theradial monomer distribution profile is dense�corelike,with a plateau which broadens and flattens in the max�imum swelling region. The charged terminal groups,solvent molecules and counterions penetrate the den�drimer’s interior. However, the degree of penetrationof solvent molecules becomes less and of counterionsbetter pronounced in the limit of very high Bjerrumlength. In this region, due to ion condensation, theradial density profiles of counterions and terminalgroups overlap.

In their calculations the authors also consideredthe dynamics of terminally charge dendrimers in asalt�free solution. Like for the size of the charged den�drimer, the global re�orientational motion of the mol�ecule is a non�monotonous function of the Bjerrumlength. The authors argued that this is due to the factthat the global reorientational dynamics is correlatedwith the dendrimer’s size. On the other hand, thereorientation of a bond slows down with decreasingsize of the dendrimer for bonds belonging to the inner�most shells of the molecule. For the bond vectors ofthe outermost shells this observation does not hold,most likely because of condensation of counterionswhich can hinder the reorientation. The relaxation ofthe vector pointing from the dendrimer’s center ofmass to the core was analyzed. The correspondingrelaxation times were found to be very small and non�monotonous functions of the Bjerrum length with apronounce minimum in the maximum swelling range.The authors conclude that an increase in the den�drimer size results in a more mobile core and in thefaster relaxation of the considered vector. These stud�ies clearly demonstrated that including explicit coun�terions can have a pronounced effect on the structureand dynamics of charged dendrimers, and further sim�plified approaches, such as the Debye�Hückel approx�imation, at least in the low salt regime, should beapplied with care.

The limitations of the Debye�Hückel theory weredemonstrated by Giupponi et al. who performedmolecular dynamics simulations to study the proper�ties of charged dendrimers at room temperature as afunction of the generation number, spacer length andionic strength [25]. The authors considered dendrim�ers with all branched and terminal beads charged, withonly terminal beads charged, and the case of neutraldendrimers in a good solvent. This corresponds to thecase of PAMAM dendrimers at low, neutral and highpH values. The charged dendrimers were modeledusing a bead�spring model with an implicit, good sol�

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vent and explicit counterions. The excluded volumeinteractions between all beads were represented by apurely repulsive shifted Lennard�Jones potential, andthe bonded ones by the FENE potential. The long�ranged Coulomb forces were calculated using the cell

multipole method. In contrast to implicit freeion simulations based on the Debye�Hückel theorythe authors found that the conformations of dendriticpolyelectrolytes are filled�corelike rather than hollow�corelike and only weakly depend on ionic strength.According to them such observations are due to thephenomenon of local charge neutrality in dendriticpolyelectrolytes, which results in internal counterionconcentrations which are much higher than the bulkvalues. The authors pointed out that this kind of ten�dency toward local charge neutrality suggests thatcharged dendrimers are in the so called “osmoticbrush” regime where electrostatic interactions arescreened and the movable counterions trapped insidethe molecule exert the osmotic pressure on the den�drimer which swells. However, the degree of swelling ismuch weaker as compared to that observed from sim�ulations using the Debye�Hückel potential. Further�more, using the mean�field Flory�type approach andthe simulation data the authors also demonstratedquantitatively that their arguments that local chargeneutrality and the resulting osmotic pressure are themajor factors affecting the conformations of chargeddendrimers are reasonable.

The behavior of terminally charged dendrimers

and the role of counterions were investigated by K osand Sommer [55]. They examined dendritic polyelec�trolytes with flexible spacers of varying length, accom�panied by explicit counterions in an athermal solventusing lattice Monte Carlo simulations based on thebond fluctuation model in the absence of salt. In thesimulations both the full Coulomb potential and the

3P M

l

excluded volume interactions were taken into accountexplicitly with the reduced temperature (the inverseBjerrum length) as a measure of the effective strengthof electrostatic interactions. Due to periodical bound�ary conditions applied the total Coulomb energy wascalculated using the Standard 3D Ewald method.

The performed calculations demonstrated thatcounterions are localized in the molecules’ interiorand condense on the terminal groups as the reducedtemperature is lowered. This, in turn, was shown tohave an effect on the conformational properties of themolecules that swell at intermediate temperature dueto the osmotic pressure of the trapped counterions andshrink in the limit of high and low temperature,respectively, see Fig. 5a. Moreover, the simulation dataindicated that the degree of swelling is not strong incomparison with the behavior of neutral dendrimers,and is dominated by temperature. The effects of gen�eration and spacer�length play a minor role. As as con�sequence, spacer�length scaling as observed in theneutral case, was found to be approximately obeyedfor terminally charged dendrimers as well. In accor�dance with this conclusion, the extension of spacers isnearly invariant with respect to the generation numberand temperature. Thus, as argued by the authors,swelling is mainly due to unfolding of the branchedstructure rather than stretching of spacers. Further�more, the simulations proved that as for neutral den�drimers, terminally�charged dendrimers display adense�core distribution of monomers and backfoldingof terminal groups into the interior of the molecules’volume, see Fig. 5b. The calculated instantaneousshape of the dendrimers displays only weak anisotropy.

In order to study the effect of pH on weak dendriticpolyelectrolytes with flexible spacers and explicitcounterions over a wide range of the reduced temper�ature in an athermal solvent, calculations were per�

τ

100

Rg/Rg0

τ

1.3

1.0

(a)

101

1234

0.6

10

ρmS3ν − 1

r/Sν

1234

0 15

0.3

510−1

1234

1.1

1.2

(d)

100 155

(e)

0.6

80 12

0.3

4

(b)

100 155

(c)

Fig. 5. (a) Aspect ratio, , between the radius of gyration of terminally charged and neutral dendrimers vs. the reducedtemperature for variable spacer length S = (1) 1, (2) 2, (3) 4, and (4) 8. Open simbols correspond to G = 5, closed to G = 6.(b)–(e) Rescaled radial monomer density, , vs. the rescaled distance from the terminally charged dendrimers’ center of massat various t for G = 5. = (b) 0.05, (c) 0.1, (d) 0.2, and (e) 0.3.

/ 0g gR Rτ

mρτ

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COARSE GRAINED SIMULATIONS OF NEUTRAL AND CHARGED DENDRIMERS 139

formed for molecules where only the terminal beadscarry charges, and for ones where both the branchedunits and the terminal groups are charged (low pHregime for PAMAM�like dendrimers) [56]. The simu�lations clearly indicated that as the temperature is low�ered most of counterions are localized inside the den�drimers’ volume and their density there is higher thanthe bulk value. Moreover, charging of dendrimersleads to swelling whose degree depends non�monoto�nously on temperature and displays a pronouncedmaximum, see Fig. 6a. The maximum volume degreeof swelling of dendritic polyelectrolytes was found toincrease in going from high to low pH. As argued bythe authors, the simulations revealed that it is an inter�play of condensation of counterions, trapping ofcounterions inside the dendrimer’s volume and evap�oration of counterions into the surrounding solutionwhich gives rise to the non�monotonous electrostaticswelling of dendrimers with temperature.

Extending the idea of Giupponi et al. [25], theobserved swelling behavior can be rationalized by con�sidering two major contributions due to electrostaticinteractions: The osmotic pressure of non�condensedcounterions which are trapped within the dendrimer’svolume, and uncompensated charges of the dendrimerwhich result from evaporation of counterions into thesolution and give rise to long�range electrostatic repul�sion. Using the generalized Flory�type approachallows to recover qualitatively all the aspects of thedependencies of the charged dendrimers’ size on tem�perature, pH and spacer length. It was assumed thatthe effect of swelling is primarily caused by the “resid�ual counterions” which are neither condensed nor faroutside the dendrimer’s volume. The osmotic pressureexerted by the residual counterions is approximated byan ideal gas law. Another source of swelling are coun�terions which escape the gyration volume of the den�drimer and thus lead to a residual, uncompensatedcharge inside the dendrimer. The generalized Flory�

type free energy per dendrimer with an extension R isgiven by

(0.11)

The first two terms are equivalent with the freeenergy of neutral dendrimers under good solvent con�ditions as given in Eq. (0.1), and correspond to theGaussian elasticity of treads (paths from the core tothe terminal groups) and to the mean excluded volumeinteractions, where v is the excluded volume parame�ter, respectively. The third term denotes the free energyof an ideal gas of n residual counterions trappedwithin a spherical volume of radius around thecenter of mass of the dendrimer, where is the calcu�lated radius of gyration of the molecule. The fourthterm describes the free energy of a homogeneouslycharged sphere by m ions, and refers to m counterionsoutside the spherical volume, i.e., to the residualcharges of the dendrimer.

Without the effect of counterions minimization ofthe free energy leads to the result for neutral dendrim�ers, , and to spacer scaling as was shown in Ref.[54] and in the section above. Including the counte�rion pressure the minimization problem for the freeenergy with respect to R leads to a polynomial equa�tion of order five. Since the swelling due to counteri�ons is a rather small effect, for dendrimers underathermal solvent conditions a first order perturbationapproximation can be applied which leads to the fol�lowing result

(0.12)

with

(0.13)

λ⎛ ⎞= + + + .⎜ ⎟π⎝ ⎠π

B22

2

2 3 3

33ln20( ) 4

mF N N nR nkT RGS R R

v

2 gR

gR

0gR

( )1/5

0

1gCI RC

g

R

R= ,+ Δ + Δ

302

0

gCI

nR

NΔ = ,

v

100

Rg/Rg0

τ

1.0101

1234

10−1

1234

1.2

1.4

100

(1 + ΔRC + ΔCL)1/5

τ

1.00101

1234

10−1

12341.02

1.04

Fig. 6. (a) Aspect ratio, , between the radius of gyration of charged and neutral dendrimers vs. the reduced temperature tfor variable spacer length S = (1) 1, (2) 2, (3) 4, and (4) 8. Open simbols correspond to low pH, closed to neutral pH. The arrowindicates the pH�driven maximum swelling for molecules with long spacers. (b) Total contribution of the residual counterions andresidual charges to swelling.

/ 0g gR R

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and

(0.14)

The total contribution of the residual counterionsand charges to the swelling behavior is displayed inFig. 6b with a value of read off for neutral moleculesstudied before in Ref. [54]. Although the swelling pre�dicted by the Flory�type argument remains muchlower as compared to the simulation results in Fig. 6a,the curves display many common features. In detail, asignificant shift of the swelling maximum with spacerlength is observed. An increase in the spacer lengthshifts the maximum to lower values of temperature andincreases its absolute value. This effect is most pro�nounced for low pH. At higher temperatures (right tothe maximum) the swelling dependence on spacerlength is inverted: dendrimers with smaller spacersshow a slightly higher degree of swelling.

Quantitatively the electrostatic swelling effect isunderestimated when using the excluded volume con�stant calculated from the data for the uncharged den�drimer (which displays very good agreement with theFlory�type prediction and corresponding spacer�length scaling). A possible origin for the discrepancy isthe well�known overestimation of Gaussian elasticityand mean�field excluded volume interaction [14].Other contributions to the free energy, which arequantitatively more accurate are thus underestimated.Scaling�up counterion and electrostatic contributionsby a factor of about 20 leads to quantitative results forthe swelling effect. However, it turns out that even withfree fitting parameters for both contributions a perfectagreement between the Flory�type prediction and thesimulated data cannot be achieved. Finally, the radialdistribution profiles, as provided by the calculations,were found to be always dense�corelike for all themonomers and bell�shaped, delocalized throughoutthe whole molecules’ volume for the terminal groups.However, increased swelling of the charged dendrim�ers at low pH corresponds to a certain “opening” ofthe conformations and separation of dendrons.

Lin et al. performed molecular dynamics simula�tions on neutral and charged dendrimers withlong spacers in dilute solutions at constant Bjerrumlength [73]. They modeled the molecules as collec�tions of monomers with a monovalent charge fraction

accompanied by explicit, monovalentcounterions in a good and �solvent. In the calcula�tions the excluded volume interactions between everypair of monomers were represented by the truncatedshifted Lennard�Jones potential, the bonds by theFENE potential, and the electrostatic interactionswere computed using the smoothed particle meshEwald (SPME) algorithm. The focus of the investiga�tion was both on the static and dynamic properties ofdendritic polyelectrolytes. With respect to the latter,the authors observed that the relaxation time of the

2 20

2020

gRC

m R

NΔ = .

π τv

0v

2 7G G−

/0 1 2f≤ ≤

θ

internal structure of neutral dendrimers with longspacers follows the Rouse model, whereas the relax�ation time of charged dendrimers reveals a weakdependence on generation or molecular weigth withincreasing the charge density. Based on their simula�tion data, the authors also proposed an empirical mas�ter function for the radius of gyration of charged den�drimers with different charge densities and long spac�ers, which however, turned out to deviate from theirtheoretical predictions for the two limiting cases cor�responding to the polyelectrolyte and osmoticregimes. The former regime was defined as one inwhich counterions of charged dendrimers are distrib�uted evenly throughout the whole system. The latter asone where counterions are trapped in the dendrimer’volume and screen the electrostatic interactionsbetween charged monomers. According to the Flory�type theory developed by the authors, swelling of den�dritic polyelectrolytes in this regime is due to therepulsive Coulomb interactions inside the moleculewhich can be described as a short�range binary repul�sion with an effective second virial coefficient. As theauthors pointed out, there is a crossover from the poly�electrolyte regime to the osmotic regime as generationand charge density increase. Furthermore, the simula�tions indicated homogeneous spatial distributions ofcounterions inside the dendrimers, except for the moststrongly charged molecules for which the counteriondensity profiles are non�monotonous. The calculatedform factors demonstrated the spherical structure ofcharged dendrimers.

By performing molecular dynamics simulations inthe framework of the bead�spring model, Huißmannet al. examined the effect of bond stiffness andmonovalent salt concentrations on the properties ofdendrimers bearing monovalent charges [38]. Thesimulation parameters were chosen appropriately torepresent water at room temperature. In the study fivekinds of dendrimers were considered with differentdegrees of charging mimicking varying pH�values ofthe solution and with two types of bonds referred to asrigid and soft. The purely repulsive, truncated andshifted Lennard�Jones and FENE potentials wereused to model the excluded volume interactions andthe connectivity between monomers, respectively. Thelong�range Coulomb interactions were handled by theEwald summation method. The calculations demon�strated that, due to the electrostatic repulsion, thedendrimers swell monotonously upon charging. Ascompared with the neutral molecule, for the fullycharged dendrimer with rigid bonds the degree ofswelling is about 15%, whereas for the one with softbonds which stretch more easily, it is about 50%. Asshown by the simulation data, no matter the degree ofcharging, stretching of bonds depends on their loca�tion, and is the largest for the innermost bonds. Theradial monomer density profile measured with respectto the center of mass of the dendrimer was found to beshell�like with a pronounced maximum at a small dis�

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COARSE GRAINED SIMULATIONS OF NEUTRAL AND CHARGED DENDRIMERS 141

tance from the center in the case of rigid bonds,whereas the ability of bonds to stretch more easily forthe soft�bonded dendrimer was shown to results indense�corelike density profiles. Furthermore, the cal�culated radial counterion density profiles provide evi�dence for diffusion of counterions into the dendrim�ers’ volume, which is better pronounced for a largeroverall charge of the dendrimer. As displayed by thedensity profiles, due to more freedom of monomers toarrange themselves in the soft�bonded dendrimers,counterions occupy all the space of the molecule. Onthe other hand, the observed significant increase inmonomer density near the center of the molecule withrigid bonds prevents counterions from entering thatarea. Furthermore, the authors compared the counte�rion distribution profiles obtained by their simulationswith predictions of the Poisson�Boltzmann theorybased on the monomer profiles from the simulations.They found an excellent agreement between the tworesults for the dendrimer with soft bonds. However, asignificant discrepancy occurs for the molecule withrigid bonds, which according to the authors, is due tothe fact that the theory neglects the short�range repul�sion between monomers and counterions.

As clearly demonstrated by the above works, ionsplay a very important role in determining the size andstructure of dendritic polyelectrolytes. It became evi�dent that only the explicit treatment of their degrees offreedom provides a microscopic explanation for thebehavior of charged dendrimers in various environ�ments. This is due to the fact, well established bymeans of computer simulation studies, that under cer�tain conditions ions are not scattered all over the avail�able space, but penetrate the interior of the moleculesinstead. As a consequence, in that volume their den�sity is much higher than the bulk value. In particular,some of them are condensed on the dendrimer,whereas others are free to move in the molecules'domain. Furthermore, ions screen the Coulomb inter�actions between the charged monomers, and the mov�able ones exert the osmotic pressure on the dendrimer,which in turn, results in swelling of the molecule. Theeffect of swelling of charged dendrimers caused bymonovalent ions is much weaker than that found withthe use of the Debye�Hückel approximation. Asalready pointed out this regime is referred to as the“osmotic brush” regime. Note that as indicated by anumber of simulations there also exist at least twoother regimes of charged dendrimers’ behavior. Actu�ally, in the limit of very high values of the Bjerrumlength practically all counterions are condensed onthe charged monomers and the dendrimer shrinksconsiderably. This effect was explained to occur due tomultipole attraction between pairs of electric dipolsthat appear in dendrimers upon counterion conden�sation. On the other hand, in the limit of low Bjerrumlength the conformational entropy dominates thedendrimers’ behavior. As a result of that their confor�

mational properties are the same as of neutral den�drimers.

Once the importance of ions was established, otherissues connected with them that were expected toaffect dendrimers were taken up as well. For instance,the simulations we have discussed so far were carriedout for dendrimers and their monovalent counterionsin salt�free solutions only, or with only monovalentsalt. On the other hand, the influence of valence ofions and their concentration are known to be tremen�dous for linear polyelectrolytes and spherical polyelec�trolyte brushes. For this reason it was reasonable tocarry out analogous studies on dendritic polyelectro�lytes with the same respect. In the following part of thissection we present the recent coarse�grained simula�tions devoted to the above mentioned questions.

Blaak et al. performed molecular dynamics simula�tions on bead�spring dendrimers with rigid bonds andmonomers either neutral, monovalent or divalentaccompanied by monova�lent or divalent counterionsat room temperature [5]. The short�range repulsiveinteraction between beads was represented by a sim�ple, shifted, and purely repulsive Lennard�Jonespotential and the bonds between jointed beads by aFENE one. In order to include the long�range Cou�lomb interactions, the authors applied periodicboundary conditions along with the Ewald summationmethod. The simulations showed that for the simplestcase of monovalent monomers and counterions nodramatic changes in the dendrimer conformation andsize take place as compared to the neutral molecule,though some swelling of dendrimers is observed. Aspointed out by the authors, this is at odds with theresults obtained by Welch and Muthukumar [154]whose calculations based on the Debye�Hückelpotential lead to swelling of about 70%. The calcula�tions indicated that the effect of varying valences ofboth the monomer and counterions on the den�drimer’s size is not monotonous. For instance, theradius of gyration of the divalently charged dendrimeraccompanied by divalent counterions was found to besmaller than that of the monovalent case. According tothe authors this suggests that the swelling of the den�drimer caused by the increasing monomer charge isreduced by counterions residing inside the dendrimerand screening the electrostatic interactions. This, inturn, results in still weaker swelling of the molecule.Furthermore, a major increase in the dendrimer’s size,due to stiffening and stretching of the dendrimer’sbonds, takes place when the monomers bear divalentcharges and release two monovalent counterions. Thecalculated radial density profiles possess an oscillatingshape with pronounced local maximums indicatingthat charged monomers form a layered structurewithin the dendrimer.

By means of molecular dynamics simulations Tianand Ma inspected the behavior of a G4 terminallycharged dendrimer with spacers S = 1 in multivalentsalt solutions in a good solvent at room temperature

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[141]. In the calculations they modeled the excludedvolume interactions using a purely repulsive, trun�cated�shifted Lennard�Jones potential and the bondsby the FENE potential. The Coulomb interactionsalong with periodical boundary conditions in threedirections were handled by the particle�particle parti�

cle�mesh ( ) method. Moreover, in order to simu�late a terminally charged dendrimer, all the terminalmonomers were set to carry negative, monovalentcharges accompanied by their monovalent counteri�ons, and solvent was treated implicitly as a medium ofdielectric constant. Salt ions were represented byspherically�shaped Lennard�Jones tetravalent cationsand monovalent anions, respectively. The main focusof the study was on how the concentration of multiva�lent salt affects the dendritic polyelectrolyte. The sim�ulations indicated that the dendrimer takes dense�corelike conformations for all multivalent salt concen�trations studied, but the radial monomer density pro�files were found to exhibit oscillations due to the pres�ence of counterions and salt ions, which affect themonomer distributions and promote layering. Theauthors observed ion exchange phenomenon: Multi�valent salt ions replace the condensed monovalentcounterions and get trapped in the dendrimer’sdomain due to stronger electrostatic attraction. It wasalso demonstrated that with the addition of multiva�lent salt the dendrimer undergoes a structure transfor�mation: It first collapses significantly as a result of adecrease in the osmotic pressure inside the molecule,and subsequently weakly swells due to an enhancedexcluded volume effect of the adsorbed salt ions. Themost compact conformation of the dendrimer wasobserved at the critical value of salt concentrationwhere the total charge of multivalent salt neutralizesthat of the dendrimer. Also, overcharging of the den�drimer with increasing salt concentration wasdetected. The sign of the mean effective charge of thedendrimer was observed to shift from negative to posi�tive as the salt concentration increases beyond the crit�ical value.

Using the same method Tian and Ma continuedtheir study on the terminally charged dendrimer inmonovalent (1:1), divalent (2:1), trivalent (3:1), andtetravalent (4:1) salt solutions [143]. The calculationsdemonstrated that the conformation of the dendrimervery weakly depends on the ionic strength in themonovalent salt case. On the other hand, with increas�ing concentration of multivalent salt the dendrimerwas shown to pass from an extended state through acollapsed state to a weakly swollen state due to thechange of osmotic pressure inside the molecule. Fur�themore, the calculations clearly indicated that at agiven salt concentration the size of the dendrimer issmaller for higher valence of the salt ions. As alreadypointed out, according to the authors this kind ofeffect is due to the fact that the condensed multivalentcations replace the monovalent counterions with theaddition of salt and neutralize the dendrimer more

3P M

effectively, which in turn, gives rise to a reduction inthe electrostatic repulsion between the terminalgroups and in osmotic pressure inside the dendrimer.The monomer density profiles measured with respectto the center of the dendrimer display valence depen�dent oscillations, which according to the authors, pro�vide evidence of layering of monomers which occursmost likely because of the presence of ions. Actually,the spatial distributions of ions demonstrate that ionsoccupy nearly all the dendrimer’s domain except forthe vicinity of the center of the dendrimer where themonomer density is so high that ions are preventedfrom penetrating that area. According to the behaviorof the total charge density profiles, the authors alsoobserved the charge layering phenomenon, which onthe one hand shows no difference at different salt con�centrations in the monovalent salt case, whereas onthe other, displays multistate transitions with the addi�tion of multivalent salt. The transitions also occuramong different valences of salt ions at a given saltconcentration. Overcharging of the molecule wasfound in multivalent salt solutions at high salt concen�trations.

COMPLEXES BETWEEN CHARGED DENDRIMERS AND LINEAR

POLYELECTROLYTES

Simulations of complexation between dendriticand linear polyelectrolytes are mainly inspired byexperimental works on dendrimer�DNA binding andits importance to biosciences [33, 39, 44, 45, 64, 103,124, 127, 131, 133, 137]. Like for the case of isolatedcharged dendrimers, in a number of coarse�grainedsimulations the electrostatic interactions were firsttreated within the Debye�Hückel approximation.Although, as we have emphasized, this kind ofapproach is not completely correct, it lead to valuableobservations for systems where the extent of screeningof electrostatic interactions by the added salt is large.

Using the same method as in Ref. [154] Welch andMuthukumar studied the complexation between the

terminally charged dendrimers and the fully,oppositely charged linear chains at room temperature[155]. The authors proposed a molecular level descrip�tion of such aggregates and the conditions necessaryfor forming them. The calculations demonstrated thatat a given value of the inverse Debye length the chainsreduce their size in the complex considerably as com�pared to that in free solutions, whereas shrinking of thedendrimer is minor, and as the inverse Debye length isincreased the size of the molecules in the complexgradually approaches its value in a free solution. Theauthors formulated the analytical criterion for com�plexation in terms of the chain’s and dendrimer’s vari�ables. They found that molecular weigth and chargedensity determine not only the conditions on whichcomplex formation takes place but also the type ofcomplex which is formed. Actually, the simulations

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COARSE GRAINED SIMULATIONS OF NEUTRAL AND CHARGED DENDRIMERS 143

revealed three types of complexes: If the chain is shortenough it is encapsulated inside the dendrimer. In thecase of longer chains the second type of complexesoccurs in which the chain penetrates the dendrimerand releases a relatively short tail outside of the den�drimer. In the limit of very long chains it is the den�drimer that actually penetrates the chain. The lattermolecule is coiled near the dendrimer only while itsrest remains extended. Moreover, in this case theauthors observed an interesting phenomenon of thedendrimer’s, “walk” along the chain from one end ofthe chain to the other.

Using the Debye�Hückel approximation Lyulin etal. carried out Brownian dynamics computer simula�tions of a bead�rod freely jointed model of a terminallycharged dendrimer and an oppositely charged chain incomplexes at the inverse Debye length correspondingto 2.2 mM aqueous salt concentrations at room tem�perature [79, 85, 86]. By varying the molar mass of thechain and of the dendrimer the authors found that thesize of the chain reaches a minimum when charge ofboth molecules is exactly compensated. Furthermore,the number of adsorbed chain monomers changesnon�monotonously with chain length. For longerchains in the complex this number becomes greaterthan the dendrimer’s charge (overcharging). Theauthors argue that this effect is in qualitative agree�ment with the predictions of the correlation theory fora hard�sphere model. The maximum of chain adsorp�tion was found to occur at some critical chain lengthwhich corresponds with the first order phase transitionof the chain from coiled conformation to one withreleased tails. It was demonstrated that the phase tran�sition is accompanied by the non�monotonous behav�ior of the relative fluctuations of the linear polyelectro�lyte’s size which reach maximum at the point of max�imum adsorption. With respect to dynamical behaviorof the molecules, the authors also found that longchains in complexes exist in forms with long�lastingone long and one very short tails. Exchanges betweenthe tails are very rare, and tails of equal length cancoexist for a very short time only. For this reason theauthors argued that the dendrimer performs a veryslow “non�random�walk” along the chain since in thecase of a random walk all possible tail lengths would beequally probable. The static structure factor of thedendrimer in the complex was found to be indepen�dent of chain length and to be very similar to the struc�ture factor of a single neutral dendrimer.

The most recent computational studies show thatexplicit treatment of counterions and the phenome�non of ion condensation connected with them is nec�essary for the proper, theoretical explanation of thestructure and conformations of charged dendrimersand their complexation with linear polyelectrolytes.

Using the same simulation method as for singlecharged dendrimers, see Ref. [34], Lyulin et al. per�formed molecular dynamics simulations of complexescomprised of a generic, terminally charged G4 den�

drimer with spacer S =1 and a short polyelectrolytechain of 10 oppositely charged, monovalent and diva�lent beads in a good solvent [88]. Monovalent anddivalent counterions of both the dendrimer and thechain as well as solvent molecules were explicitlyincorporated in the model. The main goal of the studywas to examine the influence of the strength of elec�trostatic interactions on the conformational propertiesof dendrimer�chain complexes as well as the role ofcounterions in determining the properties. The calcu�lations showed that the size of the short chain in thecomplex is rather insensitive to the Bjerrum length,whereas the dendrimer shrinks considerably as thestrength of electrostatic interactions increases due tovalence dependent condensation of dendrimer coun�terions. On the other hand, it was concluded that thedendrimer screens the poly�electrolyte chain from itscounterions efficiently. It was found that the degree ofscreening is affected by ion valence as well. Since,along with the screening of the chain from its counte�rions, the simulations also demonstrated that com�plexation with the cationic dendrimer leads to a con�siderable dehydration of the linear polyelectrolyte, theauthors concluded that charged dendrimers efficientlyprotect short, linear polyelectrolytes of oppositecharge from the surrounding medium.

K os and Sommer using Monte Carlo simulationsbased on the bond fluctuation model studied com�plexes comprised of a single G4 dendrimer with =32 positively charged terminal groups, spacer lengthS = 1 and an oppositely, fully charged linear polyelec�trolyte accompanied by neutralizing counterions in anathermal solvent [57]. In the study both the full Cou�lomb potential and the excluded volume interactionswere included with the reduced temperature andchain length as the main simulation parameters.Due to periodical boundary conditions applied thetotal Coulomb energy was calculated using the Stan�dard 3D Ewald method. The simulations showed thatcomplexation between the two polyelectrolytes occursin three stages of decreasing temperature. Binding ofchains with the dendritic polyelectrolyte occurs at

first, and depending on the chain length, is fol�lowed by selective localization of counterions insidethe complex volume at . Finally, atτ < counterions start to condense on the com�plex.

It was found that for chains above the compensa�tion point, , only the chain counterionslocalize and condense on the chain, whereas for Nch <

the dendrimer counterions condense on the den�drimer. At the compensation point, , no ioncondensation occurs at all, see Fig. 7. The authorspointed out that this phenomenon might be caused bythe entropy gain due to evaporation of counterionsduring complex formation. Moreover, the simulationsindicated overcharging, which was found to occur at

l

tN

τ

chN

τcomplex

τ ≤ τloc complex

τcond

ch tN N>

tN

ch tN N=

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the lowest temperatures considered for chains longerthan the compensation point. The size of dendrimersand chains was demonstrated to change non�monoto�nously both with the temperature and with chainlength. Actually, as shown by the calculations, chargeddendrimers and long chains swell due to electrostaticeffects at intermediate temperatures. However, theswelling of the dendrimer is considerably weakened bycomplex formation because of the reduction of thenumber of counterions trapped in the dendrimer’s vol�ume. This is in full accord with out theoretical analysisgiven in the previous section, where we have shownthat residual, i.e., trapped but non�condensed counte�rions drive swelling of charged dendrimers. Withrespect to the chain length the dendrimer size wasfound to reach a minimum at the compensation point.The size of short chains, on the other hand, onlyweakly reacts to changes in the temperature.

The simulations indicated that the adsorbed chainstake conformations with a number of beads condensedon the terminal groups of the dendrimer and with twotails released. For short chains in the complex the tailsare of nearly the same length. For long chains,

, different scenarios were observed: At verylow temperatures two tails of equal length are pre�ferred, while at intermediate temperatures a long anda short tail is formed. Furthermore, at higher temper�atures random fluctuations of tail lengths take place.The tails of longer chains are stretched at � τ �τloc due to repulsion between their beads, and subse�quently shrink as temperature is further decreasedbelow due to multipole attraction between themcaused by chain counterion condensation.

It should be stressed that in the above works linearpolyelectrolytes were considered fully flexible polymerchains. However, real bioactive agents such as DNA,whose binding with charged dendrimers is of consider�

ch tN N�

τcond

τcond

able practical importance, are characterized by vari�ous rigidities. For this reason the influence of stiffnessof linear polyelectrolytes on complexes with dendriticpolyelectrolytes is of interest.

Within the framework of the freely jointed bead�spring model, Tian and Ma employed moleculardynamics simulations to inspect the effect of stiffnessof linear polyelectrolytes on the properties of com�plexes between an anionic linear polyelectrolyte and acationic dendritic one in a good solvent [142]. In thestudy the authors considered a G4 dendrimer withspacers S = 1 and a chain composed of 16 monomersalong with explicit monovalent counterions and sol�vent molecules. Apart from the usual terms in theenergy function such as the harmonic spring potentialbetween the nearest�neighbor monomers, purelyrepulsive, short range truncated�shifted Lennard�Jones potential between all the particles, and the elec�trostatic interactions handled by the particle�particle

particle mesh ( ) method, they also included theharmonic bond angle potential

(0.15)

to represent the degree of stiffness of linear polyelec�trolytes. In Eq. (0.15) is the spring constant, Θ isthe bond angle between three consecutive monomersand is the equilibrium bond angle equal to . In thepaper the authors focused on the influence of chainstiffness on the size, shape and effective charge of themolecules in the complex. The calculations indicatedthat as stiffness of linear polyelectrolytes is increasedtheir radius of gyration increases and shape changesfrom oblate to prolate. The latter effect is associatedwith the chain’s transformations from “coil”�like to“U”�like or “V”�like and finally to “rod”�like micro�conformations. The calculations also showed that the

3P M

Θ= Θ − Θ ,angle2

01 ( )2

U K

KΘ

0Θ π

40

fch

Nch

0 80

1

0.3

0.6(a)

40

fd

Nch

0 80

0.3

0.6(b)

2345678910

Fig. 7. Fraction of condensed: (a) chain counterions on the chain , (b) dendrimer counterions on the dendrimer vs. chain

length at various values of the reduced electrostatic temperature = (1) 0.05, (2) 0.1, (3) 0.2, (4) 0.4, (5) 1.0, (6) 2.0, (7) 3.0,(8) 4.0, (9) 5.0 and (10) 10.0. The dashed vertical line indicates the compensation point.

chf df

chN τ

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COARSE GRAINED SIMULATIONS OF NEUTRAL AND CHARGED DENDRIMERS 145

size of dendrimers is affected by chain stiffness. Actu�ally, it increases at small values of the chain stiffnessparameter and becomes very weakly sensitive to it inthe limit of high chain stiffness. Furthermore, at thesmall Bjerrum length the dendrimer only slightlydeparts from the spherical shape with increasing chainstiffness, whereas at the larger Bjerrum length thechange in shape is pronounced.

The question of the influence of chain stiffness oncharged dendrimer�linear polyelectrolyte complexeswas also considered in Ref. [58]. The authors carriedout Monte Carlo simulations of complexes comprisedof a terminally�charged dendrimer and an oppositelycharged linear polyelectrolyte with finite rigidityaccompanied by neutralizing counterions in an ather�mal, implicit solvent. The simulated systems werecontrolled by the reduced electrostatic temperature(inverse of the Bjerrum length) and the flexibilityparameter of the chain which were varied over a widerange including both flexible and stiff chains as well asstrong and weak counterion condensation effects.

The calculations indicated that chain stiffness has avery weak effect on the stability of dendrimer�chaincomplexes: No matter chain rigidity such complexesare stable if the strength of electrostatic interactions issufficiently high. The influence of the chain flexibilityparameter on the number of condensed chain beadswas found to be weak as compared to that of the elec�trostatic contribution, though some desorption of thebeads at high chain rigidity and reduced electrostatictemperature was observed. The latter can be explainedby the interplay between the electrostatic attractionbetween the two polymers that promotes condensed,compact conformations of the chain and the bendingforce that prefers locally linear conformations of thechain. For this reason short chains in the complex takeoblate and spherical shaped conformations, whereasthe longest one takes rodlike conformations as it getsstiff at relatively low reduced electrostatic tempera�

tures. The calculations showed that chain rigidity hasno effect on counterion condensation: No matterchain rigidity selective counterion localization withinthe complex and condensation occur as the reducedelectrostatic temperature is lowered.

Chains longer than the compensation point displayinteresting conformational changes at low tempera�ture with increasing rigidity: They pass from coil�liketo rod�like conformations by a sharp release of twotails of similar length, see Fig. 8. This effect is also sig�naled by sharp changes in the number of condensedchain beads, lengths of loops, trains and tails as well asin the size and anisotropy of the chain.

ADSORPTION OF DENDRIMERS

Due to the complexity of the problem, adsorptionof dendrimers caused only by short range surfaceattraction was theoretically investigated mainly bymeans of computer simulations. Below we present therecent developments.

Mansfield carried out Monte Carlo simulations oflattice model dendrimers with spacers S = 7interacting with an adsorbing planar surface in a widerange of the interaction strength A in an athermal sol�vent [98]. The molecules were modeled on a diamondlattice, and trial conformations were generated usingthe so�called “wiggle” moves. The calculations clearlyshowed that dendrimers spread out and flatten downon the surface with increasing interaction strength.The authors interprete their results in the spaceby defining regions corresponding to different behav�ior of the molecules. In the desorption region charac�terized by small values of A the dendrimer desorbsfrom the surface. In the weak adsorption region withslightly higher A�values the dendrimer stays in contactwith the surface, though the number of contacts is verysmall. At larger A�values the authors report about

1 8G G−

G A−

(a) (b) (c)

Fig. 8. Tail release of the longest chain of = 60 monomers caused by decreasing chain flexibility parameter, , at the reducedelectrostatic temperature . Snapshots show the representative conformations at = (a) 2000, (b) 0.6, and (c) 0.1. Thefree spheres represent chain counterions on the complex or close to it.

chN ζ0 0 5τ = . ζ

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stages where a different number of dendrons areadsorbed, which suggests a picture of partial adsorp�tion where part of the dendrimers stay in a ratherunconstrained conformation with respect to the bulk.Only for higher values of the adsorption strength allparts of the dendrimers were found in an adsorbedstate.

In the framework of the freely�jointed�hard�spheremodel the ensemble�growth algorithm was applied byStriolo and Prausnitz to study adsorption of branchedhomopolymers (including dendrimers) onto animpenetrable surface made up of hard spheres [132].The interactions between the polymer and the wallwere represented with a square�well potential betweeneach polymer bead and each wall sphere, and the goodsolvent was treated implicitly as a continuum. Theauthors focus on the calculation of mean segment�density profile and distribution function as functionsof the normal distance from the surface at various val�ues of the polymer�wall interaction parameter. Thesimulations indicated that in the case of a non�attrac�tive surface, near the surface a polymer�segmentdepletion layer occurs. In that region the segment�density profile is affected by the surface roughness andpolymer flexibility. At intermediate distances from thesurface the mean segment�density profiles depend onthe polymer’s flexibility and size in the bulk solution.At distances larger than the radius of gyration of themolecule in the bulk solution the profile reaches unityasymptotically. Furthermore, the authors argue thatdendrimers adsorb more easily as compared to linearchains because of experiencing a smaller entropic pen�alty upon adsorption. With respect to conformationsof adsorbed dendrimers the calculations indicated thatthe segments at the end of the branches are likely to beadsorbed, while the core is located away from the sur�face. The number of adsorbed monomers is reportedto increase with the wall�segment attractive interac�tion for all the dendrimers considered except for G4ones. For the latter, the number of contacts with thesurface in their most probable conformations was thesame for each value of the interaction parameter con�sidered. Moreover, adsorbed dendrimers were found tobe globular in shape with their core at the center of theglobule at each wall�monomer attractive interaction.

Ratner et al. carried out Brownian dynamics simu�lations of dendrimers of third generation with singlespacers attracted by a structureless surface [123]. Inthe calculations they employed a freely jointed bead�rod model with excluded volume and with a Lennard�Jones potential to represent the surface�dendrimerinteractions. Detailed analysis of the data lead theauthors to the conclusion that there exist three regionsof adsorption of the dendrimer, referred to as the week,transition and strong adsorption regions, respectively.In the week adsorption region the outer segments ofone or two dendrons are in contact with the surface,while the dendrimer’s core and the other dendronsstay away from it. Subsequently, as the attraction

strength is increased the molecule enters the transitionregion where all of the dendrons are in contact withthe surface. Here, both inner and outer beads of thedendrimer are adsorbed and the molecule starts tospread over the surface. As the interaction between thepolymer and the surface grows further, in the strongadsorption region, the mean number of adsorbedbeads reaches the state of complete adsorption. In thisstate the number of adsorbed beads fluctuates by des�orption of the outermost beads.

In the framework of a freely jointed bead�rod chainmodel Suman and Kumar used Brownian dynamicssimulations to inspect isolated hydrophobic den�drimer adsorption onto a flat surface in an athermalsolvent [135]. Bead�bead, bead�surface hydrophobicinteractions and excluded�volume interactions weremodeled via a Lennard�Jones potential. In the studythey focused on the effect of the strength and distribu�tion of hydrophobicity and dendrimer generation onthe dendrimers’ conformation and bead distributionswithin the dendrimer and near the surface. The simu�lations demonstrated that no matter hydrophobicgroup distribution adsorbed dendrimers take a disklikeconformation which is flat in the direction normal tothe surface and expanded in the direction parallel to it.As the strength of hydrophobicity is lowered, the diskexpands in the normal direction and shrinks in theparallel one. In the limit of a very weak interactionwith the surface, the molecule takes a spherical con�formation. The terminal groups of adsorbed dendrim�ers are localised a bit away from the center of mass ofthe molecule. Furthermore, the obtained monomerdensity profiles relative to the surface show that theadsorbed dendrimer forms a two�layer structure: Onelayer corresponds to the adsorbed groups, and theother, less dense layer, to beads in free solution.

Lenz et al. performed Monte Carlo simulationsbased on the bead�spring model to investigate thebehavior of G2 amphiphilic dendrimers with spacerlength S = 1 in solutions as well as in the vicinity of aplanar wall [65]. In the considered model the terminalgroups of the dendrimers formed the shell and weresolvophilic. The core, on the other hand, consisted ofunits which were solvophobic. The bonds between themonomers were modeled by the FENE potential andthe nonbonded interactions by the so�called Morsepotential. The solvent was treated in an implicit way.The wall was represented by monomer�like sphericalparticles arranged in an infinitely extended surfacewith homogeneous density. An attractive potentialbetween the wall and the dendrimer beads dependedonly on the distance to the wall and was described bythe 9–3 potential. In the paper two types of walls areconsidered. One which attracts the core monomersand repels the shell ones, and the reverse. Among oth�ers, the calculations indicated that unlike dendrimersin the bulk which take spherical symmetric conforma�tions, the wall considerably affects the dendrimers’conformations. Actually, in the vicinity of the surface

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COARSE GRAINED SIMULATIONS OF NEUTRAL AND CHARGED DENDRIMERS 147

depending on the wall�dendrimer interaction the mol�ecules exist in two characteristic conformationsreferred to as the dead spider and the living spider,respectively. In the first conformation, which occursfor dendrimers with the attractive core and repulsiveshell, the core monomers prefer to stay close to thewall and the shell monomers to stay further away fromit. In the case of dendrimers with the repulsive coreand attractive shell the behavior is opposite. Also, thesimulations allowed to determine the effective interac�tions between an isolated dendrimer and another den�drimer as well as with the wall.

In contrast to adsorption of dendrimers, theadsorption behavior of linear chains was theoreticallyunderstood in terms of crossover phenomenon close toa continuous adsorption phase transition [7, 17, 19–21]. The scaling analysis developed for linear polymerscan be formally extended to branched polymers andcan be applied to the results of coarse grained com�puter simulations. This concept was used in the recentstudy using the bond fluctuation model. Here, westudied adsorption of dendrimers in a wide range ofthe generation number and spacer length onto a flatsurface. Simulations based on the method indicated atwo�step adsorption scenario for dendrimers with longspacers similar to the findings in Refs. [123, 132] notedabove: Below a critical temperature of adsorption, ,spacers in direct contact with the surface are adsorbedwhile the dendrimer as a whole is only weakly per�turbed and keeps up its spherical shape. At a charac�teristic temperature, , the molecules undergo ajump�like transition into a flat conformation. Adsorp�tion of spacers which are in contact with the surfacefollows crossover�scaling as for linear chains and stars.The estimate for the critical temperature of adsorp�tion, 1, is in good agreement with the behav�ior of star polymers. The two�step adsorption behaviorof dendrimers explains the rather poor crossover�scal�ing of the overall order parameter which is defined asthe fraction of all monomers which are in contact withthe adsorbing surface. While the corresponding orderparameter of the adsorbed spacers does display cross�over�scaling, the fraction ofspacers which areadsorbed at the surface display a jump�like transition.Related with this jump�like transition the shape of thedendrimer becomes highly anisotropic.

In the flatly adsorbed state the lateral extension ofthe dendrimer follows the scaling behavior for a 2D�object. According to the mean�field prediction a scal�

ing of is obtained. The results of the calcu�lations suggest discriminating between two states ofadsorption for dendrimers with long spacers: A weaklyadsorbed regime close to the critical point of adsorp�tion, , which is indistinguishable from the criticalpoint of adsorption of star polymers with low func�tionality. Here, the dendrimer as a whole retains itsshape but sticks to the surface. In this state the den�

cτ

* cτ < τ

1 01cτ = .

1/4gR N� ∼

cτ

drimer resembles an elastic sphere with a small impactat the surface. Below a spacer�length�dependent tem�perature, , the dendrimer collapses in the direc�tion perpendicular to the adsorbing surface. This canbe called the strongly adsorbed state of the dendrimer,and corresponds to the temperature of shape�transition. The simulations showed that flexible spac�ers have a major impact on the adsorption behavior ofdendrimers, in particular on the point of shape�transi�tion which correspond to a collapse of the dendrimerfrom an isotropic into a quasi�2D state. Long spacerslead to a very sharp, well defined collapse of the den�drimer at adsorbing surfaces, which might have inter�esting applications for responsible surfaces usinggrafted dendritic molecules.

CONCENTRATION EFFECTS

In comparison to isolated dendrimers the under�standing of concentration effects for these molecules isincomplete, experimental results partially controver�sial, and systematic computational studies are stillmissing. Actually, to a large extent such issues as theeffect of flexible spacers in dendrimer solutions, den�drimers’ size and conformational changes as a func�tion of concentration as well as their ability to inter�penetrate and to self�assemble still remain unex�plored. Little is known about dendrimer�dendrimerinteractions in concentrated solutions, it is not clear

whether the overlap concentration, , plays asimilar role in dendrimer solutions as for linear chains.Moreover, the few, existing reports provide differentresults and interpretations that contradict each other.Size changes of dendrimers in concentrated solutionswere examined with small angle neutron (SANS) scat�tering over a wide range of dendrimer mass fraction[148]. It was suggested that the behavior of dendrimersin solutions is consistent with a model of non�inter�penetrating collapse�like behavior of individual den�drimers. In particular, two concentration regimes ofdendrimer solutions, referred to as dilute and concen�trated were found. While in the first regime dendrimersbehave as soft, spherical objects, whose size is notaffected by the solution concentration, in the secondtheir size decreases as the number density increases. Itwas argued that dendrimers collapse to maintain a vol�ume fraction, which corresponds to the volume frac�tion of random close packing of hard spheres. It wasfurther argued that dendrimers are completely non�interpenetrable, and that they have to shrink at highconcentrations due to packing effects, which in turn,leads to a concentration�dependent radius of the mol�ecules. Similar conclusions concerning the absence ofinterpenetration of dendrimers in close proximity wasalso drawn from other experiments based on Trans�mission Electron Microscopy [40], steady shear rhe�ometry [149] and small�angle X�ray scattering [117].By contrast SANS experiments on poly(propylene

*( )Sτ

*( )Sτ

* / 3gc N R∼

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imine) dendrimer solutions let to the conclusion thatthe internal structure of these molecules is unaffectedby interactions between them. It was pointed out thatthe poly(propylene imine) dendrimers behave as softmolecules with possible interpenetration at higherconcentration but without substantial impact on theconformational properties of individual dendrimers[119]. As indicated by EPR and fluorescence depolar�ization study of intermolecular interactions of den�drimers, such as aggregation and degree of entangle�ment, in solutions at high concentrations, the inter�molecular dendrimer interactions depend on themolecules’ generation. More specifically, as comparedwith high generation molecules the interactions areslightly stronger for lower generation dendrimers dueto a more open structure they posses [42].

Apart from the static properties of dendrimers insolutions of variable concentration, their dynamicalbehavior is of interest as well. For instance, self�diffu�sion of dendrimers was studied over a wide range ofconcentrations using the NMR method. It was shownthat over the whole range of the concentrations stud�ied, the generalized concentration dependence of thedendrimer self�diffusion coefficient coincides with theanalogous concentration dependence of the proteinself�diffusion coefficient in aqueous suspensions[129]. For linear polymers dynamic scaling can beapplied to understand the behavior in semi�dilutesolutions. Similar attempts for dendritic polymers tothe best of our knowledge have not yet been carriedout.

The properties of dendrimers in solutions and theproblem of interactions between individual moleculeswere considered by combining liquid integral equationtheory and mean�field arguments. Here an effectiveinteraction potential of a Gaussian form between thedendrimers’ centers of mass was derived. By applyingthis theory to concentrated dendrimer solutions theo�retical structure factors were calculated and comparedwith experiments. Agreement with experiment forconcentrations below the overlap concentration couldbe achieved by fitting the free parameters of the model[28, 70, 71]. Using Gaussian effective pair potentialsalong with the mean field density functional theory thestructure, phase behavior, and inhomogeneous fluidproperties of binary dendrimer mixtures were alsostudied [30]. However, the question of interpenetra�tion and possible scaling in the semi�dilute regimecannot be answered by this approach. Moreover, theGaussian approximation might be challenged byexcluded volume effects which lead to correlationeffects of the dendrimers conformation, in particularfor long spacers [6, 54].

Monte Carlo simulations were employed by Götzeand Likos to examine the role played by many�bodyeffective interactions in concentrated dendrimer solu�tions. By analyzing the radial distribution functionsand the scattering functions between the centers ofmass of dendrimers at various concentrations it was

found that the effects of the many�body forces aresmall up to the overlap density and they might beignored for open dendrimers with long bond lengths[27]. The same simulation technique was used for acomparison between the structural and thermody�namic properties of linear chains and dendrimers insolution. At low concentrations, due to the more com�pact architecture of dendrimers, solutions of dendriticpolymers have a lower osmotic pressure than those oflinear chains of the same molecular weight. In theextreme case of concentrated solutions low�genera�tion dendrimers behave similarly to linear chains,whereas the pressure of solutions containing high�generation dendrimers increases more rapidly withconcentration. A generation dependent shrinking ofthe dendrimers with increasing concentration wasfound as well [3, 75]. A rational description of thesefindings, however is missing. Molecular dynamicssimulation addressed such crucial issues as the degreeof interpenetration of dendrimers, their morphologyand organization as well as the dynamic behavior ofdendrimers in the melt, respectively [46, 158]. Theanalysis of the dependence of the bulk density andmolecular packing on the dendrimer molecular weightand intrinsic stiffness was done with this technique too[10]. Among others, it was found that in the melt ofdendrimers as the generation number increases, themolecules assume a more spherical shape, and thedegree of interpenetration of individual dendrimersdecreases [158].

In conclusion experimental observations contra�dict about the question of possible inter�penetrationeffect in higher concentrated dendrimer solutions.Although a few simulation results exist a unifyingmodel to rationalize the concentration effects in den�drimers, similar to our understanding of linear chains,has not been developed yet. Moreover, the role of flex�ible spacers of various lengths has not been consideredso far.

As compared to neutral solutions the presentknowledge about concentration effects in chargeddendrimer solutions is even more limited. This is incontrast to the fact that many interesting applicationsof dendrimers involve aqueous environments and(usually weak) polyelectrolytic dendrimers. Experi�mental studies of solutions of charged dendrimers aremostly based on scattering techniques. For instance,SANS was used to examine charged poly(amidoam�ine) and poly(propyleneimine) dendrimer solutions indeuterium oxide. The studies showed that ionicstrength, surface charged density, and concentrationhave a major effect on the degree of structural organi�zation of the solution. An increase in ionization of ter�minal groups leads to inter�particle interactions whichcause local, liquid�like ordering. These interactionscan be screened by adding salt in large quantities [107,120]. Also, it was shown that upon changing the solu�tion pH, significant counterion association/conden�

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COARSE GRAINED SIMULATIONS OF NEUTRAL AND CHARGED DENDRIMERS 149

sation occurs which strongly mediates the inter�den�drimer interactions [11].

The effect of electrostatic screening lengths wasinspected with coarse�grained molecular dynamicssimulations of charged dendrimers under low pHinteracting via the Debye�Hückel approximation andLenard�Jones potentials. These calculations showedthat with larger electrostatic screening lengths, den�drimers form a crystalline structure. Dendrimer crys�tals melt when the screening length is decreased andthe molecules reveal a tendency to aggregate [140].The same simulation technique was used to study theproperties of many�body interactions among chargeddendrimers with explicit counterions. Among others,it was found that the triplet force among charged den�drimers becomes repulsive, in contrast to that incharged, hard�sphere systems [139]. Therefore, thestructural formation of the self�assembly of chargeddendrimers in a dense solution may differ considerablyfrom those of hard–sphere suspensions.

It was shown that at room temperature multi�valent counterions can induce attraction between like�charged dendrimers [144]. Molecular dynamics wasemployed to describe self�organization in dilute solu�tions of charged dendrimer molecules upon variationof the strength of the electrostatic interactions in thepresence of explicit solvent and mono�valent counte�rions. It was found that increasing the Bjerrum lengthdevelops a counterion�mediated attraction betweenthe dendrimers, which triggers a self�organization ofdendrimers in cubic phases. The role of dendrimerinterpenetration, trapping of counterions and den�drimer concentration in the ordering process in bothsingle component systems and binary mixtures ofcharged dendrimers was also dealt with [47, 48, 50].Apart from the static properties of charged dendrimersin solution, recently some aspects of the dynamicbehavior of monovalent counterions in such systemswere taken up with respect to ion diffusional motion aswell as residence lifetimes of pairs formed by chargeddendrimer beads and condensed counterions in differ�ent electrostatic regimes. In particular, a scaling lawwas found to describe the counterion diffusion coeffi�cient as a function of the Bjerrum length in the strongelectrostatic regime, independent of the size of thedendrimer molecules at the examined volume frac�tions [49]. In conclusion previous works demonstratedthe potential for self�organization of charged den�drimers in solution driven by counterions. In particu�lar, multi�valent salt ions can lead to complex andstructure formation. The underlying physical princi�ples, however, are to a large extend not yet explored.

CONCLUSIONS AND OUTLOOK

In this review we have presented the overal pictureof dendrimer molecules as seen from coarse�grainedcomputer simulations. Our main focus has been onconformational properties which are directly accessi�

ble in computer simulations and can help establishsimple models to understand the properties of den�drimer�systems. We have considered isolated, neutraldendrimers and dendritic polyelectrolytes as well asadsorption of the former onto surfaces and complex�ation of the latter with linear polyelectrolytes. We havealso discussed the state�of�the�art knowledge of thebehavior of dendrimers at finite concentrations whenintermolecular interactions influence dendrimers’conformations.

Contrary to the first analytical theory but in agree�ment with experiments, most of the lattice and off�lat�tice simulations reveal the dense�core picture of iso�lated dendrimers related with backfolding of dendriticthreads. In particular, the most recent calculationsconfirm the scaling formula derived within the mean�field, Flory�type approach for the radius of gyration ofdendrimers in solvents of different quality [Eq. (0.2)].These results emphasize the role of conformationalentropy on the properties of dendrimers. The conceptof spacer�length scaling, which maps the properties ofdendrimers with different spacer�length to a universalbehavior could be proven in simulations up to a highnumber of generations of dendrimers. For this reason,at present the scaling relation Eq. (0.2) can be consid�ered as appropriate for dendrimers with flexible spac�ers under various solvent conditions.

In our review we have emphasized the crucial roleplayed by counterions in determining conformationsof isolated, charged dendrimers. As clearly indicatedby the most recent coarse grained simulations ions dif�fuse into the dendrimers' interior, which results in aconsiderable increase in their density there and inscreening the electrostatic interactions. According tothe mean�field, Flory�type approach applied for den�dritic polyelectrolytes [Eq. (0.11)] some of the trappedions are condensed on the dendrimers, whereas othersare free to move within the dendrimers’ volume (resid�ual counterions). Residual counterions exert osmoticpressure on the molecules which gives rise to swelling.However, swelling of charged dendrimers caused bythe osmotic pressure of residual counterions is lesspronounced than that observed from simulationswithout explicit ions. In the latter case, the effect isexplained to occur because of repulsion between likecharged monomers of the dendrimer only. Simulationsalso show that, apart from swelling, in the limits ofhigh and low Bjerrum length dendrimers shrink. In theformer case shrinking is due to pronounced condensa�tion of ions on the dendrimer, in the latter the behaviorof dendrimers is determined by the entropy itself andis the same as of neutral molecules. The conforma�tional changes of dendrimers are also affected by ionvalence. In particular, in comparison with monovalentmonomers and counterions, swelling observed fordendrimers with monomers bearing multivalentcharges and accompanied by monovalent counterionsis much stronger. On the other hand, as proved by anumber of simulations the size of charged dendrimers

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can be considerably reduced with the addition of mul�tivalent salt. In this case multivalent salt ions replacemonovalent counterions, neutralize the dendrimermore effectively and reduce the osmotic pressureinside the molecule.

The effect of explicit counterions is crucial also forthe description of the properties of complexes betweenoppositely charged dendrimers and linear polyelectro�lytes. The release of counterions due to complexationcan be taken as the main driving force for complex for�mation, which can even counterbalance overchargingeffects. At the compensation point at which both mol�ecules carry the same absolute charge, counterions donot localize inside the complex at all. For long chainsin the complex only chain counterions condense onthe chain, whereas dendrimer counterions evaporate.For this reason, due to a reduction in the osmotic pres�sure inside the dendrimer its swelling is considerablyweakened as compared to complexes with shortchains. On the other hand, the behavior of the chains’size is strongly non�monotonous and exhibits a pro�nounced maximum with varying Bjerrum length. Thesimulations show that adsorbed chains take conforma�tions with some of their monomers condensed on thedendrimers’ charged groups and with two tailsreleased. Depending on the Bjerrum length and chainlength conformations with equally long tails or onelong and one short tail are preferred.

With respect to adsorption behavior of dendrimersonto flat surfaces the number of simulation works isstill rather limited. The up�to�date picture of this phe�nomenon obtained from coarse grained simulations issuch that there exist three distinct regions of dendrim�ers’ behavior near adsorbing surfaces. Above a criticaltemperature of adsorption dendrimers are in the non�adsorbed state where, on average, only few monomersare in contact with the surface. For this reason theconformational properties of dendrimers remainunaffected by the surface and are indistinguishablefrom those of free, isolated molecules. Below a criticaltemperature of adsorption, in the weakly adsorbedstate, spacers in direct contact with the surface areadsorbed while dendrimers as a whole are only weaklyperturbed. In particular, they keep on their sphericalshape. At a characteristic temperature below the criti�cal temperature of adsorption, dendrimers undergo ajump�like transition into the strongly adsorbed statewhere they exist in flat conformations. Related withthis jump�like transition the shape of adsorbed den�drimers becomes highly anisotropic and density pro�files change from spherical into flat shapes. In theflatly adsorbed state the lateral extension of dendrim�ers follows the scaling behavior for 2D�objects.

We have also outlined the recent progress made inthe emerging field of neutral and charged dendrimersat finite concentrations. Actually, the understanding ofconcentration effects for these molecules is poor ascompared to isolated dendrimers. While, on the onehand, the existing experimental results are partially

controversial, systematic simulation studies are stillmissing on the other. To name but a few dendrimer�dendrimer interactions, dendrimers’ interpenetrationand ability to self�assemble as well as their size proper�ties and conformational changes with varying concen�tration still remain unrevealed. With this respect thereare many open questions which, as computer technol�ogy is improved, need to be taken up in the near future.

ACKNOWLEDGMENTS

Support from the Deutsche Forschungsgemein�schaft (DFG) contract numbers SO�277/2�1 and KL2470/1�1 is gratefully acknowledged.

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