s caling properties in the molecular structure of three ... · s caling properties in the molecular...

Carbon 41 (2003) 2269–2283

S caling properties in the molecular structure ofthree-dimensional, nanosized phenylene-based dendrimers as

studied by atomistic molecular dynamics simulations*Sabrina Pricl , Maurizio Fermeglia, Marco Ferrone, Andrea Asquini

Computer-Aided Systems Laboratory, Department of Chemical Engineering-DICAMP, University of Trieste, Piazzale Europa 1,I-34127 Trieste, Italy

Received 17 April 2003; accepted 29 May 2003

Abstract

Three-dimensional polyphenylene dendrimers (PDs) can be prepared in ways that enable control of their shape. Theirstructures may be used as scaffolds with a wide variety of functionality, enabling them to be used as functional nanoparticleswith a large range of possible applications, ranging from light emitting devices to biological sensors or drug delivery tools.As PDs have been synthesized only recently, their structural and chemico-physical characterization is still in its infancy.Accordingly, in this paper the shape and internal organization of three PD families based on three different cores wereprobed by accurate, atomistic molecular dynamics simulations (MD). Particular care was taken to ensure complete structuralequilibration by implementing an MD simulated annealing protocol prior to evaluation of the molecular structure anddynamics. All dendrimer families were found to be characterized by molecular dimensions in the nano-range, and by ashape-persistent, non-spherical structure, of molecular fractal dimension around 2.5–2.6, and of surface fractal dimensionpractically constant and almost equal to 2 with increasing generations in all cases. The MD analysis revealed also that, forthis type of dendrimers, the starburst limited generation is presumably located in correspondence of the third generation. 2003 Elsevier Ltd. All rights reserved.

Keywords: A. Non-graphitic carbon; B. Molecular simulation; D. Biocompatibility; D. Chemical structure; D. Surface areas

1 . Introduction materials are just a few examples of the plethora of toolsavailable nowadays to the chemical engineer for the design

Polymers, with their large spatial extent and chemical and production of novel polymeric materials.variety, afford materials scientists the opportunity to be End-use properties, which are the ultimate measure ofarchitects at the molecular level. Once the architecture is the efficacy of a polymer manufacturing operation, aredecided, however, the task of construction moves to the generally affected as much by the chemical and physicalchemical engineer to build the material in a faithful, stable processing as by the molecular design. Polymer productionand efficient manner. This involves assembly not only the is not simply a matter of executing the molecular blueprint.of molecular structure but also of the larger scale internal There is a level of interaction between the process and themicro- and meso-structure of the material. New processes product with polymers which is much richer, more com-are continually becoming available to the chemical en- plex and more intricate that for most chemical products.gineer to accomplish this task. Innovative catalysts, origi- Polymer properties are not intrinsic, as are the properties ofnal macromolecular building blocks, reactive processing, simple chemicals, but rather they can be manipulatedself-assembly, manipulation of phase behavior, application widely by polymerization and processing conditions. Forof strong orienting fields and genetic engineering of this reason, chemical engineers producing polymers must

be conversant with molecular design, and molecular de-signers must be aware of processing considerations.*Corresponding author. Tel.:139-040-558-3750; fax:139-

An intriguing development in the last decade has been040-569-823.E-mail address: [email protected](S. Pricl). the burst of activity in the design, synthesis and applica-

0008-6223/03/$ – see front matter 2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0008-6223(03)00254-9

mailto:[email protected]

2270 S. Pricl et al. / Carbon 41 (2003) 2269–2283

tions of dendrimer polymers[1–6]. The term functional designed solvophilicity at the surfaces (even with con-polymer is often used to describe polymers that carry trolled, local variation over the surface of the samereactive functional groups that can participate in chemical molecule).processes without degradation of the original polymeric Polyphenylene-compounds containing benzene ringsunit. Functional polymers are abundant in nature. Molecu- linked bys-bonds form an increasingly important class oflar architecture has significant effects on the characteristics organic materials, as the benzene ring is an extremelyof functional polymers. Most polymers consist of largely flexible modulus for the construction of a wide range oflinear chains that are randomly coiled and entangled with structures and can bear a large range of active functionalitytheir neighbors. Introduction of substantial amounts of [22]. Also, the delocalizedp-systems on adjacent rings canbranching is always accompanied by large changes in overlap to form extended conjugate assemblies, which canthermophysical properties, such as viscosity and density. be of considerable importance in material science, as theirFrom the point of view of functionalization, a major effect optical and electrical properties make them suitable for useof branching is the multiplication of reactive chain ends. as active compounds in a variety of electronic devices,

Dendrimers are highly branched, three-dimensional including light-emitting diodes (LEDs)[23], optically-macromolecules with branch points at every monomer, pumped lasers[24], and field-effect transistors (FETs)leading to a structure that has essentially as many end- [25].groups as it has monomeric units. When synthesized by Tetrahedral polyphenylene dendrimers (PD), alias cas-controlled, convergent growth, these dense, regular struc- cade molecules with four successively branched armstures tend to adopt compact, well-defined shapes and, in made of phenyl rings only and emanating from a centralinitial explorations, have begun to exhibit physical prop- core, are a new class of dendritic materials that have been

¨erties not seen even in other more traditional forms of recently synthesized by Mullen and coworkers[26,27]. Inhighly branched polymers[1]. In this introduction, we can particular, they introduced 3,4-bis(4-triisopopylsilyl-just briefly summarize some of the exciting, new applica- ethynyl-phenyl)-2,5-diphenylcyclopenta-2,4-dienone as ations for dendrimers actually being explored in industry building block for the synthesis of a new type of nanosizedand university. These include uses as polymeric catalyst hydrocarbon dendrimers. The key step of their dendrimerparticle and nanoscale reactors[7,8], molecular mimics of synthesis is the [214] cycloaddition of the building blockmicelles[9,10], delivery vehicles for drugs[11], magnetic to a core, such as 1,3,5-triethynylphenyl (1a), 3,39,5,59-resonance imaging agents[12], immuno-diagnostic and tetraethynylbiphenyl (1b) or tetra-(4-gene therapy[13,14], synthetic mimics of protein active ethynylphenyl)methane (1c) (Fig. 1).sites [15,16], surface-coating dendritic sensors[17,18], After cleavage of the triisopropylsilyl groups, the ob-energy funnels[19,20], and building blocks for more tained oligoethynyl is ready for further Diels–Alder re-elaborate supermolecular structures[21]. actions. Thus, by this repetitive cycloaddition-deprotection

The molecular features enabling these applications are sequence, the construction of dendrimer generations, that ispersistent and controllable nanoscale dimensions in the various well-defined, radial symmetrical layers of repeat-range from 1 to 100 nm, some control over shape via ing units G1, G2, . . . Gn, can be accomplished.molecular design of the dendrimer core, precise masses The surface of theses dendrimers can be coated with athat can approach 100 000 (with practically no polydis- wide variety of functionalities, including halo, cyano,persity, i.e.M /M ¯1), chemically reactive surface func- carboxy, amino, hydroxyl and thiomethyl groups[28]. Thew n

tionality, interiors that can be specifically tailored to amine and carboxy substituents especially should not onlyhydrolytically or thermally demanding environments, and provide a means for obtaining water solubility but also for

Fig. 1. Tri- and tetrafunctionalized cores leading to PDs.

S. Pricl et al. / Carbon 41 (2003) 2269–2283 2271

attaching a wide range of other groups to the dendritic molecule was built and its geometry optimized via ananoparticles, e.g. peptides, antigens or other biological combined steepest descent–conjugate gradient algorithm,substrates. using as a convergence criterion for the energy gradient the

Due to their very dense intramolecular packing, these root-mean-square of the Cartesian elements of the gradient24 ˚monodisperse polyaromatic dendrimers are of particular equal to 10 kcal /(mol A). Long-range non-bonded

interest with respect to the design of nanostructures interactions were treated by applying suitable cut-offcharacterized by an invariant shape[29]. Besides their distances, and to avoid the discontinuities caused by directsignificantly enhanced thermal and chemical stability, their cut-offs, the cubic spline switching method was used. vanpostulated rigidity as compared to aliphatic dendrimer der Waals distances and energy parameters for non-bondedsystems, coupled with the wide variety of possible func- interactions between heteronuclear atoms were obtained bytionalizations, provide the basis for their potential applica- the 6th-power combination rule proposed by Waldman andtions, such as support for catalysts, dyes or biological / Hagler[38].biomedical active substances in human diagnosis and Such a straightforward molecular mechanics scheme ismedicine. All these application potentials, however, will likely to trap the simulated system in a metastable localnot be realized before the understanding of their physical high-energy minimum. To prevent the system from suchproperties is considerably advanced. entrapments, the relaxed structures were subjected to a

Notwithstanding the fact that both scattering and micro- combined molecular mechanics/molecular dynamics simu-scopic methods have successfully been applied to the lated annealing (MDSA) protocol[16,39,40].Accordingly,structural characterization of polyaromatic dendrimers the relaxed structures were subjected to five repeated[30], attempts to understand their dynamic behavior have temperature cycles (from 298 to 1000 K and back) usingbeen confined to limited, preliminary molecular dynamics constant volume/constant temperature (NVT) MD con-simulations[31–33]. Since knowledge of the global shape ditions. At the end of each annealing cycle, the structures

24and extent of flexibility are among the most important were again energy minimized to converge below 10˚criteria for the design of well defined architectures, in this kcal /(mol A), and only the structures corresponding to the

work we investigated the dynamics of amorphous bulk minimum energy were used for further modeling.dendrimers based on cores1a, 1b and 1c and up to The calculation of molecular surfaces was performedgeneration 3 by means of accurate molecular dynamics using the so-called Connolly dot surfaces algorithm[41–simulations. 43]. Accordingly, a probe sphere of given radiusp ,r

representing the solvent molecule, is placed tangent to theatoms of the molecule at thousands of different position.

2 . Computational details For each position in which the probe does not experiencevan der Waals overlap with the atoms of the molecule,

All molecular mechanics and dynamics simulations were points lying on the inward-facing surface of the probe2performed using the program packages Cerius (v. 4.2), sphere become part of the molecule solvent-accessible

Discover and Materials Studio (v. 2.2) (all from Accelrys, surface (SAS). According to this procedure, the molecularSan Diego, USA), and in-house developed codes (stand- surface generated consists of the van der Waals surface ofalone and add-on to the commercial software). the atoms which can be touched by a solvent-sized probe

All calculations were carried out using the Compass sphere (thus called contact surface), connected by aforce field (FF) [34]. The Compass FF is an augmented network of concave and saddle surfaces (globally calledversion of the CFF series of forcefields[35–37] and is the reentrant surface), that smoothes over crevices and pitsfirst ab initio forcefield that has been parameterized and between the atoms of the molecule. The sum of the contactvalidated using condensed-phase properties in addition to and the reentrant surface forms the so-called molecularvarious ab initio and empirical data for molecules in surface (MS); this surface is the boundary of the molecularisolation. The bond terms of the Compass FF potential volume (MV) that the solvent probe is excluded from if itenergy function include a quartic polynomial both for bond is not to undergo overlaps with the molecule atoms, whichstretching and angle bending, a three-term Fourier expan- therefore is also called solvent-excluded volume. Finally,sion for torsions and a Wilson out-of-plane coordinate performing the same procedure by setting the probe sphereterm. Six cross-terms up through 3rd order are present to radius equal to zero, the algorithm yields the van der Waalsaccount for coupling between the intramolecular coordi- surface (WS).nates. The final two non-bonded terms represent the The details of the isolated dendrimer structures at 298 Kintermolecular electrostatic energy and the van der Waals were obtained by performing MD simulations underinteractions, respectively; the latter employs an inverse 9th isochoric / isothermal (NVT) conditions. Each molecularpower term for the repulsive part rather than the more dynamics run was started by assigning initial velocity forcustomary 12th power term. atoms according to Boltzmann distribution at 23 T. Tem-

The generation of accurate PD model structures was perature was controlled via weak coupling to a temperatureconducted as follows. For each dendrimer generation, the bath[44], with coupling constantt 50.01 ps. The NewtonT


molecular equations of motion were solved by the Verlet dendrimer progressively evolves towards a more compact,leapfrog algorithm[45], using an integration step of 1 fs skewed structure, characterized by a flat, central zoneand for a total simulation time of 3 ns. resembling a Spanish fan (Fig. 3, center and right). The

evolution of the structure is intimately related to thedynamics of the core and of the G1-1b tetraphenylbenzeneunits. In fact, although the dynamics reveal a tendency of

3 . Results and discussion the two units to get closer due to mutual attractiveinteractions, a strong coupling between the G2-1b units of

Figs. 2, 3 and 5show the molecular models of gene- these two branches cannot take place due to the stericration 1, 2 and 3 for the1a-, 1b- and1c-based dendrimers, constraints in the G1-1b layer that limit the molecularrespectively, obtained as a result of the first structural mobility. The modification of the form is actually driveninvestigations performed with MD simulations under NVT by a change in the torsion angleu , which decreases to an1

conditions. An accurate analysis of the molecular trajec- average value of 378 (678). Correspondingly, stabilizingtories reveals that, in spite of intramolecular dynamics van der Waals and Coulomb interactions then lead to theinvolving rearrangements of the dendritic arms, as well as bending and association of the branches. As a result,

˚of the benzene units, the global shape of all dendrimeric distance 1–2 increases to|38 A while distance 3–4˚series does not change throughout the simulation, and the lengthens to|29 A. An utterly analogous situation is

systems retain their overall nano-architecture. encountered in the next generation G3-1b, where theThe first generation of the1a-based PDs is characterized molecule is characterized by almost the same average

by a rather stiff structure with radial symmetry (Fig. 2, value of the torsion angleu (386108), and again by two1˚left). This result could be anticipated, since the core is different values of the distances 1–2 (48 A) and 3–4 (38

˚planar, with the three active sites at 1208 from each other. A).The freedom of movement of the lateral branches is For the dendrimer with a tetrahedral core1c, if the mostnoticeably reduced, if compared with the corresponding stable conformation of the first generation is also char-generations of the other two PD families (see below). This acterized by an almost symmetrical shape, with an averagecan be attributed to the short distance between the branch-value of the angle at the central, tetrahedral carbon of 1098

es which, in turn, is a direct consequence of the small (638) (Fig. 5, left), upon increasing generation the forma-dimension of the core (i.e. a single phenyl ring). The next tion of pairs of tetraphenylbenzene units takes place.generation, G2, tends to assume a conformation in which Indeed, passing from G1-1c to G2-1c and G3-1c, thetwo dimensions prevail over the third, and such a behavior pairing of the branches has important consequences for thebecomes even more manifest with G3 (Fig. 2, center and morphology of the molecule since (i) it leads to theright). Accordingly, the molecules of G2 and G3 tend to formation of an asymmetric diabolo-like molecular shape,assume a flower-like form, in which one side, the one formed by a concave part having a high packing density ofwhere the branches protrude from, exposes a highly phenylene rings (Fig. 5, center and right), and (ii) weirregular and fringed surface, whereas the other one is observe an asymmetric growing of the molecular structure,rather flat. characterized by a larger increase in the length with respect

The 1b-based dendrimers grow in a fan-like shape, as to the width of the dendrimer (see below). These corre-evidenced byFig. 3. Indeed, all three generations preserve spond to a bimodal distribution of the angle values arounda flat, central zone around which the structure develops in the core carbon atom centered, for G2-1c, around thea somewhat regular fashion. This structural feature is average values of 1038 (638) and 1138 (638), and forexpected to affect the dynamics of the molecule; this can G3-1c around 1018 (638) and 1148 (638), respectively.be followed by the evolution of the torsion angleu , which The asymmetry in higher generation structures gives rise to1

defines the relative orientation of the benzene rings of the different mobility of the branches, which are essentiallycentral core (see labeling inFig. 4). governed by steric constraints imposed by the relative

Such analysis reveals that, for higher dendrimer genera- orientations of the two branches in a given pair. The MDtions and during the entire runs of the dynamic simula- simulations show that this dendrimer family has goodtions, the peculiar fan-like structure is preserved, and the shape persistence; despite the fact that the global shape ofmajor movements interest the outer groups. In fact, for the the dendrimer does not evolve throughout the simulation,first generation G1-1b, the torsion angle valueu varies local rearrangements of the G2 and G3 tetraphenylbenzene1

with amplitude of 138 around its average value of 598. groups can take place in the arms, leading to exchange inAccordingly, the distances 1–2 and 3–4 that govern the the inner versus outer positions of the peripheral units.overall shape of the molecule are only slightly affected by This pictorial evidence can be quantified by the aspect

˚such torsions, being typically of the order of 22 A and ratio of principal moments of inertia for various genera-˚varying by 1–5% (0.3–1.1 A) during the entire course of tions G of the three PD series, reported inFig. 6.

the simulation. In passing to G2-1b and G3-1b, from a For the series of molecules based on core1a, therelatively open and symmetrical structure (Fig. 3, left) the tendency to assume a structure in which one dimension


Fig. 2. Optimized molecular models of G1 (left), G2 (center) and G3 (right) of PD with core1a.


Fig. 3. Optimized molecular models of G1 (left), G2 (center) and G3 (right) of PD with core1b.


molecular dynamics simulations for all generations of thethree series of PD molecules are plotted inFig. 7 as afunction of the dendrimer molecular weightM .w

The linear relation between logR and log M is ag w

further indication of compact space filling structures withfractal dimensionalityd of |2.15 in the case of1a-PDs,f

2.59 for 1b-PDs, and 2.43 in the case of1c-baseddendrimers. Fractal geometry is a mathematical tool fordealing with complex systems that have no characteristiclength scales. Scale-invariant systems are usually char-acterized by non-integer (i.e. fractal) dimensions, andhence the objective of any fractal analysis is to find arelationship of some kind of power-law:

scaling exponentphysical property~ variable (1)

where the variable and the exponent are related to thefractal dimensiond . This relation is obviously one thatfFig. 4. Two-dimensional projection of G1-1b. The distances usedcan cover a very broad range of molecular structures;are labeled, and the torsion angleu is indicated in bold.1

however, this kind of power law requires some symmetryin these structures. Interestingly enough, the values ofd f

definitely prevails over the other two is clearly evident in calculated for all PD series considered are close to theFig. 6: indeed, the values ofI andI are at least twice that average mass fractal dimension exhibited by the vastx y

of I , and the ratioI /I is approximately equal to 1 for majority of proteins[41], i.e. 2.6–2.7, and to the exponentz x y

each dendrimer generation. In the case of the1b-based characterizing the distribution of masses of an ideallynanoparticles, the main aspect ratio (I /I ) remains con- branched polymer (d 52.5). Arteries and veins in mam-x z f

stant and approximately equal to 2 with increasing G, since malian vascular systems, too, have been found to obey thethe planar, rigid biphenylenic nucleus does not undergo scaling law (1) over a range of 20 bifurcations betweenconsiderable distortions as the number of lateral ramifica- heart and capillaries. Estimates of the relevant scalingtions increases. Similarly, theI /I and theI /I ratios only exponent[48] give values again near 2.7. Another interest-z y y x

very slightly increase and decrease with respect to average ing example is the branching pattern of arterial kidneyvalues of 0.76 and 0.63, respectively. In the case of the vessels, which is characterized by fractal geometry with1c-based series, the behavior is somewhat similar, although fractal dimensions between 2.0 and 2.5[49]. This is atwo ratios, i.e.I /I and I /I , are almost coincident and reasonable value for biological evolution to have attained,x z z y

slightly increase from 1 to 1.4, with an average value of given the requirement that arteries and veins should come1.2, whereas the third ratioI /I is practically constant to close to every point of the body that needs nourishmenty x

0.67. and waste disposal. But the ideal value ofd 53 for thisf

For the three dendrimeric series, the overall molecular purpose is, of course, unattainable, because a space-fillingdimensions deduced from the MD trajectories are reported vascular system leaves too little tissue for other tasks.in Table 1.Such data are in excellent agreement with the One of the important problems in supramolecularcorresponding, experimental observations by atomic force chemistry is the origin of specificity and recognition inmicroscopy, transmission electron microscopy and light molecular interactions. An essential step in this process isscattering measurements[22,23].Further, in the case of the complementary contact between approaching molecularmolecular class based on core1b, our MD measurements surfaces. Surface representations of macromolecules suchsupport the values proposed by Morgenroth et al. for G2 as, in our specific case, proteins or dendrimers, have

˚ ˚(|36 A) and G3 (|55 A), as estimated on the basis of provided a powerful approach for characterizing the struc-molecular mechanics calculations[46]. Also for the 1c- ture, folding, interactions and properties of such molecules.based dendrimers we find a good accord with the prelimin- A fundamental feature of surfaces that has not been

˚ ˚ary values published by Wind et al. (i.e. 23 A for G1, 37 A characterized by these representations, however, is the˚for G2 and 50 A for G3, respectively)[33]. texture (or roughness) of polymer surfaces, and its role in

The radius of gyrationR is a fundamental tool for the molecular interactions has not been defined. Anotherg

characterization of the structural properties of dendrimers. important, related issue deals with a basic property ofThis quantity is defined as the square root of the second surface fractals: their accessibility to incoming moleculesinvariant of the first order tensorS, which accounts for the depends on their size. We analyze here the implications ofspatial distribution of the atom chains by mediating over these properties on the derivatization of PD dendrimerall N molecular components. TheR values estimated by surfaces. Derivatization of surface is a key process, giveng


Fig. 5. Optimized molecular models of G1 (left), G2 (center) and G3 (right) of PD with core1c.


Fig. 5. (continued)

the ability to fine-tune the type of surface–adsorbate locus with which they react. The second factor is theinteraction by a suitable choice of the derivatizing agents. surface irregularity, tortuosity, and connectivity, and theGenerally, there are three main factors that determine the degree to which these surface features shield the reactiveaccessibility of derivatizable spots to reaction. The first is points from an incoming reagent molecule. The thirdthe umbrella effect of the reagent, as most reagents have parameter is surface heterogeneity, from the point of viewcross-sectional areas that shield more than the molecular of both the possible clustering of reactive sites and the


Fig. 6. Aspect ratios for all PD series. Filled symbols: PD-1a; open symbols: PD-1b; shadowed symbols: PD-1c. Diamonds:I /I ; squares:x z

I /I ; triangles;I /I . Lines serve as a guide for the eye.y x z y

difference in reactivities of the terminal groups due to ters which determine the diffusion kinetics of a smallinductive effects of the surrounding moieties. substrate on the surface of the macromolecule or into it.

It has been long known in surface science that apparent The method of determining theD value of dendrimerssurface areas decrease with increase in adsorbate size, due[16,50] has been to apply Eq. (2) by ‘covering’ the surfaceto the parallel decrease in geometric accessibility. It was of the PDs with a monolayer of model spherical moleculesfound that, in many instances, the relation between these of cross-sectional areas. The areaA thus determined is:two parameters is:

(2–D ) / 2A ~ s (3)– –D / 2m ~ s (2)

Such a relationship is illustrated inFig. 8 for the threewherem is the monolayer value of a physisorbed molecule generations of the1b-based series of dendrimers, as anof cross-sectional areas, and D is usually interpreted as example.the fractal dimension of the surface available to adsorption. As we may infer from this graph, the slopes of the plotsSince it is found that, usually, 2#D,3, it seems that the show a tendency to reach a plateau (corresponding tosimplest interpretation ofD is that it reflects mainly the D52) in the limit of both small and large probe sizes.geometrical nature of the surface, i.e. from a flat surface Small probes predominantly interact with the smooth van(D52) up to extreme volume-like irregularity (D53). Eq. der Waals spheres describing the dendrimer atoms, where-(2) may be applied, in principle, for the evaluation of the as large probes are sensitive to the overall shape of thesurface roughness and surface accessibility of macromole- molecule. For probes in the intermediate range (i.e. 1–4

˚cules, such as proteins and dendrimers. The degree of A), however, the value ofD increases from 2.06 of G1-1bgeometric irregularity of a dendrimer is one of the parame- to 2.20 of G3-1b. Table 2 lists the calculated surface

fractal dimensions for all dendrimer series.It is worthwhile noticing that the calculated fractalT able 1

˚ dimensions for all dendrimer families are quite close toOverall molecular dimensions (A) of the first three generations ofPDs based on cores1a, 1b and 1c those characterizing various protein and biomacromole-

cules, which vary, just to cite a few, between 2.05 forG1 G2 G3DNA to 2.18 for retinol binding protein[47].

PD-1a 1560.3 3260.4 4860.5 The results of this analysis are of much value, especiallyPD-1b 2360.3 3560.5 5360.7 in the molecular design of pharmaceutical and/or biocom-PD-1c 2660.2 3960.3 5560.7 patible molecules, as well as in surface molecular recogni-


Fig. 7. Relationship between radius of gyrationR and molecular weightM for all PD series. Diamonds: PD-1a; squares: PD-1b; triangles:g w

PD-1c. Symbols: calculated values; lines: best-fit.

Fig. 8. Double logarithmic plot of the molecular surface areaA as a function of the probe cross-sectional areas. for the PD-1b series.Diamonds: G3-1b; triangles: G2-1b; squares: G1-1b.


T able 2 mind that, in this preliminary investigation, solvent effectsSurface fractal dimensionD of the first three generations of PDs have been neglected, these results reflect only the effect ofbased on cores1a, 1b and 1c molecular structure variations upon density emphasizing

the relative differences between different generations. ThatG1 G2 G3is, the data clearly show that there is no change in density

PD-1a 2.10 2.11 2.12with increase in generation number, in accordance with thePD-1b 2.06 2.14 2.20shape-persistence of all dendrimeric series.PD-1c 2.01 2.02 2.04

As mentioned earlier, parameters such as size, shape andmultiplicity are transcribed, and displayed throughout thedendrimer development. These variables can have dramaticeffects on the ultimate shape, the interior topology and the

tion based on similar fractal dimensions. In particular, the exterior surface properties (alias congestion) of the de-surface pattern of both paraphenylene dendrimer series veloping molecule. Mathematically, we can appraise den-seems to be flat and regular, being the relevant fractal drimer surface congestion as a function of generation fromdimension practically constant and almost equal to 2 with the following relationship:increasing generations. This suggests that, in principle, all

2A Rthese starburst molecules should be easily derivatized, with] ]]A 5 ~ (4)z Ga few errors and defects even at generation 3. N N NZ c b

Other molecular issues deriving from the collected dataare the values of van der Waals surface areas (A ) and in whichA is the surface area per terminal group Z,A isvdW Z

volumes (V ), and the so-called molecular solvent the dendrimer surface area andN is the number ofvdW Z

accessible surfaces (SAS). The estimated values for the terminal groups Z per generation[1].three PDs series are reported inTable 3. From this relation we can see that, at higher generations

In the case of SAS, the relevant values refer to the G, the surface area per Z group should become increasing-molecular surface available to water as the solvent, repre- ly smaller and experimentally approach the cross-sectional

˚sented as an equivalent probe sphere of radius 1.4 A. area of the van der Waals dimension of the surface groupUsing the molecular weight and the van der Waals volumes Z. The generation G thus reached is referred to as thereported inTable 3, we could estimate an approximate starburst dense-packed (limited) generation. As predictedvalue for the density of each dendrimeric structure. This by de Gennes and Hervet[51], ideal starburst growthapproach allows comparison of the density between differ- without branch defects is possible only for those genera-ent generations of dendrimers. The calculated values for tions preceding the dense-packed state. This critical den-the three dendrimeric series reveal that, in all cases, the drimer property gives rise to self-limiting starburst dimen-density does not change in passing from G1 to G3, being sions, which are a function of the branch-segment lengthl,

3approximately equal to 1.1, 2.0 and 1.7 g/cm for the1a-, the core multiplicityN , the branch-juncture multiplicityNc b

1b- and the1c-based molecules, respectively. Keeping in and the sterical dimensions of the terminal group Z. Sincethe dendrimer radiusR in the expression above is depen-dent on l, larger l values will delay this congestion,whereas largerN and N values and larger Z dimensionsT able 3 c b

2 3˚ ˚ will dramatically hasten it.van der Waals surface areas (A ) (A ), volumes (V ) (A ) andvdW vdW2˚molecular solvent accessible surfaces (SAS) (A ) for the first Computer-assisted molecular simulations allowed us to

three generations of PDs based on cores1a, 1b and 1c determine the surface area per Z group,A , as a functionZ

of generation for all PD series (Fig. 9).G1 G2 G3For all dendrimer families,A increases from generationZ

PD-1a 1 to 3, but the rate of increase progressively diminishes,A 1291 3672 8351vdW owing to the beginning of surface congestion. Quite aV 1160 3317 7617vdW similar behavior seems to characterize other starburstSAS 1057 2713 5682

molecules of different nature[1,16,50,52]. Accordingly,this analysis allows us to conclude that, for this type ofPD-1bdendrimers, the starburst limited generation, that is theA 1748 4860 11 067vdW

V 1584 4444 10 165 generation at which the molecule should exhibit (i)vdW

SAS 1340 3310 6600 sterically inhibited reaction rates, (ii) sterically inducedstoichiometry, and, quite possibly, (iii) a critical phase

PD-1c change due to surface cooperativity, is presumably locatedA 2104 5388 11 964vdW in correspondence of G3. Such a conclusion has beenV 1878 4869 10 861vdW ¨theoretically anticipated by Grimsdale and Mullen[22],SAS 1751 4027 8313

and this study seems to confirm their hypothesis.


Fig. 9. Surface area per terminal group versus generation for the three PD series. Diamonds: PD-1a; squares: PD-1b; triangles: PD-1c.Symbols: calculated values; lines serve as a guide for the eye.

4 . Conclusions which branch defect and surface congestion phenomenabegin to appear, is presumably located in correspondence

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