NANO LETTERS
The Effects of Crowding in Dendronized Polymers
2006 Vol. 6, No. 9 1922-1927
Paul M. Welch* and Cynthia F. Welch Theoretical DiVision and Materials Science and Technology DiVision, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 Received May 8, 2006; Revised Manuscript Received June 30, 2006
ABSTRACT We present a comprehensive numerical study of the effect of degree of polymerization, generation of dendron growth, length of dendron tether, and dendron grafting density on the conformational statistics of dendronized polymers. This class of supramolecular assembly promises to find application in a number of nanotechnological devices in which their dimensions and conformations are key. We find that the radius of gyration estimates obtained from Brownian dynamics simulations yield to a “Flory” scaling argument in the high degree of polymerization regime and that these data from a range of topologically distinct molecules collapse onto a single curve in this limit. The size of the tethered dendrons serve as the key parameter in the scaling theory. Close examination of the dendrons also reveals some curious trends. In particular, we observe that as the grafting density is increased, spatial packing constraints around the main chain backbone force the dendrons further away from the backbone and compress them, significantly affecting the spatial distribution and accessibility of terminal groups; in contrast to dendrimers, the terminal groups of these molecules display a tendency to partition near the surface at high dendron grafting densities.
Many exponents of nanotechnology tout supramolecular assemblies as potential structural and functional building blocks in a host of devices. The envisioned constructions require materials with several physical characteristics, including a well-defined shape, tunable dimensions, slow relaxation dynamics, the possibility of possessing a large number of chemically active sites, a propensity for self-assembly, and stimuli-responsive local conformations. Dendronized polymers, linear chains with grafted dendrons as schematically pictured in Figure 1, have recently garnered much attention in the synthetic chemistry literature1-3 because they offer many of these traits. Experimental studies2 indicate that for low degrees of polymerization, the polymer adopts a spherical conformation. However, as the chain’s contour length increases, the steric crowding of the side-chain dendrons forces the polymer backbone into an extended or possibly helical conformation,3 giving the molecule an overall tubular or rodlike shape. The length and radial extent of these polymers may be controlled by the number of repeats along the backbone and the generation of growth of the attached dendrons. As recently demonstrated by Ecker, Das, and their respective co-workers,4 these assemblies behave like singlemolecule glasses with slow relaxation dynamics; thus, once assembled into a device, they may be expected to stay in place rather than hopping about as would be expected for more common synthetic polymers. Because their girth derives from the basic building blocks of dendrimers, dendronized polymers may offer many of the same benefits as dendrimers, including a large number of functional moieties at the * Corresponding author.
[email protected]. 10.1021/nl061036d CCC: $33.50 Published on Web 07/27/2006
© 2006 American Chemical Society
Figure 1. (A) The topology studied and nomenclature applied here, as exemplified by a G ) 5, NC ) 1, S ) 2, DP ) 4 dendronized polymer. (B) A representative configuration observed in simulations of G ) 5, NC ) 1, S ) 2, DP ) 100 dendronized polymers. Dendrons are colored to highlight segregation.
dendron termini and a tunable density profile.5 Applications envisioned for these molecules include roles in nanoelectronics,6,7 catalysis,8 biotechnology,2,9 and molecular device fabrication.10 The large size and often-poor solubility of dendronized polymers make their characterization challenging, however, and many basic questions regarding their conformational
statistics stand open. Though some single-chain studies11 have been performed, our understanding of the effect that the topological parameters have on the physical properties of these materials remains incomplete. We lack a clear picture of how properties such as the size-mass scaling, the cross-sectional size, the terminal group accessibility, the extent of dendron interpenetration, and the polymer conformation are affected by dendron generation (G), degree of polymerization (DP), dendron tether length (NC), and the spacing between dendrons along the backbone (S). A better understanding of these trends will improve our ability to tune the polymer’s physical properties through chemical design. Thus motivated, we report here the results from computer simulations performed to investigate in a controlled manner the effects of G, DP, NC, and S on both the local and global conformation of locally flexible dendronized polymers. Our results verify some speculation and conjecture found in the literature as well as offer new insight into the importance of steric crowding in these materials. Significantly, we find that the dendrons become extended away from the main chain backbone as S decreases, accompanied by a compression of the dendrons and a higher occupancy of the terminal groups near the surface of the molecule. We employed a coarse-grained model that faithfully captures the salient features of a typical dendronized polymer in a good solvent. Every bead interacts through the LennardJones potential ULJ(rij) ) [(σ/rij)12 - 2(σ/ri)6] where rij is the distance between beads, σ is the location of the potential minimum, and is the strength of the interaction. The latter sets the energy scale for the simulations. The temperature simulated here, kT/ ) 5.0, falls well within the good solvent limit for linear chains of this model. A simple harmonic potential between bonded pairs Us(rij) ) k(l - l0)2 maintains the topological connectivity, with bond length l ) |rij|, an equilibrium bond length of l0, and a spring constant k with k/ ) 1000. We report all length scales relative to l0 and set σ/l0 ) 1.14. To prevent the occurrence of bond crossing, the simulation code enforces that the instantaneous reduced bond length l/l0 falls below a maximum value of 1.43. The specific values for the parameters employed are important only in that they maintain the physical constraints of the topology and reflect the chosen solvent conditions. Results from simple models such as this one frequently withstand the scrutiny of more detailed and computationally expensive “atomistic” simulations. As noted above, four topological parameters describe the connectivity of each molecule. The number of beads along the linear backbone DP determines the contour length of the chain. Every S beads along the backbone serve as a grafting site for a dendron of generation G. The branch points in the dendrons consist of beads bound to three other branching sites. Each dendron is tethered to the main chain backbone by a leash composed of NC bonds. Values for these parameters in our study ranged from 1 to 5 for G, 4 to 100 for DP, 1 to 11 for NC, and 1 to 10 for S. To make the study of these slowly relaxing molecules tractable, S was fixed to unity as NC was varied; conversely, NC was fixed to unity Nano Lett., Vol. 6, No. 9, 2006
when S was varied. Figure 1A illustrates a schematic of a DP ) 4, G ) 5, NC ) 1, S ) 2 topology in our nomenclature. We applied the Brownian dynamics12 simulation technique to collect conformational statistics on single dendronized chains in an implicit solvent. The Langevin equation of motion r¨i ) -∇Ui - Γr˘ i + ξi(t) describes the dynamics of the chains in a thermal bath. The total potential Ui acting on bead i at ri is the sum of pairwise and bonded interactions described above. We set the frictional drag coefficient Γ to unity. The random “noise” force ξi was drawn from a Maxwell-Boltzmann distribution appropriate for the temperature simulated and generated using the Box-Muller algorithm. Our simulation code numerically propagates the Langevin equation forward in time with the velocity Verlet integrator.12 The reduced time step used was ∆t/t* ) 0.004, where t* is the basic unit of time in the simulation, xmσ2/, with the mass m set to unity. Typically, several thousand uncorrelated samples were saved from individual simulation trajectories to calculate the statistical averages reported here.13 Due to the slow total-polymer relaxations of these molecules, individual simulations ran for as long as 51 days on a single 2 GHz G5 processor. Dendrimers do not interpenetrate, thus their Newtonian rheological behavior.14 Indeed, even individual dendrons segregate within dendrimers, as shown by Mansfield.15 This exclusiveness should carry over to the dendronized polymers and is speculated to be the central conformational feature responsible for many of their physical properties. This trait also holds significance for device fabrication; if the dendrons do not interpenetrate, then each dendronized polymer can be viewed as an independent building block. Figure 1B contains an image of a typical conformation from the simulations of a DP ) 100, G ) 5, NC ) 1, S ) 2 molecule. All of the dendrons are identical but are colored to more easily illustrate dendron segregation. The dendrons do not appear to interpenetrate. As shown below, the generation G and spacing S of the dendrons as well as their tether length NC dictate the dimensions and physical layout of these molecules as a result of this segregation. Atomic force microscopy imaging of synthetic dendronized polymers indicates that they transition with increasing DP from what appear to be globules to self-avoiding random walkers. Lu¨bbert et al.2 suggested the presence of this transition in their studies. Observation of this transition implies a corresponding change in the apparent size-mass scaling exponent, υ, defined by the expression R ∝ DPυ, where R is some characteristic size and DP is proportional to the molecular weight.16 Globules typically display υ ) 1/3 behavior while self-avoiding random walkers in three dimensions, such as linear polymers in good solvents, usually have υ approximately 3/5. Fo¨rster et al.17 and Ouali et al.18 employed scattering techniques to extract estimates for υ. Fo¨rster and co-workers found their polystyrene chains with benzyl ether (“Fre´chet-type”) dendrons have υ approximately 3/5. In contrast, Ouali’s study showed that their carbosilanebased dendronized polymers are more like rigid rods with υ approaching 1. These seemingly contradictory results leave the value of υ for this class of molecules open to debate. 1923
However, υ should be expected to intimately depend on the topological parameters and the intrinsic stiffness of the chemical constituents, as pointed out by Connolly et al.19 To probe the size-mass scaling, we constructed a “Flory” argument16 relating the radius of gyration Rg of the entire molecule to the DP. For a conventional linear polymer in a good solvent, the free energy F consists of two components such that F ∝ Rg2/Nl2 + wN2/Rg3. The first term is an elastic contribution and the second is a pairwise overlap penalty where N is the number of statistical repeats (∼DP), l is the statistical step length, and w is the excluded volume parameter. Minimizing this expression with respect to the Rg yields Rg ∝ N3/5w1/5l2/5. This expression is appropriate for any self-avoiding random walk. We envision the dendronized polymer to be composed of a chain of blobs each with diameter lb. An individual blob is composed of S backbone beads and, therefore, one tethered dendron. There are Nb ) DP/S blobs in each dendronized polymer. The interior of each blob is assumed to follow self-avoiding walk statistics. Thus, lb ∝ S3/5w1/5l02/5, where w is the excluded volume parameter for the interaction between the Lennard-Jones beads and l0 is the equilibrium bond length. The excluded volume parameter can be obtained from the binary-cluster integral over the interaction potential and is found numerically to be 0.42l03. It is, however, only a proportionality constant whose precise value does not affect the results of our analysis. Now, the entire dendronized polymer should also follow self-avoiding walk statistics for a range of topological parameters such that its radius of gyration is described by Rg ∝ Nb3/5wb1/5lb2/5. We assert that the excluded volume swept out by the blob wb is proportional to the cube of the radius of gyration for the dendron dRg3. Applying our ansatz for wb and substituting the expressions above for Nb and lb yields Rg ∝ DP3/5Z3/5l00.16w0.08 where Z ) dRg/S3/5. Note that the primary consequence of assuming self-avoiding random walk statistics within the blobs is that Rg ∝ S-0.36. Several factors could invalidate this assumption, including small values for S and local chemical stiffness along the main chain backbone. If globular, simple random walk, or rodlike behavior more correctly describes the interior of the blobs, then the -0.36 value for the exponent becomes -0.47, -0.40, or -0.20, respectively. Nevertheless, as shown below, our assumption of self-avoiding walk statistics describes our data well. Figure 2 contains a plot of Rg/(Z3/5l00.16w0.08) versus DP for dendron generations 3 through 5 and the full range of NC, DP, and S values studied. The values for dRg were obtained from direct measurement on the simulated structures and include both the dendron and the corresponding leash. Clearly, the scaling reduction works well in the limit of DP greater than roughly 20. Below DP ) 20, the points tend to spread depending upon the local topological parameters such as NC, S, and G. The lines represent the various limiting values of the scaling exponent υ. In the high DP regime, υ ≈ 3/5. This value is comparable to that obtained by Connolly and co-workers19 from Monte Carlo simulations on DP ) 48 dendronized polymers. Note that while we recover selfavoiding walk statistics for these locally flexible molecules 1924
Figure 2. The root-mean-square radius of gyration Rg normalized by the topological parameters prescribed by the scaling analysis plotted as a function of the degree of polymerization, DP. Data for G ) 3-5 and the full range of S, DP, and NC are presented. The solid lines represent the limiting power laws obeyed by the data, and the annotations are the corresponding slopes.
in a good solvent, these results do not preclude the possibility of rodlike (υ ) 1) behavior under different circumstances. Indeed, our cursory examination of lower kT/ values indicates that helical bundles form that may be expected to yield rodlike scaling. Similarly, an intrinsically rigid backbone may produce this characteristic as well. However, we leave this as a subject for future study. In the lower DP limit, the values for υ range from 1/5 to 1/3. In this regime, the molecules are essentially dendrimers and this range of exponents agrees with the observation of Wong20 and coworkers. Note that while the transition is around DP ) 20, the actual number of particles in the molecule can be substantially larger, ranging from as little as 50 to as high as 1480 at this value of DP. The images in Figure 3 graphically illustrate the effect of the different topological parameters on the molecule. The top central image taken from simulations of G ) 5, NC ) 1, S ) 1, DP ) 7 molecules appears qualitatively globular. Increasing the DP to 51 while maintaining G ) 5, NC ) 1, and S ) 1, typified by the central image, demonstrates a transition to a more tubular random walker. The lower righthand image of a DP ) 100, S ) 10, NC ) 1, G ) 5 molecule shows that, even though the DP is twice that of the molecule in the center, their overall sizes are comparable. This arises from the difference in the overall stiffness of the two molecules, stemming from the changes in their dendron packing densities, as evidenced by the difference in the accessibility of the main chain backbones. Finally, the lower left-hand image of a DP ) 100, S ) 1, NC ) 10, G ) 5 molecule demonstrates that increased tether length produces a larger molecular girth, as will be further illuminated below. Dendron packing constraints dictate the transformation from globular to tubular conformations. In the globular regime, doubling the mass only results in a 2-fold increase in volume. However, in the tubular regime, increasing the mass by two more than triples the volume. The tubular conformation therefore permits greater entropy to the new mass relative to the globular. One would expect that the ability of the molecule to deform to a lesser or greater extent due to topological parameters such as NC might affect the value of DP at which this transition takes place. However, Nano Lett., Vol. 6, No. 9, 2006
Figure 4. The root-mean-square radius of gyration of the dendrons plus tether dRg normalized by the equilibrium bond length l0 plotted as a function of DP. Data for G ) 5, S ) 1, and six different values of NC are presented.
Figure 3. Images representing typical conformations observed in simulations of G ) 5 dendronized polymers with NC ) 1, S ) 1, DP ) 7 (top); NC ) 1, S ) 1, DP ) 51 (center); NC ) 1, S ) 10, DP ) 100 (bottom right); and NC ) 10, S ) 1, DP ) 100 (bottom left). The dendrons are colored for contrast while the backbone chain beads are shown in black.
the data in Figure 2 indicate that NC plays an undetectable role in determining the crossover region within the precision of our simulation results. Interestingly, a competition between tether stretching and dendron compression appears to be operative in this transition region. Increasing the DP produces a corresponding compression of the dendrons for all values of G, S, and NC studied. The extent depends on the values of S and NC. We examine the effect of S below, but holding S constant at unity and varying NC highlights the balance between dendron compression and tether stretching. Figure 4 contains a plot of the radius of gyration of both the dendron and corresponding tether, the dRg value used in our scaling analysis, as a function of DP for G ) 5 and a range of NC values. In all cases there is a variation in dRg at low DP values that asymptotes around DP ) 20. For molecules with NC less than or equal to 5, dRg decreases. Molecules with NC greater than 5 display an increase in dRg with DP. In the limit of NC ) 1, the dendrons simply compress as there is no slack in the tether. However, at the higher NC values, the dendrons compress to a lesser extent and the tether is tightened as DP increases, producing an overall increase in the value of dRg. One would expect that an increase in the value of G would result in an increase in the radius of the tube occupied by Nano Lett., Vol. 6, No. 9, 2006
Figure 5. The normalized dimensions of the effective tube occupied by the dendronized polymers and the corresponding sizes of the dendrons plotted as a function of grafting spacing S. The solid symbols represent the root-mean-square distance RT between terminal groups of the dendrons and their corresponding dendron grafting sites along the chain, normalized by its asymptotic value of RT∞ at high S. The open symbols indicate the values of the dendron root-mean-square radius of gyration dRg normalized by the free dendron radius of gyration dRg∞. Data for NC ) 1 and three generations of dendrons are shown.
the dendronized chain. We estimated this radius by the rootmean-square distance RT between the grafting-point along the main chain backbone and all of the terminal groups on the attached dendron for a range of S values and G spanning 1 through 5. We found that RT increased linearly with G at a rate dependent upon the value of S. Fo¨rster and coworkers17 also noted this linear trend in their experimental study and Christopoulos et al.21 found indication of this in their simulations of poly(p-phenylene) chains with grafted benzyl ether dendrons. However, examination of how RT varies with S reveals a surprising trend; the dendrons are not only forced further away from the main chain backbone with decreasing S but also become compressed as a result of the steric crowding. We explore this by holding NC ) 1 and DP ≈ 50. Plotting RT divided by RT∞, its value in the limit of large S, as a function of S shows that the tube radius can change as much as 20%. Figure 5 illustrates this with RT/RT∞ represented by solid symbols. The radius of gyration of the dendrons dRg normalized by the free-dendron radius of gyration dRg∞ captures their corresponding compression and is shown in the plot with open symbols. This compression could prove significant in electrical applications where 1925
Figure 6. The radial density F(r) of terminal groups about their respective attachment points along the main chain backbone versus the radial distance r normalized by the equilibrium bond length l0. Data for G ) 5, NC ) 1, and DP ≈ 50 shown.
densely packed dendrons may provide enhanced insulation around a conductive backbone.7 This relegation of the dendrons further away from the chain backbone also affects the accessibility of any chemical functionality present. Plotting the spherical distribution of terminal groups of a dendron about its attachment point along the backbone, F(r), for G ) 5, NC ) 1, and DP ≈ 50 for a range of S values illustrates this,22 as shown in Figure 6. For large values of S, the terminal groups distribute themselves in a roughly Gaussian fashion in the effective tube. However, as S decreases and steric crowding begins to force the dendrons further away from the tube axis, the distribution of terminal groups changes dramatically; they segregate preferentially toward the tube surface. This surface segregation of the terminal groups has been proposed to exist in dendrimers, though most experimental and theoretical studies seem to negate this possibility without the introduction of dendron stiffness.23 This has been the source of much debate within the dendrimer literature over the past 2 decades.24 Here, however, we observe this sought-after trait in fully flexible molecules. Moreover, while the molecular weight, and thus the number of terminal groups, that can be synthetically achieved in their dendritic precursors is limited, dendronized polymers can continue to add terminal groups by merely increasing DP while holding S constant. In summary, we have presented the results of the first comprehensive study of the effect that the topological parameters have upon the physical properties of a promising class of molecular building blocks, dendronized polymers. These results illuminate a number of key trends, knowledge of whose details is required for use of these materials in their many envisioned applications. We find that the constituent grafted dendrons segregate to such an extent that the size and shape of the molecule as well as the distribution of chemical functionality depend in a systematic fashion upon how closely the dendrons are packed along the chain backbone and their generation of growth. These trends, supported by experimental observations found in the literature, provide a prescription for specifying the desired properties with straightforward chemical design. In particular, we find that the molecules undergo an apparent glob-to-coil transition with increasing degree of polymerization and that 1926
the radii of gyration obey a simple scaling theory whose key parameter is the size of the tethered dendron. The dendron size varies in a curious fashion to produce some unexpected trends. The diameter of the effective tube occupied by a dendronized polymer, typified by the average distance of the terminal groups away from the main chain backbone, increases linearly with generation of dendron growth but nonlinearly with grafting density. Further, we note that, in addition to moving away from the chain backbone, the dendrons are also compressed as they are more closely packed, resulting in an increased probability of finding the terminal groups closer to the effective tube’s surface. We therefore predict that these molecules will find use in many of the applications requiring terminal group access originally proposed for dendrimers. We hope this study stimulates further investigation into these possibilities. Acknowledgment. We thank Thomas Sewell and Kim Rasmussen for their helpful suggestions during preparation of this paper. This work was carried out under the auspices of the National Nuclear Security Administration of the U.S. Department of Energy at Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396. References (1) Kaneko, T.; Horie, T.; Asano, M.; Aoki, T.; Oikawa, E. Macromolecules 1997, 30, 3118. Yin, R.; Zhu, Y.; Tomalia, D. A.; Ibuki, H. J. Am. Chem. Soc. 1998, 120, 2678. Schlu¨ter, A.-D. Top. Curr. Chem. 1998, 197, 165. Frey, H. Angew. Chem., Int. Ed. Engl. 1998, 37, 2193. Schlu¨ter, A.-D.; Rabe, J. P. Angew. Chem., Int. Ed. 2000, 39, 864. Zhang, A.; Shu, L.; Bo, Z.; Schlu¨ter, A.-D. Macromol. Chem. Phys. 2003, 204, 328. Zhang, A.; Okrasa, L.; Pakula, T.; Schlu¨ter, A.-D. J. Am. Chem. Soc. 2004, 126, 6658. Hassan, M. L.; Moorefield, C. N.; Newkome, G. R. Macromol. Rapid Commun. 2004, 25, 1999. Malkoch, M.; Carlmark, A.; Woldegiorgis, A.; Hult, A.; Malmstro¨m, E. E. Macromolecules 2004, 37, 322. Andreopoulou, A. K.; Carbonnier, B.; Kallitsis, J. K.; Pakula, T. Macromolecules 2004, 37, 3576. Fu, Y.; Li, Y.; Li, J.; Yan, S.; Bo, Z. Macromolecules 2004, 37, 6395. Yoshida, M.; Fresco, Z. M.; Ohnishi, S.; Fre´chet, J. M. J. Macromolecules 2005, 38, 334. (2) Lu¨bbert, A.; Nguyen, T. Q.; Sun, F.; Sheiko, S. S.; Klok, H.-A. Macromolecules 2005, 38, 2064. (3) Percec, V.; Ahn, C.-H.; Ungar, G.; Yeardley, D. J. P.; Mo¨ller, M.; Sheiko, S. S. Nature 1998, 391, 161. Lee, C. C.; Fre´chet, J. M. J. Macromolecules 2006, 39, 476. (4) Ecker, C.; Severin, N.; Shu, L.; Schlu¨ter, A. D. Rabe, J. P. Macromolecules 2004, 37, 2484. Das, J.; Yoshida, M.; Fresco, Z. M.; Choi, T.-L.; Fre´chet, J. M. J.; Chakraborty, A. K. J. Phys. Chem. B 2005, 109, 6535. (5) Welch, P.; Muthukumar, M. Macromolecules 1998, 31, 5892. (6) Bao, Z.; Amundson, K. R.; Lovinger, A. J. Macromolecules 1998, 31, 8647. Malenfant, P. R. L.; Fre´chet, J. M. J. Macromolecules 2000, 33, 3634. Setayesh, S.; Grimsdale, A. C.; Weil, T.; Enkelmann, V.; Mu¨llen, K.; Meghdadi, F.; List, E. J. W.; Leising, G. J. Am. Chem. Soc. 2001, 123, 946. Pereverzev, Y. V.; Prezhdo, O. V.; Dalton, L. R. Chem. Phys. Lett. 2003, 373, 207. (7) Sato, T.; Jiang, D.-L.; Aida, T. J. Am. Chem. Soc. 1999, 121, 10658. (8) Hu, Q.-S.; Sun, C.; Monaghan, C. E. Tetrahedron Lett. 2002, 43, 927. (9) Go¨ssl, I.; Shu, L.; Schlu¨ter, A.-D.; Rabe, J. P. J. Am. Chem. Soc. 2002, 124, 6860. (10) Stocker, W.; Schu¨rmann, B. L.; Rabe, J. P.; Fo¨rster, S.; Lindner, P.; Neubert, I.; Schlu¨ter, A.-D. AdV. Mater. 1998, 10, 793. Barner, J.; Mallwitz, F.; Shu, L.; Schlu¨ter, A.-D.; Rabe, J. P. Angew. Chem., Int. Ed. 2003, 42, 1932. (11) Shi, W.; Wang, Z.; Cui, S.; Zhang, X.; Bo, Z. Macromolecules 2005, 38, 861. (12) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon: Oxford, 1987.
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(13) The time correlation function (Rg(t)Rg(t + ∆t) - 〈Rg〉2)/(〈Rg2〉 - 〈Rg〉2) for the radii of gyration Rg was calculated, and averages were taken over snapshots separated in time by a value sufficient for this function to have fallen to zero. (14) Uppuluri, S.; Keinath, S. E.; Tomalia, D. A.; Dvornic, P. R. Macromolecules 1998, 31, 4498. (15) Mansfield, M. Polymer 1994, 35, 1827. (16) de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979. (17) Fo¨rster, S.; Neubert, I.; Schlu¨ter, A. D.; Lindner, P. Macromolecules 1999, 32, 4043. (18) Ouali, N.; Me´ry, S.; Skoulios, A.; Noirez, L. Macromolecules 2000, 33, 6185. (19) Connolly, R.; Bellesia, G.; Timoshenko, E. G.; Kuznetsov, Y. A.; Elli, S.; Ganazzoli, F. Macromoelcules 2005, 38, 5288. (20) Wong, S.; Appelhans, D.; Voit, B.; Scheler, U. Macromolecules 2001, 34, 678. (21) Christopoulos, D. K.; Photinos, D. J.; Stimson, L. M.; Terzis, A. F.; Vanakaras, A. G. J. Mater. Chem. 2003, 13, 2756.
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(22) The density profiles are constructed by subdividing space around a graft point into concentric shells of thickness ∆r/l0 ) 0.14. The average number of terminal groups for the dendron attached at the graft point that fall in each shell is obtained from the simulations. The profiles are assembled from examining all grafting sites and their corresponding dendrons except for those sites that lie at the chain ends. The profiles are then normalized such that the total area under the curves is unity. (23) See, for example, Ballauff, M. Top. Curr. Chem. 2001, 212, 177 and references therein. (24) For examples, see: de Gennes, P. G.; Hervet, H. J. Phys. Lett. 1983, 44, L351. Naylor, A.; Goddard, W.; Kiefer, G.; Tomalia, D. J. Am. Chem. Soc. 1989, 111, 2339. Lescanec, R. L. Muthukumar, M. Macromolecules 1990, 23, 2280. Topp, A.; Bauer, B. J.; Klimash, J. W.; Spindler, R.; Tomalia, D. A.; Amis, E. J. Macromolecules 1999, 32, 7226. Zook, T. C.; Pickett, G. T. Phys. ReV. Lett. 2003, 90, 015502.
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