Hyperbranched polymer stars with Gaussian chain statistics revisited
aa r X i v : . [ c ond - m a t . s o f t ] A ug Hyperbranched polymer stars with Gaussian chain statistics revisited
P. Poli´nska, C. Gillig, J.P. Wittmer, ∗ and J. Baschnagel Institut Charles Sadron, Universit´e de Strasbourg & CNRS,23 rue du Loess, 67034 Strasbourg Cedex, France FMF, University of Freiburg, Stefan-Meier-Str. 21, D-79104 Freiburg, Germany (Dated: September 9, 2018)Conformational properties of regular dendrimers and more general hyperbranched polymer starswith Gaussian statistics for the spacer chains between branching points are revisited numerically.We investigate the scaling for asymptotically long chains especially for fractal dimensions d f = 3(marginally compact) and d f = 2 . PACS numbers: 82.35.Lr,61.43.Hv,05.10.Ln
I. INTRODUCTION
Hyperbranched stars with Gaussian chain statistics.
Regular exponentially growing starburst dendrimers, assketched in fig. 1, and more general starlike hyper-branched chains [1] with Gaussian chain statistics havebeen considered theoretically early in the literature [2–8]and have continued to attract attention up to the recentpast [9–16]. One reason for this is that hyperbranchedstars [17, 18] with sufficiently large spacer chains betweenthe branching points (as indicated by the filled circles)are expected to be of direct experimental relevance undermelt or θ -solvent conditions [19–21]. Assuming a tree-likestructure and translational invariance along the contour,the root-mean-square distance R s between two monomers n and m , as shown in panel (a), is thus given by R ≡ (cid:10) ( r m − r n ) (cid:11) = b s ν with ν ≡ / s the curvilinear distance along the tree between bothmonomers and b the statistical segment size of the spacerchains [20]. As a consequence, the typical distance R e between the root monomer and the end monomers of themost outer generation g = G of spacer chains, as onepossible observable measuring the star size [22], scalesas R = b SG with S being the length of the spacerchains (assumed to be monodisperse). Other momentsare obtained from the normalized distribution P ( r, s ) ofthe distance r = | r m − r n | which, irrespective of thespecific topology of the branched structure, is given by P ( r, s ) = (cid:18) d πR (cid:19) d/ exp − d (cid:18) rR s (cid:19) ! (2)with d = 3 being the spatial dimension [23]. Due to theirtheoretical simplicity such Gaussian chain stars (includ- ∗ Electronic address: [email protected] spacer ofS monomersroot monomer (f=3)branching (f=3) incompletebranching (f=2)early ending(f=1) i−1i connected spacer i i−1i i−1 d = 3 f initiator: i−1i end monomer (f=1) α−1 M ~ g g m r n (a) regular dendrimer of G=2 g=2 g=2g=2 g=1 g=2g=1 always with regular G=2 dendrimer g=1 g=2 g=3 g=2 g=3 n = n S M
S = S /nM = M n
SM S M n = 4, n = 8mix with simple generator B: generator A: α (c) self−similar fractal −star β (b) −star with G=3(d) stochastic −star γ FIG. 1: Sketch of different topologies of branched polymerstars considered: (a)
Regular dendrimer of generation num-ber G = 2 with M = 9 arms. (b) Hyperbranched so-called“ α -star” with imposed spacer chain number M g ∼ g α − for g ≤ G constructed iteratively ( g → g + 1) by restricting ran-domly the branching of the arms. Some branches may thusend at g < G . (c) Self-similar fractal “ β -stars” are gener-ated starting with regular G = 2 dendrimers and replacingiteratively the M i − spacers of length S i − by M i = M i − n M spacers of length S i = S i − /n S . The generator shown cor-responds to self-similar stars of constant density ( d f = 3). (d) Multifractal “ γ -stars” are obtained by applying randomlymore than one generator. Mixing with equal weight the gen-erator B ( n S = n M = 4) with the compact star generator A( n S = 4, n M = 8) leads to a star with d f = 2 . ing systems with short-range interactions along the topo-logical network) allow to investigate several non-trivialconceptual and technical issues, both for static [10, 13]and dynamical [6, 11, 12, 14, 24, 25] properties, relatedto the in general intricate monomer connectivity imposedby the specific chemical reaction history. Aim of current study.
We assume here that (i) thechemical reaction is irreversible (quenched), (ii) allspacer chains are monodisperse of length S and (iii) flex-ible down to the monomer scale and (iv) that the branch-ing at the spacer ends is at most three-fold ( f = 3) as inthe examples given in fig. 1. Our aim is to revisit vari-ous experimentally relevant conformational properties inthe limit where the total monomer mass N and the to-tal number M = ( N − /S of spacer chains becomesufficiently large to characterize the asymptotic univer-sal behavior and to sketch for different star architecturesthe regimes where the Gaussian spacer chain assump-tion becomes a reasonable approximation. We focus onthe large- S limit since this allows under θ -solvent [26] ormelt conditions to broaden the experimentally meaning-ful range of the generation number G of spacer chains. Fractal dimension.
One dimensionless property char-acterizing the star classes considered below is their fractaldimension d f which may be defined as [27, 28] d f ≡ lim R →∞ log( N )log( R/b ) (3)with N being the mass and R the characteristic chainsize. (Less formally, this definition is often written N ∼ R d f [28].) For the regular dendrimers shown inpanel (a) the number of spacers M and, hence, the to-tal mass N increase exponentially with the generationnumber G , while the typical chain size R ( G ) ∼ √ G onlyincreases as a power law. That the fractal dimensionthus must diverge, is denoted below by the shorthand“ d f = ∞ ”. In addition we shall consider star classesof finite fractal dimension d f , focusing especially on nottoo dense systems which should be (at least conception-ally) of experimental relevance. Specifically, we consider (i) marginally compact chains [29] of fractal dimension d f = d = 3 and (ii) stars of fractal dimension d f = 2 . Power-law stars.
As sketched in panel (b), such hy-perbranched stars of finite fractal dimension may be con-structed most readily by imposing a number of spacerchains M g per generation g such that the power law M g ∼ g α − holds. Hence, M ∼ N ∼ G α . The “growthexponent” α of these so-called “ α -stars” is set by thefractal dimension α = d f ν (4)as may be seen using N ∼ R d f and R ≈ R e ∼ ( SG ) ν [23]. While being a natural generalization of the regulardendrimer case, restricting the branching of star armsdoes, unfortunately, not lead to a self-similar tree sincethe iteration g → g + 1 is not a proper self-similar gen-erator acting on all spacer chains [27, 28]. We thereforealso consider truly self-similar (multi)fractal stars, calledin the following β - and γ -stars, generated iteratively asshown in panel (c) and panel (d) of fig. 1 by the iterativeapplication of a well-defined generator (or several gener-ators) on all the spacer chains as in the recent theoreticalwork on Vicsek fractals [9]. For the latter architecturesone thus expects to observe for the intramolecular coher-ent form factor F ( q ) the power-law scaling [19, 29, 33] F ( q ) ∼ /q d f for d f ≤ d = 3 (5) in the intermediate regime of the wavevector q . Notethat eq. (5) only holds for open or marginally compactself-similar structures [29, 33]. In fact, Gaussian hyper-branched stars with higher fractal dimension, d f > d ,approach with increasing generation number and massthe Gaussian limit F ( q ) ≈ N exp (cid:0) − ( qR g ) /d (cid:1) for q ≪ /bS / (6)as shall be demonstrated below. Outline.
The paper is organized as follows: We sum-marize first in sect. II the numerical methods and specifythen in sect. III the different topologies studied. Somereal space properties are presented in sect. IV before weturn to the characterization of the intramolecular formfactor F ( q ) in sect. V. While most of this study is ded-icated to strictly Gaussian hyperbranched stars, i.e. allexcluded volume effects are switched off, we investigatemore briefly in sect.VI by means of Monte Carlo (MC)simulations [34] effects of a weak excluded volume in-teraction penalizing too large densities. Even an expo-nentially small excluded volume is seen to change qual-itatively the behavior of large regular dendrimers. Weconclude the paper in sect. VII. Neglecting deliberatelythe long-range correlations expected as for linear chains[35], we sketch the regime where the Gaussian approxi-mation for melts of hyperbranched stars should remainreasonable for sufficiently large spacers. II. SOME COMPUTATIONAL DETAILS
Settings and parameter choice.
We suppose that themonomers are connected by ideal Gaussian springs. Thespring constant is chosen such that the effective bondlength b , eq. (1), becomes unity. Also, both the tempera-ture T and Boltzmann’s constant k B are set to unity. AllGaussian spacers are of equal length S (which comprisesone end monomer or branching monomer). With M be-ing the total number of spacer chains, a hyperbranchedstar thus consists of N = 1 + SM monomers. If nothingelse is said, S = 32 is assumed. (This arbitrary choice ismotivated by simulations of dendrimer melts presentedelsewhere.) For S = 32 we sampled up to a genera-tion number G = 17 for regular dendrimers and up to G ≈ d f = 3 and d f = 2 .
5. (Even larger G obtainedusing smaller S are included below where appropriate.)Some properties of the largest system computed for eachinvestigated star architecture are listed in Table I. Local and collective MC moves.
Due to their Gaus-sian chain statistics many conformational properties canbe readily obtained using Gaussian propagator tech-niques [19] or equivalent linear algebra relations [10, 14–16, 36]. However, some interesting properties, such as theeigenvalues λ i of the inertia tensor, can be more easilycomputed by direct simulation which are in any case nec-essary if long-range interactions between the monomersare switched on (see below). As shown in fig. 2, we use star type d f G N/ N e / h s i s max r.f. R e R g Dendrimer ∞
17 12.6 197 0.87 0.11 23 22 α -star 6 50 22.6 41.6 0.74 0.19 40 34 α -star 5 80 10.4 10.0 0.70 0.22 51 42 α -star 4 200 7.2 2.2 0.63 0.29 80 64 α -star 3 2000 16.2 0.4 0.47 0.45 253 138 α -star 2.5 2000 2.4 0.05 0.36 0.61 253 108 β -star 3 2048 8.4 1.2 0.49 0.51 256 179 β -star 2.5 4096 1.1 0.03 0.45 0.56 362 171 γ -star 2.5 8192 11.1 0.3 0.47 0.54 512 351TABLE I: Various properties for different hyperbranched startypes of spacer length S = 32: fractal dimension d f , largestgeneration number G , total mass N , number of end monomers N e in the last generation shell g = G , rescaled Wiener in-dex h s i /s max with s max = 2 GS being the largest curvilin-ear distance between pairs of monomers, relative root mean-square fluctuation p h s i − h s i / h s i (r.f.) of the normalizedhistogram w ( s ), root mean-square end distance R e betweenthe root monomer and the end monomers of the generationshell g = G and radius of gyration R g .
12 3 45 6 78 9 (a) Pivot MC move (b) suitable data structureroot r o t a t i on ax i s
112 3 pivot mon i j j i root g = 1g = 2
1. Get pivot monomer i 2. j = last(i). If i< j:6. Goto to 13. Get rotation axis4. Get rotation angle5. Turn all k between i and j
FIG. 2: Sketch of pivot MC move (a) and data structure(b). A monomer i (filled square) is selected randomly and allattached monomers k closer to the ends (within thin circles)are rigidly turned by an angle θ . A suitable data structureconsists in ordering the spacer arms (their index indicated bythe numbers) and the monomers such that all monomers k become neighbors in the monomer lists ( i < k ≤ j ). pivot moves with rigid rotations of the dangling chainend (as shown by the monomers within the thin cir-cles) below a randomly chosen pivot monomer i . Themonomers are collectively turned (using a quaternion ro-tation [34]) by a random angle θ around an also randomlychosen rotation axis through the pivot monomer. As il-lustrated in panel (b) of fig. 2, it is useful to organize thedata structure such that arms and monomers which areturned together are also grouped together. This allowsto rotate all monomer k with i < k ≤ j . The tabulatedmonomer j = last ( i ), the last monomer to be turned,must be an end monomer. A pivot move does leave un-changed the distances between connected monomers. (Ifthe connectivity of the monomers is the only interaction,a suggested move is thus always accepted.) To relax the local bond length distribution simple local MC jumps areadded [34]. The root monomer at the origin never moves. Excluded volume interactions.
Due to excluded vol-ume constraints the volume fraction occupied by a real-istic chain can, obviously, not exceed (much above) unity.One simple way to penalize too large densities is to in-troduce an excluded volume energy through the latticeHamiltonian E = ǫ X r n ( r ) ( n ( r ) −
1) (7)using the monomer occupation number n ( r ) of a simplecubic lattice. For all examples presented below we set δx = δy = δz = 1, i.e. the grid volume δV = δx δy δz is unity and n ( r ) = ρ ( r ) δV measures the instantaneouslocal density. The Hamiltonian is similar to the finite ex-cluded volume bond-fluctuation model for polymer meltson the lattice described in [35, 37], however, the particlepositions are now off-lattice and only the interactions aredescribed by the lattice. A local monomer or collectivepivot move is accepted using the standard Metropolis cri-terion for MC simulations [34]. Note that the collectivepivot moves are best implemented using a second latticefor the attempted moves. III. CHARACTERIZATION OF IMPOSEDINTRACHAIN CONNECTIVITY
Introduction.
We assume that the hyperbranchedstar topology is not annealed, i.e. not in thermal equilib-rium, but irreversibly imposed by the chemical reaction.The first step for the understanding of such quenchedstructures is the specification and characterization of theassumed imposed connectivity, often referred to as “con-nectivity matrix” [10, 16]. A central property charac-terizing the monomer connectivity is the normalized his-togram of curvilinear distances w ( s ) = 1 N N X n,m =1 δ ( s − s nm ) (8)with s nm being the curvilinear distance between themonomers n and m . Trivially, w ( s = 0) = 1 /N and w ( s ) ≈ N/N = 2 /N for 0 < s ≪ S since the samemonomer pair is counted twice. Note that the histogram w ( s ), sampled over all pairs of monomers of the chain,may differ in general from the similar distribution w ( s )of the curvilinear distances between the root monomerand other monomers. We remind also that for a linearpolymer chain [35] w ( s ) = 2 s max (cid:18) − ss max (cid:19) for 0 < s ≤ s max (9)with s max = N − ≈ N . For most of the star archi-tectures considered the largest curvilinear distance s max is given by s max = 2 SG . The histogram w ( s ) will be g M g regular dendrimer α -star with ρ =1 β -star with n S =4,n M =8 and i=6 iterationsaverageroot-mean-square fluctuation4g α -1 with α =3/2 d f =3, α =3/2fractal: relative fluctuation of order 1d f = ∞ FIG. 3: Number of spacer chains M g for dendrimers (boldsolid line) and power-law stars of fractal dimension d f = 3( α = 3 / α -star, the smallfilled triangles to a β -star constructed as shown in fig. 1(c).The logarithmically averaged number of arms (thin line) andthe root-mean-square fluctuations (circles) are of same order. G N / S d f = ∞ d f =6d f =5d f =4d f =3d f =2.5 y = G G G D L A - li k e filled symbols: β -stars G G G c o m p a c t crosses: γ -stars for d f =2.5 FIG. 4: Number of monomers
N/S ≈ M vs. generation num-ber G for different imposed topologies and fractal dimensions d f . Dendrimers are indicated by d f = ∞ (circles), α -stars bythe other open symbols. The filled triangles corresponds to β -stars of d f = 3 ( n S = 4, n M = 8) and d f = 2 . n S = 16, n M = 32) dimensions, the crosses to γ -stars of d f = 2 . used below for the determination of experimentally rele-vant properties such as the radius of gyration R g and theintramolecular form factor F ( q ). The first and secondmoments of w ( s ) are given in Table I for the different ar-chitectures studied. We remind that N h s i is sometimescalled “Wiener index” W [3, 16]. Regular dendrimers.
Let us first summarize severalsimple properties of the regular dendrimers sketched infig. 1(a). As already mentioned above, the number M g of spacer chains per generation shell g ≤ G increases ex-ponentially as M g = 3 · g − as shown by the bold linein fig. 3. Since we assume monodisperse spacer chainsof length S , this implies w ( s ) ≈ s/S for S ≪ s ≤ SG x = s/s max y = w ( s ) s m ax d f = ∞ d f =6d f =4d f =3d f =2.5 x -4 -2 ys=32(a) lin-linlin-logd f = ∞ , S=8 li n ea r c h a i n -5 -4 -3 -2 -1 x = s/s max -2 -1 y = w ( s ) s m ax -5 -4 -3 -2 -1 -2 -1 d f =3.0 α -starsd f =3.0 β -starsd f =2.5 α -starsd f =2.5 β -starsd f =2.5 γ -stars s=32 x x (b) FIG. 5: Histogram w ( s ) measuring the number of monomerpairs at curvilinear distance s along the branched chain: (a) Dendrimers for G = 20 (bold line) and α -stars for differentfractal dimensions (open symbols). The dashed line indicatesthe histogram for a linear chain of length N ≈ s max . Inset:Half-logarithmic representation for dendrimers. (b) Doublelogarithmic representation for d f = 2 . d f =3 (lower data). As emphasized by the solid and dashed linesa power law x α − is observed only for β - and γ -stars. and that the mass N at generation number G must alsoincrease exponentially, as shown in fig. 4. The histogram w ( s ) of curvilinear distances s for dendrimers is given inpanel (a) of fig. 5 (bold solid lines). The main panel givesa linear representation of the dimensionless rescaled his-togram w ( s ) s max as function of s/s max , the inset on theleft-hand side a similar half-logarithmic representation.As one expects, the histogram increases exponentiallyfor curvilinear distances S ≪ s ≪ s max due to the expo-nential increase of alternative paths of length s startingfrom an arbitrary monomer. Using simple combinatoricsit can be seen that the histogram must become w ( s ) ≈ N ( s/S − / for 1 ≤ s ≪ s max . (10)The cutoff observed for large s ≈ s max is due to the finitemass of the star and the finite length of its branches, justas the finite length of a linear chain gives rise to eq. (9).As seen from Table I, the reduced first moment h s i /s max approaches unity for dendrimers and the relative fluctu-ations are the smallest for all architectures considered. Hyperbranched α -stars. As already noted in the In-troduction, a simple way to generate stars of a finite frac-tal dimension d f is to impose a power law M g = cg α − for the number of spacers per generation shell with c being a constant [38]. This is done by randomly attach-ing M g spacer chains to the end monomers of generation g − α -star with α = 3 / N ≈ SG α as a functionof G is shown for α = 6 / α = 5 / α = 4 / α = 3 / α = 2 . / w ( s ) of curvilinear distances from the rootmonomer increases as w ( s ) ∼ s α − for S ≪ s ≤ s max as implied by the M g -scaling (not shown). The curvilin-ear histograms w ( s ) over all pairs of monomers are pre-sented in the main panel of fig. 5(a). The histograms areagain non-monotonous increasing first due to the branch-ing and decreasing finally due to the finite length of thestar arms. The latter decay becomes the more markedthe weaker the branching, i.e. the smaller α , getting simi-lar for the smallest exponent α = 2 . / α -stars cannot be described bya simple power law or exponential behavior for w ( s ) [40]. Self-similar β -stars. This is different for self-similarfractals created starting from a G = 2 dendrimer ofspacer length S (as specified below) as initiator and iter-ating a generator as the one shown in fig. 1(c). At everyiteration step i a spacer of length S i − is replaced by n M spacers of length S i = S i − /n S . Hence, S i = S /n iS , M i = 9 n iM , N i − S i M i = 9 S ( n M /n S ) i and G i = 2 n iS for, respectively, the spacer length, the number of spac-ers, the total mass and the generation number of thestar. Importantly, the arms added laterally to the origi-nal spacer can always be distributed such that the root-mean square end-to-end distance of the original spacer(filled circles) still characterizes the typical size of the re-placed spacer. Since S i G i = 2 S for the curvilinear dis-tance between the root monomer and the end monomersin the largest generation shell g = G i , the typical chainsize R , thus remains by construction constant as we shallexplicitly verify in sect. IV. Note that the spacer length S i of the final iteration step is set by S ! = S i = S /n iS , (11)which fixes the mass N ≈ S ≈ Sn iS of the initiator star.Using N i ∼ R d f ∼ N νd f this implies n M = n βS with β = d f ν (12)relating thus both numerical constants n S and n M . Asshown for d f = 3 ( n S = 2 , n M = 2 ) by the small filledtriangles in fig. 3, such a self-similar construction leads to a strongly fluctuating number M g of spacers. However,as shown by the thin solid line the (logarithmically) av-eraged number of arms still increases as M g ∼ g α − with α = β = d f ν in agreement with eq. (4). Interestingly, thecorresponding (also logarithmically averaged) root-meansquare fluctuations (as indicated by open circles) are ofthe same order, i.e. the relative fluctuations of spacernumber M g per generation shell are of order one. Theimportant point is here that all monomers are statisticallyequivalent and that the root monomer does not play anyspecific role which would break the self-similarity. (As wehave verified, this implies w ( s ) ≈ w ( s ).) Averaging overall spacer chains, the total mass N scales, as expected,again as N/S ≈ G α with α = d f ν as shown in fig. 4 byfilled triangles for d f = 3 and d f = 2 .
5. The latter ar-chitecture, constructed using n S = 2 and n M = 2 , ismotivated by the fractal dimension d f ≈ . d = 3dimensions [28, 30–32]. In our view this is one interest-ing universal limit of (at least conceptional) experimentalrelevance [24]. Being self-similar all monomers are equiv-alent and since the number of monomers at a curvilineardistance s must increase on average as ( s/S ) α − , one ex-pects for S ≪ s ≪ s max the power-law scaling w ( s ) ≈ N × N ( s/S ) α − ≈ s max ( s/s max ) α − (13)with N ≈ S ( s max /S ) α ≈ SG α . This is confirmed by thehistograms (filled symbols) shown in fig. 5(b). Stochastic two-generator multifractals.
Since theDLA limit is of some importance we have sampled asecond system class of fractal dimension d f = 2 . γ -stars” arein fact multifractals [28, 32]. (We remember that DLAclusters are also multifractal [32]. No multifractal anal-ysis [28] is required here, however.) For a given spacerwe apply the generator A with a probability f A and thegenerator B with a probability f B = 1 − f A . By choos-ing different values of f A any fractal dimension between d f = 2 and d f = 3 can be sampled using both generators.By reworking the arguments leading to eq. (12) it can beseen that f A = f B = 1 / d f = 2 .
5. While β -stars are deterministic, the γ -stars have a stochastic topology due to the random mixing of both generatorsand an ensemble average over several stars is thus taken.As may be seen from the crosses in fig. 4 and fig. 5(b), theproperties of β - and γ -stars are, however, rather similar. IV. REAL SPACE CHARACTERIZATION
End distance R e . There are several ways to charac-terize the typical star size R all being equivalent fromthe scaling point of view. A double-logarithmic repre-sentation of the reduced mean-square end distance R /S x = N/S y = R e2 / S d f = ∞ d f =6d f =5d f =4d f =3d f =2.5 s l op e l og a r i t h m i c s l o p e . S=32 s l o p e / s l o p e / s l o p e / s l o p e / filled symbols: β -starscrosses: γ -stars (a) G -2 -1 ρ = N / R e3 d f = ∞ d f =6d f =5d f =4d f =3d f =2.5 S=32 y ~ G / G / ρ =const=1 G -0.25 G G ρ ~G α -d/2 G (b) FIG. 6: Root-mean square end distance R e for different im-posed topologies: (a) Double-logarithmic representation ofthe reduced mean-squared end distance R /S vs. reducedmass N/S . (b) Density ρ ≡ N/R vs. total generation num-ber G for a spacer length S = 32. vs. the reduced mass N/S is presented in panel (a) offig. 6. Note that the values of R e obtained by directMC simulations are within statistical accuracy identicalto R = b SG . Both data sets are lumped together. Theregular dendrimer size increases, of course, logarithmi-cally with the mass (circles and bold solid line). Thepower-law slopes indicated for finite- d f systems are con-sistent with the definition N ∼ R d f . As one measure ofthe overall density of a star one may define ρ ≡ N/R d e .(Obviously, a suitable order-one geometrical factor, suchas 4 π/
3, might be included in this definition.) As can beseen from panel (b) of fig. 6, the density for regular den-drimers exceeds already at G = 10 an unrealistic order of10 monomers per volume element. As indicated by thevarious power-law slopes, ρ ∼ G α − dν for power-law starsof finite fractal dimension, i.e. the density increases for d f > d and decreases for d f < d as it should [8, 23]. Radius of gyration R g . The radius of gyration R g pre-sented in fig. 7 has been determined with identical results(lumped again together) either from the MC sampled G y = ( R g / R e ) d f = ∞ d f =6d f =4d f =3d f =2.5 S=32 filled symbols: β -starscrosses: γ -stars FIG. 7: Reduced radius of gyration y = ( R g /R e ) vs. genera-tion number G . The ratio y becomes constant only for β -stars(filled symbols) and γ -stars (crosses). configuration ensembles or by means the formula [41] R ≡ N N X n,m =1 (cid:10) ( r n − r m ) (cid:11) = 12 s max X s =0 w ( s ) R (14)using the histogram of curvilinear distances w ( s ) dis-cussed above and the Gaussian chain property R = b s .Measuring thus the first moment of w ( s ), the radius ofgyration is equivalent for Gaussian chains to the Wienerindex W . The reduced radius of gyration y = ( R g /R e ) is plotted as a function of G . Since the end monomersdominate the mass distribution of regular dendrimers forlarge G , R g becomes similar to R e . As expected, y ap-proaches unity from below (circles). Interestingly, theratio y is constant for the self-similar β - and γ -stars, i.e. R e and R g are similarly rescaled by the iterative applica-tion of the generators. This confirms the choice of gen-erators discussed in sect. III. We note finally that otherobservables characterizing R , such as the hydrodynamicradius [20], have been found to scale similarly as the enddistance R e and the radius of gyration R g . Density profiles.
Figure 8 presents various normalizeddensity profiles p ( r ) with r being the radial distance fromthe root monomer. The rescaled distribution y = p ( r ) R d is plotted as a function of the reduced distance x = r/R with R = R e in panel (a) and R = R g in panel (b)and panel (c). The distribution of the end monomers forregular dendrimers ( G = 12, S = 32) shown in panel(a) is a reminder of eq. (2), i.e. of the trivial fact thatthe distances of all pairs of monomers have a Gaussiandistribution (dashed line). The rescaled density ρ ( r ) = p ( r ) N of all monomers is shown in panel (b) of fig. 8(using a half-logarithmic representation) for the largeststar of each topology class. Note that the distribution p ( r ) has been either obtained for masses up to N ≈ from our MC simulations or for larger systems using p ( r ) = s max X s =0 w ( s ) P ( r, s ) (15) x = r/R -12 -10 -8 -6 -4 -2 y = p ( r ) R d -7 -6 -5 -4 -3 -2 -1 regular dendrimer G=12 -3 -2 -1 x -1 y end monomer: R=R e all monomers: R=R g G a u ss i a n d f =2.5 s l op e - . (a)(b) (c) FIG. 8: Density distributions y = p ( r ) R d with r/R being thereduced distance from the root monomer: (a) End monomerdistribution with R = R e showing the expected Gaussianity(dashed line), (b) total monomer distribution rescaled with R = R g using the same symbols as in fig. 7, (c) double-logarithmic representation for three architectures with d f =2 .
5. The slope indicates the exponent d f − d = − . with w ( s ) being the already mentioned normalized his-togram of monomers of same curvilinear distance fromthe root monomer and P ( r, s ) the size distribution of asubchain of arc-length s given by eq. (2). Since the den-sity distribution of large regular dendrimers (circles) isdominated by the end monomers, p ( r ) becomes essen-tially Gaussian (dashed line). We shall come back to thispoint at the end of sect. V. The histograms get natu-rally broader with decreasing d f . Panel (c) on the right-hand side gives a double-logarithmic representation ofthe total monomer density distribution for three topolo-gies with d f = 2 .
5. As explained in de Gennes’ book[19], the density should decrease as n ( r ) /r d ∼ /r d − d f with n ( r ) ∼ r d f being the mass distributed within thevolume r d . The same power-law exponent is obtainedusing w ( s ) ∼ s α − and integrating eq. (15) for d f < d and x ≪
1. Even the not self-similar α -star (open trian-gles) is seen to follow the predicted slope (solid lines). Itis sufficient for this property that w ( s ) has a power-lawasymptotics albeit w ( s ) has not. Center of mass fluctuations.
Albeit spherically aver-aged density profiles may reasonably characterize some aspects of the conformational properties of our hyper-branched polymer stars [42] it is important to empha-size that a given instantaneous configuration may not be spherically symmetric and depending on the propertyprobed experimentally or in a computer experiment theseaspherical fluctuations become crucial. This issue is ad-dressed in fig. 9. The main panel compares the true ra-dius of gyration R = N P n h ( r n − r cm ) i with a spher-ical approximation of the mass distribution defined by R ≡ N P n h r n i assuming the center of mass r cm ofthe star to be set by the root monomer at the originfor all configurations, i.e. r cm ! = 0. The main panel of G y = ( R g / R s p ) d f = ∞ d f =6d f =3d f =2.5 -6 -5 -4 -3 -2 x = 1/N ∆ S=32 filled symbols: β -starscrosses: γ -stars FIG. 9: Aspherical fluctuations: ratio ( R g /R sp ) (main panel)and rescaled largest eigenvalue ∆ ≡ h λ i /R − / α -stars (inset). fig. 9 presents ( R g /R sp ) as a function of G for differenttopologies. The ratio is always smaller than unity. Theratio is seen to approach unity from below for regulardendrimers and α -stars with d f > d . While the sphericalapproximation r cm = 0 becomes thus better with increas-ing size, stars with an incredible huge molecular massare required to reach R g ≈ R sp . Interestingly, the ratio decreases for α -stars with d f = 3 and d f = 2 . Asphericity.
The asphericity of the stars may be(also) characterized by computing the three eigenvalues λ ≥ λ ≥ λ of the inertia tensor of each star and aver-aging over the ensemble. Since R = h λ i + h λ i + h λ i ,the rescaled eigenvalue ∆ ≡ h λ i /R − / h λ i = h λ i = h λ i .We have plotted ∆ as a function of the inverse mass forseveral topologies in the inset of fig. 9. As expected fromthe consideration of R sp , ∆ is seen to vanish in the large- N limit for regular dendrimers and α -stars with d f > d .(As shown by the solid line, ∆ decays only logarithmi-cally with mass.) The opposite behavior is found forsmaller fractal dimensions as shown by the open trian-gles. Whether for these systems ∆ becomes constant for N → ∞ (as for linear chains) cannot be confirmed yetfrom our numerical data. V. FORM FACTOR
Introduction.
Conformational properties of branchedand hyperbranched star polymers can be determined ex-perimentally by means of light, small angle X-ray or neu-tron scattering experiments [33, 43]. Using appropriatelabeling techniques this allows to extract the coherentintramolecular form factor F ( q ) defined as N F ( q ) = (cid:10) ˆ ρ ( q )ˆ ρ ( − q ) (cid:11) = * || N X n =1 exp (cid:0) i q · r n (cid:1) || + (16)with ˆ ρ ( q ) being the Fourier transform of the instanta-neous density and q the wavevector. The average is sam-pled over the ensemble of thermalized chains. For suffi-ciently large N and small q ≡ || q || the radius of gyration R g , as one measure of the star size, becomes the onlyrelevant length scale. The form factor thus scales as [19] F ( q ) = N f ( Q ) with Q = qR g (17)being the reduced wavevector and f ( Q ) a universal scal-ing function with f ( Q ) = 1 − Q /d in the “Guinierregime” for Q ≪
1. The opposite large- q limit probesthe density fluctuations within the spacer chains and theform factor becomes [20] F ( q ) = 12( bq ) for 1 bS / ≪ q ≪ b . (18)For even larger wavevectors correlations on the monomerscale are probed. In the following we shall focus on theintermediate wavevector range 1 /R g ≤ q ≪ /bS / be-tween the Guinier regime and the large- q limit. Dendrimers.
Focusing on dendrimers, fig. 10 presentsa Kratky representation [33] of the form factor y ≡ ( F ( q ) /N ) Q as a function of the reduced wavevector Q = qR g . Panel (a) shows stars of different spacer length S for a generation number G = 12, panel (b) differentgeneration numbers G for a fixed spacer length S = 128.The increase of the rescaled data for very large wavevec-tors q ≫ /b observed in both panels is caused by thediscrete monomeric units used in our simulations (see be-low). The scaling observed for different S in panel (a) forthe intermediate wavevector regime, where the Gaussianspacer chains are probed, is due to the fact that both themass N and the radius of gyration R are linear in S .The corresponding failure of eq. (17) in panel (b) showsthat there is more than one characteristic length scale.Note that the strong decay after the Guinier regime above Q ≈ G . The bold solid lines in both pan-els indicate the expected asymptotic limit for G → ∞ asdiscussed at the end of this section. Note that the den-drimer with G = 20 (large circles) shown in panel (b) israther close to this limit. The form factor of this hugechain has not been obtained by MC simulations but bycomputing numerically the equivalent discrete sum F ( q ) = s max X s =0 w ( s ) P ( q, s ) (19)with w ( s ) being the curvilinear segment histogram dis-cussed above and P ( q, s ) the Fourier transform of the seg-ment size distribution P ( r, s ). Since for Gaussian chains P ( q, s ) = exp( − ( aq ) s ) with a ≡ b/ √ d , the form factor -1 Q = q R g -2 -1 y = ( F ( q ) / N ) Q -1 -2 -1 S=8S=16S=32S=64S=128S=256G ⇒∞ G u i n i e r P o r od regular dendrimer with G=12 r esca l e d G a u ss i a n Gaussian chain continuous d i sc r e t e (a) q ≥ / b -1 Q = q R g -2 -1 y = ( F ( q ) / N ) Q -1 -2 -1 G=1G=2G=3G=4G=5G=6G=7G=8G=9G=10G=11G=12G=20G ⇒∞ d i sc r e t i z a t i on G u i n i e r y = / Q S=128regular dendrimer (b)
FIG. 10: Kratky representation of the form factor y =( F ( q ) /N ) Q as a function of the reduced wavevector Q = qR g for dendrimers: (a) G = 12 for different spacer length S , (b) S = 128 for different generation number G . The dash-dottedlines indicate the Porod power law [33], the bold solid linesthe predicted asymptotic behavior, eq. (6). is readily computed yielding, as one expects, the sameresults as obtained from the explicitly computed config-uration ensembles. This can be seen from the dashedline in panel (a) of fig. 10 for a spacer length S = 32.To compute numerically the form factor using w ( s ) hasthe advantage that the already mentioned discretizationeffect at q ≫ /b can be eliminated. To do this the dis-crete sum eq. (19) is replaced by a continuous integralfor s > s = 0-contribution to the form factor isadded. As shown by the thin solid line in panel (a), thisallows to get rid of the irrelevant discretization effect. Marginally compact stars.
Figure 11 presents theform factor obtained using the continuous version ofeq. (19) for self-similar fractals of marginal compactness( d f = 3). As one expects according to eq. (5), the dataapproach with increasing generation number the power-law slope 2 − d f = − w ( s ) ∼ s α − forself-similar fractals. Interestingly, eq. (5) does not holdfor the (not self-similar) α -stars as may be seen from -1 Q = q R g -2 -1 y = ( F ( q ) / N ) Q -1 -2 -1 i=0: G=2i=1: G=8i=2: G=32i=3: G=128i=4: G=512i=5: G=2048i=6: G=8192 Gaussian spacer G u i n i e r P o r od y = / Q β -stars with d f =3 N=257N=2049N=16385N=131073N=1048577N=8388609N=67108868 s ph e r i ca l p r eave r a g i ng FIG. 11: Kratky representation for β -stars with d f = 3. Thereduced form factor approaches with increasing G the power-law slope − N is indi-cated for each iteration i . The dashed line has been obtainedaccording to eq. (20) by Fourier transformation of the spher-ically averaged density ρ ( r ) for i = 6. -1 Q = q R g -3 -2 -1 y = ( F ( q ) / N ) Q -1 -3 -2 -1 d f = ∞ d f =6d f =5d f =4d f =3d f =2.5 / Q G u i n i e r P o r od / Q . - s=32 asy m p t o t i c G a u ss i a n li m i t FIG. 12: Rescaled form factor y ( Q ) = ( F ( q ) /N ) Q for thelargest stars available obtained using eq. (19). The self-similar β - and γ -stars (filled symbols and crosses) decay, as expected,with a power law Q − d f in the intermediate wavevector regimeas shown by the thin solid line for d f = 3 ( i = 6) and by thedash-dotted line for d f = 2 .
5. The dashed line indicates thepreaverage approximation using eq. (20) for α -stars of d f = 5,the bold solid line the expected large-dendrimer limit. fig. 12. Note also that the large- q plateau of the rescaledform factor in fig. 11 only decays as R /N ∼ /N / extremely slowly with mass. This makes the numericalconfirmation of the power-law slope demanding. For realexperiments this implies that the determination of a frac-tal dimension d f ≈ Comparison of different architectures.
The rescaledform factors for the largest chains considered for eachstudied topology are compared in fig. 12. As expected,all data sets collapse in the Guinier regime below Q ≈ q ≫ /bS / . (The discretization effect for large q is againavoided using the continuous version of eq. (19).) Thedecay of the reduced form factor in the intermediatewavevector is seen to become systematically strongerwith increasing fractal dimension d f . For the self-similarstars this decay is described by eq. (5) as emphasizedby the solid and the dash-dotted power-law slopes for,respectively, d f = 3 and d f = 2 .
5. All other architec-tures decay stronger than a power law. Note that it isthe shape of this decay which is the most central prop-erty to be tested experimentally to characterize, at leastapproximatively, the structure of hyperbranched stars.
Spherical preaveraging.
As reminded at the beginningof this section, the intramolecular form factor is the en-semble average of the squared Fourier transform ˆ ρ ( q )of the fluctuating instantaneous monomer density. Fol-lowing the recent work by Likos et al. [45], this begsthe question of whether in the limit of large and densestars, where density fluctuations should become suffi-ciently small, one may replace ˆ ρ ( q ) by the Fourier trans-form ρ ( q ) of the averaged density profile ρ ( r ) discussed insect. IV. Due to the spherical symmetry of our stars thissuggests using eq. (6.54) of ref. [33] the approximation F ( q ) ≈ N (cid:18)Z d r p ( r ) sin( q · r ) q · r (cid:19) (20)with p ( r ) = ρ ( r ) /N being known from eq. (15). As seenin fig. 11, eq. (20) is not useful for open ( d f < d ) andmarginally open ( d f ≈ d ) architectures for which the den-sity fluctuations are yet too large. The approximationbecomes systematically more successful, however, withincreasing fractal dimension as seen in fig. 12 for α -starsof fractal dimension d f = 5. Note that the striking decayof the rescaled form factor above the Guinier regime isaccurately described by the approximation. As we haveseen in fig. 8, the distibution p ( r ) becomes systemati-cally more Gaussian with increasing star size and fractaldimension since the end monomers of the largest gener-ation shell dominate the total density. Since the Fouriertransform of a Gaussian is again a Gaussian, this im-plies finally eq. (6) as already stated in the Introduction.As seen by comparing the solid bold lines in fig. 10 andfig. 12 with the form factors computed using eq. (19) forour largest dendrimers (circles), the asymptotic behavioreq. (6) gives an excellent fit to our numerical data. VI. WEAK EXCLUDED VOLUME EFFECTS
Introduction.
Up to now we have only considered ef-fects of the imposed monomer connectivity assuming allother interactions (persistence length, excluded volume,0 -7 -6 -5 -4 -3 -2 -1 ε R g G=5G=6G=7G=8G=9G=10G=11 -2 v( ε )/v * ( R g / R * ) γ = / d S=32, ε≤ FIG. 13: Radius of gyration R g for dendrimers vs. excludedvolume energy ǫ for generation number G as indicated. Mainpanel: Unscaled raw data for S = 32. Inset: Data collapseof rescaled radius of gyration ( R g /R ⋆ ) as a function of thereduced excluded volume v ( ǫ ) /v ⋆ with R ⋆ = R g ( ǫ = 0) and v ⋆ = R d⋆ /N . The bold slope corresponds (approximately) tothe compact limit N ∼ R d g . . . . ) to be switched off. Since essentially all proper-ties (apart the eigenvalues λ i of the inertia tensor) canbe obtained analytically or numerically using the Gaus-sian chain statistics, the presented MC simulations wereless crucial. Direct simulations are, however, essential fortesting the influence of (albeit weak) excluded volume in-teractions computed using the lattice occupation numberHamiltonian, eq. (7), described at the end of sect. II. Scaling of chain sizes.
Figure 13 presents the ex-cluded volume dependence of the radius of gyration R g for regular dendrimers. (Similar behavior is found forother characterizations of the typical chain size R .) Asreveiled in the main panel, the excluded volume effectsare the more marked the larger the mass N ( G ): The ra-dius of gyration increases already at ǫ = 10 − for G = 11while it has barely changed at ǫ = 0 . G = 5. A suc-cessful data collapse is seen in the inset of fig. 13 wherethe rescaled radius of gyration ( R g /R ⋆ ) is plotted asa function of the reduced excluded volume v ( ǫ ) /v ⋆ with R ⋆ ≡ R g ( ǫ = 0) ≈ ( SG ) ν being the typical size of theGaussian dendrimer star and v ( ǫ ) ≡ δV (1 − exp( − βǫ )) ≈ βǫδV for βǫ ≪ β denoting the inverse temperature). The charac-teristic excluded volume v ⋆ below which the star shouldremain Gaussian is set by v ⋆ ≡ R d⋆ /N . This scaling is adirect consequence of Fixman’s general criterion [20]1 ≫ vρ R d⋆ ≈ vN /R d⋆ (22)for the Gaussian chain approximation with ρ ≈ N/R d⋆ theoverall density for Gaussian stars. That the stars remainGaussian for v/v ⋆ ≪ γ = 1 /d (bold line) for large reduced excluded volumesis only an approximative guide to the eye not taking intoaccount logarithmic corrections. This can be seen (i) from the usual power-law ansatz [19] R ≈ R ⋆ ( v ( ǫ ) /v ⋆ ) γ , (ii) neglecting the weak logarithmic N -dependence of R ⋆ (fig. 6) and (iii) assuming that the dendrimers becomeessentially marginally compact, N ∼ R d g , for large ǫ inagreement with ref. [46]. The latter point has explicitlybeen checked. For finite- d f stars a similar scaling hasbeen found (not shown). Spacer chain length criterion.
We note finally that interms of the generation number G and the spacer length S , Fixman’s criterion may be rewritten remembering that N ≈ S G for dendrimers and N ≈ S G d f ν for power-lawstars [38]. Hence, the Gaussian approximation must holdfor S ≪ S ⋆ with an upper critical spacer length [23] S ⋆ ≈ (cid:0) ( v/b d ) 2 G /G dν (cid:1) − / (2 − dν ) and S ⋆ ≈ (cid:0) ( v/b d ) G α − dν (cid:1) − / (2 − dν ) , (23)respectively, for dendrimers ( d f = ∞ ) and finite- d f hy-perbranched stars. In both cases S ⋆ ≈ ( b /v ) in d = 3 dimensions (while four-dimensional stars are onlymarginally swollen). VII. CONCLUSION
Summary.
We have revisited by means of direct an-alytical calculation, using for instance eq. (19), and MCsimulations (sect. II) several conformational properties ofregular (exponentially growing) dendrimers and power-law hyperbranched stars (fig. 1) assuming Gaussian chainstatistics ( ν = 1 / w ( s ) of curvilinear dis-tances s between monomer pairs (fig. 5). Focusing onexperimentally measurable observables such as the ra-dius of gyration R g (fig. 7) and the intramolecular formfactor F ( q ) (figs. 10-12), we investigated the scaling forasymptotically long stars with different fractal dimen-sions d f . Due to their topological simplicity regular den-drimers ( d f = ∞ ) have played a central role in our pre-sentation (fig. 10) as in other recent computational stud-ies [42, 45–51]. Being (in our view) experimentally andtechnologically more relevant, we have also focused onstochastic architectures with d f = 3 (marginally com-pact) and d f = 2 . α -stars” constructed by imposing M g ∼ g α − arms per generation with truly self-similarso-called “ β -stars” and “ γ -stars” for which M g becomes astrongly fluctuating quantity (fig. 3). As shown in fig. 12,only the latter two topologies show the power-law decay,eq. (5), of the form factor in the intermediate wavevectorregime expected for open self-similar systems [19, 29, 33].While large compact ( d f > d ) stars may roughly be seenas dense colloidal spheres in agreement with Likos et al. [45], the instantaneous aspherical fluctuations cannot beneglected for experimentally relevant properties for the1 G -2 -1 S * d e nd r i m e r d f = ∞ d f =4marginal compactness d f =3 D L A - li ke d f = . S >> S * : Gaussian star R~N R~N G * ( S ) FIG. 14: Sketch of critical spacer length S ⋆ for melts. TheGaussian star assumption holds above the bold lines. Notethat the scaling argument does not allow to fix the scale ofthe vertical axis. If the generation number G is increased atconstant spacer length S , as indicated by the dashed arrows,ideal chain behavior is expected for small G ≪ G ⋆ ( S ), whilethe star becomes colloid-like for larger G ≫ G ⋆ ( S ). Thenumber of chains interacting with a reference star should havea maximum at ≈ G ⋆ ( S ). smaller fractal dimensions studied (fig. 9, dashed linein fig. 11). We have commented briefly on the effectsof gradually switching on an excluded volume potential.Coupling the (off-lattice) monomers by means of a (lat-tice) MC scheme (sect. II), we have sketched for differentarchitectures the regime ( ǫ ≪ ǫ ⋆ , S ≪ S ⋆ ) where theGaussian star approximation can be assumed to be rea-sonable (fig. 13). Conjectures for melts of hyperbranched stars.
As al-ready pointed out, the Gaussian star assumption shouldbe relevant under melt conditions assuming a large spacerlength S ≫ S ⋆ . That this holds can be seen by rewritingFixman’s Gaussian chain criterion, eq. (22), for melts1 ≫ vN ρ R d⋆ ≈ v N/R d⋆ (24)remembering that the bare excluded volume v ∼ ǫ hasto be rescaled by the total chain mass N [19, 35, 52]. The hyperbranched stars should thus remain Gaussianfor interaction energies ǫ ≪ ǫ ⋆ ≈ k B T R d⋆ / ( N δV ). Since ǫ is not a parameter which can be readily tuned experi-mentally over several orders of magnitude, it is of someimportance that eq. (24) sets equivalently a lower bound S ⋆ ≪ S depending on the generation number G . Follow-ing the discussion at the end of sect. VI, this implies S ⋆ ≈ (cid:0) G /G dν (cid:1) / ( dν − for d f = ∞ and S ⋆ ≈ (cid:16) G ( d f − d ) ν (cid:17) / ( dν − for finite- d f stars. (25)This scaling prediction is sketched in fig. 14 for severalarchitectures. Hyperbranched stars should remain thusGaussian (albeit with a renormalized effective statisti-cal segment length [20, 35, 53]) as long as S ≫ S ⋆ , ifthe interaction parameter βǫ is switched on as in the re-cent study of linear chain polymer melts [35]. Detailsmay differ somewhat, of course, since the spacer chainsmay not be rigorously Gaussian due to long-range corre-lations related to the overall incompressibility of the melt[35]. It is thus possible that even self-similar stars of im-posed d f = 2 . ǫ = 0) mayswell somewhat. We do conjecture, however, that this“swelling” for interacting large- S hyperbranched stars inthe melt remains perturbative as long as d f < d = 3 [54].Considering the dynamical properties of strongly inter-penetrating hyperbranched stars for S ≫ S ⋆ sampled us-ing standard molecular dynamics [34], it will be of someinterest to characterize the mean-square displacement ofthe star center of mass or, even better, the associateddisplacement correlation function [35]. As for the centerof mass motion of linear polymer melts [35, 55], strongdeviations from the Rouse scaling are to be expected [56]. Acknowledgments
P.P. thanks the IRTG Soft Matter for funding. We areindebted to A. Blumen and C. Friedrich (both, Freiburg)and A. Johner (ICS, Strasbourg) for helpful discussions. [1] For simplicity, we call “stars” or “branched stars” all tree-like branched architectures, “regular dendrimers” thedeterministic and exponentially growing stars shown inpanel (a) of fig. 1, and “power-law stars” hyperbranchedstars of finite fractal dimension d f .[2] B. Zimm and W. Stockmayer, J. Chem. Phys. , 1301(1949).[3] H. Wiener, J. Am. Chem. Soc. , 17 (1947).[4] W. Burchard, K. Kajiware, and D. Nerger, J. Polym.Sci., Polym. Phys. Ed. , 157 (1982).[5] B. Hammouda, J. of Polymer Science: Part B: PolymerPhysics , 1387 (1992). [6] P. Biswas and B. Cherayil, J. Chem. Phys. , 3201(2001).[7] S. Obukhov, M. Rubinstein, and T. Duke, Phys. Rev.Lett. , 1263 (1994).[8] Hyperbranched stars or trees with fractal dimension d f = 4 have been discussed in the context of randomlybranched polymers (often called “lattice animals”) [2],dilute rings in a gel of topological obstacles [7] and asa possible model describing the topological interactionsof unconcatenated melts of rings [7, 29]. All these mod-els have in common that on the local scale the branchedstructure is described by Gaussian spacer chains. [9] A. Blumen, A. Jurjiu, T. Koslowski, and C. von Ferber,Phys. Rev. E , 061103 (2003).[10] M. Dolgushev and A. Blumen, Macromolecules , 5378(2009).[11] M. Dolgushev and A. Blumen, Macromolecules ,044905 (2009).[12] M. Dolgushev and A. Blumen, Macromolecules ,124905 (2010).[13] M. Dolgushev, G. Berezovska, and A. Blumen, Macro-molecular theory and simulations , 621 (2011).[14] F. F¨urstenberg, M. Dolgushev, and A. Blumen, J. Chem.Phys. , 154904 (2012).[15] A. Kumar and P. Biswas, Macromolecules , 7378(2010).[16] A. Kumar and P. Biswas, J. Chem. Phys. , 124903(2012).[17] B. Duplantier, J. Stat. Phys. , 581 (1989).[18] Stars play a central role in polymer theory in generalsince the partition function of a sufficiently large polymerof any topology can be decomposed as a product of a starand a ring partition function as shown by Duplantier [17].[19] P. G. de Gennes, Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca, New York, 1979).[20] M. Doi and S. F. Edwards,
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