Dynamics of chains and dendrimers with heterogeneous semiflexibilityMaxim Dolgushev and Alexander Blumen Citation: The Journal of Chemical Physics 132, 124905 (2010); doi: 10.1063/1.3366662 View online: http://dx.doi.org/10.1063/1.3366662 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/132/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Interchain coupled chain dynamics of poly(ethylene oxide) in blends with poly(methyl methacrylate): Couplingmodel analysis J. Chem. Phys. 135, 194902 (2011); 10.1063/1.3662130 Dynamics of semiflexible treelike polymeric networks J. Chem. Phys. 131, 044905 (2009); 10.1063/1.3184797 Morphology and rheology of compatibilized polymer blends: Diblock compatibilizers vs crosslinked reactivecompatibilizers J. Rheol. 52, 1385 (2008); 10.1122/1.2995857 Viscoelastic properties of dendrimers in the melt from nonequlibrium molecular dynamics J. Chem. Phys. 121, 12050 (2004); 10.1063/1.1818678 Viscoelastic properties of a cross-linked polysiloxane near the sol–gel transition J. Rheol. 45, 995 (2001); 10.1122/1.1378027

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Dynamics of chains and dendrimers with heterogeneous semiflexibilityMaxim Dolgusheva� and Alexander BlumenTheoretical Polymer Physics, University of Freiburg, Hermann-Herder-Str. 3, D-79104 Freiburg, Germany

�Received 28 January 2010; accepted 22 February 2010; published online 31 March 2010�

Based on our recent model for the dynamics of semiflexlible treelike networks �M. Dolgushev andA. Blumen, J. Chem. Phys. 131, 044905 �2009��, we study the dynamical properties of chainpolymers and of dendrimers whose junctions display different stiffness degrees �SD�. In thesepolymers the functionality f of the inner junctions is constant, being f =2 for the linear chains andf =3 for the dendrimers. This allows us to focus on the effects caused by the heterogeneities due todifferent SD. For this we study alternating, diblock, as well as random arrangements of the SD. Eachof these cases shows a particular, macroscopically observable behavior, which allows to distinguishbetween the different microscopic SD arrangements. © 2010 American Institute of Physics.�doi:10.1063/1.3366662�

I. INTRODUCTION

Nowadays, polymer science focuses on large classes ofmacromolecules, with very different topologies and chemicalstructures.1,2 Especially, compounds built from severalmonomer species turn out to demonstrate remarkable staticaland dynamical properties. From a theoretical point, the dy-namics of flexible polymeric networks of arbitrary architec-ture may be approached using the generalized Gaussianstructure �GGS� model.3 The GGS model extends the classi-cal Rouse beads-and-springs picture4 by allowing the beads�the monomers� to be connected to more than two neighbors.The GGS concept was successfully applied to a large numberof structures, see, e.g., Ref. 5.

Another modern direction of study is the treatment ofnetworks built from different monomer species. Investiga-tions in this field were reported for linear chains,6–11 fordendrimers,12 for dendrimers built from stars,13,14 and forspecifically cross-linked polymers.11,15,16 In all these casescopolymers built from monomer units of different mobilitiesdisplay very rich dynamics. Furthermore, a recent work fo-cused on elastomeric networks in which the junctions alter-nate regularly in their functionality.17 Also of importance isthe flexibility of the subunits; recent developments consid-ered rodlike particles incorporated into flexible polymerchains,18 into flexible polymer networks,19–21 and also net-works built from anisotropic nematic chains.22

In this article we study the dynamics of semiflexible co-polymeric structures using a recently developed model forsemiflexible treelike networks �STN�;23 the model extendsearlier approaches for polymer chains24,25 to arbitrary tree-like architectures. We note that the pioneering approach ofRef. 24 was applied extensively,26–32 see also Ref. 33. Theapproach given in Ref. 23 is quite general, since it allows notonly to treat arbitrary treelike arrangements but it also per-mits to specify explicitly the stiffness degree �SD� at eachsingle junction, and thus to account for heterogeneous situa-tions �as in the case of copolymers� throughout the sample.

In order to highlight the effects of different SD we restrictourselves in this work to very regular topologies, namely, topolymer chains and to dendrimers of functionality f =3.

The structure of the paper is as follows. First, we recallthe main points of the theory of STN. Then we considersemiflexible chains whose junctions display different SD; wefocus on junctions of alternating SD for chains and for den-drimers, on diblock situations for chains, and also on randomdistributions of the SD for chains and for dendrimers. Weclose in Sec. IV with our conclusions.

II. THE MODEL

In order to study the dynamics of chains and dendrimerswith heterogeneous stiffness we use our recent model forSTN.23 The basic idea of this model is to implement bond-bond correlations into the GGS formalism3,5 for treelike net-works. As we have shown in Ref. 23, our formalism extendsthe generalized Langevin equation approach of Bixon andZwanzig, originally applied to semiflexible chains.24 In fact,the formalism also follows from a maximum entropyprinciple.23,25,34

Now, the GGS formalism is the generalization of theRouse model for linear polymer chains4 to arbitrary architec-tures, where the polymer network is represented by beadslocated at ri �i=1, . . . ,N� connected by springs �bonds�, sayda=ri−r j. In the simplest case the potential VGGS betweenthe GGS beads is purely harmonic, so that it is diagonal inthe bonds’ variables,

VGGS��da�� =K

2 �a

da2. �1�

Here K=3kBT / l2 denotes the spring constant, l2 the mean-square length of each bond, and kB the Boltzmann constant,and the sum runs over all the bonds. In the absence of angu-lar correlations between the different bonds, as is the case forflexible structures, the averages �da ·db, taken with respectto the Boltzmann distribution exp�−VGGS /kBT�, vanish.a�Electronic mail: dolgushev@physik.uni-freiburg.de.

THE JOURNAL OF CHEMICAL PHYSICS 132, 124905 �2010�

0021-9606/2010/132�12�/124905/7/$30.00 © 2010 American Institute of Physics132, 124905-1

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On the other hand, when the polymers are semiflexiblethe bonds are correlated. As discussed in Ref. 23 for STNone may introduce semiflexibility using the generalized po-tential,

VSTN��da�� =K

2 �a,b

Wabda · db, �2�

where one assumes the bonds to be oriented. Now, the evalu-ation of �da ·db with respect to the Boltzmann distributionexp�−VSTN /kBT� and under assumption that the �da� areGaussian-distributed gives

�da · db = l2�W−1�ab. �3�

Defining now �da ·db allows to compute the matrix W−1,which after inversion specifies all Wab in Eq. �2�. As shownin Refs. 23 and 24, a physically reasonable choice of the set�da ·db is as follows: First, the mean-squared lengths are

�da · da = l2. �4�

Second, adjacent oriented bonds, say a and b, which areconnected by a bead, say i, obey

�da · db = � l2ti. �5�

Here the parameter ti reflects the stiffness of the junction i.The plus sign holds for a head to tail configuration of theoriented bonds a and b and the minus sign otherwise. Inthree dimensions, the SD ti is bounded by ti�1 / �f i−1�,35

where f i is the functionality of the ith bead. The lower boundfor ti is ti=0, which corresponds to the fully flexible case.Moreover, for nonadjacent bonds, say a and c, one may set23

as in the freely rotating chain model,

�da · dc = �da · db1�db1

· db2 ¯ �dbk

· dcl−2k, �6�

where �b1 ,b2 , . . . ,bk−1 ,bk� is the shortest path that connects awith c, which for STN is unique.36 The choice of Eq. �6� alsofollows automatically from the maximum entropy principleapplied to STN.23

We now recall the results of Ref. 23 for the matrix W�Wxy�. For nonadjacent bonds, say a and c, the correspond-ing elements of W vanish,

Wac = Wca = 0. �7�

For adjacent bonds, say a and b, which hence have a bead,say i, in common, one obtains23

Wab = Wba = �ti

�f i − 1�ti2 + �f i − 2�ti − 1

, �8�

where ti is the SD and f i is the functionality of the ith junc-tion; again the plus sign holds for head to tail configurationsand the minus sign otherwise. The diagonal element of Wcorresponding to the bond b, which connects beads i and j, is

Wbb = 1 −�f i − 1�ti

2

�f i − 1�ti2 + �f i − 2�ti − 1

−�f j − 1�tj

2

�f j − 1�tj2 + �f j − 2�tj − 1

, �9�

with the corresponding �ti , f i� and �tj , f j� parameters. We re-

mark that Eqs. �7�–�9� give the potential, Eq. �2� in an ana-lytically closed form. In the limit �ti→0� one is led to Wab

=Wba=0 and to Wbb=1, and one recovers thus the potential,Eq. �1�, of flexible treelike networks.

Classically, the dynamics of polymeric structures can bedescribed by a set of Langevin equations for the beads’positions.5 The transformation from the bonds’ to the posi-tions’ variables, da=ri−r j, can be written in terms of theincidence matrix G,36

da �k

�GT�akrk. �10�

Here the elements of G= �Gia� are Gja=−1 and Gia=1, whenthe bond a connects the beads i and j, and are zero other-wise. Any treelike network consisting of N beads has �N−1� bonds. Therefore, G is a rectangular N� �N−1� matrix.Substitution of Eq. �10� into Eq. �2� gives

VSTN��ri�� =K

2 �k,n

�GWGT�knrk · rn. �11�

The Langevin equation, say for the x-component of theposition vector ri= �xi ,yi ,zi�, is given by5,23

��

�txi�t� +

�

�xiVSTN��rk�� = f i�t� . �12�

Here f i is the x component of the usual Gaussian force actingon ith bead, for which �f i�t�=0 and �f i�t�f j�t��=2kBT��ij��t− t�� hold. Let us define the matrix ASTN

= �AijSTN� through

ASTN = GWGT. �13�

In the case of vanishing stiffness values, ti→0, the matrix Wis the identity matrix, W=1, and ASTN turns into the usualconnectivity matrix A of the GGS,5,36 ASTN��ti→0��=GGT

A.After substituting Eq. �11� into Eq. �12� and taking Eq.

�13� into account we obtain

��

�txi�t� + K�

j=1

N

AijSTNxj�t� = f i�t� . �14�

The set of Langevin equations, Eq. �14�, can be solvedby diagonalizing the matrix ASTN. Now, the elements ofASTN are known in closed form, see Ref. 23. Moreover, ASTN

is a sparse matrix, a fact which considerably simplifies itsdiagonalization. The eigenvalues ��k� of ASTN are fundamen-tal for determining the mechanical relaxation pattern of theSTN considered.5,32 Looking at the response of STN to ap-plied harmonic strain fields, one is led to the complex shearmodulus G����=G����+ iG����. Dividing the G���� andG���� by �kBT renders them dimensionless. For the reducedvariables one has5,32

�G����� = G����/�kBT =1

N�k=2

N��0/2�k�2

1 + ��0/2�k�2 �15�

and

124905-2 M. Dolgushev and A. Blumen J. Chem. Phys. 132, 124905 �2010�

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�G����� = G����/�kBT =1

N�k=2

N�0/2�k

1 + ��0/2�k�2 . �16�

In Eqs. �15� and �16� the ��k� are the nonvanishing eigenval-ues of the matrix ASTN and we have set 0=� /K.

The mechanical relaxation moduli have universal scalingrelations which hold for all finite networks: for very small �one has �G�������2 and �G������� and for very large �one has �G�������0 and �G�������−1. As in all relatedcases, the particular structure of the network can be seenonly in the in-between region.

III. RESULTS

A major advantage of the formalism presented in Sec. IIis its generality, namely, its ability to allow for different SDvalues at each junction, see Eq. �5�. Thus we can considernot only treelike structures of any given complexity but alsotreat heterogeneous polymers whose junctions’ stiffness var-ies through the structure. In order to see clearer the effects ofsuch stiffness heterogeneities, we will focus on STN ofhighly symmetric topology, namely, on linear chains and ondendrimers of functionality f =3. Note that the elements ofmatrix W, Eqs. �7�–�9�, depend on the functionalities �f i� andon the SD values �ti� at each junction i.

A. Polymer chains

Polymer chains are special STN, in which all beads�apart from the end beads� have functionality f =2. When allthe ti of the internal beads are equal, ti t, one recovers thehomogeneous case, which was already discussed in Refs. 24and 32. The matrix W has then a simple structure,

W =1

1 − t2�1 − t 0 ¯ 0

− t 1 + t2 − t � ]

0 � � � 0

] � − t 1 + t2 − t

0 ¯ 0 − t 1 . �17�

Here we treat the general case, study how local changesin flexibility affect the dynamics of the chain, and report thereduced loss moduli �G�����. First, we consider the role of asingle impurity, i.e., having tj� t for a single site j, see Fig.1�a�. Second, we consider chains with alternating SD values,t2k t and t2k+1q for all k, see Fig. 1�b�. Then we studydiblock chains �DC�, see Fig. 1�c�. Finally, we considerchains with random distribution of the SD, see Fig. 1�d�.

In the case of a single site, say j, having a distinct SDfrom the rest, tj� t, the set of eigenvalues ��i� of the corre-sponding matrix ASTN differs only slightly from its homoge-neous counterpart. If tj t one notes that the highest eigen-value �N grows and separates significantly from the othereigenvalues, which differ only slightly from those of the ho-mogeneous chain. In the opposite case, tj � t, the differencesin the spectra are even weaker, the higher eigenvalues beingslightly smaller than those of the homogeneous chain. There-fore, in both cases the effect on �G����� is rather weak. Fortj t the �G����� widens somewhat, whereas for tj � t it getsslightly narrower.

1. Chains with alternating SD values

We consider now chains in which the SD changes alter-nately from junction to junction, say, by taking the values tand q, see Fig. 1�b�. Here we fix N=30, let t=0 and vary q.We plot in Fig. 2 the eigenvalue spectrum by arranging theeigenvalues in ascending order. For q=0 we recover thespectrum of the completely flexible Rouse chain. With grow-ing q the spectra show the appearance of two types of modes.In the spirit of solid state physics one might call them acous-tical and optical; they are due to the periodic chainlike ar-rangements of structured subunits consisting of two mono-mers. In the spectrum, the eigenvalues corresponding tolarge-scale motions decrease with growing q, whereas theeigenvalues corresponding to small-scale motions increase,which leads to the formation of a gap. Inside the gap there isa single eigenvalue �in Fig. 2 this is �16� whose value staysconstant �here �16=2�. Moreover, the spectral region widenswith growing q. We note that similar behaviors were alsoobserved for chains built from rodlike particles and springsin an alternating way;18 comparable findings were also re-ported in Ref. 20 for polymer networks which included rod-

FIG. 1. Semiflexible chains with different SD distributions of theirjunctions.

0 5 10 15 20 25 300

5

10

15λ

i

t=0.0, q=0.0t=0.0, q=0.25t=0.0, q=0.5t=0.0, q=0.75

FIG. 2. Eigenvalues of polymer chains with alternating SD, plotted in as-cending order for the SD values given in the inset.

124905-3 Dynamics of chains and dendrimers J. Chem. Phys. 132, 124905 �2010�

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like particles and in Ref. 16 for cubic networks with differentfriction parameters ��i�, where the relaxation spectrum alsosplits into acoustical and optical branches.

The reduced loss moduli �G����� corresponding to thespectra of Fig. 2 are plotted in Fig. 3. With growing q thespectral region widens, which is reflected in the widening ofthe �G����� curves. In the intermediate domain the typicalscaling �1/2 for flexible linear polymers changes to an effec-tive �0.43 scaling behavior. For large q, namely, q=0.75 inFig. 2, the gap between the two eigenvalue domains is verywide. In Fig. 3 this manifests itself in the appearance of asmall minimum in �G�����. Such a minimum was also ob-served in Ref. 11 for flexible, alternating chains, in which themobilities of the beads differ; such a situation also leads tothe splitting of the corresponding low and high frequencymodes and to the appearance of a minimum in �G�����.

2. Diblock chains „DC…

Here we consider DC, in which one part is flexible andthe other part is semiflexible, and choose for the SD valueq=0.75, see Fig. 1�c�. Again we take N=30 for the length ofthe chain. In terms of the STN model such block structureswere encountered in an earlier study of a star polymer with12 arms,26 for which the SD of the junctions was varied, byassuming the junctions closer to the center of star to bestiffer than the rest.

The relaxation spectra for DC is presented in Fig. 4,where we vary the length of the semiflexible part, whilekeeping the overall length constant. All the eigenvalues ofthe DC are bounded by the corresponding eigenvalues for thefully flexible and for the fully semiflexible chains of thesame length. We note that for all DC the low eigenvalues areclose to the corresponding eigenvalues of the fully flexiblechain, whereas the high eigenvalues are close to the ones ofthe fully semiflexible chain. A similar behavior is observedfor flexible DC with different friction coefficients, say �A and�B, for which for �A /�B=104 the relaxation times interpolatein a comparable fashion between the pure �A and the pure �B

cases.6

The reduced loss moduli �G����� corresponding to theeigenvalue spectra of Fig. 4 are plotted in Fig. 5. Flexiblechains show in the intermediate region the typical scalinglaw of �1/2,5 whereas short semiflexible chains display in the

same region a behavior close to �1/4;32,37 for long stiff chainsa domain with �1/2 and a domain with �1/4 scaling arepresent.38–40 For the depicted situation, with rather shortchains �N=30� and a SD value of q=0.75 the exponent of �in the intermediate domain varies between 1/4 and 0.3; infact, for lower SD values, the intermediate behavior getseven smoother.32 Generally, one has in the intermediate re-gion a roughly �G�������� scaling behavior. The exponent� for DC with different lengths of the semiflexible parts isbounded by its values for fully flexible and for fully semi-flexible chains. As a rough scaling behavior for �G����� wefind for DC containing a semiflexible part of 1/4, of 1/2, andof 3/4 as approximate exponents �=0.43, �=0.39, and �=0.34, respectively. Moreover, in the intermediate region offrequencies one finds with an increasing DC semiflexiblepart a broadening32 of the �G����� curves.

3. Chains with random stiffness

Finally, we study chains whose junctions display randomSD values �see Fig. 1�d��; we assume these values to behomogeneously distributed in the interval �0;0.8�. For achain of N=30 monomers we simulate 5000 distinct realiza-tions using the standard random number generator of theMATHEMATICA7 package. For each realization we calculate

-4 -3 -2 -1 0 1 2log

10(ωτ

0/2)

-3

-2

-1lo

g 10[G

’’(ω

)]

t=0.0, q=0.0t=0.0, q=0.25t=0.0, q=0.5t=0.0, q=0.75

ω0.43

ω1/2

FIG. 3. Reduced loss moduli of polymer chains with alternating SD, plottedfor different SD values, see text for details.

0 5 10 15 20 25 300

5

10

15

20

25

λi

flexible3/4 flexible, 1/4 semiflexible1/2 flexible, 1/2 semiflexible1/4 flexible, 3/4 semiflexiblesemiflexible

FIG. 4. Eigenvalues of polymer chains with a diblock structure of their SDvalues, plotted in ascending order for different lengths of the semiflexiblepart, see text for details.

-4 -3 -2 -1 0 1 2log

10(ωτ

0/2)

-3

-2

-1

log 10

[G’’

(ω)]

flexible3/4 flexible, 1/4 semiflexible1/2 flexible, 1/2 semiflexible1/4 flexible, 3/4 semiflexiblesemiflexible

ω1/2

ω1/4

ω0.3

FIG. 5. Reduced loss moduli of chains with a diblock structure of their SDvalues, plotted for different lengths of the semiflexible part, see text fordetails.

124905-4 M. Dolgushev and A. Blumen J. Chem. Phys. 132, 124905 �2010�

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the respective relaxation spectrum as well as the reduced lossmodulus �G�����; we average then the �G����� over all re-alizations. The averaged ��G����� curve is presented in Fig.6, where we compare it to the loss modulus �G����� for achain whose junctions have all the value t=0.4. As is evidentfrom Fig. 6, the averaged loss modulus ��G����� of chainswith random SD values from the interval �0;0.8� behaves inthe low and intermediate frequency domains very similarlyto the �G����� for a homogeneous semiflexible chain with t=0.4; the only noticeable difference occurs in the high fre-quency domain, where the two curves differ slightly, the��G����� curve extending to higher frequencies.

B. Dendrimers

In Fig. 7 we recall the topology of dendrimers; depictedis a dendrimer whose inner beads have functionality f =3, avalue which we will assume in the following. The generationof the dendrimers depicted in Figs. 7 and 8 is g=3. Becauseof their construction, dendrimers are STN.23 We let the aver-age mean-squared length of each bond be l2 and focus on

different patterns with different SD distributions, as indicatedschematically in Figs. 7 and 8, and as explicitly discussedfurther on.

Starting from the fully flexible situation, based on har-monic bonds �the simple GGS�, we consider first a singlejunction having a SD distinct from zero. Taking for it a largeparameter value, say qj =0.45, the relaxation spectrumchanges �similarly to the corresponding situation for a poly-mer chain�, in that the highest eigenvalue gets separatedfrom the rest of the spectrum. Given that for dendrimers thenumber of beads grows exponentially with the generation,the effect of this eigenvalue on the reduced loss modulus�G����� is, in general, very weak; thus, already for genera-tion g=6 one can hardly notice any effect on the�G�����-curve.

1. Dendrimers with alternating SD values

First, we consider dendrimers whose SD values changegenerationwise and we let these values alternate. Thus wetake for the central node the SD value to be q, then to be t forthe junctions connecting the first generation with the secondgeneration bonds, then again to be q, and so forth. We haveindicated this by the different coloring of the beads in Fig. 7.

In Fig. 9 are plotted the reduced loss moduli �G����� fordendrimers with alternating SD values for different genera-tions. The corresponding values are set to q=0.4 and to t=0.0. One can see significant differences between the�G����� for odd �continuous lines� and for even �dashedlines� generations: the curves for odd generations have two,for even generations only one peak. Such differences werepreviously found for another type of heterogeneity, namely,for dendrimers built from beads whose mobility alternatedwith the generation;12 in that case, however, it was found inboth cases that �G����� displayed two peaks. The reason forthe behavior which we witness here is rooted in the expo-nential growth of the number of beads with each generation;thus the beads and the junctions of the last generation are

-4 -2 0 2log

10(ωτ

0/2)

-3

-2

-1lo

g 10<

[G’’

(ω)]

>

random in averagehomogeneous, t=0.4

FIG. 6. Averaged reduced loss modulus of polymer chains with random SDvalues compared to that of a homogeneous semiflexible chain with a SDvalue of t=0.4, see text for details.

FIG. 7. Dendrimer of generation g=3 and of functionality f =3, with gen-erationwise changes in its SD values.

FIG. 8. Dendrimer of generation g=3 and of functionality f =3, whosejunctions display random SD values.

124905-5 Dynamics of chains and dendrimers J. Chem. Phys. 132, 124905 �2010�

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very numerous and play a fundamental role. In our case, atodd generations the layer of junctions most distant from thecore has the SD value q=0.4, which leads to the pronouncedintermediate minimum in �G����� �typical for fully semiflex-ible dendrimers28,29,32�. At even generations, that last layer ofjunctions has a SD value t=0.0, for which no intermediateminimum is visible.

In Fig. 10 we plot additionally �G����� for dendrimers ofgeneration g=11, in which we keep for all junctions the SDvalue equal to zero, except for the three last layers of junc-tions �8, 9, and 10�, for which we assume for the correspond-ing SD triples �t8 , t9 , t10� three sets of distinct values, namely,�0.0, 0.0, 0.4�, �0.4, 0.0, 0.4�, and �0.0, 0.4, 0.0�. For the firstset of SD values, �0.0, 0.0, 0.4�, �G����� displays a shoulderon the right-hand side of the curve, for the sets �0.0, 0.4, 0.0�and �0.4, 0.0, 0.4�, �G����� shows one and two peaks, re-spectively; the same features are also observed for dendrim-ers with alternating distribution of SD values, see Fig. 9. Fora more direct comparison we display in Fig. 10 �G����� fordendrimers of generation g=11 with alternating distributionsof the SD values, namely, we consider first q=0.0, t=0.4 andthen q=0.4, t=0.0. The �G����� for q=0.0 and t=0.4 differsonly slightly from the �G����� plotted for the set �0.0, 0.4,0.0�; the same is true for �G����� for q=0.4 and t=0.0 com-

pared to the �G����� plotted for the set �0.4, 0.0, 0.4�. Thus,for an alternating arrangement of SD values the last layers ofjunctions play indeed in dendrimers the main role.

2. Dendrimers with random stiffness

As a last example we study here dendrimers whose SDvalues are randomly distributed in the interval �0;0.4�, seeFig. 8. We consider dendrimers of generation g=8, for whichwe generate 10 000 realizations using the standard randomnumber package of MATHEMATICA7. For every structure wecalculate its relaxation spectrum as well as its reduced lossmodulus �G�����, from which we compute the averaged��G�����, which is displayed in Fig. 11. On the low fre-quency side the averaged reduced loss modulus ��G�����behaves like the �G����� of a homogeneous semiflexibledendrimer with a SD value of q=0.2 �the two curves areindistinguishable in this range�; the curves differ on the highfrequency side, where the �G����� for the random case ex-tends toward somewhat higher frequencies.

IV. CONCLUSIONS

In this work we have considered semiflexible chains anddendrimers whose junctions have different SD values, beingarranged in distinct ways. Here, we have explicitly calculatedthe loss moduli for mechanical relaxation for several typicalcases. We treated stiffness effects in the framework of thetheory of STN,23 in which one is able to consider specificconstraints on the level of each single junction.

In several instances we could show, both for the linearchains and for the dendrimers, that specific SD distributionslead to very typical �G����� forms. The approach used by ushere is not limited to polymer chains or to dendrimers, butcan be readily implemented to arbitrary STN. We expect thatboth our method and the findings described here may beuseful in the analysis of relaxation experiments on STN, andespecially on hyperbranched polymeric structures.

-2 -1 0 1 2log

10(ωτ

0/2)

-1,4

-1,2

-1

-0,8

-0,6

log 10

[G’’

(ω)]

g=3g=4g=5g=6g=7g=8g=9g=10g=11

FIG. 9. Reduced loss moduli of dendrimers of functionality f =3 with gen-erationwise alternating SD values, plotted for different generations, see textfor details.

-2 -1 0 1 2log

10(ωτ

0/2)

-1.2

-1

-0.8

-0.6

log 10

[G’’

(ω)]

t8=0.0, t

9=0.0, t

10=0.4

t8=0.4, t

9=0.0, t

10=0.4

t8=0.0, t

9=0.4, t

10=0.0

q=0.4, t=0.0q=0.0, t=0.4

FIG. 10. Reduced loss modulus of dendrimers of functionality f =3 andgeneration g=11 for particular values of the SD of their three last genera-tions of junctions. Also plotted �for the sake of comparison� are the corre-sponding alternating dendrimers with SD values of q=0.4, t=0.0 and q=0.0, t=0.4, respectively, see text for details.

-4 -2 0 2 4log

10(ωτ

0/2)

-4

-3

-2

-1

log 10

<[G

’’(ω

)]>

random in averagehomogeneous, q=0.2

FIG. 11. Averaged reduced loss modulus of a dendrimer of generation g=8 and functionality f =3 with random SD values compared with a homo-geneous semiflexible dendrimer with a SD value of q=0.2, see text fordetails.

124905-6 M. Dolgushev and A. Blumen J. Chem. Phys. 132, 124905 �2010�

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ACKNOWLEDGMENTS

The authors acknowledge the support of the DeutscheForschungsgemeinschaft and of the Fonds der ChemischenIndustrie.

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124905-7 Dynamics of chains and dendrimers J. Chem. Phys. 132, 124905 �2010�

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