Dynamics of semiflexible scale-free polymer networks Mircea Galiceanu, Adriane S. Reis, and Maxim Dolgushev Citation: The Journal of Chemical Physics 141, 144902 (2014); doi: 10.1063/1.4897563 View online: http://dx.doi.org/10.1063/1.4897563 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/14?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Hydrodynamic effects on scale-free polymer networks in external fields J. Chem. Phys. 140, 034901 (2014); 10.1063/1.4861218 Dynamics of end-linked star-polymer structures J. Chem. Phys. 123, 034907 (2005); 10.1063/1.1942490 Deformation of crosslinked semiflexible polymer networks AIP Conf. Proc. 708, 364 (2004); 10.1063/1.1764177 Scaling behavior: Effect of precursor concentration and precursor molecular weight on the modulus and swelling of polymeric networks J. Rheol. 44, 897 (2000); 10.1122/1.551120 Semiflexible polymer brushes: A scaling theory J. Chem. Phys. 109, 7017 (1998); 10.1063/1.477338

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THE JOURNAL OF CHEMICAL PHYSICS 141, 144902 (2014)

Dynamics of semiflexible scale-free polymer networks Mircea Galiceanu,1,a) Adriane S. Reis,1 and Maxim Dolgushev2 1 2

Departamento de Fisica, Universidade Federal do Amazonas, 69077-000 Manaus, Brazil Theoretische Polymerphysik, Universität Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg, Germany

(Received 6 May 2014; accepted 30 September 2014; published online 10 October 2014) Scale-free networks are structures, whose nodes have degree distributions that follow a power law. Here we focus on the dynamics of semiflexible scale-free polymer networks. The semiflexibility is modeled in the framework of [M. Dolgushev and A. Blumen, J. Chem. Phys. 131, 044905 (2009)], which allows for tree-like networks with arbitrary architectures to include local constrains on bond orientations. From the wealth of dynamical quantities we choose the mechanical relaxation moduli (the loss modulus) and the static behavior is studied by looking at the radius of gyration. First we study the influence of the network size and of the stiffness parameter on the dynamical quantities, keeping constant γ , a parameter that measures the connectivity of the scale-free network. Then we vary the parameter γ and we keep constant the size of the structures. This fact allows us to study in detail the crossover behavior from a simple linear semiflexible chain to a star-like structure. We show that the semiflexibility of the scale-free networks clearly manifests itself by displaying macroscopically distinguishable behaviors. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4897563] I. INTRODUCTION

Scale-free networks (SFNs) are structures constructed from nodes, which show a power-law degree distribution for high functionalities (degrees). SFNs were used with great success to model various real networks, such as protein-protein interaction network,1 metabolic networks,2 river networks,3, 4 food webs,5 internet,6, 7 the author collaboration networks of scientific papers,8, 9 financial networks,10 transport networks,11 airport networks,12 or reaction-diffusion processes.13, 14 Since the first algorithm to create an SFN, introduced by Barabási and Albert,15 many other models were developed.14, 16–18 Here, we build the networks by using an algorithm used with great success in our previous works.19–21 The advantage of this growth procedure is that the construction of an SFN never stops by itself because the minimum allowed degree is 2. This means that the newly added nodes have at least one branch to grow. Now, for studying polymers with complex architectures the method of generalized Gaussian structures (GGS), which extends the well-known Rouse model22–24 for linear chains to branched topologies, turned out to be very useful. Many theoretical investigations of the GGS model have been devoted to complex polymers, such as dendrimers25, 26 and their dual structures,27, 28 star-based structures,29, 30 regular hyperbranched structures,31, 32 fractals,33, 34 small world networks,35–37 and scale-free networks.20 However, the GGS model has important limitations: does not include the hydrodynamic interactions, the excluded volume interactions, or stiffness. Despite all these restrictions the GGS approach represents a very important step in a theoretical understanding of the dynamics of complex polymer networks. The simplicity of the GGS model often allows theoretical solutions even for a) Electronic mail: [email protected]

0021-9606/2014/141(14)/144902/9/$30.00

very intricate topologies. This great advantage is lost when we consider interactions which the GGS disregards; such additions certainly make the model more realistic, but an analytical solution is improbable. The hydrodynamic interactions, which are solvent-mediated interactions, may be included explicitly by a preaveraged Oseen tensor, the so-called Zimm model.21, 22, 28, 38, 39 Also the stiffness, on which we focus here, can be included in the theory. Semiflexible polymers show an unquenchable, steady interest to them.40–57 The exact focus to semiflexible objects is not surprising, since many biological macromolecules like proteins and DNA have high persistence lengths.58–60 Usually, the semiflexibility is modeled just through fixing the angles between the nearest-neighboring bonds, whereas the orientations between all other bonds are given from the assumption that the bonds rotate “freely.” This approach allows one to gain in a simple way the basic properties related to the semiflexibility of polymers and therefore it is widespread in the theory.40–56 Moreover, recently the method enabled to put forward the theory of semiflexible treelike polymers (STP) with very general architectures.48–51, 56 The important aspect for the STP-model is the use of a discrete picture, in which a macromolecule is represented through Gaussian-distributed bonds connected to each other in such a way that the structure does not have any loops. In this article we extend the study of tree-like semiflexible structures in the framework of the STP model to scale-free polymer networks. For this choice it is possible to monitor a transition from predominant star-like networks (low values of γ ) to linear-like networks (high γ s), where γ is a parameter which controls the connectivity of the SFNs. Finally, we recall that the hyperbranched polymers fall into different universality classes, depending whether their dynamical characteristics scale or do not scale in the intermediate frequency or time domains.52, 53, 55, 56 As we proceed to show here the semiflexible SFNs belong for many values of

141, 144902-1

© 2014 AIP Publishing LLC

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γ to the second class: Apart from the extreme values (related to linear chain), we find only the value γ = 2.0 which leads to a scaling behavior. The article is structured as follows: In Sec. II we recall the basic features of the STP method for arbitrary semiflexible branched (treelike) polymers. In Sec. III we briefly describe the algorithm used to construct the polymer networks with a scale-free topology. In Sec. IV we study the relaxation patterns of the polymer networks modeled in Sec. III. Here, we focus on the following aspects: the influence of the network’s size on polymer dynamics, the role of the stiffness parameter, and we also monitor the influence of γ on the dynamics. The article ends with the conclusions.

a and c, one may set as in the freely-rotating chain40 model da · dc  = da · db db · db  . . . db · dc l −2k , where (b1 , 1 1 2 k b2 , . . . , bk − 1 , bk ) is the shortest path that connects a with c, which for treelike polymers is unique.62 Under the choice of da · db  described above, the matrix W is sparse and can be expressed analytically.49 Now, the set of Langevin equations, Eq. (1), contains the potential VST P ({rk }) which depend on the beads’ positions. The transformation from the bonds’ to the positions’ variables, da = ri − rj , can be written in terms of the incidence matrix G,62  da ≡ (GT )ak rk . (4) k

Here G denotes transpose of G = (Gia ), whose elements are Gja = −1 and Gia = 1, when the bond a connects the beads i and j, and zero otherwise. Given that any treelike network consisting of N beads has (N − 1) bonds, G is a rectangular N × (N − 1) matrix. Substitution of Eq. (4) into Eq. (2) leads to K (GWGT )kn rk · rn . (5) VST P ({ri }) = 2 k,n T

II. THE THEORETICAL MODEL

Here we recall briefly the model of STP of Refs. 48 and 49 which we then apply to SFNs in the following. Let the polymer network be represented by beads located at ri (i = 1, . . . , N) connected by springs (bonds), say da = ri − rj . The dynamics of such a network is described through a set of Langevin equations. For the x-component of the position vector ri = {xi , yi , zi } the Langevin equation is given by24, 49 ∂ ∂ ζ xi (t) + V ({r }) = f˜i (t). ∂t ∂xi ST P k

(1)

Here f˜i is the x component of the usual Gaussian force acting on ith bead, for which f˜i (t) = 0 and f˜i (t)f˜j (t  ) = 2kB T ζ δij δ(t − t  ) hold (ζ is the friction coefficient and T is the temperature). Moreover, the structure of the network is described through the potential VST P ({rk }), which accounts both for the connection of beads in the network and for the network’s semiflexibility. In order to determine the potential VST P one starts with the bond picture in which one has49 K VST P ({da }) = W d ·d . (2) 2 a,b ab a b In Eq. (2) the constant K = 3kB T/l2 and the matrix W = (Wab ) is determined through the bond-bond correlations {da · db }. Indeed, the evaluation of da · db  with respect to the Boltzmann distribution exp(−VST P /kB T ) and under assumption that the {da } are Gaussian-distributed gives da · db  = l 2 (W−1 )ab .

(3) −1

Defining now da · db  allows to compute the matrix W , which after inversion specifies all Wab in Eq. (2). Following the traditional choice of Refs. 40–45 and 48–56 we have first for the mean-squared lengths da · da  = l 2 . Second, for adjacent oriented bonds, say a and b, which are connected by a bead, say i, we take da · db  = ±l 2 qi . Here the parameter qi reflects the stiffness of the junction i. The plus sign holds for a head to tail configuration of the oriented bonds a and b and the minus sign otherwise. In three dimensions, the qi is bounded by qi < 1/(fi − 1),61 where fi is the functionality of the ith bead. On the other hand, the value qi = 0 corresponds to the fully flexible case. Finally, for nonadjacent bonds, say

P Let us define the matrix AST P = (AST ij ) through

AST P = GWGT .

(6)

With Eqs. (5) and (6), Eq. (1) is given by  ∂ 1 ˜ P f (t), AST xi (t) + σ ij xj (t) = ∂t ζ i j =1 N

(7)

where we have set σ = K/ζ . P For STP, the matrix AST P = (AST ij ) is known in a closed 49 analytical form. Let us consider, say, bead i, to which we correspond the stiffness parameter qi . Moreover, bead i has fi nearest neighbors ik (k = 1, . . . , fi ) with corresponding functionalities fi and stiffness parameters qi . Thus, any bead k k ik has as nearest neighbors the bead i and fi − 1 beads iks k (s = 1, . . . , (fi − 1)). In these notations the diagonal elek ments of AST P are given by P AST ii

(fi − 1)qi2  fi k k = + , 2 1 − (fi − 1)qi 1 − (f − 2)q − (f i i i − 1)qi i k

k

the elements of AST P are P AST =− ii k

k

k

k

(8) related to the nearest-neighboring beads 1 − (fi − 1)(fi − 1)qi qi k

k

(1 − (fi − 1)qi )(1 − (fi − 1)qi ) k

and those of the next-nearest-neighboring beads are qi k P AST . iiks = 1 − (fi − 2)qi − (fi − 1)qi2 k

k

k

,

(9)

k

(10)

k

All other elements of AST P vanish. We note that Eqs. (8)–(10) are also correct when some of beads i, ik , or iks are peripheral, i.e., of functionality f = 1. In such a case, due to the fact that corresponding (f − 1) factor vanishes, the final expression simplifies, so that with the

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J. Chem. Phys. 141, 144902 (2014)

peripheral beads no stiffness parameter is associated. Indeed, if, say, bead i has functionality fi = 1, then the corresponding diagonal element for a network with N > 2 from Eq. (8) yields P =1+ AST ii

(fi − 1)qi2 1

1

1 − (fi − 2)qi − (fi − 1

1

1

1)qi2 1

.

(11)

The off-diagonal element related to the bead i and to its nearest-neighboring bead i1 follows from Eq. (9), P AST =− ii k

1 , (1 − (fi − 1)qi ) 1

(12)

1

and those related to the next-nearest-neighboring beads of i remain as in Eq. (10), where k is set to k = 1. Thus, if bead i is peripheral, there is no stiffness parameter (such as qi ) associated to it, see Eqs. (10)–(12). The same conclusion holds if i is nonperipheral (fi > 1), but m ≤ fi of its nearest neighboring beads, say is with s = fi − m + 1, . . . , fi , are peripheral (fi = 1). In this case Eq. (8) reads s

P AST = ii

fi 1 − (fi − 1)qi fi −m

+

ik =1

k

k

k

(17)

1 . =− (1 − (fi − 1)qi )

N  l2  1 Rg2 = . N k=2 λk

 (14)

Thus, again the peripheral beads {is } do not receive any stiffness parameter (such as qi ). s Here we note that the theory49 allows to model the dynamics of polymers with arbitrary architectures as well as to treat heterogeneous polymers with distinct stiffness parameters at each junction.50, 51 In this work, being interested on the role of the scale-free topology, we focus on the homogeneous stiffness. Hence we assume that the stiffness parameter of any bead i (which exists only for fi ≥ 2) is given by qi = q/(fi − 1), which implies that qi = 0 for q = 0 and qi = 1/(fi − 1) for q = 1. In this way, changes in q allow to reach the flexible and the rigid limits for all beads (junctions) simultaneously41, 42, 48, 56, 61 and we can change stiffness of the polymeric network based on a single stiffness parameter q. Now, the solution of the set of Eq. (7) requires diagonalization of AST P . The ensuing eigenvalues {λi } are fundamental for many important in polymer physics quantities.24 One of them is the mechanical relaxation form, namely, the complex dynamic modulus G*(ω) or, equivalently, its real G (ω) and imaginary G (ω) components (known as the storage and the loss moduli).22, 63 One has namely, N ω2 1  2 N i=2 ω + (2σ λi )2

N 1  r. N i=1 i

(18)

In a Gaussian framework, it has been shown (both in the case of flexible64–67 and of semiflexible polymers51 ) that the radius of gyration obeys

and for is Eq. (9) reads

G (ω) = νkB T

N  1  Rg2 = (r − RC )2 , N i=1 i



(13)

k

ik =is

P AST iis

(16)

In (15) and (16) ν is the number of polymer segments (beads) per unit volume and {λi } are the eigenvalues of the matrix AST P . In these equations the vanishing eigenvalue (λ1 = 0) corresponds to the translation of the system as a whole and it does not contribute to the moduli. We are mainly interested in the intermediate scalings of G (ω) and G (ω), thus we will give our results in terms of reduced storage and loss moduli by setting νkB T/N = 1 and σ = 1 in (15) and (16). While the G (ω) and G (ω) reflect the dynamical behavior of the polymeric system, also the static properties can be determined based on the eigenvalues {λi }.51, 64–67 Here, we choose the radius of gyration, which shows how the beads are positioned relatively to each other and is given by

RC =

1)qi2 k

1 − (fi − 2)qi − (fi − 1)qi2 k

N 2σ ωλi 1  . G (ω) = νkB T N i=2 ω2 + (2σ λi )2 

where the position of the center of mass is (fi −



and

(15)

(19)

III. SCALE-FREE POLYMER NETWORKS

Here we extend the study of scale-free polymer networks by introducing semiflexibility in the framework described in Sec. II. From the wealth of algorithms13, 15–18 which construct an SFN we choose an algorithm used with great success in our recent works.19–21 The functionality f (also named as degree k in theory of networks) of a node represents the number of bonds emanating from it or, equivalently, the number of its nearest neighbors. All the models of SFNs show a power law in the degree distribution pf ∝ f −γ ,

(20)

where p f is the probability that the degree (or functionality) of a node is f and γ is a parameter that measures how densely connected a network can be. There are many ways to postulate that such a degree distribution obeys Eq. (20) for large f. Here, in our algorithm Eq. (20) holds starting only from f = 2; we assume that p1 = 0. The main property of SFNs is the existence of hubs (nodes with high degree), which allows us to focus on such nodes, neglecting the nodes with low degree, i.e., f = 1. The probability that the degree of a node is

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J. Chem. Phys. 141, 144902 (2014)

f (f ≥ 2) is given by f −γ . pf =  ∞ −γ f =2 f

conditions:22 polymers in solutions at θ -temperature or linear polymer melts. (21)

The sum on the denominator is used to keep the total probability equal to 1. More precise, the probability to create a node with degree 1 is zero during the construction process, but when the construction stops the networks will contain nodes with degree 1, namely, the peripheral nodes. In order to construct SFNs with finite sizes we consider a cut-off: the construction stops when it reaches a preset number Nmax of nodes, even if there still exists open nodes, which did not receive a degree from the chosen degree distribution, Eq. (21). However, these open nodes will not alter the shape of the degree distribution of our constructed networks, which for f > 1 obeys the power-law given by Eq. (21). In Fig. 1 we display realizations of SFNs with N = 50 nodes obtained from distribution (21), where γ is varied from 1.5 to 4.0 with the stepsize equal to 0.5, from up to down and left to right. We show by filled circles the sites which received a degree number from a random number generator, while by open circles we show the vertices which did not receive the degree from the distribution (21) until the construction stops. The numbers are given according to the chronological order in which the nodes were created. Now, we give a short explanation of the way an SFN was created. In order to do this we choose the case γ = 1.5, the up left drawing in Fig. 1. We start with vertex 1 and pick its degree randomly according to the distribution (21). In this case the degree turns out to be 11, thus we create eleven new open vertices, labeled 2, 3, . . . , 12. Then we pick at random one of the open vertices (at this moment there are eleven open vertices). It turned out to be the vertex 5 and its degree was obtained from the degree distribution pf , Eq. (21). In this case the chosen degree was 3, meaning that we have to add two new vertices, labeled 13 and 14, since the vertex 5 already has one link: the link with node 1. The procedure is iterated by picking another open vertex randomly. Due to the limited time and available computer memory resources, we restrict the size of every constructed SFN to a preset number of vertices Nmax (in this figure Nmax = 50). When we reach this value we stop the growth by assigning to all remaining open vertices the degree one. From the drawings of Fig. 1 one can notice the influence of γ on the topology of the networks. SFNs with low γ s show a more star-like topology (or coupled stars) and for high γ s a more linear-like topology is encountered.20 To prove this and in order to help the reader in distinguishing the two limiting topologies we depicted in Fig. 1 by blue big circles the nodes with degree higher than 4 and with a continuous red line the longest linear-like path. In general, by increasing the value of γ the number of vertices with a high degree is decreasing while the length of the longest linear path is increasing. Each vertex of the constructed SFN will correspond to a monomer of the GGS and the links between vertices show the interactions between monomers. Here, we consider that the polymers are ideal and behave as phantom polymers: there are no interactions between monomers that are far apart along the chain, even if they approach each other in space. This situation is also encountered for real polymers in some special

IV. RESULTS A. Scale-free networks with fixed γ

First we study the influence of the number of monomers N on the dynamics of semiflexible scale-free polymer networks (SSFPNs). In Fig. 2 we plot the eigenvalue spectrum for SSFPNs with γ = 2.5. In order to study more efficiently the influence of the stiffness we choose relatively high value for the stiffness parameter q, i.e., q = 0.8. Moreover, in Fig. 2 the number of monomers N is varied from N = 100 to N = 4000. For each value of N we took different number of realizations, S, such that the product N · S keeps constant, equal to 106 . Thus, for each value of N we took an average over the same number of eigenvalues. The width of the bins is not kept constant, due to the fact that N is variable. We took the width equal to ξ = (2 + lg N )/50, meaning that for N = 100 the width is ξ ≈ 0.080 and for N = 4000 we have ξ ≈ 0.112. For all Ns, it was observed that the eigenvalue’s distribution, ρ(λ), follows a power-law behavior for low eigenvalues, with a constant slope equal to 0.88 and another power-law behavior, slope −1.62, for high eigenvalues. For 0.2 ≤ λ ≤ 1.0 we observe some features which are related with the value of γ , more precise they start to disappear when γ increases, corresponding to predominant linear-like polymers. However, higher stiffness parameter accentuates these behaviors. For 10 ≤ λ ≤ 20 we notice features that are strictly related to the stiffness parameter, as we will also show in Fig. 4. Remarkably, by comparing the results with scale-free polymer networks in the Rouse model (q = 0.0), shown in Ref. 20, we observe only little differences in the scalings. A similar feature has been also found for regular semiflexible hyperbranched polymers56 and it shows that for branched polymers the global scaling behavior does not depend very much on the local stiffness conditions. Also from Fig. 2, one can notice a clear cut-off in the region of high values of λ, due to the fact that the highest eigenvalue is limited to λ ≈ N, the size of the structures. Now we turn to the mechanical relaxation forms, Eqs. (15) and (16). We note that both relaxation forms, G (ω) and G (ω), show a similar behavior in the intermediate frequency domain,24 therefore we focus here only on the loss modulus G (ω). In Fig. 3 we plot in a double logarithmic scale the G (ω), Eq. (16), for SSFPNs with γ = 2.5 and variable number of monomers N. Here the parameter N is equal to the values 100, 200, 500, 1000, and 4000, while the number of realizations S equals 10000, 5000, 2000, 1000, and 250, respectively. In this way the product N · S keeps the same for all the curves. The stiffness parameter is equal to q = 0.8. Here, we plot the results for semiflexible polymers, shown by continuous lines in the figure, and also the results for flexible SFPNs in the Rouse model (dashed lines), for the same N, S, and γ , but with q = 0.0. For all the curves we recover the expected behaviors: for very low frequencies a power-law with exponent equal to 1 and for very high frequencies another power-law with exponent −1. However, the curves for

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44

16

9 8 44 43 42 4140 39 4645 38 47 37 36 48 35 49 34 25 50 33 24 32 20 19 21 23 31 18 30 22 17 8 7 9 16 10 6

2

48 47

6 46

10 11

12

13 14

19 20 21 22 23 24 25 26

38 39 40 4142 43

28 29

15

26

7

49 50

2

3

14

17 5

18 27

12

4

13

4

1

11 1

5

3

3132 30 2728 29

37 36 35 34 33

45

47

33 36

33 32

31 32

37 11

15 16

38 22

6

15

14 35 34

9

3 5

40 41 42 43 44

29 19 20

2

13

28

21 22 23

10

1 8 7

12

48

5

24 34

19

37 30 29 28

35

49

1 25

18

21 20 44 43 42 41

27 26

24 25 26

31 30

50 48 49

44

41

46

41

36

39 38 37 48 49

38 39 40

46

27

47

2

14

1112 13

45

3

4

9 10

45

47 46

36

23

8

7 6

50 4

17

39

16 17 18

40

29

28 17

33 34

32 31 30

45 29

33

40

26

15

26 27

25

24

25

15 5

9

18 5

4

42 43 13

20

21

3 2

46 19

38 37

8

34 2 3

36 35

1

42

44

39 47

31 43

16

49

7 6

4

14

21 13

6 7

30

17

8

10

45 19 20

11 12

32

28

50

18 23

16

27

22

1 9

22 48

10 11

12 24

35 50

23

14

FIG. 1. Examples of scale-free networks with the same size, N = 50, and γ variable: 1.5 and 2.0 in the upper row, 2.5 and 3.0 in the middle row, and 3.5 and 4.0 in the lower row.

semiflexible SFPNs are considerably broader than those of the flexible SFPNs, which is related, as we proceed to show, to a broader eigenvalue spectrum. Such a broadening leads to a pseudo-gap in the spectrum, so that one can also notice in

the loss moduli of SSFPNs a local minimum around ω ≈ 10. Turning now to the role of the monomers number N, in the intermediate region, which is related to the particular geometry of the structure, we do not notice big differences between

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Galiceanu, Reis, and Dolgushev

J. Chem. Phys. 141, 144902 (2014)

1

1000 N=100 S=10000 200 5000 500 2000 1000 1000 4000 250

0.1 .88

100 10

s

1

λi

0.01

p slo

ρ(λ)

=0

e lop

q=0.0 0.4 0.8

e=

100

6 -1.

0.1

2

λi

144902-6

0.001

0.01

0.01

0.0001 0.01

0.1

1

λ

10

100

1000

FIG. 2. Eigenvalues spectrum for semiflexible scale-free polymer networks of various sizes, but the same parameter γ , equal to 2.5. Here the parameter of stiffness is q = 0.8.

SSFPNs of different sizes. Thus, we conclude that for the relaxation dynamics of the SSFPNs with the same value of γ the size of the network does not influence too much. From Fig. 3 one can easily notice a very different behavior between polymers without stiffness (Rouse model), q = 0, and polymers with a high stiffness value, q = 0.8. Thus, it is very important to study the influence of the stiffness strength, q, on the dynamics of polymers. First we will check the influence of q on the eigenvalue spectra. In Fig. 4 we present the eigenvalue spectra for SSFPNs with N = 4000, S = 250, and γ = 2.5 as a function of the stiffness parameter. For each curves we show the eigenvalues in ascending order. Given that the difference between the smallest and the largest eigenvalues is very high, we plot the results in semilogarithmic scales. With growing stiffness the largest eigenvalues increase, while the small eigenvalues decrease a little bit. The eigenvalue with the highest degeneracy, λ = 1, encountered for the case without stiffness q = 0 (Rouse model) will be also recovered with a value approximately equal to 1 when we switch on the stiffness. This eigenvalue is situated in the intermediate part of the spectra and it occupies more than half of the spectrum. Moreover, in the inset of Fig. 4 we present the eigenvalue spectrum of a semiflexible dendrimer with generation number G = 7

dendrimer G=7 f=4

0.0001 0

0.001 0.0001 0

1

1000

1000 2000 3000 4000

N

2000

3000

N

4000

FIG. 4. Spectra for semiflexible scale-free polymer networks with N = 4000, γ = 2.5 and three different degrees of stiffness: q = 0.0, 0.4, and 0.8.

and functionality f = 4, given a total of N = 4373 monomers, while the stiffness parameter equals q = 3qdend = 0.8. Here, one can notice a similar behavior as for the curve of SSFPNs with the same q, with the difference that the dendrimer’s spectrum is clearly more degenerate (visible by the size of steps in the eigenvalue spectrum) and the most degenerate eigenvalue equals 3/3.8 ≈ 0.79. One can also notice a gap between this eigenvalue and the larger eigenvalues. This is related to the fact that for dendrimers with growing stiffness the eigenvalues related to the large scale eigenmotions (spatially exponential) become higher (since stiffer structures are larger) and the eigenvalues related to the small-scale ones (spatially periodic) become smaller.55 Now we turn our attention to the relaxation dynamics of the polymers and we determine the influence of q on the loss modulus. We display in Fig. 5 the loss modulus, Eq. (16), for SSFPNs with the same size, N = 4000, the same γ , equal to 2.5, but variable q. We choose three values of q: 0.0 (no stiffness), 0.4, and 0.8. For all the cases we consider an average over S = 250 realizations. Immediately apparent are the limiting cases: for very low frequencies we get a ω1 behavior and for very high frequencies we obtain a ω−1 behavior. We encounter the most interesting situation in the intermediate frequencies’ region. In this case we found out that the loss modulus starts to loose its power-law behavior

4

3

-2 -4

N=100 S=10000 200 5000 500 2000 1000 1000 4000 250

-2

0

log10ω

2

2

1 α"

0

log10G"

log10G"

2

q =0.0 0.4 0.8

0

1 -1 -6 -4 -2 0

4

FIG. 3. The loss modulus for SSFPN of variable sizes N, ranging from N = 100 to 4000, from below. The parameter γ is 2.5 and q = 0.8. The dashed lines show the results for flexible polymers: q = 0.0.

log10ω

0 -4

-2

0

log10ω

2

4 2

4

FIG. 5. Loss modulus for SSFPNs of fixed N = 4000 and γ = 2.5. Here the stiffness degree is varied: q = 0.0, 0.4, and 0.8.

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Galiceanu, Reis, and Dolgushev

J. Chem. Phys. 141, 144902 (2014)

B. Scale-free networks with variable γ

Now we study the influence of γ on the dynamics of semiflexible SFPNs. We fixed the size of the networks to N = 4000 and we varied γ from 1.5 (more star-like structures) to 4.0 (more linear-like structures). For a better comparison we plot the results for the loss modulus, G , and additionally for the radius of gyration. First we study the influence of γ on the eigenvalue spectra, for SSFPNs with the same size, N = 4000, each with S = 250 realizations, and the same stiffness value, q = 0.8. In Fig. 6 we display the eigenvalues in ascending order for SSFPNs with γ variable from 1.5 to 4.0, with step size equal to 0.5, as a function of the stiffness parameter. We plot the results in semilogarithmic scales because the difference between the smallest and the largest eigenvalues is very high. We notice that for γ = 1.5 the eigenvalue λ ≈ 1 is highly degenerate and by increasing the γ value its degeneracy decreases. The degeneracy of this eigenvalue is strongly related with the star-coupled topology of the network, more pronounced for small γ s. In fact, for a perfect semiflexible star consisting of N beads, there are only three distinct eigenvalues: two nondegenerate (0 and N(1 − q)−1 ) and one (N − 2)-fold degenerate ([1 + q/(N − 2)]−1 ≈ 1).55, 56 For high γ s, which corresponds 1000 100 10

λi

1 γ=1.5 2.0 2.5 3.0 3.5 4.0

0.1 0.01 0.001 0.0001 0

1000

2000

N

3000

4000

FIG. 6. Spectra for SSFPNs with N = 4000 and q = 0.8, while γ is variable.

3 2

γ=1.5 2.0 2.5 3.0 3.5 4.0 star linear

1

1 0.5 "

0

α

observed in the Rouse case (q = 0.0) by increasing the value of q. The loss modulus gets more wide towards the region of large frequencies and presents a small local minimum only for high values of stiffness parameter. It is worth to mention that this minimum is not as sharp as for the dendritic semiflexible polymers55 or dendrimers45, 48 since the (pseudo)-gap in the region of high eigenvalues observed for these structures is not so pronounced for SSFPNs. To have a better insight in the intermediate domain, we plot as inset graph the derivad(log G ) tive α  = d(log10 ω) for the three curves. Here, we notice a 10 small decrease of the slope in the intermediate frequency region (until ω ≈ 1) by switching on the stiffness parameter. Also a small peak around ω ≈ 10 appears for high qs (here q = 0.8), corresponding to the local minimum above mentioned. Although not shown here, it is worth to stress that the same qualitative behavior was observed also for the storage modulus.

log10G"

144902-7

0

-0.5

-1 -2 -6

-1 -6

-4

-2

-4

-2

0

0

log10ω

log10ω

2

4

2

4

FIG. 7. Loss modulus for SSFPNs with N = 4000, q = 0.8, and γ ranging from 1.5 to 4.0.

to a more linear-like topology, its degeneracy decreases, making the spectrum more continuous. It is worth to say that the same qualitative behavior was observed for all studied qs. Now, with these eigenvalues we are able to calculate the loss modulus. In Fig. 7 we plot in double-logarithmic scale G (ω), Eq. (16), for SSFPNs with N = 4000, S = 250, q = 0.8, and γ is variable. One can clearly notice the limiting cases: a ω1 behavior for very low frequencies and a ω−1 behavior for very high frequencies. In the intermediate frequencies’ region the topology of the network will come into play. Having this in mind we plot the results for the two limiting cases: a pure star topology, more precise a core with N − 1 arms, shown by square symbols, and a linear polymer chain, shown by circles. We note that for many values of γ there is no intermediate scaling observable, in line with the experiments on hyperbranched polymers in dilute solutions.68 For our choice of parameters we encountered a broad power-law behavior for SSFPNs with γ = 2.0 (G (ω)∝ω0.82 ), differently than the value of γ = 2.25 observed for SFPNs in the Rouse case (q = 0.0), see Ref. 20 for more details. This aspect bed(log G ) comes more evident by plotting the derivative α  = d(log10 ω) , 10 see the inset graph. In this graph one can notice a pronounced local maximum around log10 ω ≈ 4 for SSFPNs with γ = 1.5, the magnitude of this peak being higher for networks with more star-like segments. The highest magnitude is encountered for an N − 1 arms star polymer, depicted by squares in Fig. 7. Additionally, for higher γ s the size of this peak diminishes and it appears for smaller frequencies, in the region between ω = 1 and ω ≈ 50. The value of α  corresponding to this maximum approaches the value of α  ≈ 1/4, which was encountered for semiflexible linear chains48 and SSFPNs with high γ s. This is clearly related to an increase of linear segments in the networks. However, the existence of this local maximum is related with the increase of the stiffness parameter since it was not observed for small values of q and it was explained in detail in the text corresponding to Fig. 5: loss modulus for SSFPNs with γ = 2.5. In Fig. 8 we display in double-logarithmic scale the normalized radius of gyration, Rg2 / l 2 , Eq. (19), for SSFPNs with variable N and S in such a way the product N · S = 106 ,

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144902-8

Galiceanu, Reis, and Dolgushev q=0.8 γ=1.5 2.5 3.5 linear star q=0.0 γ=1.5 2.5 3.5

4

3

2

2

log10(/l )

J. Chem. Phys. 141, 144902 (2014)

2

1

0 1

2

3

log10N FIG. 8. Radius of gyration for SSFPNs with q = 0.8 and γ variable (lines with symbols). The results are compared to those of fully flexible SFPNs (lines).

q = 0.8 and γ equal to 1.5 (circle symbols), 2.5 (diamond symbols), and 3.5 (triangle left). For a better visualization of the transition between the two limiting cases, namely, a pure star topology and a linear polymer we show also these results by star symbols and square symbols, respectively. In order to check the influence of the stiffness parameter q we also plot the result for flexible scale-free polymer networks (q = 0.0), corresponding to same set of parameters: N, S, and γ = 1.5, shown by red dashed lines, 2.5 with dotted-dashed lines, and 3.5 with dotted lines. We know from the literature69 that the radius of gyration for ideal linear polymers is Rg2 / l 2 = N/6 N (3 − f2 ), and for an ideal f-arms star polymer is Rg2 / l 2 = 6f which for our case (an N − 1 arms star) and in the limit of very large N equals 1/2. The values of the radi of gyration for SSFPNs in the region of high N-values are situated between the above mentioned two limiting cases: an N-dependence for very high γ s and a constant for very low γ s. For q = 0.8 we observed the following behaviors: N0.015 for SSFPNs with γ = 1.5; N0.15 for SSFPNs with γ = 2.5, and N0.22 for SSFPNs with γ = 3.5. For comparison, the radi of gyration for flexible SFPNs (q = 0.0) we encountered the following scaling exponents: N0.04 for γ = 1.5; N0.18 for γ = 2.5, and N0.22 for γ = 3.5. As it is evident from the figure, the SSFPNs have larger Rg2  than their flexible counterparts. However, for the small value of γ , γ = 1.5, this difference is rather small for high N. This feature can be traced back to the eigenvalue λ ≈ 1 which dominates the behavior of SFPNs (both flexible and semiflexible) in case of small γ related to the topologies close to a star-graph. On the other hand, the gyration radii of flexible and of semiflexible SFPNs with γ ≥ 2.5 are very distinct in the whole range of N. V. CONCLUSIONS

In this work we have studied the influence of semiflexibility on the dynamics of scale-free networks. These networks were constructed by using an algorithm for which its degree distribution starts with degree 2. In this way the construction of an SFN never stops by itself: there are no nodes with degree 1, except the peripheral nodes, which are still open when the growth finishes.

The stiffness has been taken into account through correlations between bonds, modeled by assuming freely rotating segments. In this framework one has a set of Langevin equations for the beads dynamics, which can be written in the analytically closed form. As in the case of totally flexible generalized Gaussian structures, the eigenvalue spectrum of the corresponding dynamical matrices of semiflexible polymers is crucial for determining many static and dynamic physical quantities. From the wealth of applications related to polymer physics we showed the eigenvalue spectrum, the results for the loss modulus and the radius of gyration. First we have studied the influence of network size and of the stiffness parameter on the chosen quantities. In the second part of this article we have investigated the influence that γ , a parameter which controls the topology of SFNs, has on dynamics. We have noticed clear differences between semiflexible scalefree polymer networks with the same size and γ , but different stiffness parameter. A smooth decrease of the slope was observed in the intermediate frequency region. We have also monitored the transition from a predominant star-like polymers, obtained for low values of γ , to a linear-like polymers, with high γ s. This was done by keeping constant the size N and the stiffness parameter q of SSFPNs, but varying γ . We have noticed a mild crossover between two limiting behaviors: a star polymer with N − 1 arms and a pure linear chain. For some values of the parameter set (q = 0.8, γ = 2.0) we have encountered a region with board constant slope in the intermediate frequency domain. We expect these findings to be useful in distinguishing between semiflexible polymer networks with complex (random) topologies which display linear-like or/and star-like segments.

ACKNOWLEDGMENTS

The authors are grateful for fruitful discussions with Professor Alexander Blumen and with Dr. Denis A. Markelov. M.G. acknowledges the financial support of the international cooperation program PROBRAL - DAAD/CAPES. M.D. acknowledges the support of the Deutsche Forschungsgemeinschaft through Grant No. Bl 142/11-1 and through IRTG “Soft Matter Science” (GRK 1642/1), of the DAAD through the PROCOPE program (Project No. 55853833), and of the Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Program SPIDER (PIRSES-GA-2011-295302). 1 H.

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Galiceanu, Reis, and Dolgushev

J. Chem. Phys. 141, 144902 (2014)

9 M.

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