Computer Physics Communications 182 (2011) 1949–1953

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Computer Physics Communications

www.elsevier.com/locate/cpc

The structure factor of dense two-dimensional polymer solutions

H. Meyer ∗, N. Schulmann, J.E. Zabel, J.P. Wittmer

Institut Charles Sadron, CNRS UPR22, 23 rue du Loess, BP 84047, 67034 Strasbourg Cedex 2, France

a r t i c l e i n f o a b s t r a c t

Article history:Received 29 July 2010Received in revised form 25 November 2010Accepted 2 December 2010Available online 4 December 2010

Keywords:FractalsMacromolecular and polymers solutionsPolymer melts

According to the generalized Porod law the intramolecular structure factor F (q) of compact objects withsurface dimension ds scales as F (q)/N ≈ 1/(R(N)q)2d−ds in the intermediate range of the wave vector qwith d being the dimension of the embedding space, N the mass of the objects and R(N) ∼ N1/d theirtypical size. By means of molecular-dynamics simulations of a bead-spring model with chain lengthsup to N = 2048 it is shown that dense self-avoiding polymers in strictly two dimensions (d = 2) adoptcompact configurations of surface dimension ds = 5/4. In agreement with the generalized Porod law theKratky representation of F (q) thus reveals a nonmonotonous behavior with q2 F (q) ∼ 1/(N1/2q)3/4. Usinga similar data analysis we argue briefly that melts of non-concatenated rings in three dimensions becomemarginally compact with ds = d = 3, i.e. q2 F (q) ∼ N0/q, for asymptotically long chains.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

1.1. The generalized Porod law

Conformational properties of polymers or polymer-like aggre-gates can be determined experimentally by means of light, smallangle X-ray or neutron scattering experiments [1]. Using appro-priate labeling techniques this allows to extract the coherent in-tramolecular structure factor F (q) = 1

N 〈‖∑Nn=1 exp(iq · rn)‖2〉 [1,2].

Here N stands for the number of monomers of a chain, rn for theposition of monomer n and q for the wave vector. The average 〈. . .〉is sampled over the ensemble of thermalized chains. The struc-ture factor is of interest, of course, since it allows to compare realexperiments with theoretical predictions and numerical data. Forsmall wave vectors, in the so-called Guinier regime, the structurefactor scales quite generally as [1,2]

F (q)/N = 1 − Q 2/d for Q � 1 (1)

with Q ≡ qRg(N) being the reduced wave vector, Rg2(N) ≡

12N2

∑Nn,m=1〈(rn − rm)2〉 the squared radius of gyration charac-

terizing the typical chain size and d the spatial dimension. Ifsufficiently small q-vectors are available the gyration radius canthus in principle be determined experimentally from the structurefactor of labeled chains.1 We also remind that the inverse fractaldimension ν of a chain is defined by the chain length dependenceof the typical chain size, Rg(N) ∼ Nν , in the limit of asymptotically

* Corresponding author.E-mail address: hendrik.meyer@ics-cnrs.unistra.fr (H. Meyer).

1 The experimental challenge is here to cover a sufficiently small q-region to al-low the determination of the radius of gyration.

0010-4655/$ – see front matter © 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.cpc.2010.12.003

long chains [2,3]. For dilute self-avoiding walks in d = 2 dimen-sions ν is known to take the value ν0 = 3/4 [2]. For “open” chainswith 1/ν < d the fractal dimension determines the structure factorin the intermediate wave vector regime,

F (q) ∼ N0q−1/ν for1

Rg(N)� q � 1

σ, (2)

with σ being a monomer scale [1,2].2 The “Kratky representation”of the structure factor, q2 F (q) vs. q, thus corresponds to a plateaufor strictly Gaussian chains with 1/ν = 2 [1].

Obviously, Eq. (2) does not hold any more if the chain becomescompact (1/ν → d), i.e., if Porod-like scattering due to the composi-tion fluctuation at a well-defined surface S(N) becomes possible.3

The surface dimension ds is defined by the asymptotic scaling [3],

S(N) ∼ Rg(N)ds ∼ Ndsν = N1−νθ , (3)

where we have introduced the exponent θ ≡ 1/ν − ds � 0 to markthe difference between the fractal dimension and the surface di-mension. Obviously, for open chains S(N) ∼ N , hence θ = 0. Sincethe scattering intensity N F (q) of compact objects is known to beproportional to their surface S(N) and since F (q) must match theGuinier limit, Eq. (1), for Q ≈ 1 it follows for asymptotically longchains that N F (q) = N2 f (Q ) ∼ S(N) with f (Q ) being a univer-

2 For simplicity of the presentation we assume here that no additional lengthscale, such as the blob size ξ(ρ) of semidilute solutions [2], is relevant. Obviously,there may be more than one power-law regime if, say, Rg(N) � ξ(ρ) � σ as is thecase for the semidilute 2D solutions presented in Fig. 4(b).

3 The “surface” of compact 2D objects is often called “perimeter”.

1950 H. Meyer et al. / Computer Physics Communications 182 (2011) 1949–1953

sal function. Using standard power-law scaling [2] this implies the“generalized Porod law” [1,4,5]

F (q)/N = f (Q ) ≈ 1/Q 2/ν−ds = 1/Q 1/ν+θ (4)

for the intermediate wave vector regime.4,5 As one expects, Eq. (4)yields for a smooth surface (1/ν = d, ds = d−1, θ = 1) the classicalPorod scattering

F (q) ∼ N/(N1/2q

)3(5)

in d = 2 dimensions [1]. Interestingly, Eq. (4) implies that F (q) de-pends in general on the chain length.

1.2. Specific system classes considered

In our research group in Strasbourg we are currently investigat-ing numerically two classes of dense polymer solutions where thechains become compact due to topological constraints and Eq. (4)should be relevant:

• two-dimensional (2D) self-avoiding walks without monomeroverlap and chain crossings [6–14];

• unconcatenated and unknotted rings in d = 3 dimensions[15–21].

Surprisingly, the fact that for compact objects Eq. (2) should notbe used [9,7,10] or should be used with care [18,20] appears tohave been overlooked in the recent literature. For this reason wesummarized above the scaling relations leading to Eq. (4), well-known to the polymer community at the end of the last century.Interestingly, assuming sufficiently long chains the structure factorof both system classes is given by [13]

F (q) ≈ 1

N

N∫

0

ds 2(N − s)G(q, s) (6)

using the Fourier transform G(q, s) of the normalized size distribu-tion G(r, s) of a subchain of arc-length s � N . In general, the sizedistribution scales as [2,6]

Rd(s)G(r, s) = xθ2 fc(x) (7)

with R2(s) = ∫drd r2G(r, s) being the second moment of the dis-

tribution, θ2 the “contact exponent”, x = r/R the scaling variableand fc(x) a cutoff function with fc(x) = const for x � 1 [22]. Com-paring the power-law exponents of Eqs. (4) and (6) yields readily[13]

θ!= θ2, hence, ds = 1/ν − θ2. (8)

In the remainder of this paper, we focus on the first system classfor which the contact exponent θ2 = 3/4 has been predicted byDuplantier [6] and the fractal surface dimension is thus given byds = 5/4. We comment briefly on the second system class at theend of the paper.

2. Dense two-dimensional self-avoiding walks

2.1. Computational issues

As in previous numerical work on dense strictly 2D self-avoiding walks [12–14] we use the Kremer–Grest bead-spring

4 See footnote 2.5 Additional logarithmic corrections to the leading power-law behavior are not

ruled out.

Fig. 1. Snapshot of a typical chain out of a large system of self-avoiding walks oflength N = 2048 and monomer number density ρ = 0.875 in strictly two dimen-sions (d = 2). The model parameters chosen do not allow chain crossing. Due tothis topological constraint the chains are expected to become compact [2] as clearlyrevealed by the presented chain. The compactness implies Porod-like scattering dueto the composition fluctuation at the well-defined chain contour [1]. However, com-pactness does apparently not imply a finite line tension of the contour which wouldminimize the perimeter length S(N). It will be shown here that the contour is infact fractal with a surface dimension ds = 5/4 > 1.

model [23]. Although we treat only static properties in this pa-per, molecular dynamics was used to be able to analyse also thedynamics [14]. For computational details see Ref. [13]. Lennard-Jones units are employed throughout this paper. We focus onmonodisperse systems at temperature T = 1 with chain lengthsranging up to N = 2048 and number densities up to ρ = 0.875.The largest system computed corresponds to 96 chains of lengthN = 2048, i.e. 196 608 monomers, contained in a square simula-tion box of linear length L = 474.02. As obvious from the snapshotgiven in Fig. 1 of one typical chain taken from this large systemthe chains adopt compact configurations. However, compactnessdoes apparently not imply a disk-like shape which would mini-mize the chain perimeter S(N).6 This can be better seen from thesnapshot in Fig. 2 where we only represent the contour monomersof the chains interacting at least with one monomer from anotherchain [13].

2.2. Mean subchain size and perimeter length

The main part of Fig. 2 presents the typical size and perimeterlength of subchains between the monomers n and m = n + s − 1as indicated by the sketch. We average over all pairs (n,m) pos-sible in a chain of length N . Open symbols refer to subchains oflength s � N of chains of length N = 2048, full symbols to totalchain properties (s = N). The subchain size may by characterizedby its radius of gyration Rg(s). In agreement with various experi-mental and numerical studies reviewed in [13], the presented dataconfirms that the chains are compact, i.e. ν = 1/2 (solid line), onall scales s. The same scaling is also found if the typical end-to-endvector of the subchain is measured [12,13].

A perimeter monomer of a subchain is defined as a monomerbeing within a (slightly arbitrary) distance 1.2 to a monomer notbelonging to the same chain segment [13] just as in the snap-shot given in the top inset. The mean number S(s) of these con-tour monomers measures the typical perimeter of the (sub)chain.As predicted by Eqs. (3) and (8), the perimeter increases with apower-law exponent 1 − νθ = 5/8 (dashed line). Note that this re-sult holds for arbitrary segment lengths provided that the subchain

6 See footnote 3.

H. Meyer et al. / Computer Physics Communications 182 (2011) 1949–1953 1951

Fig. 2. A snapshot of a melt configuration can be seen in the left panel only pre-senting the perimeter monomers interacting with other chains. The numbers referto an arbitrary chain index used for computational purposes. The “chain 1” is thesame one already presented in Fig. 1. The main figure presents the radius of gy-ration Rg(s) and the perimeter length S(s) of subchains containing s = m − n + 1monomers as indicated by the sketch on the right. The full symbols refer to over-all chain properties (s = N). The solid line confirms the exponent ν = 1/2 for the(sub)chain size and the dashed line the exponent θ = 3/4 suggested by Eq. (8).

Fig. 3. Intramolecular structure factor F (q) as a function of wave vector q forρ = 0.875 and for different chain lengths N as indicated. F (q) becomes chain lengthindependent only for wave vectors corresponding to the monomer scale (“Braggpeak”). The intermediate wave vector regime is compared to three power-law expo-nents −2, −11/4 and −3 indicated by the thin, the bold dashed line and the dash-dotted line, respectively. The first exponent corresponds to Eq. (2) with 1/ν = 2,the second exponent to Eq. (4) with ds = 5/4 and the last exponent to Eq. (5) fords = 1.

is sufficiently large (s > 50), i.e. the fractal structure is self-similaron all scales.

2.3. Scaling of structure factor

Fig. 3 presents the unscaled structure factors F (q) as a func-tion of the wave vector q for one density, ρ = 0.875, and for abroad range of chain lengths N as indicated. As one expects, F (q)

becomes constant for very small wave vectors (F (q) → N), de-creases in an intermediate wave vector regime and shows finallythe non-universal monomer structure for large q comparable tothe inverse monomeric size (“Bragg peak”). The first striking resultof this plot is that F (q) does not become chain length independentin the intermediate wave vector regime as it does for linear chains

Fig. 4. Kratky representation of the structure factor F (q) tracing y = (F (q)/N)Q 2 asa function of the reduced wave vector Q = qRg(N) using the measured radius ofgyration Rg(N): (a) the same data and symbols as in Fig. 3, (b) different densities ρas indicated in the panel for N = 2048. The Debye formula (thin lines) correspondsto a plateau for Q � 1. At variance to this a strong nonmonotonous behavior isrevealed by our data which approaches with increasing N or ρ a power law expo-nent −θ = −3/4 (dashed lines). The Fourier transform of the Redner–des Cloizeauxapproximation is shown by the solid lines. The classical Porod scattering, Eq. (5), isindicated by the dash-dotted line in the first panel. The thin dashed line in panel (b)corresponds to Eq. (2) for dilute self-avoiding walks with ν0 = 3/4.

in three dimension [2]. The second observation to be made is thatwith increasing chain length the decay becomes stronger than thepower-law exponent −2 indicated by the thin line correspondingto Gaussian chain statistics. Since dense 2D chains are not Gaus-sian but compact (Fig. 2), this is to be expected according to thegeneralized Porod law, Eq. (4), stated above.

Motivated by Eq. (4) we have plotted in Fig. 4 the struc-ture factor using a Kratky representation with vertical axis y =(F (q)/N)Q 2 and reduced wave vector Q = qRg(N). The Debyeformula for Gaussian chains [2] is given by the thin line whichbecomes constant in this representation for Q � 1 in agreementwith Eq. (2). Using the measured gyration radius Rg(N) this scalingobviously allows to collapse all data in the Guinier regime, Eq. (1).The observed collapse is, however, much broader in Q increas-ing systematically with chain length N and density ρ as shownin panels (a) and (b), respectively. Note that the striking decay ofy(Q ) observed for our data implies the chain length dependenceseen for the unscaled structure factor in Fig. 3. The deviationsfrom the Q -scaling observed for large wave vectors are due tothe (chain length independent) physics on scales corresponding tothe monomer size already seen in Fig. 3. Interestingly, within theso-called “Redner–des Cloizeaux approximation” [22] of the cutofffunction fc(x) defined in Eq. (7) the structure factor can be com-puted exactly from Eq. (6). This yields a lengthy analytic formulagiven in Ref. [13] represented by the solid lines indicated in bothpanels of Fig. 4. For wave vectors corresponding to the power-law

1952 H. Meyer et al. / Computer Physics Communications 182 (2011) 1949–1953

regime of Eq. (4) this formula reduces to the simple power lawy ≈ 1.98/Q θ2 [13] which is indicated by the bold dashed lines.As can be seen in Fig. 4, our data approaches systematically withincreasing chain length and density this power-law slope. Evenlonger chains are obviously warranted to unambiguously show thepredicted asymptotic exponent −3/4 in a computer experiment.

3. Conclusion

3.1. Summary

As emphasized in the Introduction, the intramolecular structurefactor F (q) of compact polymer-like structures without (relevant)surface tension must scale in the intermediate wave vector regimeaccording to the generalized Porod law, Eq. (4). Assuming a suf-ficiently broad range of wave vectors q this allows, in principle,to determine in a real experiment as in a computer simulationthe surface dimension ds [1]. In this paper we have investigatedby means of molecular-dynamics simulations of a standard bead-spring model dense solutions of 2D self-avoiding walks. As ex-pected theoretically [2,6,11], the chains are shown to adopt com-pact configurations with ds = 5/4. The latter exponent is clearlydemonstrated from the directly measured (sub)chain perimeters(Fig. 2) and concurs with the observed nonmonotonous behaviorof the Kratky representation of the structure factor with q2 F (q) ∼1/(N1/2q)3/4 (Fig. 4). Our study shows that rather long chains athigh densities are required to verify ds using Eq. (4). Interestingly,the direct visualization of 2D chain configurations is possible forDNA molecules [7] or brush-like polymers [8] adsorbed on stronglyattractive surfaces or confined in thin films by means of fluores-cence microscopy [7] or atomic force microscopy [8]. Since thedirect experimental determination of the chain perimeter S(s) isthus conceivable for these systems it is this route which should bechosen first to determine ds. Deviations from the large-N behav-ior we focused on are to be expected in view of the typical molarmasses currently used and considering the high rigidity of theselarge-monomer macromolecules.

3.2. Outlook

Interestingly, Eqs. (2) and (4) match for chain configurationswhich are only marginally compact such that 1/ν → d and θ =1/ν − ds → 0 for asymptotically long chains and, hence,

F (q) ≈ N/Q d ∼ N0/qd. (9)

Such a behavior is known for various biological systems, such asthe lungs of mammals, attempting to maximize the surface atconstant overall embedding volume [24]. As mentioned in the In-troduction, we are currently also investigating dense solutions ofunknotted and unconcatenated polymer rings in d = 3 dimensionsusing the same computational approach as for the 2D linear poly-mer chains presented above. As summarized in Fig. 5 preliminarynumerical data suggest to us the possibility that asymptoticallylong rings may become marginally compact the sense of Eq. (9).7

That the rings should adopt compact configurations is expecteddue to the mutual repulsion caused by the topological constraints[15–18,20,21]. As shown, e.g., by radius of gyration Rg(N) pre-sented in the inset of Fig. 5, our largest rings become indeed com-pact (1/ν ≈ 3) confirming recent computational studies [18,20,21].Assuming that 1/ν = 3 this begs the question how to characterizethe surface of these objects. Since there is no obvious reason fora finite surface tension, the surface should be fractal characterizedby a yet unknown surface dimension ds which may be determined

7 See footnote 5.

Fig. 5. Conformational properties of dense solutions of unknotted and unconcate-nated rings in d = 3 dimensions obtained using the same Kremer–Grest model[23] as in our study of 2D linear polymers. Main panel: The structure factor F (q)

for ρ = 0.68 is represented as in Fig. 4. At variance to the Debye formula (thinline), strong nonmonotonous behavior is again observed. The dashed line indicatesa possible power-law slope, y ≈ 1/Q , for asymptotically long chains. Inset: Radii ofgyration Rg(N) used for the rescaling of the data in the main panel. As expected[18,20,21] our largest rings are fitted by a fractal dimension 1/ν ≈ 3 (dashed line).

in principle using the generalized Porod law of the structure fac-tor, Eq. (4). Using the same Kratky representation as in Fig. 4for linear chains, the main panel of Fig. 5 reveals a similar non-monotonous behavior but with a power-law envelope, y ≈ 1/Q ,as suggested for marginally compact rings, Eq. (9). Obviously, evenour largest chain N = 4096 is yet too short to demonstrate clearlythis asymptotic limit. From the theoretical point of view it is ap-pealing that while reducing the topological interactions betweenchains a marginally compact structure allows to keep all monomersevenly exposed to the topological constraints imposed by otherrings. Just as for 2D linear polymers all ring subsegments are thusruled by the same statistics. We will corroborate this conjectureelsewhere.

Acknowledgements

We thank T. Kreer (Mainz), S.P. Obukhov (Gainesville), A. Johner(Strasbourg) and J. Baschnagel (Strasbourg) for useful discussions.N.S. acknowledges financial support by the Région d’Alsace, J.E.Z.a grant by the ANR Blanc (FqSimPIB). We are grateful for gener-ous grants of computer time by GENCI-IDRIS (Nos. i2009 091467,i2010 091467) and by the Pôle Matériaux et Nanostructures d’Al-sace (PMNA).

References

[1] J. Higgins, H. Benoît, Polymers and Neutron Scattering, Oxford University Press,Oxford, 1996.

[2] P.G. de Gennes, Scaling Concepts in Polymer Physics, Cornell University Press,Ithaca, New York, 1979.

[3] B. Mandelbrot, The Fractal Geometry of Nature, W.H. Freeman, San Francisco,California, 1982.

[4] H.D. Bale, P.W. Schmidt, Phys. Rev. Lett. 53 (1984) 596.[5] P.Z. Wong, A.J. Bray, Phys. Rev. Lett. 60 (1988) 1344.[6] B. Duplantier, J. Stat. Phys. 54 (1989) 581.[7] B. Maier, J.O. Rädler, Phys. Rev. Lett. 82 (1999) 1911.[8] F.C. Sun, A.V. Dobrynin, D.S. Shirvanyants, H.I. Lee, K. Matyjaszewski, G.J. Ru-

binstein, M. Rubinstein, S.S. Sheiko, Phys. Rev. Lett. 99 (2007) 137801.[9] I. Carmesin, K. Kremer, J. Phys. France 51 (1990) 915.

[10] A. Yethiraj, Macromolecules 36 (2003) 5854.[11] A.N. Semenov, A. Johner, Eur. Phys. J. E 12 (2003) 469.[12] H. Meyer, T. Kreer, M. Aichele, A. Cavallo, A. Johner, J. Baschnagel, J.P. Wittmer,

Phys. Rev. E 79 (2009) 050802(R).[13] H. Meyer, J.P. Wittmer, T. Kreer, A. Johner, J. Baschnagel, J. Chem. Phys. 132

(2010) 184904.

H. Meyer et al. / Computer Physics Communications 182 (2011) 1949–1953 1953

[14] J.P. Wittmer, H. Meyer, A. Johner, T. Kreer, J. Baschnagel, Phys. Rev. Lett. 105(2010) 037802.

[15] M.E. Cates, J.M. Deutsch, J. Phys. 47 (1986) 2121.[16] S.P. Obukhov, M. Rubinstein, T. Duke, Phys. Rev. Lett. 73 (1994) 1263.[17] M. Müller, J.P. Wittmer, M.E. Cates, Phys. Rev. E 53 (1996) 5063–5074.[18] M. Müller, J.P. Wittmer, M.E. Cates, Phys. Rev. E 61 (2000) 4078.[19] K. Hur, R.G. Winkler, D.Y. Yoon, Macromolecules 39 (2006) 3975.

[20] T. Vettorel, A.Y. Grosberg, K. Kremer, Phys. Biol. 6 (2009) 025013.[21] J. Suzuki, A. Takano, T. Deguchi, Y. Matsushita, J. Chem. Phys. 131 (2009)

144902.[22] R. Everaers, I. Graham, M. Zuckermann, J. Phys. A 28 (1995) 1271–

1288.[23] K. Kremer, G. Grest, J. Chem. Phys. 92 (1990) 5057–5086.[24] G.B. West, J.H. Brown, B.J. Enquist, Science 284 (1999) 1677.