Ring polymers in melts and solutions: scaling and crossover
aa r X i v : . [ c ond - m a t . s o f t ] A p r Ring polymers in melts and solutions: scaling and crossover
Takahiro Sakaue
1, 2, ∗ Department of Physics, Kyushu University 33, Fukuoka 812-8581, Japan PRESTO, Japan Science and Technology Agency (JST),4-1-8 Honcho Kawaguchi, Saitama 332-0012, Japan
We propose a simple mean-field theory for the structure of ring polymer melts. By combining thenotion of topological volume fraction and a classical van der Waals theory of fluids, we take intoaccount many body effects of topological origin in dense systems. We predict that although thecompact statistics with the Flory exponent ν = 1 / ν = 2 / PACS numbers: 61.25.H-,36-20.-r,83.80.Sg
Statistics of melts and concentrated solutions ofring polymers is a longstanding problem in polymerphysics [1–14]. Unlike linear chain systems, their physi-cal properties crucially depends on the preparation his-tory during which the topology of the system is frozen.Then, the non-crossing requirement creates topologicalconstraints which impose nontrivial restrictions on thephase space of the system, hence, have a strong influenceeven in statics.In the linear polymer counterpart, a well-known Florytheorem states that the chain conformation is Gaussiancharacterized by the Flory exponent ν = 1 / ν = 1 /
3) and Gaussian ( ν = 1 /
2) chains [1]. Specif-ically, they proposed a conjecture on the scaling expo-nent ν = 2 / ν = 1 / ∗ Electronic address: [email protected]
It is important to keep in mind that the topological ef-fect manifests itself in the scale larger than some charac-teristic length ξ . Individual rings in concentrated solu-tions of small molecular weight N < g ≡ ( ξ /a ) φ / ( N is the number of monomers in each ring, a is the monomersize and φ is the monomeric volume fraction) thus showGaussian behaviors with the size R ≃ ξ ( N/g ) / ≃ aN / φ − / where ξ ≃ ag / ≃ aφ − / is the correla-tion length of concentration fluctuation. In the presentpaper, we introduce the notion of the topological volumefraction and construct a mean-filed theory for the con-centrated solution of noncatenated long ring polymers.Requiring the theory to be compatible with the abovefeature associated with the length scale ξ , we show thatthe scaling exponent for the long chain limit is given by ν = 1 /
3. However, extremely long chains
N > N ∗ = Cg with a large numerical factor C ≃
10 are required for thisasymptotic to be reached, thus, most practical cases fallinto a broad crossover region ( g < N < N ∗ ) where theapparent exponent is given by ν = 2 /
5. At various stageof the paper, we will look into the physics behind the C-Dtheory in the light of the present argument. We also sug-gest a connection to the CG model through a topologicalblob which adjusts its own size in an intriguing way inthe crossover region. This view provides a natural bridgebetween otherwise conflicting two lines of previous con-jectures (C-D and CG).
Mean field theory — A rigorous analysis of the topo-logical effects may rely on the mathematical knot invari-ance, which, however, seems to be yet formidable. Weinstead seek for a physically motivated coarse-grainedphenomenological description. A successful prototype ofsuch an approach is already found in the dynamics andrheology of high molecular weight linear polymer melts inwhich the tube model provides a bridge between molecu-lar level topological constraints and macroscopic materialproperties [15, 16]. In the present problem, we also pos-tulate an intrinsic length scale ξ which is an analogueof the tube diameter in the linear chain solutions, belowwhich the topological effects are irrelevant. At this stage,what is needed is an element to treat large scale static behaviors of unconcatenated rings, just as the reptationtheory provides a basic framework to treat the dynamicalproperties of entangled linear chain solutions beyond thetube size.A key observation from previous studies is that thetopological constraints could be effectively representedas excluded-volume effects. Notable topics in this line in-clude an entropic repulsion between untangled rings [19],a topological swelling of isolated random knots [20, 21]and the anomalous bond correlation function of planarrings [22]. It is expected that the above conjecture, firstproposed by des Cloizeaux [23], would be equally usefulfor the present problem, too.Let us consider a concentrated solution of ring poly-mers. The equilibrium spatial size R (such as a radius ofgyration) of each ring is a function of N and the monomervolume fraction φ . The number of total monomers in theregion of volume ∼ R of single ring is ∼ R φ/a , thus,the number of rings there is N R ≃ R φa N (1)Using a “self density” φ s = a N/R , it can be written as N R = φ/φ s .The topological contribution to the free energy consistsof terms arising from (i) noncatenation constrains amongdifferent rings and (ii) intra-ring effects associated withself-knotting [1]: F = F inter + F intra (2)In C-D theory, the first term is evaluated by assigning ∼ k B T to each of N R neighboring rings, leading to the es-timate F inter /k B T ≃ ( R φ ) / ( a N ). One may notice thatthis amounts to the evaluation of binary interactions. Inconcentrated solutions of long rings, however, the many-body correlation effects should become progressively im-portant, which may be taken into account in line withthe excluded-volume analogy as followings. The effectiveexcluded-volume v R of the ring with the spatial size R scales its volume v R ≃ R Y (3)where a dimensionless factor Y (independent of N ) shallbe fixed later to be consistent with the presence of thelength scale ξ mentioned earlier. To evaluate the re-sultant repulsive interaction, we adopt a classical vander Waals theory of fluids whose free energy densityfor a one component fluid with the volume fraction φ and the excluded volume v is given by f ( φ ) / ( k B T ) =[ φ/v ] ln [ φ/ (1 − φ )] − ǫφ . In our case, this may be trans-formed as f inter / ( k B T ) ≃ − [ φ R /v R ] ln [(1 − φ R )] wherewe set ǫ = 0 (athermal), and as usual for the polymersolution theory, the ideal gas term is irrelevant here [17],and φ R ≡ v R N R R = N R Y (4)is the ring’s “volume fraction” of topological origin. Thefree energy per ring F inter = f inter × R /N R is thus F inter ( φ R ) k B T = − ln (1 − φ R ) (5) An inspection of eq. (5) indicates that the larger size R costs free energy due to the noncatenation constraint.Thus, this topology effect leads to the squeezing of thering toward a globular state, which should be negotiatedwith the unknotting constraint. A scaling analysis sug-gests that this free energy cost for the confinement shouldbe written as F intra k B T ≃ (cid:18) R R (cid:19) δ = (cid:18) aN / φ − / R (cid:19) δ ≃ (cid:18) N Y φ / φ R (cid:19) δ/ (6)where R = ξ ( N/g ) / = aN / φ − / is the size of un-perturbed Gaussian ring (more precisely, the Gaussianring of the blobs of size ξ = ag / = aφ − / ), and usehas been made of eqs. (1) and (4) in the last equality.One then requires F intra to be extensive ( ∼ N ) with φ R given, which leads to δ = 6 [25]; F intra ( φ R ; N ) k B T = φ − R N Y φ / (7)Note that this differs from a naive guess δ = 2 adopted inthe C-D theory which, however, is no longer correct fornon-ideal chains confined in closed cavity (see ref. [24]for a general discussion on it). Indeed, eq. (7) is nat-urally expected in the context of the excluded-volumeanalogy [17, 24] and applies when rings feel squeezing R < R ⇔ N > g . It is noted that while F inter solelydepends on the topological volume fraction φ R , F intra linearly increases with the chain length N at a given φ R .This reflects different aspects of the constraints betweennonconcatenation and unknotting. F inter F intra F φ R F/k B T φ R F/k B T F inter F intra F FIG. 1: Plots of free energy as a function of φ R for short ring B N/g = 0 . B N/g = 10 (right). a. Short scale behavior Figure 1 shows typical freeenergy profiles for short and long rings. For small N , thefree energy minimum is attained at low φ R <
1. Onemay find a situation F inter ( ˜ φ R ) ≃ F intra ( ˜ φ R ; g ) ≃ k B T at ˜ φ R . This indicates that, in short length scale r <ξ = ag / φ − / , the topological effect is not effective,and the ideal ring behavior is observed. From the abovecondition, we find Y ≃ g − / φ − / B / ( ˜ φ R ) (8)where B ( ˜ φ R ) ≃ ˜ φ R ≃ .
5. We shall shortly argue thatthis numerical factor B is crucial for the characterizationof the crossover regime. The confinement free energyeq. (7) can now be rewritten as F intra ( φ R ; N ) k B T = B φ − R Ng = φ − R Ng (9)where g ≡ g / B . b. Free energy minimization Now let us seek for thespatial structure of rings in large scales
N > g . Here,the energy scale is much larger than the thermal energy(see Fig. 1 ), thus, the equilibrium size would be readilydetermined through the free energy minimization withrespect to φ R . The result of numerical solution shown R/ ξ N/g N* g FIG. 2: Normalized plot of the ring size
R/ξ as a functionof the chain length N/g (double logarithmic scale). in Fig. 2 is summarized as follows; (i) sufficiently longrings ( N > N ∗ ) obey a compact statistics with ν = 1 / g < N < N ∗ ) arewell characterized by the effective exponent 2 /
5. There-fore, one may say that the statistics of concentrated ringpolymers has well-defined two regimes (ideal and com-pact statistics for short
N < g and long N > N ∗ rings,respectively) which are separated by a wide crossover re-gion spanning over about one order of magnitude. Toget an insight into this crossover, we note that fromeqs. (1), (4), (5), (8),(9), a mean-field solution for N > g can be generally written in the form R = ξ N φ ( eq ) R ( N/g ) g ! / = ξ ♮ ( N ) (cid:18) Ng ♮ ( N ) (cid:19) / (10)where ξ ≡ aφ − / g / = ξ B − / and φ ( eq ) R denotes theequilibrium value which depends solely on N/g . Thisindicates that upon renormalization a large scale spa-tial organization of individual rings is formally charac-terized by a space-filling curve with fractal dimension N/g N R /N R ( φ R =1 ) φ s / φ s ( φ R =1 ) N R /N R ( φ R =1 ) φ s / φ s ( φ R =1 ) g N * FIG. 3: Dependence of the self density φ s and the number ofneighboring rings N R on the chain length N (semi-logarithmicplot). φ s decreases from φ s ≃ φ / g − / at N = g toward φ ( φ R =1) s ≃ φ / g − / at N > N ∗ , whereas N R increases from N R ≃ φ / g / at N = g toward N ( φ R =1) R ≃ φ / g / at N > N ∗ . ν − = 3. Unlike usual cases, however, a renormaliza-tion factor, which is invoked in an earlier study as topo-logical blob [3], ξ ♮ ( N ) = ag / ♮ ( N ) φ − / is not constant,but rather depends on N , i.e., starting from g ♮ = g at N = g , it slowly (logarithmically) increases toward g ♮ = g at N = N ∗ , then saturates. It is this slow evo-lution toward a topological dense-packed limit ( φ R → φ s and N R (roughly corre-sponding to the number of neighboring rings) are plottedagainst the chain length. A point N = N ∗ at which theseobservables saturate corresponds to the onset of the com-pact statistics, i.e., φ R → N > g which leads to theonset length N ∗ = Cg with C ≃
10 (Fig. 3). To get anumber, let us adopt an entanglement length as a mea-sure of g , and substitute a typical value ∼ N ∗ ∼ F inter can beexpanded in a virial series F inter /k B T ≃ φ R + φ R / · · · .Retaining only a lowest order term (second virial approx-imation), one finds F inter /k B T ≃ R φY / ( a N ) whichcorresponds to the free energy adopted in the C-D the-ory aside from a factor Y [27]. Our analysis suggests,however, that higher order terms are essential. Sec-ond, a view of the compact ring of size R ( < R ) asa dense piling of topological blob implies ξ ♮ ≃ ξ ( g ♮ /g ) / and a g ♮ /ξ ♮ = φ s . One thus finds ξ ♮ ≃ aφ / φ − s ≃ aφ − / N R and g ♮ ≃ φ − / N R . Assigning ∼ k B T pertopological strand, we obtain F intra / ( k B T ) ≃ N/g ♮ ≃ R /ξ ♮ , which coincides with the scaling derivation of F intra (eq. (7)) [24]. Moreover, the deduced relations ξ ♮ ≃ ξN R and g ♮ ≃ gN R indicate that the number ofunits (correlation blobs) required to form a topologicalconstraint is on the order of ∼ N R .The optimum free energy for large N ( ≫ g ) is F/ ( k B T ) ≃ N/g ♮ , i.e., an order of the magnitude of ther-mal energy per topological strand. This translates intothe osmotic pressure Π top ≃ F N R /R ≃ k B T φ/ ( a g ♮ ) ≃ ( N R ) − k B T φ / /a . Adding this to the usual excluded-volume contribution Π linear ≃ k B T φ / /a [15], the totalosmotic pressure thus can be written asΠ = (1 + N − R )Π linear (11)where one may identify the small factor as the increaseof the local excluded-volume effect, i.e., N − R ↔ δv/v inaccordance to the des Cloizeaux conjecture [23].To summarize, the present attempt strongly suggeststhe applicability of the excluded-volume analogy to theproblem of dense ring solutions. It sets up a way to han-dle immense topological constraints, thus allowing one tocapture the essential static properties in such systems. In particular, it provides us with a fairly accurate descrip-tion of the ring dimension over the range of semidiluteto concentrated solution regimes. Other key quantitiessuch as φ s , N R as well as the estimated onset length N ∗ of the compact statics are also in line with reported nu-merical observations [9–14]. It should be noticed thatour starting point is similar in spirit to that in C-D the-ory, i.e., the identification of two competing contribu-tions in eq. (2). Rather, it is the difference in the basicstructure of the free energy that leads to deep insightsinto the hierarchical spatial structure of ring solutions,hence a bridge to the CG concept. Further careful stud-ies are awaited to examine the validity and the limitationof the present phenomenological approach as well as theusefulness of the concept such as the topological volumefraction, etc. As stated in ref. [13], there are various sit-uations in molecular systems, i.e., the collapse of a gel,the existence of chromosome territories, the compatibilityenhancement [4] etc. where topological constraints mat-ter. We hope that the present analysis provides valuableinsights into such cases, too. T.S. thanks H. Nakanishiand A. Yoshimori for useful comments and discussions. [1] M.E. Cates and J.M. Deutsch, J. Physique , 2121(1986).[2] A.R. Khokhlov and S.K. Nechaev, Phys. Lett. A ,156 (1985).[3] A.Yu. Grosberg, S.K. Nechaev and E.I. Shakhnovich, J.Phys. France , 2095 (1988).[4] A.R. Khokhlov and S.K. Nechaev, J. Phys. II France ,1547 (1996).[5] S.P. Obukhov, M. Rubinstein and T. Duke, Phys. Rev.Lett., , 1263 (1994).[6] V. Arrighi, et. al., Macromolecules , 8057 (2004).[7] A. Takano, Polym. Prepr. Jpn. , 2424 (2007).[8] T. Pakula and S. Geyler, Macromolecules , 1665(1988).[9] M.M¨uller, J.P. Wittmer and M.E. Cates, Phys. Rev. E , 5063 (1996).[10] S. Brown, G. Szamel, J. Chem. Phys. , 6184 (1998).[11] M.M¨uller, J.P. Wittmer and M.E. Cates, Phys. Rev. E , 4078 (2000).[12] M.M¨uller, J.P. Wittmer and J.-L. Barrat, Europhys.Lett. , 406 (2000).[13] T. Vettorel, A.Yu. Grosberg and K. Kremer, Phys. Biol. , 025013 (2009).[14] J. Suzuki, A. Takano, T. Deguchi and Y. Matsushita, J.Chem. Phys., , 144902 (2009).[15] P.-G. de Gennes, Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca, 1979).[16] M. Doi and S.F. Edwards,
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15 for melts ( φ ≃≃