THE JOURNAL OF CHEMICAL PHYSICS 139, 184904 (2013)

Chain conformations of ring polymers under theta conditions studiedby Monte Carlo simulation

Jiro Suzuki,1,2,a) Atsushi Takano,3 and Yushu Matsushita3

1Computing Research Center, High Energy Accelerator Research Organization (KEK), Oho 1, Tsukuba,Ibaraki 305-0801, Japan2Information System Section, J-PARC Center, 2-4 Shirane Shirakata, Tokai-mura, Naka-gun,Ibaraki 319-1195, Japan3Chemical and Biological Engineering, Graduate School of Engineering, Nagoya University, Furo-cho,Chikusa-ku, Nagoya 464-8603, Japan

(Received 25 July 2013; accepted 23 October 2013; published online 12 November 2013)

We studied equilibrium conformations of trivial-, 31-knot, and 51-knot ring polymers with finitechain length at their θ -conditions using a Monte Carlo simulation. The polymer chains treated in thisstudy were composed of beads and bonds on a face-centered-cubic lattice respecting the excludedvolume. The Flory’s critical exponent ν in Rg ∼ Nν relationship was obtained from the dependenceof the radius of gyration, Rg, on the segment number of polymers, N. In this study, the temperaturesat which ν equal 1/2 are defined as θ -temperatures of several ring molecules. The θ -temperatures fortrivial-, 31-knot, and 51-knot ring polymers are lower than that for a linear polymer in N ≤ 4096,where their topologies are fixed by their excluded volumes. The radial distribution functions of thesegments in each molecule are obtained at their θ -temperatures. The functions of linear- and trivial-ring polymers have been found to be expressed by those of Gaussian and closed-Gaussian chains,respectively. At the θ -conditions, the excluded volumes of chains and the topological-constraints oftrivial-ring polymers can be apparently screened by the attractive force between segments, and the〈R2

g〉 values for trivial ring polymers are larger than the half of those for linear polymers. In the finiteN region the topological-constraints of 31- and 51-knot rings are stronger than that of trivial-ring, andtrajectories of the knotted ring polymers cannot be described as a closed Gaussian even though theyare under θ -conditions. © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4829046]

I. INTRODUCTION

Topology of loops in two dimensions must be trivial, be-cause loops, closed-strings, cannot be knotted in this dimen-sion. In four or higher dimensions, making knots in loopsis meaningless. In three dimensions we can generate loopswith any topology, and their topologies are fixed if they haveexcluded volumes. Fig. 1 shows three types of topologiesin three dimensions, trivial, 31-knot, and 51-knot rings. Wecan observe knotted loops in “soft-matters”; deoxyribonucleicacids (DNA) and polymers, etc. Dean et al.1 and Wassermanet al.2 observed knots in DNAs with electron micrograph andalso studied topology changes in the knots. Takano et al.3, 4

synthesized trivial ring polystyrene and purified them by size-exclusion chromatography and liquid chromatography at thecritical condition. The linear polymer contamination was 4%or less, and the molecular weight of ring polystyrene M is573k. They observed the θ -temperature at which Flory’s crit-ical exponent ν goes to 1/2 in cyclohexane is 300.9 K, whichis 6.8◦ below the θ -temperature of linear polystyrene. The ex-perimental result of the θ -temperature depression agrees withthe simulational5, 6 and theoretical7, 8 studies. Ohta et al.9 syn-thesized knotted ring polymers and separated them from thetrivial ring fraction by the chromatography technique.

The scaling behavior of radius of gyration Rg of polymerchains can be classified by the ν values,10 Flory’s critical ex-

a)E-mail: jiro.suzuki@kek.jp

ponent, with the segment number of the molecule N by

Rg ∝ Nν. (1)

If trajectories of linear polymers are described as a self-avoiding walks (SAW) in three dimensions: chain crossingis not permitted, the ν values are approximately 3/5. If tra-jectories of linear polymers are described as a random walk,the ν value in Eq. (1) is 1/2. In this cases the excluded vol-ume of chains is screened by attractive force between seg-ments, and the magnitude of chain conformation entropy ismaximum. Dimension of linear polymers in melt and at θ -temperature can be described by the random walk model. Forexample, the θ -temperature for linear polystyrene is 307.7 Kin cyclohexane.11

The topological constraint in flexible rings was discussedtheoretically,12 and the effect of the constraint is the same asthe excluded volume interactions. If trajectories of ring poly-mers are described as a closed random walk, Gaussian ran-dom polygon (GRP), the ν value is 1/2.13, 14 The chain topol-ogy of rings cannot be fixed: they are random knots, becausechain crossing is allowed. Katritch et al.15 generated 9 × 105

GRPs and the polygons were classified by topology with theAlexander polynomial.16 The most frequent minimal sizes forthe 31- and 51-knot domains are 6-8 and 20 segments, respec-tively. They concluded that the knotted-domains in the GRPsare localized. Orlandini et al.17 studied numerically the con-formations of 31-knot ring polymers absorbed on a plane at

0021-9606/2013/139(18)/184904/7/$30.00 © 2013 AIP Publishing LLC139, 184904-1

184904-2 Suzuki, Takano, and Matsushita J. Chem. Phys. 139, 184904 (2013)

FIG. 1. Schematic illustrations of polymer molecules, (a) linear, (b) trivial-,(c) 31-knot-, and (d) 51-knot-ring polymers.

the high and low temperatures. The knot size dependency onring length L and temperature was obtained, and the resultswere extrapolated at L → ∞. The knotted domains in 31 ringare localized in the high temperature, but delocalized in thelow temperature. At their θ -temperature, the collapse transi-tion, the knots are weakly localized. Ercolini et al.18 obtainedthe ν values and the persistence length of DNAs adsorbed on asurface by atomic force microscope. The localization of knotsin two-dimensions was observed under the weak absorptionconditions. Mansfield19 observed the collapse transition ofpolymer chains by a dynamic Monte Carlo simulation fromgood- to poor-solvent conditions. The relaxation of the knotspectrum is much slower than that of the radius of gyration.Orlandini et al.20 discussed the size of knots in polymers un-der good-, poor-, and θ -conditions. In three dimensions knotsin good-solvent are weakly localized, and they are delocalizedbelow the θ -temperature. Mansfield and Doublas21 obtainednumerically equilibrium dimensions of knotted ring polymerson three classes of polymers, self-avoiding-, theta-state-, andGaussian-rings. The simulation result extrapolated to large Ndemonstrated knot localization in all three classes of chains.But at large N, 104, the data have not reached to the asymp-totic domain of N.

A large number of conformations of GRPs generated inN ≤ 600 were classified by topology, and their averaged di-mensions and the ν values were obtained.14 The ν values fortrivial-, 31-, and 51-knot rings are 0.589, 0.590, and 0.596, re-spectively. Their ν values for each topology in the finite N val-ues are the same as that for linear polymers in good solvents,because the rings are swollen by the topological constraints.Suzuki et al.5, 6 reported the topological constraint in trivial,31-knot, and 51-knot rings in the finite N region of N ≤ 2048under the θ -conditions with Monte Carlo simulation. In thiscase topology of ring polymers is kept, because chain crossingis prohibited by the excluded volumes. The θ -temperaturesfor linear-, trivial-, 31-knot, and 51-knot ring polymers areobtained, where their ν values were 1/2. As a result, the θ -temperatures for four molecules in the finite N region show

the following relative relationship,

θlinear > θtrivial > θ31 ≥ θ51 . (2)

This means that the ν values for trivial-, 31-knot, and 51-knot ring polymers are larger than 1/2 at the θ -temperaturefor linear polymers. Trajectories of linear polymer under θ -conditions and random-knots can be described as a random-walk and a closed-random-walk, respectively. The confor-mational statistics of their chains is expressed by Gaussian,which is well known and shown in the Appendix.

Miyaki et al.11 obtained the 〈R2g〉 value dependence on

molecular weight, M, for linear polystyrene in the θ -solvent,cyclohexane at 307.7 K, with the light scattering, as⟨

R2g

⟩ = 8.8 × 10−4M [nm2]. (3)

Properties of polystyrene is well studied,22 and the Kuhnlength of polystyrene is 1.67 nm and it can be accountedfor 5.07 monomeric units.11, 23 The Kuhn length of linearpolymer at the θ -temperature in the simulator employed inthis study is 1.759,6 and the chain length of N = 4096 inthe simulator corresponds to M = 1.4 × 106 of polystyrene.It is very difficult to perform polymerization to reach sucha large M with monodispersed molecular weight distribu-tion. In most case experiments are performed in the range ofM < 106, and molecular weight of ring polymers in the Mregion does not reach the asymptotic region. In this paper,conformational statistics of trivial-, 31-knot, and 51-knot ringpolymers at ν = 1/2 are obtained with the simulator and com-pared with those of the Gaussian in the range of N ≤ 4096.The analogy between the excluded volume effects of chainsand the topology-driven swelling of ring polymers is dis-cussed in the finite N region.

II. SIMULATION

A simple and an efficient simulation algorithm is requiredbecause we try to obtain averaged conformations of linear,trivial- and knotted-ring polymers at large N. The simula-tion algorithm used in this study was introduced in a previousstudy,6 where polymer chains were placed on a face-centered-cubic (fcc) lattice. Because the motions of beads and bondsare defined on fcc lattice, excluded volume of polymer chainsis associated with bead contact only. Since chain flexibilityis required to discuss the topological constraints in the rings,polymer beads can move not only with the local motions butalso with the trans-locations in the algorithm. We employ theMetropolis Monte Carlo method in obtaining averaged R2

g

values of polymer molecules vs. N for linear, trivial-, 31-, and51-knotted ring polymers in dilute solutions. Since only di-lute solutions are studied in this paper, linear and the threering polymer molecules with different topology isolated in athree-dimensional lattice are treated. The procedure of the al-gorithm in detail was explained in the previous papers.5, 6 Apolymer beads is selected randomly for the transition frominitial to trial positions in one Monte Carlo (MC) step. Ifthe trial position is occupied by the other beads, the trial isrejected. Probability p of transition and temperature Tα are

184904-3 Suzuki, Takano, and Matsushita J. Chem. Phys. 139, 184904 (2013)

defined as

p ={

1 if �E < 0,

exp (−�E/Tα) otherwise ,(4)

where �E is the energy change between initial and trial con-formations, �E = −(Etrial − Einitial). The numbers of beadscontacting the selected beads at initial and trial position areEinitial and Etrial, respectively.

The ensemble average of the square of the radius of gy-ration of a chain, 〈R2

g〉, is defined as

⟨R2

g

⟩ =⟨

1

N

N∑i=1

|rcm − ri |2⟩, (5)

where ri and rcm are the position vector of the ith bead in achain and that of the center of mass of the chain, respectively,where mutual relationship is defined in Eq. (6).

rcm = 1

N

N∑i=1

ri . (6)

Fig. 2 shows the double-logarithmic plots of 〈R2g〉 and N

for linear, trivial-, 31 knot-, and 51 knot-ring polymers at(a) high temperature and (b) Tα = 7.614. The ν values forlinear and trivial-rings at a high temperature are approxi-

FIG. 2. Double-logarithmic plots of 〈R2g〉 and N for linear, trivial-, 31-knot,

and 51-knot ring polymers at (a) high temperature and (b) the θ -temperatureof linear polymers, Tα = θ linear = 7.614. The error bars express ±σ , whereσ is the standard deviation in evaluating 〈R2

g〉.

mately 0.6, which is the same value as those in a good sol-vent. The 〈R2

g〉 values at the θ -temperature for linear polymer,Tα = θ linear = 7.614, are shown in Fig. 2(b). Under this tem-perature the slope, the 2ν value, for linear polymer is 1.0 inthe large but finite N region, 512 ≤ N ≤ 2048. The Tα val-ues for trivial-, 31-, and 51-knotted ring polymers are θ triv

= 7.199, θ31 = 6.998, and θ51 = 7.001, respectively, at whichtheir 2ν values become unity. In this paper, these are definedas the θ -temperatures. Because knotted domains in the ringsare localized at the θ -temperature in the large N limit, the ν

values for rings may be the same values. The simulation datashown in Fig. 2 and the θ -temperatures are quoted from theprevious paper.6

III. RESULTS AND DISCUSSION

A. Segmental distribution functions

Radial distribution functions of the distance of the twosegments in a molecule separated by λN steps, W (r, λ), canbe given by

W (r, λ) = Br2 E(r, λ)

n(r)(7)

from the simulator, where r is the distance between segments,E(r, λ) is the frequency of appearance and n(r) is the numberof lattice points with the distance r from the origin. The λ

value is defined and explained in Fig. 3, and λL is occasionallyabbreviated as λ in the following formulas. The E(r, λ) valueswere acquired with obtaining the 〈R2

g〉 values. The W (r, λ)function for each λ is normalized by B in Eq. (7) as∫ ∞

0W (r, λ)dr = 1. (8)

Radial distribution functions for linear, trivial-, 31-knot-, and51-knot-ring polymers are obtained at the θ -temperatures, Tα

= θ linear, θ triv, θ31 , and θ51 , respectively. Fig. 4 shows the radialdistribution functions at N = 4096 under the θ -conditions ac-companied by their standard deviation σ with respect to r forlinear (λL = 1.0) and three-types of rings (λ = 0.5). The val-ues are the ensemble average of the 24 simulation jobs (seebelow), which are confirmed to be essentially the same asthose of the 12 jobs. The distribution of end-to-end distancefor linear polymers at the θ -temperature can be fitted with

FIG. 3. Definition of the distance of r between segments and parameters λL

and λ with 0 ≤ λL ≤ 1 and 0 ≤ λ ≤ 0.5. The number of segments in amolecule is N. (a) A linear polymer chain divided into three partial chains byO and B. Their number of segments are (1 − λLN)/2, λLN, and (1 − λLN)/2,respectively. The end-to-end distance of the middle partial chain is defined asr. (b) The two segments, at O and C, are connected by the two chains whosenumber of segments are λN and (1 − λ)N, respectively.

184904-4 Suzuki, Takano, and Matsushita J. Chem. Phys. 139, 184904 (2013)

FIG. 4. Radial distribution functions for linear (λL = 1.0) and ring polymers(λ = 0.5) with N = 4096. The error bars express the standard deviation inevaluating the probability. The solid curves correspond to the best fits of thesimulated data, which are evaluated by applying the formula in Eq. (9).

the Gaussian,6 and therefore the excluded volume of linearchains is screened completely by the attractive force betweensegments.

The computations in this study were executed on the Cen-tral Computing system hosted by the Computing ResearchCenter of the High Energy Accelerator Research Organiza-tion. Two days each were required to complete data acqui-sition for N = 4096 with an Intel Xeon processor for eachsimulation job.

B. Universal scaling function of probabilitydistribution dunctions

A universal scaling formula of a radial distribution func-tion f(r) is introduced24 as is shown in Eq. (9), which is ap-plied to fit the simulation data obtained in this study,

f (r) = Csr2+θ exp[−(Dsr)δ]. (9)

In Eq. (9), r is the end-to-end distance of a polymer chain,and Cs and Ds are constants. Flory’s critical exponent νd

can be obtained from the δ value, according to the relationνd = (δ − 1)/δ. The f(r) at r ≥ 0 is an upward convex func-tion, and it has a maximum value at r = rmax,

rmax = δ−1/δ(2 + δ)1/δ

Ds. (10)

The simulated data for each λ at N = 4096 are evaluated withthe least square method by applying the formula, Eq. (9), andthe best fits at λL = 1.0 and λ = 0.5 are shown in Fig. 4.Fig. 5 shows the dependence of the νd values on λL and λ atN = 4096. The νd values are increasing with decreasing thesectional chain length, λL and λ, because of the short-rangeinteraction between segments. The trend of the curve for thetrivial ring is the same as that of linear polymers, but those forthe 31- and 51-knot-rings are different from the others. The ν

values for linear, trivial-, 31-knot, and 51-knot ring polymersare all 1/2 with the same simulation parameters, while the νd

values for 31- and 51-knot-rings at λ = 1/2 are much smaller

FIG. 5. Variation of the νd values of the sectional chains at their θ -conditionsagainst λL for a linear polymer and λ for trivial-, 31-knot, and 51-knot ringpolymers with N = 4096. The νd values are obtained from the δ value inEq. (9).

than 1/2, indicating that their trajectories cannot be describedas a closed random walk.

Segmental distribution functions for a random walk anda closed random walk have been studied and established well,which are shown in the Appendix. The topology of a closedrandom walk cannot be fixed, because their chains can becrossed each other.

The polymer chains treated in this study have the ex-cluded volumes which are screened by the attractive forcesbetween segments. Accordingly, topology of each ring is keptby their excluded volumes when they are at θ -temperatures.In Fig. 6(a) the distribution functions for trivial-rings at θ triv

are compared to those for linear chains at θ linear. The distri-bution functions of the end-to-end distance of linear polymerand the distance between the opposite segments of trivial-ringare shown. From Eqs. (A2) and (A6), for example, it can berecognized that the distribution function of the random walkmodel for a linear molecule at N = 2048 and λL = 1/2 is iden-tical to that of closed-phantom-chain at N = 4096 and λ = 1/2.The νd value for a trivial-ring in Fig. 5 is approximately 1/2at λ = 1/2, which is the same value as that for a linear poly-mer. Figs. 5 and 6(a) show that the segmental distribution oftrivial-ring polymers under θ -conditions can be described asa closed-random walk. Thus Fig. 6(b) shows the distributionfunctions of trivial-, 31-knot-, and 51-knot-ring polymers un-der the θ -conditions. The distribution functions are shifted tosmaller r region with increasing the topological constraints inthe ring polymers.

Fig. 7 compares the normalized ensemble average val-ues of r2: 〈r2

nor(λ)〉, and the square of rmax: r2max,nor(λ) at

N = 4096 for linear and ring polymers. The functionsof 〈r2

nor(λL)〉 and r2max,nor(λL) for linear polymers overlap

and are almost proportional to λL, because they are under

184904-5 Suzuki, Takano, and Matsushita J. Chem. Phys. 139, 184904 (2013)

FIG. 6. Radial distribution functions for (a) linear and trivial ring polymersand (b) trivial-, 31-knot, and 51-knot ring polymers at their θ -conditions. Thesolid curves correspond to the best fits of the simulated data.

θ -conditions. The similarity of 〈r2nor(λL)〉 and r2

max,nor(λL) oflinear polymers satisfy the relationships of Eqs. (A3) and(A4), and it is not influenced by the λL values. It is naturalthat conformation of sectional chains of a linear polymer canbe described as a random walk, since the ν value should be1/2. In Fig. 7 the functions for the ring polymers are not pro-portional to λ any more, and the magnitudes of curvatures areincreased with increasing the topological constraint.

In Fig. 8 two functions for ring polymers at (a) high tem-perature, (b) θ linear, (c) θ triv, and (d) three θ -temperatures forthree topology are compared. The broken lines in Fig. 8 ex-press the functions associated with the closed random walkmodel, which are proportional to λ(1 − λ) from Eqs. (A7)and (A8). At high temperature, Fig. 8(a), both of the func-tions for trivial-, 31-knot, and 51-knot ring polymers are allconvex downward. As shown in Fig. 8(b), at θ linear, all theplots are increased comparing with the data in Fig. 8(a), par-ticularly the functions for 31-knot ring are very close to thebroken lines, though the ν value for 31-knot at θ linear is not 1/2.On the other hand as shown in Fig. 8(c), at θ triv, merely thetrivial ring molecule gives the identical values as those for theclosed random walk model, and the others are convex upward.Fig. 8(d) compares the functions for three ring chains at theirown θ -temperatures. Equations (A7) and (A8) suggest that thefunctions of rings in Fig. 8(d) are linear function if they areregarded as a closed random walk. The r2

max,nor(λ) functionsof 31- and 51-knot rings are apparently larger than their own〈r2

nor(λ)〉 functions, because their νd values shown in Fig. 5 are

FIG. 7. The normalized ensemble average of r2, 〈r2nor(λ)〉, and the square of

rmax, r2max,nor(λ), at ν = 1/2 against λL and λ for a linear and rings with N of

4096.

much smaller than 1/2. Thus the statistics of trivial-ring poly-mers at N = 4096 under θ -conditions can be described by theclosed random walk model.

The expansion factor β introduced in the previous paper6

was defined as

β(N, Tα) =⟨R2

g

⟩(N, Tα)⟨

R2g

⟩linear(N, Tα = θlinear)

, (11)

where the denominator in Eq. (11) is the ensemble average ofR2

g for linear polymers at θ linear and should be proportional toN. At the θ -temperatures for trivial-, 31-knot or 51-knot ringpolymers the 〈R2

g〉(N, Tα) values are also proportional to N,and the N-dependence of the β-values vanishes, therefore theβ-values at ν = 1/2 depend on topology. It is well knownthat the β value for the closed random walk model is 1/2,while those for trivial-, 31-knot, and 51-knot ring polymersat N = 4096 are 0.525, 0.404, and 0.349, respectively, andthey evidently decrease with increasing degree of topologicalconstraint.6 The β-value for trivial rings is larger than 1/2, andtrivial ring polymer at ν = 1/2 is only swollen by preventingthe topological transitions. The chain conformation entropy ismaximum with the swollen chain conformations described bya closed random walk, whereas the others are squeezed due totheir strong topological constraints.

Not only the N dependence of 〈R2g〉 values but also the

segmental distribution function for trivial rings at ν = 1/2keep Gaussian statistics, because the functions of 〈r2

nor(λ)〉 andr2

max,nor(λ) for trivial ring polymers at θ triv are proportional toλ(1 − λ) value, which is shown in Figs. 8(c) and 8(d). Thisproperty for trivial-rings is the same as that of linear polymersunder the θ -temperature. The trajectories of trivial rings atθ triv in the finite N region which are swollen by the topologicalconstraint can be described as the closed random walk model.

184904-6 Suzuki, Takano, and Matsushita J. Chem. Phys. 139, 184904 (2013)

λλ

λ(1−λ)

α θ

α θ

α θ

λ

λ

λλ

λ(1−λ)

α θ

α θ

α θ

λ

λ

λλ

λ(1−λ)

θ ν

α θ

α θ

α θ

λ

λ

λλ

λ(1−λ)

λ

λ

ννν

FIG. 8. 〈r2nor(λ)〉 and r2

max,nor(λ) against λ(1 − λ) with N of 4096 for the ring polymers at (a) high temperature, (b) θ linear, (c) θ triv, and (d) their θ -temperatures.The broken lines in the figures show the functions for closed-random walk.

Both of the excluded volumes of chains and the topologicalconstraints in a trivial ring are screened by the attractive forcebetween segments. If the topological constraint in the trivial-ring can be ignored, the θ -temperature for the trivial-ring isthe same as that for the linear polymers. This situation is real-ized only for a trivial-ring, because the 〈r2

nor(λ)〉 and r2max,nor(λ)

values for 31- and 51-knots in Fig. 8(d) are shifted up with in-creasing the topological constraints. This result is obtainedfrom the simulation performed in the finite N region, and thetopological constraint in the rings can only be ignored at N→ ∞. The influence of the topological constraint can be ob-served in experiments with ring polymers, because the N val-ues used in this study is large enough for polymer molecules.

C. Conclusion

We obtained the segmental distribution functions for ringpolymers under the conditions of ν = 1/2 in the finite N re-gion, where the topologies of rings are fixed. The distribution

function for trivial rings is merely described with the Gaus-sian, the closed random walk model, and the 〈R2

g〉 values fortrivial ring polymers are larger than the 〈R2

g〉/2 values for lin-ear polymers. The excluded volume effects of chains and thetopological constraints in a trivial ring can be screened com-pletely by the attractive force between segments in the finiteN region.

ACKNOWLEDGMENTS

We would like to thank Professor Tetsuo Deguchi atOchanomizu University for his insightful suggestions and dis-cussions. The simulation work in this study was executed onthe Central Computing System of KEK. This work was finan-cially supported by Grant-in-Aid for Scientific Research forYoung Scientists B (Grant No. 22740281) from Japan Soci-ety for the Promotion of Science.

184904-7 Suzuki, Takano, and Matsushita J. Chem. Phys. 139, 184904 (2013)

APPENDIX: SEGMENTAL DISTRIBUTION FUNCTIONSOF CLOSED RANDOM WALK

1. Dimension of random walk

A three-dimensional random walk with N-steps from theorigin O to the point E can be described with probability ofp(R, N), where b and R are the step length and the distancebetween O and E, respectively,

p(R,N ) =( 3

2πNb2

)3/2exp

[−3R2

2Nb2

]. (A1)

The radial distribution function of Eq. (A1) is written by

w(r,N ) = 4πr2 p(r,N ), (A2)

where r is the end-to-end distance of random walks with N-steps. In Eq. (A2), w(r,N ) at r ≥ 0 is a upward convex func-tion, and the function w(r,N ) has a maximum value at

rw,max(N ) =[2

3Nb2

]1/2. (A3)

In Eq. (A3),[rw,max(N )

]2is apparently proportional to N. The

ensemble average of the square of end-to-end distance of ran-dom walks with N-steps can be expressed by

〈r2(N )〉 =∫ ∞

0r2w(r,N)dr = Nb2, (A4)

where 〈r2(N)〉 is proportional to the number of steps, N,namely, molecular weight.

The radial distribution functions of end-to-end distanceof linear polymers which is also employed in this study wereobtained from the simulator in the previous paper.6 The trajec-tories of linear polymers at θ -temperature can be described asa random walk; the distribution curve under the θ -conditionis represented by the Gaussian statistics as expressed byEq. (A1). The simulation result at the θ -temperature can befitted well with Eq. (A2), therefore the excluded volume oflinear polymers is screened completely by the attractive forcebetween segments at the θ -temperature.6

The radial distribution function of the distance betweentwo segments, at O and B, in an ideal chain can also be ob-tained from Eqs. (A1) and (A2), if the number of steps be-tween O and B is defined as λLN shown in Fig. 3(a), where λL

is in the range of 0 ≤ λL ≤ 1.

2. Dimension of closed random walk

The segmental distribution function of the closed randomwalk model is shown in this section. Topology of a closed ran-dom walk cannot be fixed: they possess random knots. Theycan have any type of topology including trivial-, 31-, and 51-knots, and so on, and the number of topologies is infinitelylarge.

A three dimensional closed random walk with N-stepsis defined as a Gaussian random polygon, and r is the dis-tance between two segments, O and C, on the walk shown inFig. 3(b). The two segments are connected by two randomwalks with λN and (1 − λ)N steps, where the λ value is inthe range of 0 ≤ λ ≤ 1/2. If the two segments at O and C arespanned by these two random walks, the probability pring(R,

N, λ) is obtained from the product of the probabilities for eachrandom walk, as

pring(R,N, λ) ∝ p(R, λN ) p(R, (1 − λ)N )

∝ p(R, λ(1 − λ)N ). (A5)

The radial distribution function for p(R, λ(1 − λ)N) inEq. (A5) is

wring(r,N, λ) = A 4πr2 p(r, λ(1 − λ)N ), (A6)

where r is the distance between segments shown in Fig. 3(b),and A is the normalization factor. The function wring(r,N, λ)has a maximum value at

rring,w,max(N, λ) = rw,max(λ(1 − λ)N )

=[2

3λ(1 − λ)Nb2

]1/2, (A7)

and the ensemble average of the square of r is

〈r2ring(N, λ)〉 =

∫ ∞

0r2wring(r,N, λ)dr

= 〈r2(λ(1 − λ)N )〉= λ(1 − λ)Nb2. (A8)

It should be noted that [rring, w, max(N, λ)]2 and 〈r2ring(N, λ)〉 are

proportional to λ(1 − λ).

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Chain conformations of ring polymers under theta conditions studied byMonte Carlo simulationJiro Suzuki, Atsushi Takano, and Yushu Matsushita Citation: J. Chem. Phys. 139, 184904 (2013); doi: 10.1063/1.4829046 View online: http://dx.doi.org/10.1063/1.4829046 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v139/i18 Published by the AIP Publishing LLC. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors