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EPJ manuscript No. (will be inserted by the editor) 6 0 Smoothening Transition of a Two-Dimensional Pressurized 0 2 Polymer Ring n a Emir Haleva and Haim Diamant J 7 School of Chemistry,Raymond & Beverly Sackler Faculty of Exact Sciences, Tel AvivUniversity,Tel Aviv 69978, Israel 1 Received: date/ Revised version: date ] t f o Abstract. We revisit the problem of a two-dimensional polymer ring subject to an inflating pressure dif- s ferential. The ring is modeled as a freely jointed closed chain of N monomers. Using a Flory argument, t. mean-field calculation and Monte Carlo simulations, we show that at a critical pressure, pc ∼ N−1, the a ring undergoes a second-orderphase transition from a crumpled,random-walk state, where its mean area m scalesashAi∼N,toasmoothstatewithhAi∼N2.Thetransitionbelongstothemean-fielduniversality - class. Atthecritical point a newstateof polymerstatistics isfound, in which hAi∼N3/2.Forp≫pc we d use a transfer-matrix calculation to derive exact expressions for the properties of the smooth state. n o PACS. 36.20.Ey Macromolecules and polymer molecules: Conformation (statistics and dynamics) – c 05.40.Fb Random walks and Levy flights– 64.60.-i General studies of phase transitions [ 2 v 1 Introduction mer comprises a set of Gaussian springs of fixed elas- 5 tic constant and no bending rigidity. At a critical pres- 4 3 Considerable theoretical efforts were directed during the sure pc ∼ N−1 the mean area was found to diverge (i.e., 1 1980s and 1990s at random polymer rings constrained to the ring inflates to infinity). This divergence, obviously, 0 a plane, both as a fundamental problemof statisticalme- is made possible by the extensibility of the chain, i.e., 6 chanics [1,2,3,4,5] and as a highly idealized model for the ability of the springs in this model to be infinitely 0 membrane vesicles [6,7,8,9,10,11,12,13,14]. In addition, stretched. t/ there was much interest in transitions between crumpled In the current work we revisit the Rudnick-Gaspari a and smooth states of membranes [15,16]. model while imposing inextensibility of the chain, as is m Two bodies of work, in particular, concerned pressur- actually appropriate for real polymers or vesicles. This - izedtwo-dimensional(2D)rings.Fisheretal.studiedboth constraint changes the infinite inflation at p = pc into a d inflated and deflated, closed,2D self-avoidingwalksusing second-orderphasetransitionbetweenacrumpled,random- n Monte Carlo (MC) simulations and scaling analysis [6,7, walkstateandasmoothone.ThemodelisdefinedinSec- o c 8].ForafiniteringofN monomers,subjecttoaninflating tion2.WebegintheinvestigationinSection3withasim- : pressure differential p > 0, three regimes were found: (i) pleFloryargumentwhich,nevertheless,yieldsthe correct v a weak-inflation crumpled regime for 0 < pN2ν . 1 (ν = physics. In Section 4 we present a mean-field calculation i X 3/4beingtheswellingexponentof2Dself-avoidingwalks), which accurately captures the behavior below and at the r where the mean area scales as A N2νfA(pN2ν); (ii) a transition.InSection5wederiveexactexpressionsforthe a strong-inflation crumpled regimheifo∼r 1.pN2ν N2ν−1, behavioratp pc usingatransfer-matrixtechnique.The where the same scaling holds but the scaling≪function behavior foun≫d in Sections 3-5 is verified in Section 6 by f becomes a power law; and (iii) a smooth regime for MC simulations. Finally, the results are summarized and A pN2ν &N2ν−1, where A N2. The crossoversbetween discussed in Section 7. h i∼ theseregimesaregradualwithnophasetransitions.Inthe thermodynamic limit, defined as N and p 0 such that pN2ν is finite, the range of the→sca∞ling regim→e (ii) in 2 Model the case ofself-avoidingwalks(ν =3/4)becomesinfinite. Note, however, that in the random-walk case (ν = 1/2) The system under considerationis illustrated in Fig. 1. A this regime disappears; we will show below that for ran- closed,two-dimensional,freelyjointedchainofN monomers dom walks regime (ii) turns, in fact, into a second-order is subject to an inflating 2D pressure differential p > 0 phase transition between regimes (i) and (iii). between its interior and exterior. The monomers are con- Thesecondbodyofworksconsistsofanalyticalstudies nected by rigid links of length l. We define l 1 as the ≡ ofclosed,pressurized,2Drandomwalks,firstpresentedby unit length and the thermal energy k T as the unit en- B Rudnick and Gaspari [9,10,11]. In this model the poly- ergy.(Thus, p is measuredin units of k T/l2.) The chain B 2 Emir Haleva, Haim Diamant: Smoothening Transition of a Two-Dimensional Pressurized Polymer Ring is ideal,i.e., thereareno interactionsbetweenmonomers. 3 Flory Argument (Effects related to self-avoidance will be briefly discussed in Sections 3 and 7.) In addition, no bending rigidity is We begin the analysis with a simple Flory argument [17] takeninto account,i.e., the chainis freely jointed.A con- which captures most of the physics to be more rigorously figuration of the ring is defined by a set of 2D vectors treated in the following sections. The free energy of the r specifyingthe positionsofthe monomers.The ring (in units of k T) is expressedas a function of R, the j j=0...N B { } conditionthatthechainbeclosedisexpressedbyr =r . radiusofthe statisticalcloudof monomers(i.e., the root- 0 N mean-square radius of gyration). We divide it into three terms, F(R)= F +F +F el inext p F R2/N l el ∼ F R4/N3 inext ∼ F pR2. (4) p ∼ − r j+1 The elastic term, F , is the usual entropic-spring free en- el ergyofaGaussianchain[17].Thesecondtermisthelead- r ing non-Gaussian correction due to the inextensibility of j the chain [see Appendix, Eq. (A.5)], needed here to sta- O bilize the ring against infinite expansion.The last term is thepressurecontribution,wherethemeanareaofthering is taken as proportional to R2 [4,5]. Equation (4) has the form of a Landau free energy, describing a second-order transition at p = p N−1. c ∼ Since the critical pressure depends so strongly on system size,weusehereaftertherescaledpressurepˆ p/p pN, Fig. 1. Schematicillustration of thering and its parameters. ≡ c ∼ anddefinethethermodynamiclimitasN andp 0 →∞ → such that pîs finite. For pˆ< 1 F is negligible, and R inext has a Gaussian distribution with R2 N(1 pˆ)−1. For The probability of a specific configuration is pˆ> 1 we have R2 N2(pˆ 1). Th huis,∼definin−g an order ∼ − parameterM =R/N,wefindinthethermodynamiclimit N P ( r ,p) epA[{rj}] δ(r r 1), (1) { j} ∝ | j − j−1|− 0 pˆ<1 j=1 M (5) Y ∼((pˆ 1)β, β =1/2 pˆ>1. − where A is the area enclosed by the ring. As in previous At the critical point itself F = F R4/N3, and R works[2,3,9,10,11],wetakeAasthealgebraicarearather inext ∼ has a non-Gaussian distribution with than the geometrical one, R2(pˆ=1) N2νc, ν =3/4. (6) c N h i∼ 1 A[{rj}]= 2 (rj−1×rj)·ˆz, (2) Note that the competition between Fel and Fp, lead- j=1 ing to the second-order transition, is unique to 2D. In X addition, when an excluded-volume term, F N2/R2, ev where ˆz is a unit vector perpendicular to the plane of the is added to the free energy, Eq. (4), the transi∼tion is re- ring. This area may take both positive and negative val- moved. This agrees with previous studies of self-avoiding ues.At zeropressureboth signs areequallyfavorableand rings [6,7,8], which did not report any phase transition the mean area must vanish. At high pressures the prob- upon increasing pressure. ability of configurations with negative A is exponentially small in pA, and whether one takes the algebraic or ge- | | ometrical area will become statistically insignificant. (We 4 Mean-Field Theory shall further discuss this assumption in Section 7.) UsingEqs.(1)and(2),wewritethepartitionfunction In this section we calculate the partition function of the of the ring as freely jointed ring, Eq. (3), by relaxing the rigid delta- function constraints on link lengths into harmonic poten- N tials. The spring constant λ of the links is chosen so as Z(p,N)= drje12p(rj−1×rj)·ˆzδ(rj rj−1 1), (3) to make the root-mean-square length of a link equal to | − |− Z j=1 l=1.Thistype ofapproximation,firstsuggestedby Har- Y risandHearst[18],wassuccessfullyemployedinstudiesof with r =r . the Karatky-Porodworm-like-chain model [18,19], where 0 N Emir Haleva, Haim Diamant: Smoothening Transition of a Two-Dimensional Pressurized Polymer Ring 3 it was shown to be equivalent to a mean-field assumption (for the field conjugate to the rigid link-length con- 1 straints).The partition function contains now only Gaus- sian terms and, therefore, can be calculated exactly, 0.8 N Z(p,N,λ)= drje21p(rj−1×rj)·ˆz−λ(rj−rj−1)2 max0.6 Z jY=1 (7) /A 1 Np 〉A 0.4 = . 〈 λN 4λsin(Np) 4λ 0.2 (The spring constant λ is in units of k T/l2.) Details of B the calculation can be found in Ref. [10]. This result can 0 be obtained also by analogy to the quantum propagator 0 5 10 15 20 of a charged particle in a magnetic field [20]. The mean ^p = p / p area is obtained by differentiation with respect to p, c Fig. 2. ThemeanareainunitsofAmax∼N2 asafunctionof ∂lnZ 1 Ncot(Np) the rescaled pressure (in units of pc ∼N−1) as obtained from A(p,N,λ) = = 4λ . (8) the mean-field approximation. The dashed curves are calcu- h i ∂p p − 4λ lated numerically using Eqs. (10) and (11), whereas the solid curves (calculated only for pˆ≥ 1) present the approximation For λ = 1 Eq. (8) is the same as the result obtained by given by Eq. (15). Calculations were performed for N = 10 RudnickandGaspari[9,10,11],exhibitingadivergenceat (thin curves) and N =105 (thick curves). p =4π/N. (9) c cot[Np/(4λ)] [Np/(4λ) π]−1. This gives Yet, in our case λ is not fixed but is to be determined ≃ − self-consistentlytoensurethesoftenedinextensibilitycon- straint.Itisclearthat,asthepressureincreases,thesprings pˆ+1+ 1 + (pˆ 1)2+ 2(pˆ+1)+ 1 must become stiffer to satisfy this constraint. To impose λ(pˆ,N 1) N − N N2, ≫ ≃ q 2 the constraint, we calculate the mean-square link length (13) and set it to 1, wherepˆ p/p =pN/(4π)isthe rescaledpressure.Inthe c ≡ thermodynamic limit Eq. (13) reduces to the continuous 1 ∂lnZ (r r )2 = but nonanalytic function j j−1 h − i −N ∂λ 1 p 1 Ncot(Np) (10) 1 pˆ<1 = + 4λ =1, λ(pˆ,N )= (14) λ Nλ p − 4λ ! →∞ (pˆ pˆ>1. thusobtainingatranscendentalequationforλ(p,N).Equa- Equation (13) should be regarded as an asymptotic ex- tions (8)and(10) arecombinedto yielda simplerexpres- pression in the limit Np/(4λ) π, which turns out to sion for A as a function of λ, be valid for any p p . (A Tay→lor expansion around this h i 6≪ c point fails because of the nonanalyticity inferred above.) N(λ 1) Substituting Eq. (13) in Eq. (11) yields an approximate A(p,N,λ) = − . (11) h i p expression for A as a function of pând N, h i Numerical solution of Eq. (10) for λ [in the range λ > A(pˆ 1,N 1) Np/(4π)] and substitution of the result in Eq. (11) yield h 6≪ ≫ i≃ tinheFmige.a2n.)areaasafunctionofpandN.(Seedashedcurves N4π2pˆ−1+ N1 +q(pˆ−21pˆ)2+ N2(pˆ+1)+ N12. (15) For very low pressures, p p , we expand Eq. (8) to c ≪ first order in p to get Inthethermodynamiclimit,thebehaviorof A around h i and above the critical pressure is obtained from Eq. (15) 1 A(p p ,N) = N2p, (12) as c h ≪ i 48 i.e., a linear dependence on p as expected from linear re- N 1 pˆ→1− N 1 1 pˆ N−1/2 4πpˆ(1−pˆ) −−−−→ 4π1−pˆ − ≫ sponse. (Recall that at p = 0 the mean algebraic area A =N3/2 1 pˆ N−1/2 vanFisohresh.)igher pressure we obtain a good approximation h i N42πpˆ−1 pˆ→1+ N2(pˆ 1) |pˆ −1|≪N−1/2, 4π pˆ −−−−→ 4π − − ≫ forλ(p,N)inthe limit N ≫1bysubstituting inEq.(10)  (16) 4 Emir Haleva, Haim Diamant: Smoothening Transition of a Two-Dimensional Pressurized Polymer Ring revealing a continuous (second-order) transition. Below the critical point, N < N , R2 N, as in a Gaussian c h gi ∼ thetransitionwegetthesamebehaviorasintheRudnick- chain. Above it R2 N2, as in a stretched chain. As Gaspari model [9,10], A N(1 pˆ)−1. Yet, due to p decreases, thehtragnis∼ition becomes sharper. At exactly h i ∼ − the inextensibility in our model, the increase of A as N = N R2 scales as N3/2. Thus, the analysis of R2 pˆ 1− breaks at 1 pˆ N−1/2. In the tranhsiition yields thcehsagmie scaling with N as obtained for A . h gi reg→ion, 1 pˆ N| −−1/2,|w∼hich shrinks to a point in h i the ther|mo−dyn|am≪ic limit, we find A N2νc,νc = 3/4. h i ∼ Abovethetransitiontheringreachesasmoothstatewith A N2(pˆ 1)/pˆ. All of these results agree with the fihndiin∼gsofthe−FloryargumentpresentedinSection3once 1e+15 we identify A R2 . As p is increased to infinity, A h i ∼ h gi h i tends, as it should, to 1e+10 N2 hA(pˆ→∞,N)i=Amax = 4π, (17) 〈R 2〉 g which is the area of a circle of perimeter N. 1e+05 Figure2showsthedependence of A onpˆforN =10 and 105 calculated both from the nuhmierical solution of Eqs.(10) and (11), and using the approximateexpression 1 (15). For large N the critical behavior becomes appar- ent, with a transitionbetween two distinct states—one in 1 100 10000 1e+06 1e+08 which A N and, hence, in units of A N2, the N max h i ∼ ∼ meanareavanishesforN ,andanotherwithamean area proportional to Amax→∼∞N2. As can be seen in Fig. Ffixiged. v3a.luMeesaonf-psq=ua1r0e−r5a,d1i0u−s3oafngdyr1a0t−io1n(tahsicak,fudnaschtieodnaonfdNthaint 2, the approximate expression (15) is practically indistin- curves, respectively) as obtained from the mean-field theory. guishable from the numerical solution for pˆ& 1, even for For N < N = 4π/p (marked with arrows) hR2i ∼ N, while c g small N. for N >N hR2i∼N2. At thecritical point (diamonds) hR2i c g g The compressibility (defined with respect to the re- isproportionaltoN3/2.Thedottedlinesshow,frombottomto duced pressure pˆ) is obtained from Eq. (16) as top,thedependenciesforanunpressurizedring(hR2i=N/12), g for a ring at the critical pressure [hR2i=N3/2/(4π2)] and for κ= 1 ∂A pˆ→1± pˆ 1−γ, γ =1. (18) a stretched circle [hR2i=N2/(4π2)].g A ∂pˆ −−−−→| − | g At the critical point itself the compressibility divergesas 1 κ(pˆ=1)= N1/2. (19) 2 5 Transfer-Matrix Formulation Tocalculatethemean-squareradiusofgyration, R2 = h gi Since the interactions between all adjacent monomers are N−1 r2 (wherer aremeasuredwithrespecttothecen- j j identical,includingtheonebetweenthefirstandtheNth, ter of mass), we add a h r2 term to the Hamiltonian of P j the partition function may be rewritten in the following Eq. (7) and differentiate the resulting partition function transfer-matrix form: P with respect to h. This yields N 1 ∂lnZ(p,N,λ,h) Z(p,N)= Tr T(r ,r ), r =r hRg2(p,N,λ)i= N ∂h = {rj}j=1 j−1 j 0 N (21) 4λ Npcot(Np) (cid:12)(cid:12)(cid:12)h4=λ0 (20) T(r,r′)=ep2(r×Yr′)·ˆzδ(r r′ 1). = − 4λ =(cid:12) A . | − |− Np2 Nph i Solution of the associated eigenvalue problem, For pˆ 1, combining this result with Eqs. (12) and (14), weget≪hRg2i=N/12,whichisthewellknownresultforthe µΨ(r)= dr′T(r,r′)Ψ(r′), (22) mean-square radius of gyration of a Gaussian ring (e.g., Z [17]). For large pressures, pˆ 1, we have from Eq. (14) ≥ (in particular,finding the two eigenvaluesµ oflargestab- N→∞ λ pˆ=pN/(4π),thereby recoveringthe relationfor solute value) will yield the exact solution of the model. an−−a−ve−→rage circle, πhRg2i=hAi. TwopropertiesoftheoperatorT arereadilynoticed:it Figure 3 shows the dependence of R2 on N at fixed is non-Hermitian, T(r,r′) = T(r′,r), and it is rotational- h gi 6 pressure. The data were obtained by substituting the nu- invariant.Toexploittheinvariancetorotations,wechange merical solution for λ [Eq. (13)] in Eq. (20). The scaling to polar coordinates,r=(ρ,ϕ), and separate variablesas of R2 changes at the critical point N = 4π/p. Below Ψ(r) = χ(ρ)eimϕ, where m = 0, 1, 2,... to maintain h gi c ± ± Emir Haleva, Haim Diamant: Smoothening Transition of a Two-Dimensional Pressurized Polymer Ring 5 periodicity in ϕ. Equation (22) can then be integrated ofgyration R2 andmean-squareareafluctuation ∆A2 h gi h i over angles to give as a function of pressure p for different ring sizes N. ρ+1 µχ(ρ)= dρ′T˜(ρ,ρ′)χ(ρ′) Zρ−1 6.1 Numerical Scheme 2cosh pρρ′√1 ∆2/2+imcos−1∆ T˜(ρ,ρ′)= − (23) ρ√1 ∆2 (cid:0) (cid:1) − ρ2+ρ′2 1 ∆(ρ,ρ′)= − , 2ρρ′ thusreducingtheoriginaloperatortotheone-dimensional operator T˜. Unfortunately,we havenotbeenableto diagonalizeT˜ for anyp.Forp p ,nevertheless,the ringhas stretched c ≫ configurations, and we can assume that the distances of all monomers from the center of mass are much larger thanthelink length,ρ 1.We expandχ(ρ′)aroundρto zeroth order,change va≫riables according to ρ′ =ρ+sinθ, and integrate over θ to get µ(0)χ(ρ)=2πI (pρ/2)χ(ρ), (24) Fig. 4. Schematicviewofasectionofthesimulatedpolygon. 0 A randomly chosen vertex (marked by a circle) can be moved where I is the zeroth-order modified Bessel function of 0 toasinglepositiononly(markedbyadiamond)soastomain- the first kind. Thus, at this order of approximation, the tain the lengths of the two links attached to it constant. The eigenfunctionshavetheformχ(0)(ρ)=[N/(2πρ )]1/2δ(ρ area enclosed by the resulting rhomb is the difference in total k k − ρ ) with a continuous spectrum of eigenvalues. The spec- polygon area for thegiven step. k (0) trum is bounded from above by µ =2πI (pρ /2)= max 0 max 2πI [pN/(4π)], where ρ = N/(2π) is the radius of a 0 max WeconsiderapolygonofN equaledges,definedbythe perfect circle of perimeter N. (The value of ρ can also max 2D coordinates of its vertices. An off-lattice simulation be obtained from the condition that χ(ρ)2dρ 1, i.e., | | ≥ is used, i.e., the positions of the vertices are defined in that going from the center of mass outward, one must R continuous space.At each step a randomvertex is chosen cross the ring at least once.) and moved to the only other position that satisfies the Withinthezeroth-orderapproximation,andinthether- edge-length constraint (Fig. 4). The difference in energy modynamic limit, the partition function is given by between the two steps is proportional to the difference in Z(0)(p,N)=[µ(0) ]N =[2πI (pN/4π)]N. (25) total area, which in turn is simply given by the area of max 0 the rhomb composed of the two edges prior to and after Thisresulthasastraightforwardinterpretation.Asshown the move(see Fig. 4). This wayeachstep takesonly O(1) intheAppendix[Eq.(A.1)],itisidenticaltothepartition operations.Themoveissubsequentlyacceptedorrejected function of a 2D, open, freely jointed chain subject to a according to the Metropolis criterion. tensile force f = pN/(4π). This force is just the tension The initial configuration is a stretched, regular poly- associated with a Laplace pressure p acting on a circle of gon.Thisinitialconditionandthedynamicsdefinedabove radius ρ N. The mean area is obtained from Eq. max ∼ imply that the polygon angles are restricted to change in (25) as discrete quanta of 2π/N. Thus, although the algorithm ∂lnZ N2I (pˆ) ± A(pˆ,N) = = 1 , (26) is off-lattice, the simulated ring is a discrete variant of a h i ∂p 4π I0(pˆ) freelyjointedchainwhichstrictlycoincideswiththefreely which saturates, as expected, to A =N2/(4π) as p jointed model only for N . . The approach to saturation is gmiavxen by → Thesimulationswerep→erf∞ormedforN between50and ∞ 3200, and for p between 0 and 4p . Away from the crit- c 1 A(pˆ 1) /A 1 , (27) ical point, the number of steps required for equilibration max h ≫ i ≃ − 2pˆ is O(N3), but near p the simulation length must be ex- c tended due to criticalslowing down [21]. This limited our which corrects the mean-field prediction, Eq. (16), by a investigation of the transition to N .3000. factor of 2. 6 Monte Carlo Simulations 6.2 Results Since our Flory argument and mean-field theory may fail Figure 5 shows simulation results for the mean area as near the critical point, we conducted Monte Carlo simu- a function of pressure for different values of N. When p lations to obtain the mean area A , mean-square radius is scaled by p N−1 and A by N (below p ) or by c c h i ∼ h i 6 Emir Haleva, Haim Diamant: Smoothening Transition of a Two-Dimensional Pressurized Polymer Ring of the compressibility is within the standard error of the A fitwhilethatofthemeanareadiffersby4standarderrors. We also measuredfrom simulations the dependence of 0.8 N=50 the mean-square radius of gyration on N, as depicted in N=100 N=200 Fig. 6B. As predicted, below the critical pressure we find N=400 R2 N1.01±0.02, and above it R2 N1.985±0.011. At 0.6 MN=e8a0n0 Field (N→∞) ph =gip∼we get h gi ∼ c N 〉A/0.4 hRg2(p=pc)i=(0.043±0.025)N1.46±0.13. (30) 〈 0.2 A 0 0 0.2 0.4 0.6 0.8 10000 ^ p 〈A(p=p )〉 B c 1 1000 0.8 100 ax0.6 κ(p=p ) m c A 〉/ 10 A 0.4 N=50 〈 N=100 N=200 N=400 50 100 200 400 800 1600 3200 0.2 N=800 N Mean Field (N→∞) B 0 1 1.5 2 2.5 3 3.5 4 ^p 10000 p=2p c 1000 Fig. 5. Mean area as a function of pressure below (A) and 2 p=p above (B) the critical point. The pressure is scaled by pc = 〈Rg〉 c 4π/N, and the area by N in (A) and by Amax = N2/(4π) in (B). Symbols show the results of MC simulations for different 100 p=1p 2 c valuesofN.Thedashedlinesshowthepredictionofthemean- field theory in thelimit N →∞. 10 A N2 (above p ), the data below and above the 50 100 200 400 800 1600 max c ∼ N transition collapse onto two universal curves, thus con- firming the predicted scaling laws for the crumpled and smooth states (see Sections 3 and 4). However, while the data well below pc coincide with the scaling function ob- Fig. 6. (A) Mean area and compressibility at p = pc as a tainedfromthemean-fieldapproximation,thedataabove functionofN obtainedbyMCsimulations.Thefits,hA(pc)i= the critical pressure collapse onto a different curve. (0.102±0.007)N1.49±0.01 andκ(pc)=(0.56±0.09)N0.495±0.025, The simulation results for the mean area and com- are given by the solid lines. (B) Mean-square radius of gy- pressibility atp=pc asa function of N are shownin Fig. ration as a function of N at different values of p = 21pc, pc 6A. The reduced compressibility, defined in Eq. (18), was and 2pc with the best fits (solid lines) hRg2(21pc)i = (0.12± calculated from the measured mean-square area fluctua- 0.01)N1.01±0.02, hRg2(pc)i = (0.043 ± 0.025)N1.46±0.13, and tion as κ=(4π/N) ∆A2 / A . We get hRg2(2pc)i=(0.019±0.001)N1.985±0.011. h i h i A(p=p ) =(0.102 0.007)N1.49±0.01 (28) c h i ± κ(p=p )=(0.56 0.09)N0.495±0.025. (29) ForillustrationweshowinFig.7fourrandomlychosen c ± conformationsofan1600-segmentringatthecriticalpres- Hence, the mean-field exponents for the scaling with N, sure. The shapes vary in size significantly due to critical Eqs.(16)and(19),areconfirmed.Thepredictedprefactor fluctuations. Emir Haleva, Haim Diamant: Smoothening Transition of a Two-Dimensional Pressurized Polymer Ring 7 Fig.7. FourrandomconformationsofaringwithN =1600atthecriticalstateasobtainedbyMCsimulations,demonstrating the critical fluctuations. (The positions of the rings have no significance.) The spacing between gridlines is ten times the link length. The dotted circle, shown for reference, has an area equalto themean area of the ring at thisstate. The simulation results for the smooth state, p > p , 6.3 Critical Exponents c areshowninFig.8,wheretheycanbecomparedwiththe transfer-matrix calculation, Eq. (26), and the mean-field result,Eqs.(10)and(11).Onthe one hand,there is good We have confirmed the predicted exponents relating A , agreement with the transfer-matrix calculation for pˆ&5, hRg2iandκatp=pc withthesystemsizeN.Wenowthurin particularly compared to the mean-field result. On the totheexponentscharacterizingthedivergencewith pˆ 1. | − | otherhand,themean-fieldtheorysucceedsinreproducing As in any continuous phase transition, the critical fluctu- the phase transition, whereas the zeroth-order transfer- ations make it hard to accurately measure these critical matrix calculation is invalid for these low pressures. exponents from simulations. Instead, we choose the more reliable route of finite-size scaling to obtain the relations between them. Let us divide the ring at the critical state into Pincus correlation blobs [17,22] of g monomers and diameter ξ each, such that within a blob the polymer behaves as an unperturbedrandomchain,i.e.,g ξ2,whereasthechain 1 ∼ ofblobsisstretchedbythe pressure.Theperimeterofthe ring is R (N/g)ξ N/ξ. We have already established 0.8 that the p∼erimeter o∼f the ring at p = p scales as R c N3/4. (See Fig. 6B.) Thus, for a finite-size system, we ge∼t ax0.6 a correlationlength which scales at p=pc as m A 〉A/ 0.4 ξ(p=pc)∼N1/4. (31) 〈 On the other hand, close to the critical point the correla- 0.2 tion length diverges as ξ pˆ 1−ν, the compressibility as κ pˆ 1−γ, and th∼e |ord−er|parameter increases as 0 M ∼pˆ| 1−β.|Using Eq. (31) and the numerically estab- 0 5 10 15 20 lish∼ed|re−sul|ts, M = R/N N−1/4 and κ N1/2, we ^ p ∼ ∼ obtain the relations Fig. 8. Mean area(scaled byAmax)asafunction ofpressure (scaledbypc),asobtainedbythezeroth-ordertransfer-matrix β =ν, γ =2ν, (32) calculation (solid curve), mean-field approximation (dashed curve),and MC simulations for N =1600 (error bars). whichholdforthemean-fielduniversalityclass.(Notethat theexponentsαandδ areirrelevantforthissystem,since both the ordering field and temperature are incorporated in the single parameter p.) 8 Emir Haleva, Haim Diamant: Smoothening Transition of a Two-Dimensional Pressurized Polymer Ring 7 Discussion ring contains n = N/g N1/2 such blobs. Hence, the ∼ total deviation of the geometricalarea from the algebraic one is (n a2 )1/2 N3/4. In the limit N this is We have demonstrated that the swelling of a 2D freely h i ∼ → ∞ negligible compared to the mean area of the ring at the jointedringduetoapressuredifferentialexhibitsasecond- critical state, A N3/2. Thus, we conjecture that the order smoothening transition of the mean-field univer- h i ∼ samesmootheningtransitionastheonereportedherewill sality class. Below the critical pressure the ring behaves be found also in a model which considers the geometrical as a random walk, with both the mean area A and mean-square radius of gyration R2 proportionhal ito N. area rather than the algebraic one. h gi WethankM.Kroyter,H.Orland,P.PincusandT.Witten In this crumpled state the mean area obeys the scaling A = Nf<(p/p ). Mean-field theory accurately captures for helpful discussions. This work was supported by the US– thhei scalingAlaw cas well as the scaling function f<. (See Israel Binational Science Foundation (Grant no. 2002-271). A Fig. 5A.) This lies in the fact that, for an unstretched chain, the Gaussian-spring description in the mean-field Appendix: 2D Freely Jointed Chain under Ten- calculation and the actual freely jointed model coincide as N . sion →∞ Above the critical pressure A and R2 are propor- h i h gi tional to N2. For this smooth state we have found a new InthisAppendixwerecalltheresultsforthepartitionfunction scalingbehavior, A =N2f>(p/p )(Fig.5B).Themean- of a 2D, open, freely jointed chain subject to a tensile force. h i A c These results are used in Sections 3 and 5. fieldtheorycorrectlygivesthescalinglaw,yetfailstopre- dict the correct scaling function f>. This is because in a Consider a 2D freely jointed chain composed of N links of A length l ≡ 1. A chain configuration is defined by N 2D unit stretched state the entropy of a chain of Gaussian, vari- able springs is much larger than that of a freely jointed vectors, {uj}j=1,...,N, specifying the orientations of the links. Oneend of thechain is held fixed while theotheris pulled by chain of rigid links. For p p we have calculated f> ≫ c A aforce f (measured in unitsof kBT/l).Thepartition function exactly [Eq. (26) and Fig. 8]. of thechain is At the transition point itself a new state of polymer chains has been discovered, with A and R2 propor- N h i h gi Z(f,N)= du ef·uj otiuotnatol tboeNid2eνnct,iνccal=to3/t4h.atTohfeasw2eDllisneglf-eaxvpooidnienngtwνcaltku,ranls- Z jY=1 j (A.1) 2π N though the physical origins of the two exponents are un- = dθefcosθ =[2πI0(f)]N. related; the freely jointed ring in this intermediate state (cid:18)Z0 (cid:19) betweencrumpledandsmoothbehaviorscontainsnumer- Themean end-to-endvectorisobtained from Eq.(A.1)as ous intersections, as is clearly seen in Fig. 7. self-Oinutrerdseiscctuinsgsiocnhahinass. bWeeenharveestcroicmtemdetnotefdreienlySejcotiniotned3, R=∇flnZ =NII10((ff))ˆf. (A.2) that the addition of self-avoidance to the Flory free en- The mean end-to-end distance in the limit of weak force, to ergy removes the second-order transition. Thus, the cur- two leading orders, is rent work suggests a different perspective for the previ- ously studied swelling behavior of 2D self-avoiding rings R/N ≃f/2−f3/16, (A.3) [6,7]. The large-inflation scaling regime thoroughly dis- leading, upon inversion, to cussed in those works can be viewed as a broadening of a phase transitionpresent in the non-self-avoidingmodel, f ≃2R/N +(R/N)3. (A.4) the self-avoidance thus acting as a relevant parameter for the critical point discussed in the current work. Finally, the free energy for fixed end-to-end distance (to two Oneofthestrongestassumptionsunderlyingouranal- leading orders in small R) is ysis,as wellasthose ofRefs.[9,10,11], is the replacement R2 R4 of the actual geometrical area of the ring with its alge- F(R)=−lnZ[f(R)]+f·R≃ + . (A.5) braic area. 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