We determine the Poisson ratio of self-avoiding fixed-connectivity membranes,
modeled as impenetrable plaquettes, to be sigma=-0.37(6), in statistical
agreement with the Poisson ratio of phantom fixed-connectivity membranes
sigma=-0.32(4). Together with the equality of critical exponents, this result
implies a unique universality class for fixed-connectivity membranes. Our
findings thus establish that physical fixed-connectivity membranes provide a
wide class of auxetic (negative Poisson ratio) materials with significant
potential applications in materials science.Comment: 4 pages, 3 figures, LaTeX (revtex) Published version - title changed,
  one figure improved and one reference change

A. Cacciuto

A. E. H. Love

A. Travesset

B. Duplantier

B. Efron

C. F. Schmidt

D. J. Weidner

D. R. Nelson

E. Guitter

F. David

F. F. Abraham

G. Thorleifsson

J. A. Aronovitz

J. H. Fendler

J. Shao

K. E. Evans

K. J. Wiese

L. D. Landau

M. Bowick

M. Falcioni

M. Goulian

M. J. Bowick

M. Kardar

M. Paczuski

M. S. Spector

N. Keskar

P. Le Doussal

R. Lakes

T. Hwa

Y. Kantor

Z. Zhang

English

arXiv

We determine the Poisson ratio of self-avoiding fixed-connectivity membranes, modeled as impenetrable plaquettes, to be sigma=-0.37(6), in statistical agreement with the Poisson ratio of phantom fixed-connectivity membranes sigma=-0.32(4). Together with the equality of critical exponents, this result implies a unique universality class for fixed-connectivity membranes. Our findings thus establish that physical fixed-connectivity membranes provide a wide class of auxetic (negative Poisson ratio) materials with significant potential applications in materials science

Bowick, Mark

Cacciuto, Angelo

Thorleifsson, G.

Travesset, A.

Syracuse University Research Facility and Collaborative Environment

Syracuse University SURFACE Physics College of Arts and Sciences 9-16-2001 Universal Negative Poisson Ratio of Self Avoiding Fixed Connectivity Membranes Mark Bowick Department of Physics, Syracuse University, Syracuse, NY Angelo Cacciuto Syracuse University G. Thorleifsson DeCODE Genetics, Lynghalsi A. Travesset Loomis Laboratory, University of Illinois at Urbana Follow this and additional works at: https://surface.syr.edu/phy  Part of the Physics Commons Recommended Citation Bowick, Mark; Cacciuto, Angelo; Thorleifsson, G.; and Travesset, A., "Universal Negative Poisson Ratio of Self Avoiding Fixed Connectivity Membranes" (2001). Physics. 158. https://surface.syr.edu/phy/158 This Article is brought to you for free and open access by the College of Arts and Sciences at SURFACE. It has been accepted for inclusion in Physics by an authorized administrator of SURFACE. For more information, please contact surface@syr.edu. arXiv:cond-mat/0103173v2  [cond-mat.soft]  16 Sep 2001The Universal Negative Poisson Ratio of Self-Avoiding Fixed-ConnectivityMembranesM. Bowick,1 A. Cacciuto,1 G. Thorleifsson2 and A. Travesset31 Physics Department, Syracuse University, Syracuse NY 13244-1130, USA2 DeCODE Genetics, Lynghalsi 1,IS-110, Reykjavik, Iceland3 Loomis Laboratory, University of Illinois at Urbana, Urbana IL 61801, USAWe determine the Poisson ratio of self-avoiding fixed-connectivity membranes, modeled as impenetrable plaque-ttes, to be σ = −0.37(6), in statistical agreement withthe Poisson ratio of phantom fixed-connectivity membranesσ = −0.32(4). Together with the equality of critical expo-nents, this result implies a unique universality class for fixed-connectivity membranes. Our findings thus establish thatphysical fixed-connectivity membranes provide a wide classof auxetic (negative Poisson ratio) materials with significantpotential applications in materials science.Fixed-connectivity (also known as polymerized, teth-ered or crystalline) membranes are fluctuating and flex-ible fishnet-like two-dimensional surfaces with nodes offixed coordination number (for two recent reviews see[1,2]). Physical examples include such naturally occur-ring structures as polymerized Langmuir-Blodget films[3,4] and the spectrin/actin cytoskeleton [5] of erythro-cytes (mammalian red blood cells). A wide varietyof additional examples is discussed in [1]. Currentadvances in soft condensed matter experimental tech-niques suggest the likelihood of many new realizations offixed-connectivity membranes such as cross-linked DNAnetworks as well as composite structures that includefixed-connectivity membranes as fundamental ingredi-ents. One universal and remarkable feature of the low-temperature (so-called flat) phase of non-self-avoiding(phantom) fixed-connectivity membranes is that they ex-pand transversely when stretched longitudinally [6–9]. Inother words, they exhibit a negative Poisson ratio [10].Such materials have been dubbed auxetic [11].In this letter we estimate, via Monte Carlo simula-tions, the Poisson ratio of physical self-avoiding fixed-connectivity membranes. We establish that they are alsoauxetic materials with a Poisson ratio and roughness ex-ponent in statistical agreement with those of flat phan-tom membranes. Thus there appears to be a uniqueuniversality class of flat fixed-connectivity membranes,whether they arise from high bending rigidity or self-avoidance. Direct experimental measurements should beable to measure this negative Poisson ratio and test uni-versality. The remarkable properties of auxetic mem-branes suggest a rich new avenue of exploration in mate-rials synthesis.A fixed-connectivity membrane may be modeled as anelastic surface with bending rigidity and self-avoidance[12–15]. The free energy is composed of an elastic and aself-avoiding contribution:Ffcm = Fel + Fsa . (1)The elastic free-energy Fel is given byFel (u, h) =∫d2x[κ2(∂α∂βh)2 + µu2αβ +λ2u2αα](2)where u denotes phonon modes, h denotes height modes,κ is the bending rigidity and λ and µ are the classicalLamé coefficients. The self-avoiding free-energy we taketo be of the Edwards formFsa =b2∫d2x∫d2yδ3(~r(x) − ~r(y)) , (3)where b determines the strength of self-avoidance. Thestrain tensor uαβ [10] is related to the embedding ~r(x),defining the membrane by~r(x) = x + u(x) + ẑh(x) (4)uαβ =12(∂αuβ + ∂βuα + ∂αh∂βh)Combined efforts both on the analytical and numericalside have led to a complete clarification of the phase dia-gram of the phantom case (b = 0) and detailed estimatesfor the critical exponents, as shown in Fig. 1 and Table I.The phase diagram consists of a crumpled phase (associ-ated with the Gaussian fixed point GF) and a flat phase(associated with a flat phase fixed-point FL), with an in-termediate infrared-unstable crumpling transition (CT),as depicted in Fig. 1.The self-avoiding model of Eq. (1) with no bendingrigidity (κ = 0) has proven to be tractable numerically[16]. The model possesses a unique infrared fixed point(SA) describing a flat phase, with detailed results forcritical exponents given in Table I. From the analyticalstandpoint there is evidence for a unique SA [17,18], butthe calculations are inconclusive on whether it describesa flat or a crumpled phase [19–24].A look at Table I reveals that the critical exponentsand the Poisson ratio of both phantom and self-avoidingmembranes coincide within the error bars quoted, im-plying that the FL and the SA are equivalent. This is1OO00 GF                                                                                                                                                     bCT FLκSAFIG. 1. The renormalization group flows in thetwo–dimensional space of couplings for a fixed-connectivitymembrane with bending rigidity (κ) and self-avoidance (b).The phantom model (b = 0) has two infrared stable fixedpoints, the crumpled phase (GF) and the flat phase (FL),with an intermediate continuous crumpling transition as-sociated with the infrared-unstable fixed point (CT). Thepure self-avoiding model with no microscopic bending rigidity(κ = 0) has a infrared stable self-avoiding fixed point (SA).There is a line of equivalent fixed-points joining the FL andthe SA (the solid (red) line), thus defining a redundant direc-tion in κ − b space.a very surprising result since it means that the samelong wavelength limit is reached via two very differentroutes: either sufficiently large bending rigidity or strongself-avoidance in the absence of bending rigidity. Thisis even more remarkable considering the very differentshort-distance structure of the two models: a membranewith large bending rigidity is very smooth at short dis-tances while a purely self-avoiding membrane in the ab-sence of bending rigidity is extremely rough.In light of our results, we suggest the renormalizationgroup (RG) flows depicted in Fig. 1. Given these flowsand the equivalence of the SA and the FL, there must be afull line of equivalent fixed points b(κ) joining them, withno RG flow on the line (corresponding to a marginal di-rection). Since these are the infrared stable fixed pointsof the system, all the relevant physics is described bythis line. The crumpling transition present for phantommembranes may be reached only by an extremely pre-cise tuning of the parameters involved in the problem.This would make the crumpling transition in the modeldescribed very difficult to verify experimentally.We see that the combination of fixed-connectivity (in-tegrity of the lattice) and self-avoidance sufficiently re-PHANTOMMC ε-expansion SCSA Large dν 0.95(5) [25] 1 1 1ζ 0.64(2) [25] 13/25 [6] 0.59 [7] 2/3 [26]σ −0.32(4) [9] −1/5 [6] −0.33 [6]SELF-AVOIDING EXPERIMENTSMC-BS MC-IPν 1 [27] 0.97(4) [16] 0.93(5) [28]ζ 0.65 [27] 0.64(2) [16] 0.65(10) [5]σ -0.37(6)TABLE I. Critical exponents and Poisson ratio of flatfixed-connectivity membranes in the phantom (MC:MonteCarlo; SCSA:Self-consistent screening approximation) andthe self-avoiding case (MC-BS:Monte Carlo with Balls andSprings models; MC-IP:Monte Carlo with impenetrable mod-els).stricts the entropy of crumpled configurations as to de-stroy the crumpled phase. It was already observed in [29]that next-to-nearest neighbor self-avoidance, discretizedby hard-sphere potentials, induces a positive bendingrigidity. On the other hand the impenetrable plaquettemodel treated here is very flexible, since only strictly self-intersecting configurations are forbidden, and hence thephysical mechanism flattening the membrane is clearlymore general than the simple induced bending rigiditydiscussed above.We performed a Monte Carlo simulation of a suitablediscrete version of the model (Eq. (1)), as described in[16]. The Poisson ratio σ for a two-dimensional systemdeformed from its mean length l by δl is determined byσ = −δw/wδl/l, (5)where δw/w measures the fractional change in the trans-verse extent (width) of the system. In [9] it is shown thatlinear response theory givesσ = −〈uxxuyy〉c〈uyy2〉c, (6)where 〈u2〉c is the connected statistical average overMonte Carlo configurations and u is the spatial averageover the surface.Our results are presented in Fig. 2(A) and show thatthe Poisson ratio σ = −1/3 obtained in simulations offlat phantom fixed-connectivity membranes is also a goodestimate of the Poisson ratio for the pure self-avoidingmodel. Error bars have been calculated with the Jacknifemethod [30,31].To calibrate the influence of the boundary on our re-sults we calculated the Poisson ratio excluding concentric20.280.320.360.400.440.48 A−σ20 35 50 65 80 95L0.280.320.360.400.440.48 L = 95L = 65L = 49L = 33−σBFIG. 2. Poisson ratio of a self-avoiding fixed-connectivitymembrane as a function of system size (A). Poisson ratioof reduced lattices compared with non-truncated ones (B).The straight dashed line indicates in both cases the SCSAanalytical result σ = −1/3.outer shells of nodes neighboring the boundary. In par-ticular we performed the numerical analysis discardingshells of boundary nodes of increasing extent. The re-sults are presented in Fig. 2(B).The exclusion of larger shells of boundary sites slightlyincreases the absolute value of the Poisson ratio. For suf-ficiently small reduced linear size, |σ| begins to decreasedue to finite size effects. There is consequently competi-tion between boundary and finite size effects. Finite sizeeffects become important only when a sizeable fractionof nodes near the boundary are excluded. Note that wecan also make a consistent check of our analysis by sys-tematically excluding shells of nodes from the boundaryin towards the center of the lattice and comparing thelattices of reduced size thus obtained with equal volumenon-truncated lattices.Thus we can compare, for example, the L = 65 resultwith that from the reduced L = 95 lattice and likewisethe L = 49 result with that from the reduced L = 95 andL = 65 lattices. The matching given by this comparisonis consistent. In fact we are able to reproduce the L =65, L = 49 and L = 33 results simply by reducing theL = 95 lattice. The deviations found in this comparisonare another measure of boundary effects.Traditional materials get thinner when stretched andfatter when squashed, since it is typically difficult to in-(b)(a)TTTT TTTTFIG. 3. Mechanical model of an auxetic material: (a) inthe absence of applied stress and (b) under applied lateralstress T . The lateral stretching accompanying the appliedstress forces the material out in the transverse dimension.crease their volume very much when deformed.1 In theunusual world of auxetics the opposite happens, with anumber of interesting implications and potential appli-cations. There are several well known auxetic materials.The earliest example, dating from more than a centuryago, is that of a pyrite crystal [32], which has a Poissonratio in certain directions of σ ∼ −0.14. More recentlysome polyester foams under certain pressure conditionshave proved to be isotropic auxetic materials with Pois-son ratios as large as σ ∼ −0.7 [33]. A nice mechanicalmodel for auxetic materials was given in [11] (see Fig. 3).One of the rare naturally occurring auxetics is SiO2 inits α-crystobalite phase [34,35].The underlying mechanism driving fixed-connectivitymembranes auxetic has some similarities to that illus-trated in Fig. 3. Submitting a membrane to tension willsuppress its out-of-plane fluctuations, forcing it entropi-cally to expand in both in-plane directions. More physi-cally, the out-of-plane undulations renormalize the elasticconstants (the Lamé coefficients), in such a way that thelong-wavelength bulk modulus is less than the shear mod-ulus, which is the signature of a two-dimensional auxeticmaterial.Auxetic materials have desirable mechanical propertiessuch as higher in-plane indentation resistance, transverseshear modulus and bending stiffness. They have clearapplications as sealants, gaskets and fasteners. They mayalso be promising materials for artificial arteries, sincethey can expand to accomodate sudden increases in blood1A volume preserving deformation has Poisson ratio 0.5.Any value less than 0.5 involves some increase in volume un-der deformation. A negative Poisson ratio implies a very largevolume increase.3flow.We can model a realistic fixed-connectivity membranewith an elastic free energy and either large bending rigid-ity or self-avoidance. This is of practical importance inmodeling since, for example, we may replace the morecomplicated non-local self-avoidance term with a largebending rigidity. It remains an important theoreticalchallenge to verify this conclusion analytically.In this letter we have ignored the role of topologicaldefects. We think that a sufficiently large defect densitymay affect the actual value of the Poisson ratio, but adetailed discussion of this topic is beyond the scope ofthis paper.It is our hope that the results presented in this letterwill encourage materials scientists and condensed matterexperimentalists to further study the elastic and mechan-ical properties of fixed-connectivity membranes and, inparticular, to measure the Poisson ratio of these novelsystems.The work of M.B., A.C. and A.T. has been supportedby the U.S. Department of Energy (DOE) under contractNo. DE-FG02-85ER40237. A.T. acknowledges fundingfrom the materials computation center, grant NSF-DMR99-76550 and NSF grant DMR-0072783.[1] M. Bowick and A. Travesset, The Statistical Mechan-ics of Membranes, Phys. Rep. 344, 255–310 (2001); in“Renormalization Group Theory at the Turn of the Mil-lennium,” D. O’Connor and C.R. Stephens (Eds.).[2] K.J. Wiese, Polymerized Membranes, a Review, in“Phase Transitions and Critical Phenomena,” Vol. 19,253–480: C. Domb and J.L. Lebowitz (Eds.), AcademicPress (2001).[3] J.H. Fendler and P. Tundo, Acc. Chem. Res. 17, 3 (1984).[4] J.H. Fendler, Chem. Rev. 87, 877 (1987).[5] C. F. Schmidt, K. Svoboda, N. Lei, I. B. Petsche, L. E.Berman, C. R. Safinya, and G. S. Grest, Existence of aflat phase in red cell membrane skeletons, Science 259,952 (1993).[6] J.A. Aronovitz and T.C. Lubensky, Fluctuations of solidmembranes, Phys. Rev. Lett. 60, 2634 (1988).[7] P. Le Doussal and L. Radzihovsky, Self consistent the-ory of polymerized membranes, Phys. Rev. Lett. 69, 1209(1992).[8] Z. Zhang, H.T. Davis and D.M. Kroll, Molecular dy-namics simulations of tethered membranes with perioidicboundary conditions, Phys. Rev. E53, 1422 (1996).[9] M. Falcioni, M. J. Bowick, E. Guitter, and G. Thorleifs-son, The Poisson ratio of crystalline surfaces, Europhys.Lett. 38, 67 (1997): cond-mat/9610007.[10] L. D. Landau and E. M. Lifshitz, Theory of Elasticity,Vol. 7 of Course of Theoretical Physics; Pergamon Press,Oxford, UK (1986).[11] K.E. Evans, M.A. Nkansah, I.J. Hutchinson, and S.C.Rogers, Nature 353, 124 (1991).[12] Y. Kantor, M. Kardar, and D.R. Nelson, Statistical me-chanics of tethered surfaces, Phys. Rev. Lett. 57, 791(1986).[13] D.R. Nelson and L. Peliti, J. Phys. (France) 48, 1085(1987).[14] Y. Kantor, M. Kardar, and D.R. Nelson, Tethered sur-faces: statics and dynamics, Phys. Rev. A 35, 3056(1987).[15] M. Paczuski, M. Kardar, and D.R. Nelson, Landau The-ory of the Crumpling Transition, Phys. Rev. Lett. 60,2638 (1988).[16] M. Bowick, A. Cacciuto, G. Thorleifsson, and A. Trav-esset, Universality Classes of Self-Avoiding Fixed-Connectivity Membranes, Eur. Phys. J. E 5, 149 (2001):cond-mat/0006383.[17] B. Duplantier, Phys. Rev. Lett. 58, 2733 (1987).[18] M. Kardar and D. R. Nelson, ǫ expansion for crumpledmanifolds, Phys. Rev. Lett. 58, 1289 (1987).[19] T. Hwa, Phys. Rev. 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Universal Negative Poisson Ratio of Self-Avoiding Fixed-Connectivity Membranes

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Universal Negative Poisson Ratio of Self Avoiding Fixed Connectivity
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Universal Negative Poisson Ratio of Self Avoiding Fixed Connectivity Membranes

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