Kantor, Kardar, and Nelson Reply

Abstract

Kantor, Kardar, and Nelson Reply:Recently we reported some results on statistical mechanics of two-dimensional networks. 1 Because of entropy effects, a self-avoiding manifold assumes a typical radius R that grows with its linear size L as R~~L V . Monte Carlo simulations in three dimensions lead to v~0.8 in good agreement with a Flory estimate of j. For strips of width w and length L, we suggested a scaling form /£ -(wL 3 ) 1/5 which interpolates between R~~L Af5 for w~~ L and R~~L 3/5 for vv^CL. Note that j is the Flory estimate for a free polymer in 3D. The crumpled balls obtained by crushing strips of paper 2 ' 3 are nonequilibrium configurations that belong to an ensemble entirely different from the equilibrium ones mentioned earlier. Here we briefly remark on similarities and differences between these two ensembles, and present a simple explanation for the experimental results of the preceding Comment. 3 Consider a Z)-dimensional manifold (D = 1 for polymers, and Z)=2 for surfaces) crumpled under strong pressure in d dimensions. If the network collapses to a compact object whose fractal dimension equals the dimension d of space, the radius grows as R~~L Vc with v c =D/d (since mass ~~L D -R d ). A string of length L can easily be compacted in this fashion into a ball of radius R~~L Vl in d = 3. It is, however, much more difficult for surfaces to achieve this compaction without tearing. 4 This is why crumpled aluminum foils 1 and crushed pieces of paper 2 ' 3 result in a nontrivial exponent V2 « 0.8 > j. The interior of a crushed paper ball (seen by gently opening the ball) looks rather similar to pictures of free surfaces from Monte Carlo simulations, which may account for the similarity in the exponent v for the two systems.For the compacted strips of length L and width w studied by Gomes and Vasconcelos, 3 we would expect R~~w a L^ with an exponent that interpolates between compacted strings (R~~~L 1/3 ) for L»H>, and crushed surfaces (R~~L V2~~ ) for L -w. This leads to the conclusion a = y~~0.33 and P = V2-y--0.47. These ex-ponents are quite consistent with the results reported in the preceding Comment. 3 Indeed, that agreement seems to improve for samples including a larger range of L/w, and hence more likely to be compacted to a uniform density.

Randomly charged polymers, random walks, and their extremal properties

“…A reasonable compromise between stretching (which minimizes the electrostatic energy) and remaining compact (which gains in condensation energy) is for the PA to form a necklace of weakly charged blobs connected with highly charged "necks", by taking advantage of the charge fluctuations along the chain. The results of the Monte Carlo [7] and exact enumeration [8] studies qualitatively support such a picture. An example of such a low energy configuration is shown in Fig.…”

“…For the N-dependence of Q R in general space dimensions, see Ref. [8].) While the above arguments suggest that a typical PA should stretch out at low T , such stretching may lead to a loss of the condensation energy.…”

“…From a detailed study of the Q o -dependence of R g , the following picture began emerging [7,8]: Consider a dense (globular, approximately spherical) low-T state of the PA. Its energy can be roughly separated into three terms, as…”

Collapse of Randomly Self-Interacting Polymers

“…Our results were supplemented by MC simulations for L ≤ 95, using the pivoting method [17]. (The method provides excellent equilibration at high and intermediate temperatures, but does not permit equilibration of almost compact structures [18]. )…”

Dual network model for red blood cell membranes

“…In this paper, we present a mod~l for RBC membranes which draws its inspiration from computer simulations developed for both polymerized [7,81 and fluid 191 membranes. We capture what we believe to be the two essential aspects of the RBC membrane.…”