Floppy Tethered Networks

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Submitted on 1 Jan 1993

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D. Kroll, G. Gompper

To cite this version:

D. Kroll, G. Gompper. Floppy Tethered Networks. Journal de Physique I, EDP Sciences, 1993, 3 (5),

pp.1131-1140. �10.1051/jp1:1993261�. �jpa-00246785�

J. Pllys. I France 3

(1993)

i131-i140 MAY1993, PAGE i131

Classification Physics Abstracts

68.10 82.70 87.20

Floppy Tethered Networks

D-M-

Kroll(~)

_{and G.}

Gompper(~)

l~) _{I~lstitut fbr} Festk6rperforscllu~lg, KFA Jfilicll, Postfacll 1913, 5170 Jfilicll, Germany

(~) Sektio~l Pllysik der Ludo4g-Maximilians- U~liversitit Mfi~lclle~l, Tlleresie~lstr. 37, 8000 Mbnclle~l 2, Germany

(Received

22 December 1992, accepted in final form 14

January1993)

Abstract. A model for extremely flexible tethered membranes is studied by Monte Carlo

simulations and scaling arguments. In contrast to the standard string-and-bead models, _no

finite-range hard-core repulsion is used to _ensure self-avoidance.

Instead,

_{the elementary trian-} gles _are taken _to be impenetrable. Although this leads to _an extremely floppy tethered network,

the surface is found to be asymptotically flat, with _a roughness exponent ( cf 0.7, consistent

with the result of self-avoiding string-and-bead models. The orientationally

averaged

scattering intensity, _on the other hand, is found to exhibit _a nontrivial scaling behavior characteristic of

a crumpled object with _an effective fractal dimension df _cf 2.7. This result is compared with

recent experiments _on graphite oxide sheets.

1 Introduction.

Tethered surfaces _are idealized models of flexible two-dimensional solid sheets which _are free to fluctuate in three dimensional _space [1, 2]. ^{These surfaces have} a finite shear modulus _so that the statistical mechanics of tethered networks is controlled

by

the delicate

interplay

between the

in-plane

elastic modes and the

out-of-plane

undulation modes [3, 4]. ^Two realizations of

these

objects

which have

recently

^attracted _a great deal of attention _are exfoliated

graphite

oxide

crystals

_[5] and the

spectrin

network of red blood cells [6]. ^The

experiments

_on

graphite

oxide membranes [5] suggest ^{that their} conformational structure is

crumpled,

with _a fractal dimension df m 2.5.

Computer

simulations of

self-avoiding

tethered

surfaces,

_on the other

hand,

find flat

[7-9],

compact [10], ^{and folded} structures

[I Ii, but,

with _two

possible exceptions [12-14],

not the

crumpled

behavior observed in

experiment.

Current

experimental

studies of tethered networks _measure the

directionally averaged

struc- ture factor

S(q).

In _an

isotropic crumpled phase,

in which the radius of

gyration

^{scales like}

Rg

+~

L~,

where L is the

(internal)

linear dimension of the

object,

the

directionally

avera-

ged

structure function has the

scaling

form

S(q,L)

=

il(qRg)

_[5,

10,

15]. ^For _q <

2ir/Rg,

S(q)

_m I

(qRg)~/3

+

O(q~),

while for

2irla

_{> q >}

2ir/Rg,

^where a is _on the order of the

intrinsic thickness of the

membrane,

_one expects

S(q, L)

*

(qRg)~~~, (i)

with df ₌

2/v.

In the

experiments

described in reference [5] ^the _presence ^of a

scaling region

consistent with

(I)

_was

interpreted

to _mean that the membrane _was fractal

(at

the

length

scales

studied).

In contrast, ^for an

anisotropic,

flat

membrane,

for which the

eigenvalues

of the moment of inertia tensor scale _as

13 °~12

°~

L~ and

ii

_°~

L~t,

with

(

<

I,

_one expects

S(q)

+~

q~~

for

q < I

If,

where £

depends

_on the

amplitude

of the

in-plane

and

out-of-plane

fluctuations

[16].

For I

If

< _q < I

la,

_on the other

hand,

it has been shown that either

S(q)

+~

q~~+t (if in-plane

fluctuations _are either absent _or of small

amplitude),

_or

S(q)

_+~

q~~/t+~ (in

the _case of

large in-plane fluctuations)

_[16]. Since recent simulation studies of tethered networks with free

edge boundary

conditions indicate that

(

m

0.65,

there should be _{a crossover} from

q~~

to either

q~~

^~~ _or q~~.~~ ^behavior with

increasing

_q, in

disagreement

with the experimental result [5]

discussed above.

With the

exception

of the results

presented

in references [13 ^and

14],

recent simulation studies of twc-dimensional

self-avoiding

tethered networks

(without

attractive

interparticle interactions)

all indicate that the membrane is in the

anisotropic,

flat state described in the last

paragraph. Furthermore,

it is not clear at present ^{if and when the} _presence ^of attractive

interactions _can lead _to

a

crumpled phase.

Abraham and Kardar

[I

I] ^{studied tethered networks} in which non-nearest

neighbor particles

interact

through

_a Lennard-Jones

potential

truncated

at 2.5a

(where

_a is the interatomic

separation

at which the

potential

_goes

through zero).

Instead of

crumpling, they

^{found that} the membrane _goes

through

_{a sequence} of

"folding"

transitions with

decreasing

temperature ^which terminate

ultimately

in _a

symmetric collapsed

state characterized

by df

= 3.

Quite recently, however,

Liu and Plischke [12] ^claim to find _a

crumpled

state characterized

by

df SS 2.5 for _a range of temperatures for _a model tethered

membrane in which the

particles

interact

through

^hard-core

repulsion

as well _{as a}

longer-range

attractive

square-well potential.

More work is needed to

clarify

^which of these scenarios

actually

occur for _a

given

interaction

potential.

In this paper we present results for the conformation and

scaling

behavior of two-dimensional tethered networks obtained

using

_a model introduced in references [13 ^and 14] ^for

extremely

flexible

triangulated

surfaces. In this

model,

self-avoidance is

guaranteed by requiring

that the

elementary

surface

triangles

do not intersect. In this way it is

possible

to avoid the _use of

finite-range repulsive potentials

between

vertices;

this results in _{a very} flexible surface which

can fold in _on itself without _{any cost} in energy. The

resulting

surface is therefore much

rougher

than the

commonly

studied

string-and-bead

models.

Nevertheless~

_we show

(in disagreement

with references [13 ^and

14])

^that the surface is

asymptotically

flat. The

directionally averaged

structure

factor,

_on the other

hand,

exhibits _a nontrivial

scaling

behavior reminiscent of

(I),

with _an effective fractal dimension

df

_m 2.7. The outline of this paper is _as follows. In section 2

we describe the model and simulation

procedure.

Section 3 contains _an

analysis

of the

scaling

behavior of the

eigenvalues

of the _moment of inertia tensor _as well as the structure factor

S(q).

Results _are also

presented

and

compared

with

theory

for the behavior of the smallest

eigenvalue

of the moment of inertia _tensor

(which

_measures the width of the

surface) averaged

over subdomains of

increasing

diameter. We show that

although

the network is

asymptotically flat,

with

(

_t5

0.7,

^the

directionally averaged S(q)

scales _as

q~2

^? _over the whole

scaling region 2ir/Rg

_{< q} < 2ir

la,

at least for the system ^sizes we were able to study. The _paper closes with

a brief discussion in section 4.

N°5 FLOPPY TETHERED NETWORKS i133

2. Model and simulation

procedure.

The model _we consider consists of _{an open}

triangular

network

containing

N vertices with free

edge boundary

conditions. The network is

hexagonal

in

shape,

^with _a ^diameter

L,

_so that N =

(3L2 +1)/4.

The vertices _are connected

by

tethers of maximum extension £o ₌

V5.

This is the

only length

scale in the

problem.

Our Monte Carlo

procedure

consists of

sequentially updating

the vertex

positions by

_a random increment in the cube

[-s, s]~.

In contrast to the

commonly

used

string-and-bead

^models for tethered

surfaces,

_we do not utilize _a

finite-range

hard-core

repulsion

of vertex _monomers to _ensure self-avoidance.

Rather,

an

update

is

accepted

if _no vector

connecting

tethered vertices intersects

an

elementary triangle

of the surface. This guarantees ^{that there is} no intersection of

elementary

surface

triangles.

In this _way, the model resembles that studied in references [13 ^and 14].

Note, however,

that _we do not

impose

_any

constraint _on the minimum

separation

between vertices. The

resulting

network is therefore

extremely "floppy"

in that it _can fold back _on itself with

negligible

excluded volume effects.

Because of this _we have found that the models _{sweeps out}

phase

_{space very}

quickly,

_so

that,

for

a

given

system

size,

relaxation times _are

typically significantly

shorter than for the standard

string-and-bead

^models [13]. ^An

advantage

of the model _we

employ compared

to the

giant-

hole membranes studied in reference [17] ^{is the}

large

reduction in the number of

degrees

of freedom for _a

given

number of vertices. In _our

simulations,

_we have studied membranes sizes from L

= 7

(N

₌

37)

to L

= 25

(N

₌

469). Averages

_are taken

typically

over 10~ Monte Carlo steps _per vertex

(MCS), which,

because of the short relaxation times [13], ^{is sufficient to}

give good

statistical _accuracy.

3. Results.

One of the

simplest

_ways to characterize the conformation of the tethered network is to examine the

eigenvalues

11 < 12 < 13 of the moment of inertia tensor

Ta,p

=

p £ _{(r;ar;p fwfp), (2)}

;

where _a,

fl

_e

(x,

_y,

z),

and the _{sum runs over} all vertices of _a

given configuration;

f~ is the _o component of the center of _mass for that

configuration.

In the flat

phase,

^the _average

eigenvalues

are

expected

to scale _{as <} ii _>+~

Nt,

and _< 13 >+~< 12 >+~ N [7]. ^{Our results for} < ii > and

<

R(

>=

£j

_<

ii

> as a function of the system ^{size N} are

plotted

in

figure

I. The solid line

is _a

plot

of _<

R(

>+~

N°.~~,

and the dashed line

corresponds

to _<

ii

_>+~ ^N° ~~, or

(

₌ 0.63.

Both _< 12 _> and _<

13

_>

clearly

scale with _a

larger exponent

than _< ii >.

Note, however,

that there is very little curvature in the data for _<

R(

> so that

considerably larger

system sizes will be

required

before the true

asymptotic

behavior <

R(

>+~ N is attained.

Figure

2 contains results for _<

li(N~)

> obtained

by taking

_averages on subdomains _con-

taining

N~ vertices for _a membrane of size N

= 469.

Although

there is _no

simple powerlaw relationship

between <

ii (N~)

> and

N~,

^the _average

slope

of the

log-log plot

is

approximately

0.71.

The

scaling functions,

which _are indicated

by

the solid and dashed lines in

figure

2 have been evaluated _as follows.

Restricting

attention to the

out-of-plane motion,

the

leading

order

contribution to the elastic

bending

_energy of _a

thin,

flat

plate

from its _z

= 0 _zero temperature reference state is [18]

fl7i

₌

/

_d~r

16~(i7~z(r))~

^{+ it}

det[3;3jz(r)])

_,

⁽³⁾

2

<R?>

g

lo

o

,w.

,4r"

,,,'~"' _~~l~

x

10~~

10 /~ 10

Fig. i. Scaling plot of the _{mean square} radius of gyration <

R(

_> and expectation value _< Al > of the smallest eigenvalue of the _moment of inertia tensor for membranes containing N

= 37, 127, 271, and 469 vertices. The solid line has slope 0.87, ^and the dashed line is _a plot of _< Al >r- N° ~~

where,

for _a

plate

of thickness h

composed

of _an

isotropic

elastic material with three-dimensional

Young's

modulus E and Poisson's ratio _{a, K}

=

Eh~/[12(1- a~)]

and k ₌

(I a)~ [18].

The Hamiltonian

(3)

can be

diagonalized

by

expanding z(r)

in

eigenfunctions

^of the

equation i7~z

=

l~z, ₍₄₎

where for _a circular disc of radius R with free

edges,

the

boundary

conditions _are

i7~z(~=R (i «) lj

~((

⁺

)l~

= 0

(5.a)

r=R

and

&?~~

~~~

~ ~~

'~ _i TIP _W 7fil

~_ ~

" °' ~~'~~

The solutions of

(4)

_are

iInj(r, #)

= cnj

jJn(Injr)

+ BnjIn

(Injr)j ^~~~~~')

,

~°~(~#) (6)

where Jn and In _are Bessel functions. There is _a

doubly

infinite set of

eigenvalues

_>ml, with

n =

0,1,

2,..., ^and I

= 1,

2,. enumerating

the

eigenvalues

for each order

n. The

eigenvalues (as

^well _as the

Bnj)

_are determined

by

the

boundary

conditions

(5);

_cnj ^is _a normalization

constant.

Because of nonlinear

couplings

between the

out-of-plane displacements

_z and the

in-plane phonon degrees

of freedom _u, the renormalized

long-wavelength bending rigidity

and elas-

tic constants differ

considerably

from their

microscopic

values [4]. These renormalized elastic

constants enter _an effective

long-wavelength

^free _energy [10, 19] ^{for the} transformed variable

ZA,

fl7ieff

_"

~ ~R(I)l~(ZA(~, (7)

2

N°5 FLOPPY TETHERED NETWORKS i135

with

KR(I)

+~ l~n and _q

=

2(1 ().

In the flat state, ^the

expectation

value of the smallest

eigenvalue

of the _moment of inertia

tensor for _a subdomain

containing

N~ vertices is

~

>l(Ns)

_>"

m l~(~'

^~J)~ ,

s (8)

; j

where the _{sums run over} all _monomers in the subdomain. In the continuum

limit,

for _a disc of

area A~,

(8)

becomes

~ Al

(As)

>"

2

As

/~ ^d~r

^<

^Z~(r)

^>

/~

^d~r

/~ ^{d~r'< Z(r)Z(r~)} _>j ⁽⁹⁾

s > ~ ,

Using

the mean-field

eigenfunctions (6),

_{we can} transform

expression (9)

and evaluate the

resulting expectation

values

using (7).

We

thereby neglect

_any renormalization effects _on the

eigenfunctions;

the

resulting expression

for _< ii

(N~)

> is therefore evaluated in the mean-field

approximation, employing, however,

the correct value of _q. In

performing

the

resulting

_sums,

we have used q = 0.6

((

₌

0.7),

_as well _as both _a

=

1/3,

the value of the Poisson ratio for _a twc-dimensional harmonic

triangular

lattice

[20, 21],

and _{a =} 0. We have also cut off the _sums in I and _n when >ml _" 2. The result for _a

=

1/3

is

given by

the solid

line,

and that for

a = 0

by

the dashed line in

figure

2; N~ ₌

3R(R

+

I)

+ I, with R the radius of the subdomain of the disc of radius 12. The agreement ^is quite

satisfactory

for

a =

1/3.

For smaller values of _a the

scaling

function

develops

_{a more}

pronounced dip

for

large

Ns, and for

large

values of _a, the increase in _<

ii (N~)

> near the

edge

of the membrane becomes _more

pronounced.

The value for

(

_we obtain in this _way is somewhat

larger

than what _we found from _an

analysis

of the

smallest

eigenvalue

of the moment of inertia _tensor.

lo

~2 N~

Fig. 2. < Al

(Ns)

> _{as a} function of the number of vertices N~ in the subdomain. The solid line

is the scaling function obtained using _a

=

1/3

and _q

= o.6

((

=

0.7).

The dashed line _was obtained

using _a

= 0.

Further information

concerning

the

roughness

of the membrane _can be obtained

by studying

the behavior of the unit normal vectors of the

elementary triangles

of the surface. Let

Mj(o)

be the

projection

of the unit normal of

triangle

_a

along

the

eigenvector ij

^of the moment

(a)

(b)

Fig. 3. Typical configuration of _a tethered network consisting of 469 vertices

(after

^10~

MCS). (a)

Projection _{on zz} plane.

(b)

Projection _{on yz} plane.

N°5 FLOPPY TETHERED NETWORKS _i137

of inertia tensor.

Following

reference

[12],

_we ^{have evaluated} <

Ml

>e< ~

£~ Mj(a)]2

_>

(where

Na is the number of

elementary triangles)

for _a network

containing

N

= 469 vertices.

We find _<

Ml

_>=

0.056,

_<

Ml

_>= 1.3 _X

10~~,

and _<

Ml

_>= 5A _X

10~~

_As

expected

for

an

oriented,

flat

membrane,

both _<

Ml

_> and _<

Ml

_{> are}

essentially

_zero. The rather small value for <

Ml

_>

implies

that the surface is _very

rough.

A

typical configuration

of _a tethered

network with N

= 469 vertices is shown in

figure

3.

10°

g(

_~

)

^o

_j

I

i

j~ ^I

~

^{b q}

q

~~

%~

%$

i

lo~~

_{q 10~}

Fig. 4. The structure factors

53(q) (o), Si(q) (o),

and

S(q) (x)

plotted _{as a} function of _q for L

= 25

(N

=

469).

Finally,

_we have also determined the

directionally averaged

static structure factor

~~

i

~ sin(qr;>)

~~~~

q "

p

'

>,I ~~"

where r,j =

jr; rj(,

_as well _as

S>(q)

=

S(q)

=

j _L

(e'~ ^~'~~?~)

,

(l11

for _{q =}

qij, j

₌ I

(z)

and 3

(x),

where the

it

are the

eigenvectors

of the moment of inertia

tensor. The average in

(11)

is taken with respect to the frame of reference fixed to the

principle

axes of the membrane.

Our results for these three structure factors _are shown in

figure

4 for _a membrane of size L = 25

(N

₌

469).

As _can be _seen, the three _structure factors behave _very

differently

and converge

only

at

q-values

that

correspond

to the

length

of _a

single

^{bond. The} oscillations in

53(q)

arise from

scattering

off _a twc-dimensional

planar object with,

in this case, a

fairly

diffuse interface. This has been discussed in detail

by

Abraham and Nelson [10]. For

large

_q,

one expects

51(q)

_°~

^q~~/I

In the present _case,

however,

fluctuations _{are so} strong, ^that this

regime

is not yet

reached,

_even for _a network with L

= 25. In contrast, this behavior is

readily

~~o

la) I(q)

lo

1°~

~o.~~

i°~

q

10°

ib)

S

i(q)

lo

~dh

~~~

qL°'~~

~~~

Fig. 5.

(a)

Directionally averaged structure factor

S(q)

plotted _{as a} function of the scaling variable qL°.?~ for N

= 271 (o) ^{and N} ₌ 469

(o).

The solid line is _a plot of

(qL°

~~)~~.~~.

(b)

The "perpendicu-

lar" structure factor

Si(q)

for _wave vectors in the direction of the eigenvector of the smallest eigenvalue

of the moment of inertia tensor plotted _{as a} function of the scaling variable qL°.~~ for N

= 271

(o)

and N ₌ 469

(o).

observed in

string-and-bead

networks of the _same size [10]. ^As a result of the

rapid

decrease in

St (q),

the

directionally averaged scattering

function

S(q)

_appears to scale _as

q~~

^~ _{over a}

relatively large

_q-range. The small

dip

_{near q}

= I is _a

vestige

of the structure in

53(q).

Finally, scaling plots

of

S(q)

_vs

qL°

^~~ and

Si(q)

_vs.

qL°

^~~ obtained

using

data for _mem- branes of size N

= 271 and N

= 469 _are shown in

figures

5a and

5b, respectively.

The excellent

collapse

of the data for

S(q)

is consistent with _an isotropic

crumpled

membrane with the fractal dimension

df

_m 2

/0.75

_Gi 2.7.

Indeed,

not

only

does the data

collapse

for this value of

df,

but the

slope

in the

regime 2irla

> q >

2ir/Rg

is also consistent with

S(q)

SS

q~~f

The solid line

N°5 FLOPPY TETHERED NETWORKS _i139

in

figure

5a is

proportional

to

(qL° ~~)~2

^{~~ In} contrast with the behavior of

S(q), however, Si(q)

scales with

qLt,

with

(

₌ 0.63, _as

expected

for _a flat membrane. This value for

(

_agrees

with that obtained from _an

analysis

of the smallest

eigenvalue

of the _moment of inertia tensor.

4. Discussion.

In this paper we have

presented

results for the conformation and

scaling

behavior of two- dimensional tethered networks obtained

using

_a model for

extremely

flexible

triangulated

_sur-

faces. The model is constructed _so that the

resulting

network is _very flexible and _can fold in

on itself without _any cost in energy; it is therefore much

rougher

than the

commonly

studied

string-and-bead

models.

Although

the network is still

asymptotically flat,

the

large out-of-plane

fluctuations result in _a

directionally averaged

structure factor which exhibits _a nontrivial _sca-

ling

behavior reminiscent of _a

crumpled object

with _a fractal dimension df m 2.7. This result agrees with the conclusions of

a recent

study

of

"giant-hole

membranes" [17] in which the local

flexibility

of the standard

string-and-bead

model of tethered surfaces is enhanced

by replacing

the

interparticle

bonds

by

tethered

strings

several

particles long.

It must be

stressed, however,

that this result is _a finite size

effect,

and

ultimately,

for _very

large

system

sizes,

the

anisotropic

behavior described in references [10 ^and 16] ^{would be observed. An}

important

_open

question

is

just

how

large

the system ^needs to be before this _occurs.

Although

our work is restricted to membranes of diameter L < 25 bond units, the results indicate that the behavior _we observe should

persist

to

considerably larger

systems sizes.

The model _we have studied is _very

"rough"

at short

length scales,

and the

amplitude

of both the

out-of-plane

and

in-plane

fluctuations _are

substantially larger

than in the standard

string-and-bead

models for tethered surfaces. The

analysis

of reference [16] ^therefore

predicts

that there should be _a

large q-regime

in which

S(q)

scales _as

q~~/t+~

We find _a

substantially

slower decrease in

S(q)

with _q; note,

however,

that the behavior

we observe lies between the two

regimes q~~+t

and

q~~/t+~ predicted

in reference [16]. ^{It is therefore}

possible

that _our data lie in the _crossover

region

between these two

scaling regimes

and that

substantially larger

system sizes will be

required

to see the

predicted

behavior.

Finally,

the

graphite

oxide membranes studied in reference [5] are claimed _to have

an aspect ratio _on the order of1000. What is

important,

however, is the size of the membranes in terms of the

elementary

units which fold. A correct

interpretation

of the behavior observed in

graphite

oxide membranes will

require

_{a more} detailed characterization of the

microscopic

_structure and conformation of this material.

Acknowledgements.

This work _was

supported

in part

by Army

Research Office contract number DAAL03-89-C-0038 with the

Army High

Performance

Computing

Research Center at the

University

of

Minnesota,

NATO grant

CRG910156,

and the Deutsche

Forschungsgemeinschaft through Sonderforschungs-

bereich 266.

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