2D crystalline order and defects in a stack of membranes

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Submitted on 1 Jan 1993

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2D crystalline order and defects in a stack of membranes

David Morse, T. Lubensky

To cite this version:

David Morse, T. Lubensky. 2D crystalline order and defects in a stack of membranes. Journal de

Physique II, EDP Sciences, 1993, 3 (4), pp.531-546. �10.1051/jp2:1993149�. �jpa-00247852�

Classification Physics Abstracts 87.20E, 6130J, 64.70M

2D crystalline ^order and defects bla stack of membranes

David C. Morse

(~)

and T-C-

Lubensky (~)

(~) Corporate ^{Research Science} Laboratories, ^Exxon Research and Engineering Co., Annandale, NJ 08801, ^U.S.A.

(~) Department of Physics, University ^of Pennsylvania, Philadelphia ^PA 19104, U.S.A.

(Received

18 November 1992, accepted 23 December

1992)

Abstract. We consider the stability of two-dimensional

(2D)

crystalline order within the membranes of _a lyotropic Iamellar phase,

concentrating

_on the effects of 2D crystal dislocations and thermal undulations. At lamellar spacings d less than _a critical value, the 2D melting

transition is found to be essentially unaffected by the flexibility of the membranes, and thus

occurs at _a melting temperature Tm _near that of _a single membrane _{on a} rigid substrate. At

larger spacings, the interactions between membranes become weak enough to allow buckling of the membrane in _a finite region around each thermally excited dislocation. This leads to _a partial screening of the elastic interaction between dislocations, and acts to depress, but not destroy, the

melting transition. Thermal undulations act to soften the membrane and thus further depress Tm, which is predicted to vanish continuously in the limit of large d. We discuss implications

for recent experiments _on biological membranes.

The

spatial

fluctuations of _a membrane

are

strongly

aRected

by

^its

degree

of internal order:

crystalline [1-3],

^hexatic [4], ^{and fluid membranes} [5] ^{all exhibit}

qualitatively

dilserent confor- mational statistics.

Conversely,

the _presence of

spatial

fluctuations

might

be

expected

to alsect the

thermodynamic stability

of

crystalline

_or hexatic order. In the Kosterlitz-Thouless-Nelson-

Halperin-Young (IITNHY) theory

_{[6,7] of} 2D

melting, crystalline

and hexatic order both melt via _an

unbinding

^of

topological defects,

via dislocations in the

crystalline phase

and disclina- tions in the hexatic. In _a flat

membrane,

the elastic interactions between such defects grow

logarithmically

with distance. In _a flexible

membrane, however,

the form of these interactions

can be

modified,

both

by

thermal renormalization of the elsective elastic constants and

by

the mechanical

tendency

of the membrane to 'buckle~ in _response to defects. An isolated

crystalline membrane,

for

instance,

will buckle _{so as} to confine the

crystal

strain field _to finite

regions surrounding

each dislocation [8]. ^This elsect leads _{to a} finite dislocation _energy, and thus _pre-

vents formation of

quasi-long-range (QLR) crystalline

order _{at any nonzero} temperature. In

an isolated hexatic

membrane,

the

long-range

interactions between disclinations _are

essentially

unaffected

by

the membrane's

flexibility

[9],

leaving

hexatic order stable at

sufficiently

low temperatures.

Experiments

to

probe

the internal order of _a membrane

have, however,

_never been conducted

on isolated membranes. More

typical

_are

experiments

on lamellar

phases,

such _{as are} considered here. Oriented stacks of surfactant

bilayers

have been observed to

undergo phase

transitions between the

two-dimensionally

disordered La

phase

and various ordered

Lp, phases. X-ray scattering

data for the

Lp, phases

is thus far consistent with either hexatic _or

crystalline

2D order

[10].

^In a recent

study

of water-diluted stacks of

bacteriorhodopsin protein membranes,

Shen et al.

[ll]

have observed

sharp

2D

quasi-Bragg peaks (as expected

for

QLR crystalline order)

that vanish at _an

apparent melting

transition. The observed transition temperature ^in-

creases with

decreasing

^lamellar

spacing, suggesting

that the intramembrane order is stabilized

by

intermembrane interactions. There is _no

experimental

evidence of 3D

crystalline

order.

Motivated

by

these

experiments~

_we thus consider _a model

describing

_a

lyotropic

lamellar

phase,

and examine the effectiveness of intermembrane interactions in

stabilizing

2D intramern~

brane order. We find that such

interactions,

when

strong enough,

_can

effectively

_suppress both dislocation-induced

buckling

effects and the

softening

effects of thermal

undulations, leaving

the 2D

melting

transition unaffected

by

the

flexibility

of the membranes. When interactions

are

weaker,

^these elsects become

important

_over an intermediate _range of

length

scales but

are

always suppressed beyond a'flattening length' L~.

The

logarithmic

form of the dislocation interaction is thus restored for dislocations

separated by

distances

greater

than Lt. The _renor- malized elastic moduli felt at these distances _are,

however,

reduced from their bare

values,

due to

softening by thermally

excited dislocation

pairs

and

by

thermal undulations _at shorter distances. The

crystal

melts whenever the renormalized moduli become too small to prevent dislocation

unbinding.

1. The model.

The

configuration

of _a stack of

crystalline

membranes _can be described

by

3D

position

vectors

B~(x), specifying

the the

position

of _{a mass}

point

labeled

by

_a 2D internal coordinate _x and _a membrane index _n. The membrane conformations _are then described

by height

fields

hn(x)

_e I

B~(x).

Fluctuations of the fields

hn

_are controlled

by

_a smectic

elasticity

Hs ₌

~ / d~~lK(fi~hn)~

+

$(hn+i ^{hn)~l (i)}

n

where It and B _are the smectic elastic

moduli,

d is the average lamellar

spacing,

and

fi~hn

₌

fi~hn(x)/fi~~.

The

compression

modulus B is

generally

_a

decreasing

function of the lamellar

spacing d,

and

can be

primarily

either

entropic [12-15]

_or electrostatic

[15,

16] ⁱⁿ

origin.

Both of these

assumptions

about the

origin

of B lead to _a

algebraic dependence

_on d,

B(d)

_o~ d-v

,

(2)

at

large separations,

where _v

= 3 if B is

entropic

and _{v =} 2 if it is electostatic. The

bending

modulus It

originates

in the

bending rigidity

_~ of _an individual

membrane,

with K

=

~/d.

To discuss intramembrane

order,

_we construct an effective

single-membrane

free _energy for

height fluctuations,

which _we define

by formally integrating

out all fields

hn

with _n

#

0 in

equation (I).

Because Hamiltonian

(I)

is

quadratic

ⁱⁿ

hn,

this _{process can} be carried _out

exactly, giving

an effective Hamiltonian which is

quadratic

in ho- This

single-membrane

_energy

can thus be obtained

by simply inverting

the intramembrane correlation function

~~~~~~~ ~~~

_~~ ~ e(qzj~~+1(q4

' ~~~

where

h(q)

is the 2D Fourier transform of

ho(x), e(qz)

_e

2[1- cos(qzd)]/d~,

and _q is the

magnitude

of the 2D wavevector q.

Equation (3)

contains _a natural _crossover

length Lp

_~w

(d~K/B)~/~,

referred to _as the

patch length.

At

large wavenumbers, qLp

>

I, equation (3) yields

_an

essentially single-membrane

spectrum

ih(q)h(-q))

₌

$ ₍₄₎

where _~ is the

single

membrane

bending rigidity.

At smaller

wavenumbers, qLp

<

I, equation (3) yields

_a spectrum associated with the collective undulations of the

smectic,

(h(q)h(-q))

₌

( ₍₅₎

where _{r %}

24i

_is

an effective

field,

due to intermembrane

interactions,

that tends to

align

the membrane normals

along

I. The

patch length

is

conveniently expressed

in terms of the

single-membrane

parameters ~ and _{r as}

Lp

~

W ₍₆₎

The _crossover between limits

(4)

and

(5)

is somewhat

complex, and,

since it

depends

in detail

upon the

high-qz

behavior of

e(qz),

is

apparently

nonuniversal. A convenient

approximation

is

given, however, by simply taking (h(q)h(-q))

_ci

_(~q~

+

rq~)~~. Adopting

this

approximation

for the correlation

function,

and

adding

the internal elastic _energy of _a 2D

crystal ill,

_we obtain

an effective

single-membrane

Hamiltonian

H _"

d~~'l~(fi~h)~

+

~(fih)~

+

>"la

₊

2~"lbl (7)

In the

above,

_{nab +}

(fialL fiblL bob)/2

is the nonlinear strain tensor, ^{and I and} _~ are 2D Lam4 coefficients. A _more

rigorous

_way of

deriving

of

equation (7)

would be to first _use the full anharmonic

theory

for the stack [17] ^{to calculate} renormalized parameters

I(R(q), lR(q),

~R(q),

and to

only

then calculate the fluctuation spectrum ^of a

single membrane, identifying rR(q)

e 2

1(R(q)B

and

~R(q)

+

I(R(q)/d,

where B has been assumed to be unrenormalized

[17].

Note that the elsect of intermembrane

interactions, represented by

_r in Hamiltonian

(7),

is

qualitatively

different from that

produced by simply confining

the membrane between inflex- ible walls [14]. ^The

r(fih)~

term

explicitly

^breaks rotational symmetry, ^but not translational

symmetry, as would the _presence of

confining

walls. The effect of this term is

formally

identical [2, 3] ^to that of _a tensile stress,

making

all of _our conclusions

regarding

_a membrane within _a

lamellar

phase equally applicable

to _an isolated membrane under tensile stress.

The

strength

of the elastic interaction between

crystal

dislocations is controlled

by

the 2D

Young's

modulus

y ₌

4»(1

+

~)

l +

2~

~g~

Dislocation

unbinding

_occurs when the renormalized

long-wavelength

modulus satisfies

YR(q

"

0)b~

<

161rT,

where is the

magnitude

of _a dislocation

Burger's

vector. The _presence of

thermally

excited dislocations and

height

fluctuations

always

lowers the renormalized

Young's

modulus,

_so the membrane _{can never} remain

crystalline

at temperatures T _>

Yb~/161r,

where Y is the bare modulus.

In the model

presented above,

_{we assume} that the 2D translational order within _a membrane is uncorrelated with the translational order in

neighboring membranes,

and that the system,

therefore,

does _not resist

parallel

relative motion of the membranes. In reference

[13], however,

Toner showed that _a shear

coupling

between membranes is _a relevant

perturbation

of this de-

coupled phase,

in the renormalization _group

(RG)

_sense, whenever the individual membranes exhibit

QLR crystalline

order.

Physically,

Toner's argument

implies

that the intermembrane

correlations between

weakly-coupled

untethered

crystalline

membranes will exhibit _{a nonzero}

long

distance

limit,

and that _a stack of such membranes will thus behave at very

long length

scales like _a true 3D

crystal.

The

crystalline decoupled phase

considered here is thus associated with _an unstable

(but potentially physically important)

fixed

point

of the RG. Our

underly- ing physical assumption

when

considering

this

phase

is that the intermembrane

correlations, though long-range,

_can be

exceedingly

small in

magnitude,

_as is

suggested by

the

complete

absence of observable 3D order in Shen et al.'s

experiments.

The _presence of _a small

degree

of 3D

order,

and _a

correspondingly

small shear

modulus,

is

expected

to

change

elastic behavior

only

at

length

scales

beyond

_some

large

_crossover

length.

Because the

melting

transition is

relatively

insensitive to the _very

long-wavelength

behavior of the

stack,

_we expect the

melting

temperature ^of the 3D

crystal

to

smoothly approach

that of the

decoupled

system as the

degree

of intermembrane

registry

vanishes. Formation of _a weak 3D

crystal

_can also be inhibited

by non-equilbrium effects,

such _{as an} initial randomness in the

crystal alignment

of

neighboring membranes,

which

might

prevent ^{3D order from}

forming

_on

experimental

time scales

ii Ii.

The present model of

decoupled

membranes should thus

provides

_a useful

description

of the

phase behavior,

^if not the

equilibrium elasticity,

of

a real system ^of _very

weakly coupled

membranes.

2. Dislocation mechanics.

The elastic interaction between

dislocations,

in the absence of

screening by

either

height

fluctu- ations _or other dislocations, is

given by

the total, mechanical

equilibrium

_energy of _a membrane with two dislocations. This

equilibrium

_{energy can} be obtained

analytically

_[18]

only

when the

membrane is constrained _to lie in _a

plane,

^but a

qualitative description

of the interactions in _a

nonplanar

membrane _can be obtained from _a combination of

scaling arguments

^and numerical simulation.

Consider first the behavior of _an infinite membrane with _a

single

dislocation of

magnitude

b at the

origin.

To describe behavior at distances _r «

Lp

away from the

dislocation,

where

we expect ~ to dominate in

H,

_we set _{r =} 0. The

resulting single

membrane

problem

_was

considered in detail

by Seung

and Nelson [8], ^{who showed that the}

equilibrium

values of the

height

and strain fields _can be

expressed

in the

scaling

forms

h(r)

₌

/7 i()), _ua~(r)

=

( fia~(), ^{«), (9)}

where _me

I/(1+ 2p)

is the Poisson ratio and

L~

-~

</(yb) (io)

is referred _{to as} the

buckling length.

^The

corresponding

_energy ^is a function E

=

Yb~ll(Lb/b)

of

~ and

Yb~,

and is

independent

of the Poisson ratio. The

physical interpretation

of the

buckling length

is _as follows: inside _a

region

of radius

Lb

around the

dislocation,

it is

energetically

unfavorable for the membrane _to buckle

sharply enough

to

substantially

reduce the strain. At

larger distances, Lb

_{< r} <

Lp, (assuming

Lb <

Lp) buckling

becomes

favorable,

and

nab(r)

drops rapidly compared

to its value in _a flat

crystal.

At

length

scales L >

Lp,

intermembrane interactions

represented by

_{r are}

dominant,

allow-

ing

_us to

ignore

the

bending rigidity

_~. A

straightforward

modification of

Seung

and Nelson's arguments to the _{case ~ =} 0, r

#

0

(see Appendix) yields

the alternative

scaling

forms

h(r)

₌

WI()), _ua~(r)

=

) _{ia~(), «), (11)}

where

yb

(12)

Lf -~

j

will be referred to as the

flattening length.

Note that

Lr

increases _as

d(~+")/~

_with

increasing separation.

At distances

Lp

< r « Lr from the

dislocation,

intermembrane forces dominate in H but _are still too weak to suppress

buckling.

At distances _r >

Lf,

the membrane

finally

flattens out due to

interactions,

and becomes

increasingly planar

with

increasing

_r

(see Fig.

lb).

The

prediction

^of _a buckled

region

that _can extend _to

lengths

well

beyond

^the

patch length Lp

^is a result of

treating

the

neighboring

membranes _{as a} flexible

medium,

^rather than

as inflexible walls.

A buckled

region

can thus exist around each dislocation if either Lb %

Lp

or

Lp

j~ Lt.

Note, however,

that these two conditions _are

equivalent,

since LbLr

-~

L(.

The flat state thus becomes

mechanically

unstable in the presence of _a

dislocation,

and

buckling

effects becomes

important,

whenever Lb £

Lp.

^More

precisely,

dislocation-induced

buckling

_{can occur}

only

when the dimensionless ratio

g ⁺

Yb/@

-~

Lp/Lb (13)

exceeds _some critical value _gc. This

implies

that

buckling

will _occur in _a

given

system of membranes

only

when d exceeds _a nonuniveral critical value

dc,

where

dc depends

_{upon ~} and Y and _upon the function

B(d).

For d _<

dc,

the 2D

melting

temperature is thus

essentially

unalsected

by

the finite

flexibility

^{of the} membranes and becomes

roughly independent

of d.

To _{test some} of these

scaling

arguments, we have carried _out numerical simulations of dislocation-induced

buckling

_{on a} finite

triangular

lattice. Our discrete Hamiltonian contains strain and bend contributions

Hy ₌

~~~j((r;j(-a)~

4 Iii)

~~ ~(~'

_~~

' ~~~~

where r;j are vector

connecting nearest-neighbor particles,

fi; and

hi

_are unit normals _on

neigh- boring triangular plaquets,

^and a is _a

preferred

lattice

spacing.

We represent ^{the additional}

r(fih)~

contribution in H

by

_{an energy}

HT _"

/~ ~j(I

~ij)~

(15)

iiji

where I is the unit normal to the

preferred plane

of

alignment.

The numerical

prefactors

in

equations (14)

^and

(15)

are chosen _{so as} to

reproduce

^{the continuum limit of}

equation (7),

with I

= p =

(Y.

Mechanical

equilibrium

conformations and

energies

have been calculated for

hexagonal

_mem-

branes with L

particles

between the center and each _corner,

using

L <

96,

each with _a

single

dislocation of

magnitude

= a at its center

(see Fig. I).

If the membrane is forced to lie in _a

la)

16)

Fig,I.

Dislocation-induced buckling: figure la shows the equilibrium conformation of membrane of size L ₌ 16 with elastic parameters ~ = o,ois Yb~ and _r

= o, representing _an isolated membrane.

Figure

16 shows _a membrane with the _same value of

~/Y,

but _r

= 0,00 is Y, representing _a membrane confined by interactions with its neighbors. ^The dislocations

are visible _as neighboring 7-fold and 5-fold coordinated particles in the center of each membrane. For _r # 0

(i'b),

the _presence of _a bending rigidity primarily affects the membrane conformation within _a small region, of radius _r

~J

Lp,

around the dislocation, while the _presence of _{a nonzero r} controls the behavior at large distance§, _{r jS}

Lp, causing

the membrane _to flatten _{out at} distance

r jS Li >

Lp.

For _r

= 0

(la),

the

height

field apparently diverges at large distances from the dislocation. The vertical

(I)

scale is exagerrated in both pictures by _a factor of 2.

plane (I,e., h(x)

₌

0),

the strain _energy associated with the dislocation scales with the system size L _as

~

Eflat

_ci ^~~

~ln(Lla)

₊

const]

,

(16)

81r

where the constant represents a dislocation _core energy.

Simulations of

buckling

in isolated

membranes, corresponding

to the limit _{r =} 0 in the

above,

have been carried _out

previously by Seung

and Nelson [8].

Using

_a discrete Hamiltonian identical _{to our}

equation (14),

^{these authors found that the} _energy ^{E of} a buckled dislocation

approaches

_a finite value _as L _goes to

infinity,

and that this

limiting

value is well

approximated by

that of _a dislocation in _a flat

crystal

of radius

Lb

lim E _ci

~~

[ln(Lbla)

+

const] (17)

r-O

8~

As is clear from its

form,

this energy is due

primarily

to the strain energy contained within _a radius _r

-~

Lb

around the dislocation. An unconfined buckled membrane is shown in

figure

la.

When _r is _{nonzero, as} it is for _a membrane within _a

stack, scaling

arguments

suggest

^{that the}

membrane should flatten out at

large

distances. This should

produce

_{an energy} that

diverges

as

In(L)

in the limit of

large L,

but that remains less than

Eflat(L) by

_an amount that increases with

increasing flattening length

Lt. In

figure

2, _we ^{show the calculated} _energy E _{as a} function of

Y/r

_~K

Lr/b

for membranes of fixed size L

= 48 and several values of

~/Yb~

_~K

Lb16.

^For

sufficiently

small values of

Li (I.e., large r),

the membrane lies

entirely flat, giving

_{an energy}

E =

Eflat.

The membrane

begins

to

buckle,

and E

begins

to

decrease,

when

Lr

exceeds _some critical value Lrc that

depends

_upon the

magnitude

of Lb- For Lb < _a,

buckling begins

at a value Lfc

-~ a. For _a j~ Lb

L,

this

instability

_occurs at Lfc

-~

Lb,

and for Lb >

L,

the

membrane remains flat at all values of

Lf.

Once

buckling begins,

the energy decreases

steadily

with

increasing Lf

until

Lf

₂

L,

and then levels

of,

_{as r} becomes too small to alsect the energy

a a O O & &

a ° &

~ O &

j

o &

~

o

<

~

~/Bn

O ^~

Wo O

2

O

3

w

O

a

3

o

o-o m 4.o o-o e-o lo-o

In~i$/7) Fig.2.

Dislocation _energy E of membranes of radius L

= 48 _{as a} function of

Y/r

_cc

Li16.

Squares represent ^{membranes with} ~ = 0, circles represent ~ =

o.o2Yb~,

and triangles _{~ =} o.o4Yb~. _{The solid} line has _a slope of

1/8~,

_as indicated.

in this

regime.

For _{~ =} 0 and L~c j~ Lf j~ L, ^{E is found to exhibit} a

logarithmic scaling

behavior

lim E _m

j~~ ~In(L/L~)

+

const]

,

(18) reflecting

the release of strain in _area of size _{r ~K} Lf. As _~ is increased from _zero, the range

over which Lrc < Lr < L _narrows

rapidly, making

it

impossible

to observe _a clean

logarithmic scaling regime

in membranes of this size for _~ 2

0.01Yb~.

In

figure 3,

_we show the

dependence

of the critical

flattening length Lrc16

_~K

Y/r

_upon

Lbla

-~

~/Yb~.

The

slope

of this

logarithmic plot

is

expected

to

approach unity (corresponding

to _a constant value of the critical ratio _{gc =}

Ybll%)

in the continuum elastic

limit,

_a < Lb < L.

To check for finite-size

effects,

_we have determined Lrc for the three

largest

values of

Lb/b

in membranes of two

sizes,

L ₌ 48 and L ₌ 96. The results obtained from these two sizes differ

significantly only

at the

largest

value of

Lb, indicating

that the variation from

linearity

at lower

Lb

should be attributed to finite

lattice-spacing,

rather than

finite-size,

effects. We _can obtain

a

rough

estimate of the continuum value of _gc

by extrapolating

the data for size L ₌ 96 onto a line of unit

slope, giving

gc m 80

,

(19)

as indicated

by

the solid line in

figure

3.

a

a

p~ ~~

w

a

D lD

2+lJ~ ll~ ll~

K/Yb~

Fig.3.

Critical value of

Y/r

_cc Lic16 _{as a} function of

~/Yb~

_cc Lb16. Squares represent ^membranes of size L

= 48 and triangles L

= 96. The solid line is _a line of constant gc =

Ybll%,

with _an

extrapolated value of _gc

= 80.

3. Dislocation mediated

melting.

In this

section,

_we consider the effect of membrane

flexibility

_upon the 2D

melting

temperature.

We first discuss the effect of dislocation-induced

buckling,

and then include the additional effect of undulations.

The presence of

thermally

excited dislocations softens the renormalized

Young's

modulus YR, and thus tends _to destabilize the

crystal.

The

perturbative

correction _to Y due to

thermally

excited

pairs

_{on a}

triangular

lattice is calculated in reference [7], and _can be written _as

b2 oJ ~2~

~ 2

~~

~' ~

T

/~ _a( ^{ao~ ~~~~~}

^~~~ _' ^~~~~

where E is the excitation energy of _a

pair

of dislocations with

Burger's

vectors + b and

separation

_r. The

angular

factor

A,

which is

specific

to the

triangular lattice,

is

given by A(9)

₌ [~ +

( sin~

_{0], where 0 is the}

angle

between b and _r. The energy E is the mechanical

equilibrium

_energy of _a dislocation

pair

and should thus include _any effects of

buckling.

At

separations

_{r j~}

Lb,

where the membrane is forced to lie flat

by bending rigidity,

E is well

approximated by

the

corresponding expression

for dislocations in _a flat

crystal,

E _m

(~ [ln(I)

+

sin~

0] + 2Ec

,

(21)

ao

E(r)

2Eb

2E~

ao Lb Lp Lf

~(r)

FigA.

Schematic plot of the dislocation pair _energy E

as a function of

In(r),

where _r is the dislocation separation. See text for definitions of various length and energy scales.

where Ec is _a

microscopic

_{core energy.} At

separations

_{Lb % r j~ Lf,} this interaction is screened

by buckling, making

E

essentially independent

of _r.

Here,

the

pair

_energy can be

approximated

as a sum of

single-dislocation energies, E2

_"

2Eb,

where the effective

single-dislocation

_energy,

Eb _" Ec +

j~~ In(Lblao)

,

(22)

is

roughly

the _core energy

plus

the strain _energy contained within _a distance

Lb

around the defect. At

separations

_{r 2 L~,} E must

again

take _on the

logarithmic

form

(21) appropriate

to a

flat

crystal,

but with _an effective _core size _ao

- Lf and _core energy Ec _- Eb. The

dependence

of E _on

separation

is shown

schematically

in

figure

4.

The effect of

buckling

_upon the

stability

of the

crystal

_can be understood

using

_a

simple coarse-graining procedure. First,

estimate the renormalization of Y~~

arising

from dislocations with

separations

_{r <} Lf. ^Then use this renormalized value of Y~l

as the

input

in _{a coarsw}

grained theory

valid _at

length

scales L _> Lf

beyond

^which

buckling

is irrelevant.

Following

Nelson and

Halperin's

_[7] treatment of the flat

crystal,

_we define

a

coarse-grained Young's

modulus

YR(a),

and dislocation

fugacity yR(a)

e

exp(-ECR(a)/T),

where

ECR(a)

^is a coarse-

grained

core energy, as functions of _a renormalized _core size _a. It is also convenient to define _a reduced modulus

VR(a)

_e

YR(a)b~ IT.

At

length

scales _a j~

Lb

and _a 2

Lf,

where the

crystal

is

flat,

the renormalization _group

(RG)

flow of

VR(a)

and

yR(a)

is well

approximated by

the KTNHY recursion relations for _a flat

crystal.

Here _we focus _on the RG flow in the

regime

Lb % a j~ L~, ^where

buckling strongly

modifies the elastic interaction.

Following

_now standard

procedures [19],

^the

coarse-grained Young's

modulus

YR(a)

is defined

by integrating

the dislocation contribution in

equation (20)

_up to a maximum

separation

r = a, while the _core energy

(or fugacity)

is defined _{so as} to restore the form of

equation (20)

to the

remaining integral. Treating

the dislocations _as

completely non-interacting,

and

using

the

approximation

E _m

2Eb

for the

pair

_energy, we obtain

YRla)

Ci

e~~~/~l«lao)~ (23)

Vi~(a)

_Ci

n~(Lb)

₊

)lYi(a) ^Yi(Lb)1 (24)

appropriate

at

length

scales Lb _a j~ Lf. ^{For the sake of}

simplicity,

_we have included

only

the effects of dislocation

pairs,

_as described

by equation (20), ignoring higher

order effects due to

triplets

that _were included in reference [7].

The

quantity Yp~(Lb) appearing

in

(24)

_can be obtained

by integrating

the KTNHY _re- cursion relations _up to _{a =}

Lb, thereby taking

into account the contribution _to

§~

_from

dislocation

pairs

with

separations

_{r <}

Lb-

This contribution _can,

however,

be shown to be small _{at any} temperature that is

significantly

below the

melting

temperature ^{of the} flat

crystal

[7].

Temperatures

near the

melting temperature

Tm of the flexible membrane will

satisfy

this

condition whenever

buckling significantly depresses

Tm.

In

equation (23),

the

integration

of _a up ^to a = Lb has been carried _out

explicitly, using

the

assumption

that

YR(a)

_ci Y for _a < Lb-

Allowing

for _a renormalization of

YR(a)

at a <

Lb

would

simply

renormalize the effective excitation _energy

Eb, corresponding

to the _use of _a

slightly

renormalized modulus in

equation (22).

As _{we see} from the form of

equation (23), yR(a)

can be

interpreted physically

as the

probability

of

finding

an

unpaired, thermally

excited

dislocation of _energy Eb ^within _a core

region

^{of radius} _a. This

fugacity

increases

monotonically

because the strain _energy contained within _a radius _a around the dislocation has been assumed to be

independent

of _a. In _a flat

crystal,

this strain energy increases

logarithmically

with _a,

giving

_a

fugacity

which _can either increase _or decrease with _a

depending

_on the balance of the

increasing

strain _energy and the

increasing

translation entropy ^{of the} dislocation within _a

redefined _core

region.

The RG

trajectories given by equations (23)

^and

(24)

_are shown

by

the solid lines in

figure

5. The RG flow at

lengths

_{a j~}

Lb

and _a 2

Lg

^{is obtained}

^by grafting

the

trajectories

for this buckled

regime

onto the KTNHY relations for _a flat

crystal, represented by

dashed lines. The

crystal

melts if the

coarse-grained

parameters

gj~ (Lf)

and

yR(Lr)

^{fall outside the flat}

crystal's

limit of

stability.

^{Consider the}

stability

of the

crystal

in the limit of low temperature, where

Pm

_I and

Eb/T

_>

I,

and

large Lt.

In this

limit, §~(Lb)

_and

_{yR(Lb) always}

_lie

near the

origin

of

figure

5, and then follow

a

trajectory

which _crosses the flat

crystal's

limit of

stability

somewhere _near the

point

P. The

limiting low-temperature trajectory,

which connects the

origin

and the

point

P, describes the

physical

limit in which the dislocation contribution to

Yj~

dominates in

equation (20),

_so that the

Young's

modulus

YR(Lr)

is determined

primarily by

the dislocation

fugacity

rather than

by

the

microscopic

modulus Y. In this

limit, V(~(Lr)

and

yR(Lr)

take _on universal values

(corresponding

to the coordinates of

point P)

_at the

melting temperature Tm. Setting

either

yR(Lr)

or

V(~(Lr)

_{to constants at} temperature Tm then

gives

_a

melting

temperature that varies _as

)

ci

)[2 _{In(Lr lao)}

₊

_{const] (25)}

m b

with variations in Lt. This

approximation

is valid in the limit of

large Lr/Lb>

where Tm becomes

significantly

lower than the

melting

temperature ^{of the} flat

crystal.

Since the

length

Lr increases

algebraically

with

d,

this

gives

_a

dependence

Tm _~K

In~~(d) (26)

in the limit of

large

lamellar

spacing.

We _now consider the effect of thermal

height

fluctuations. In _an isolated

membrane, height

fluctuations _{cause an} anomolous renormalization of the wave-number

dependent

elastic _param-

eters

~R(q)

and

YR(q)

[1, 2]. These renormalized

quantities

become

significantly

different from

v~ia)

P

'

,$.

,

, ,' _,

'-~-" ,' ,'

,W' ,'

'

1/161r pjl(~)

Fig.5.

Schematic flow diagram Solid lines represent ^{the RG} trajectories for the buckled regime Lb _{< a <} Li, where the dislocations _are non-interacting. Dashed lines represent represent the KTNHY

trajectories for _a flat crystal with logarithmically interacting dislocations, appropriate for

a < Lb and

a > Li. The shaded _area is the the region of _a stability of the flat crystal. The connected _set of dashed

- solid

- dashed fines represent the full RG trajectory of _a particular _system. The trajectory shown represents a membrane slightly above its melting temperature.

their bare values _~ and Y

only

at

wavelengths greater

than _a non-linear

length

L~i _~

~lli ₍₂₇₎

For

q~~

2

Lj~,

the elastic parameters ⁱⁿ an isolated membrane [2] scale as

~R(q)

~W

(qLni)~°~~

YR(q)

_~W

lqLni)+°"Y (28)

where rotational invariance

gives

[2, 3] the relation _q~ +

2qh

" 2. Most numerical simulations

[20] ^have

given

_qh Ci 0.8 + 0.I. In _a

lyotropic smectic,

this

single-membrane scaling

behavior is

obeyed only

at

lengths q~l

shorter than the

patch length Lp.

At

lengths q~l

₂

_{Lp, ~R(q)}

and

YR(q)

become

roughly q-independent,

except for much weaker

logarithmic

corrections

[17].

Significant softening

of Y

by

thermal undulations _can thus _occur in the smectic

phase only

when

L~j

«

Lp.

A useful constraint _on the

magnitude

of L~i is obtained if _{we compare} L~i and

Lb, giving

Lni _~ Lb Yb2

IT.

As _we noted

earlier,

it is

theoretically

consistent to

postulate

the existence of

crystalline

order

only

at temperatures ^T <

Yb~/161r, implying

that

Lni 2

Lb (29)

at all temperatures of interest. This

inequality

has several

implications; first,

it indicates that it is inconsistent to consider _a situation in which undulation efTects _are

important (Lnj

j~

Lp)

^but

defect-induce

buckling

is not

(Lb

_j~

Lp).

^This

justifies

_our earlier statement that the

melting

temperature remains _near that of _a membrane _{on a}

rigid

substrate whenever dislocation- induced

buckling

is

suppressed, I.e.,

whenever

Lp

j~ Lb-

Second, equation (29) implies

that thermal undulations will have little effect _on the dislocation interaction at

separations

_{r j~}

Lr,

since this interaction is

already

screened

by buckling

at any

length

which is less than Lr and at which undulation effects _are

important.

Undulations _can,

however,

soften the effective

interaction felt at

separations

_r

2 Lr,

^{where the}

logarthmic

interaction is

restored,

and

thereby

further

depress

Tm.

The effects of thermal fluctuations _upon the elastic _constants of _a membrane _can be treated

by

various

analytic expansions

and

approximations

[1,

3, 21].

The best estimate of such elsects is

probably given by

the self-consistent

screening approximation

for tethered membranes

[21].

A

straightforward adaptation

of this

approximation

to include the

symmetry-breaking r(fih)~

term in H

gives

a renormalized

Young's

^modulus

j$~R

_~~~~ ~ ~

~~ /(~~2 ~R($i~~~~~k)k2]2

' ~~~~

where

P$

= bob

qaqb/q~.

The

practical

value of this

expression is,

of course, limited

by

its

dependence

_on the renormalized

quantities rR(q)

and

~R(q),

which

should,

in

principle,

be determined

self-consistently

with

YR(q).

We note,

however,

that the dominant contribution _to the

integral

_comes from wavenumbers k £

Lpl,

where

~R(q)

and

TR(q) depend only weakly

on q. We will thus estimate the

integral by replacing

these parameters ^{with their} _{q =} 0

limits, rR(0)

and

~R(0), giving

a contribution

~~~

_~~°~ ~

4'rTR~~~R(0)

~~~~

The

long-wavelength

parameters that _appear in

equation (31)

are difficult _to calculate from

microscopic

parameters, ^but _can, in favorable

circumstances,

be measured

experimentally [15].

When the undulations elsects _are

small,

_~R and _{rR can} be

replaced by

their bare values in

equation (31), yielding

the lowest-order

perturbative

correction to YR. This undulation

con-

tribution to

Yj~

_{can, as a} first

approximation, simply

be added to the

previously

calculated dislocation contribution.

It is

possible

for the undulation correction _to YR to dominate _over the correction due to

thermally

excited dislocations. This _occurs for

Lp

2 Lni ^and

large

dislocation excitation energy

Eb,

where dislocation elsects _are

suppressed by

_an

exponentially

small factor

e~~~/~

_When undulation effects _are

dominant,

the

melting

temperature for dislocation mediated

melting

may be calculated

simply by setting VR(q"0)

_to its critical value of161r. This

yields

_a

melting temperature

which decreases

algebraically

with d at

large separation.

4.

Experimental implications

and conclusions.

Given estimates of the elastic parameters Y, ~, and B, _our results _can be used to estimate the effects of membrane

flexibility

_on

crystal stability

in various

experimental

systems. Consider the

Bacteriorhodopsin (Br) membrane,

which is the

only

membrane in which 2D

crystalline

order has thus far been observed. This membrane forms _a

triangular

lattice with

a rather

large

lattice

spacing,

a ci 63

I,

and _a trimer of

protein

molecules _at each lattice site

[22].

^A

prelimary analysis

^of Shen et al.'s

X-ray scattering

data

gives

an estimated

Young's

modulus of Y

~J

500-600Tla2

at T _ci

65°C,

which is

roughly

10

degrees

below Tm. These

experiments

have thus far been carried out

using

_pure water

dilution,

without added

salt,

and the intermembrane

interactions _are believed to be

primarily

electrostatic in

origin.

The smectic

compressibility

B

can thus be estimated from the theoretical

prediction [15,

16] ^for an

electrostatically

stabilized lamellar

phase

with

large separations

and _a

single sign

counterion:

B ₌

j

,

(32)

where dw is the water

separation

between

membranes,

_e is the counterion

charge,

and le e

lre~/(eT),

with

le'~

19

I

_for

a dielectric constant e ₌ 80

appropriate

to water. Thin _approx- imation is

appropriate

for

ledwela ₂

1, ^where a is the surface

charge density

of _one side of the

membrane,

and should be

appropriate

in the Br membrane for

separations

dw _> 30

I.

_We

have _no

experimental

_measure of the

bending rigidity,

but will _assume that the

protein

^lattice

is _at least _as

rigid

_{as a}

typical single-species lipid membrane, suggesting

~ 2 ^30T.

We can use these estimates of the elastic parameters to estimate _a critical water

separation

dwc for the onset of dislocation-induced

buckling. Using

_{~ =} 30T and the estimate _{gc m} 80 for

g, we find that g =

Yb/j%

reaches _gc at _a critical water

separation

of

dwc _~w ⁷⁰⁰

I (33)

Using

_a

larger

estimate for _~ in the above

produces

_a

proportionately larger

estimate for dwc.

While

extremely rough,

this estimate does indicate that dislocations _can be forced to

lay

flat in this system _even at

fairly large

water

separations, leading

to _a rather stable

crystalline phase.

The

large

^observed value of

V,

which is believed to be

roughly

ten times its critical value at temperatures

only

10 20 °C below

Tm,

also

suggests

that the

melting

transition in Br _{may not} be

explainable

in _terms of

a

simple

dislocation

unbinding

mechanism, ^but _may involve _some

change

of structure within the Br unit cell. The critical

separation

dwc of _an

electrostatically

stabilized system can

readily

be reduced

by adding salt, thereby screening

the electrostatic interaction and

reducing B(d).

The minimum achievable value of B at _a

given separation

is obtained when the membranes interact via steric

repulsion alone, leading

to the

entropic repulsion

described

by

Helfrich

[12].

What _are the

implications

of these results for

simpler lipid membranes,

such _as those studied in reference

[10]?

One

important

difference between

a

lipid membrane,

such _as

DMPC,

and

a lattice of

large proteins,

such _as Br, ^is the difference in lattice

spacing:

_a is _a full order of

magnitude

smaller in

lipid

systems, where _{a ci} 4-7

I,

_{than in the Br lattice. The effect}

of

decreasing

the lattice

spacing

in _our

theory

is

(all

other

things being equal)

to decrease the

importance

of

buckling effects, by decreasing

the ratio _{g =}

Ybll%,

and _to

drastically

increase the

importance

of the conventional dislocation

unbinding mechanism, by decreasing

the

quantity V

₌ ^Yb~

IT.

If _{we assume} that the bare

Young's

modulus Y of _a 2D

lipid crystal (by

which _{we mean} the modulus at

sufficiently

low temperature so that thermal excitation of dislocations is

suppressed)

is the _same order of

magnitude

_as that measured in the Br

membrane,

then _we would obtain _a critical

separation

in water of several thousand

I

and _a

room temperature ^{value of}

V

_{that is}

roughly

_an order of

magnitude

below the critical value of161r. A much

higher

value of Y would thus be

required

to

produce

_{a room} temperature 2D

lipid crystal

that is either stable with respect to

simple

dislocation

unbinding

or that is

mechanically

unstable with respect to dislocation-induced

buckling.

Since _{we see no reason} for the intrinsic moduli of _a

lipid crystal

to be

drastically higher

than those in

Br,

_we

speculate

that _any such

crystal

would

probably

^melt via dislocation

unbinding

_{or some} other

purely

2D

mechanism,

rather than via dislocation-induced

buckling.

In

conclusion,

_we have shown how

crystalline

intramembrane order _can be stabilized

by

intermembrane interactions within _a lamellar

phase.

We have discussed the various mechanisms