Observation of elementary edge dislocations in phospholipid multilayers and of their annealing as a determination of the permeation coefficient

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Observation of elementary edge dislocations in phospholipid multilayers and of their annealing as a

determination of the permeation coefficient

W.K. Chan, W.W. Webb

To cite this version:

W.K. Chan, W.W. Webb. Observation of elementary edge dislocations in phospholipid multilayers and of their annealing as a determination of the permeation coefficient. Journal de Physique, 1981, 42 (7), pp.1007-1013. �10.1051/jphys:019810042070100700�. �jpa-00209076�

Observation ^of elementary edge dislocations in phospholipid multilayers

and of their annealing

^{as a}

determination of the permeation coefficient

W. K. Chan

(*)

and W. W. Webb

School of Applied ^and Engineering Physics, ^Clark Hall, ^Cornell University, Ithaca, ^N.Y. 14853, ^U.S.A.

(ReCu ^le 21 janvier 1981, accept ^{le 23 mars} 1981 )

Résumé. 2014 Nous avons observé les dislocations-coin élémentaires d’un cristal liquide smectique phospholipidique légèrement dopé par ^un analogue, fluorescent et uniformément distribué. Le balayage de l’intensité fluorescente d’un cristal en forme de coin de 8 _03BCm de résolution spatiale ^{fournit assez de} statistiques ^de photons pour la détection des modifications d’une bicouche sur cinquante. ^Les échantillons frais ont une densité de dislocations à peu près ^de

1 x 107 cm-2 ; celle-ci doit être recuite afin que les coins des bicouches deviennent bien distincts. Le dimyristoyl phosphatidylcholine (DMPC) exige d’être recuit de 2-4 semaines à 35 °C dans une phase L03B1 ^{et la} guérison ^n’inter-

vient pas après ^un recuit de 3 mois à 17 °C dans une phase P03B2’. ^Le procédé de recuit exige ^{le trans-}

port de molécules lipides entre ^bicouches voisines ; ^{aussi le} temps ^{de recuit} dépend-t-il du coefficient de perméation 03BBp qui apparaît ^{dans les} équations viscoélastiques ^d’un cristal liquide smectique, ^et offre-t-il un procédé pratique

pour la détermination de 03BBp. ^{Nous trouvons pour} 03BBp ^{une valeur} approximative ^{de 1 10-30} cm2/poise à ^{37 °C}

et cette valeur diminue considérablement à 17 °C. D’autres défauts apparaissent lorsque l’échantillon est refroidi en traversant la température de transition. Ces derniers disparaissent ^en 2 jours de recuit si la température ^est augmen- tée à nouveau, mais il faut 2-4 semaines si l’échantillon est en phase P03B2’ ^{pour plus de 24 h.}

Abstract. ²⁰¹⁴ We have observed the elementary edge dislocations in a phospholipid ^smectic liquid crystal lightly doped ^{with a} uniformly distributed, fluorescent, lipid analogue. Scanning ^the fluorescence intensity ^{of a} wedge crystal with 8 03BCm spatial resolution provided adequate photon statistics to detect changes ^{of one} bilayer ⁱⁿ fifty.

Fresh samples ^{contain a} dislocation density ^{of about 1} 107/cm2 ^{which must anneal} away before the bilayer edges ^{in the} wedge ^become clearly distinguishable. Annealing ^of dimyristoyl phosphatidylcholine (DMPC) requires

2-4 weeks at 35 °C in the L03B1 phase ^{and does not occur} during ^{3 months at} 17 °C in the P03B2’ ^phase. The anneal-

ing process requires the transfer of lipid molecules between neighbouring bilayers ; ^{thus the} annealing ^time depends

upon the permeation coefficient 03BBp which appears in the viscoelastic equations ^{for a smectic} liquid crystal and pro- vides a convenient determination of 03BBp. ^{We find} 03BBp is about 1 10-30 cm2/poise ^at 37 °C and is considerably

smaller at 17 °C. Additional defects appear when the sample ^{is cooled} through the transition temperature; ^these anneal away within 2 days ^{if the} temperature ^{is raised} again ^{within a short} time, ^but require the full 2-4 weeks if the

sample ^{is in the} P03B2’ ^{phase for more than a day.}

Classification

Physics ^Abstracts

61.30 61.70G - 66.30L

1. Introduction. ^- Above the gel transition tempe- rature,

phospholipid

^{and water form a}

lyotropic

smectic A

liquid crystal

with alternate

layers

^{of water}

and fluid-like

phospholipid

bilayers. ^{A bulk} specimen

will have various structural defects which ^can adver-

sely affect its

properties

[1]. ^The

simplest

defect is the

edge

dislocation, ^the abrupt termination of one or more

bilayers,

which has been discussed

theoretically

by ^{several authors} [2, 3, 4].

Edge

dislocations with

large

Burgers ^{vectors have}

been observed in

phospholipids using

polarization microscopy, ^{and were found to be}

extremely persistent

when

forming

^a

good specimen

[5, 6]. Screw disloca-

tions, ^{as well as} many other types ^of defects, ^have

been seen in

unaligned samples

^{of mixed}

lipids by

electron

microscopy

[7].

Except

^{for one}

questionable

case in the latter work,

elementary (Burgers

^{vector of}

one

bilayer spacing) edge

dislocations have not pre-

viously

^been reported ⁱⁿ

lyotropic liquid crystals.

Edge

dislocations with large

Burgers

^{vectors also}

have been seen in

thermotropic liquid crystals [8, 9].

Climb of

edge

dislocations in _response to a dilative strain normal to the layers ^{have been}

predicted [10,11]

and observed

[9];

these dislocations were found to have a

Burgers

^vector of about six layers. ^More recently, Meyer and coworkers [12] ^have

cleverly

used the strong

coupling

between the stress field and molecular tilt near a smectic A to smectic C

phase

transition to enhance

optical

contrast,

enabling

^them

(*) Present address : Bell Telephone Laboratories, Murray Hill, N.J. 07974, ^U.S.A.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019810042070100700

1008

to visualize an array of elementary

edge

dislocations in a narrow temperature range.

The existence of non-bilayer

lipid

structures in cell membranes recently ^{has been}

postulated [13].

^These

can be viewed as membrane fragments ^or dislocation

loops

^{inserted in the}

bilayer.

^Our observations suggest that such small scale structures occur even in a pure

lipid

system and that the

postulated

^{structure is a}

realistic

possibility.

That line defects play ^an important, and sometimes

dominating,

^{role in} diffusion in solids has been

long

established. It is reasonable to expect that defects

play a

similar role in

liquid

crystals under certain conditions. For example, fast diffusion _may occur

along defect lines

providing

^a short circuit for trans-

port ; ^because interlayer transport ^is very slow, ^mole-

cules can be

trapped

ⁱⁿ isolated lamellar

fragments

within dislocation

loops;

or, ^a tracer molecule _may

preferentially partition

^{into a defect}

[6] effectively limiting

^{diffusion to the} defect line. To relate our

diffusion measurements in

phospholipid

multilamellae to

microscopic

defects, ^we have devised a method of

detecting

^small

changes

in the number of

layers.

^It

has the obvious advantages ^{over the} Meyer ^method [12] ^{in that the} sample ^{is in a one} phase region ^{and is}

not limited to the vicinity of smectic A-C transition.

In our method, ^the

phospholipid

^is

lightly doped

with a fluorescent,

lipid

analogue molecule, ^{and the} fluorescent

intensity

excited in a small laser-illumi- nated spot is measured as a function of

position.

^To

accomplish

this, ^the

sample

is translated across a laser beam focused by ^a fluorescence

microscope,

slowly

enough

^{that the}

photon

statistics of the fluorescence resolve an

intensity change corresponding

^{to a}

single

bilayer. ^{The oriented} sample ^is

prepared

^{in a thin cell}

with a

slight

wedge ^{to it. To} accommodate the _gra-

dually increasing

sample thickness, ^{there must be} edge dislocations

periodically spaced throughout

^the sample

(Fig.

1); ^{this structure resembles a low} angle

tilt boundary ^{in a}

crystal. Scanning

^{across an ideal} sample ^should

yield

^a staircase fluorescence

intensity

pattern ^{where the} steps

correspond

^{to the}

edges

^of

single

bilayers.

Fig. ^{1. -} Multilayer ^{with an} array of edge dislocations separated by aj qJ when confined to a wedge shaped region ^with wedge angle (P.

The dislocations are forced to the centre of elastic interactions with the optical flats. This is similar to a low-angle ^tilt boundary ^{in a} crystal.

2. Experimental ^{method. - 2.1} S/N ^{AND RESOLU-}

TION CONSIDERATIONS. ^- The

sensitivity

is limited

by

^shot noise since we employ

photon counting

^elec-

tronics in which dark current and

amplifier

^{noise are}

negligible.

Suppose ^{there are c’}

photocounts

per second _per

bilayer.

Then, ^{to detect one} bilayer ^{in N}

within a time T, c’ T must be

significantly

greater ^than

the shot noise associated with all N bilayers. ^The

signal-to-noise

^{ratio is}

For c’ ⁼ 1 000

s-1,

^{T = 5} s, ^a SlVR of 10 yields ⁵⁰

bilayers

^{as the limit on total} thickness. c’

depends

^on

the

intensity

^{of the}

exciting

laser and the

dye

^concen-

tration. It cannot be made

arbitrarily large by

^increas-

ing

^{the laser} power because of the

photobleaching

^of

the

dye.

Furthermore, ^the

dye

concentration must be

kept

^low

enough

^{so that the} sample remains in a

single phase.

The spatial resolution R of a scan is determined

by

the beam diameter 2 w and the scan

velocity

^v

The thickness of the sample ^{cell limited our choice}

of

objectives

^{to one with a} long

working

^distance , which ^{has w =} 4 pm [14]. ^{A scan}

speed

^{of 0.17}

um/s

was used so the resolution was determined primarily

by

^the optics.

A third consideration is the maximum

wedge angle

T of the

sample.

^The

separation

^between steps ^is alT

where a is the

bilayer

repeat distance. If a/Q ^{is not} larger ^than R, ^the steps ^{would not be}

easily

^resolved.

The

wedge

angle ^must be less than 10-4 radians for the step

separation

^{to be ten times} greater ^{than the} spatial resolution.

2.2 SAMPLE PREPARATION. - The sample ^cell (Fig. 2) ^{consists of 15 mm} diameter, 1 ^{mm thick} quartz plates ^{flat to} better than

À/I0 (Virgo Optics,

Stirling, N.J.)

epoxied

into stainless steel holders. A low

shrinkage

epoxy ^was used (Tra-con, Medford, Mass., No.

2162D)

^to prevent warpage of the thin quartz flats as the _epoxy cured. The two steel plates ^were separated ^{with a Viton}

0-ring

^and

clamped

together

with three stainless steel screws. The

depth

^{of the}

quartz flats in the steel holder was chosen so that as

the flats touched, ^the

0-ring

^{would be} compressed ^to

its recommended

sealing

range. Two

hypodermic

needles were soldered into one of the

plates

^{to allow}

the

injection

^{of water into a} small reservoir between the

0-ring

^and

optical

^{flats. These needles were}

capped

with rubber

plugs

^{when not in use.}

Fig. ^{2. -} Diagram ^of sample cell. Details are given ^{in text.}

Before _any

lipid

^was

deposited,

^the quartz ^surfaces

were cleaned

repeatedly by placing

^a drop ^of spec-

troscopic grade

^{methanol on} the surface and

dragging

a clean lens tissue across it. The criterion for cleaniness

was a dark _gray Fizeau

fringe

^{over most of the area}

when the empty ^{cell was} assembled,

indicating

^the separation ^{was about} 0.1 um.

The

lipid

^{used was}

L-a-dimyristoyl phosphatidyl-

choline

(DMPC)

purchased ^from

Applied

^Sciences (State

College,

Penn.). A stock solution with 9.0 ^x 10-4 mole fraction dil

(3,3’-dioctadecylindo- carbocynanine

iodide, ^a generous

gift

from A. S.

Wag- goner)

ⁱⁿ spectroscopic

grade

chloroform (0.1

mg/ml)

was twice filtered

through

^{a 0.22} gum teflon

Millipore

filter and stored under

nitrogen

^{in a} freezer. To

deposit

the

lipid,

^{about 10} gl of the stock solution was

placed

on one of the flats and was

spread

back and forth with a

coverslip

^until the solvent evaporated.

Dry nitrogen

^was then blown over the deposit ^{for at least} twenty ^{minutes to} evaporate any remaining ^solvent.

The dried film was then

exposed

^{to the} vapour from

boiling

^{distilled water for a} few seconds to

slightly hydrate

^{the DMPC.}

The cell was then assembled with the screws hand

tightened

until the Fizeau

fringes

became visible. Even

though

^{there was}

light

scattering ^{in some areas,}

enough

^{of the} fringes ^{were visible to} determine the

separation ^{of the} plates. ^{The screws were} adjusted

until the

separation

^{was as small as}

possible.

^Gene-

rally,

^the separation ^was greatest ^{at the centre and}

tapered

^{off at the}

edges;

^{this made a}

wedged region

where one expects ^{to find an} array of

elementary edge

dislocations.

Any

sample ^{over 0.6} um thick at the centre was

rejected

because it would either be too thick

or have too great ^a

wedge angle,

and the order of the

fringes

became difficult to determine. The sample ^was put ^{into a 65 OC oven}

overnight.

^{It was cooled to the}

desired temperature ^{and water was}

injected. During long

anneals, ^{water was}

re-injected

every few

days

^to

replenish

any ^water that _may have

evaporated

^from

the reservoir.

Very

^small

changes

ⁱⁿ the Fizeau

fringes

were observed

during

^the

heating

^and

cooling,

^indi- cating ^the separation ^{did not}

change appreciably.

2.3 APPARATUS. ^- _{An argon} ion laser

operating

at the 5 145 A line was used to excite the dil fluores-

cence. Its _power, which was stable to better than 1

% during

^an

experiment,

^{was monitored}

by

^a

photodiode

whose output controlled a voltage controlled oscilla- tor. The laser beam was directed

through

the vertical illumination

optics

^{of a} Zeiss Universal

microscope.

The _power was attenuated with neutral

density

^filters

so that there would be no detectable

photobleaching

for longer ^{than 20} w /v, ^{i.e. ten} times the duration a

point

^is

exposed

^{to the laser} during ^{a scan. The micro-}

scope, with a 10 x

objective,

focused the beam to a

4 pm radius spot ^{on the}

sample

[14]. The fluorescence

was then detected through ^the

appropriate

^barrier

filters and pinhole ^{with a cooled}

phototube

^whose

dark count was less than 0.1 % of the count from a

single bilayer. ^The ratio of the

resulting photopulses

to the voltage controlled oscillator output ^was counted by ^{a ratio counter for five} seconds; ^this normalized the fluorescence

intensity

^{to the} laser power. The output ^{of the counter was fed into a 12 bit}

digital-to-

analogue converter, ^and

finally

^{to a} strip ^{chart recor-}

der.

The usual

microscope

stage ^was

replaced by

^a

spring

loaded

optical

translation stage ^{modified to mount}

onto the microscope. ^{A water} bath controlled the tem-

perature ^{of a hot} stage ^{bolted to} the translation stage, and the

sample

^{cell was held in} place ^{on the hot} stage without

applying

stress to the top ^steel

plate.

^The temperature ^varied

by

less than

+-0.2 OC ^during

^a

run. The stage ^was driven with a 20 RPH

synchronous

motor and differential micrometer

assembly

^{at a}

speed

^{of 0.17}

um/s.

The motion of the stage ^was monitored

by

^a balanced coil transducer

(Transtek

Model 350, Ellington,

Conn.)

^whose

voltage

output

was linear over 2 000 _J.1m and whose output voltage

noise

corresponded

^to less than _{1 J.1m.} This output

was also recorded

by

the chart recorder. We found that the translation

velocity

oscillated due to

pitch

variations inherent to differential micrometers; ^how-

ever, its

period

^was much faster than the

intensity

variations due to the

edge

dislocation _array in the

wedge

^{so did not} interfere with its detection.

To demonstrate the

reproducibility

^{of the} scans, ^we

performed

^this

experiment :

^A sample

consisting

^of

the

dye

embedded in a thin film of formvar was trans-

lated across the laser beam with its

intensity

^increased

a thousand-fold over the usual intensity ^to

photo-

bleach a straight ^{line. A second} bleaching ^{scan widened}

this line

only

very

slightly, indicating

that it retraced the first scan.

3. Results. ^- When looking ^{at the} fluorescence distribution over a

well-aligned

region ^{of the}

liquid

crystal, ^{one can see}

gradual

changes ⁱⁿ

intensity;

^the

directions of the

gradient

agree with what one would expect from the thickness distribution deduced from the Fizeau

fringes.

^{No discrete} steps ^{can be seen}

by

eye. When observed with crossed

polarizers,

^{the sam-} ple ^looks

uniformly

^dark; the thickness of the quartz

plates prohibited

^{the use of} dark field or

phase

^con-

trast.

Figure

^{3 shows a} segment ^{of the chart} recorder out-

put ^which clearly

displays

^the

periodic

steps ^{due to the} dislocations in the

wedged

region. ^{The scan was taken} along ^{a thickness}

gradient.

^The

wedge

angle, ^{as deter-}

mined by the Fizeau

fringes,

^{was 8 x 10- 5} radians;

the

predicted

average separation ^between steps ^of elementary

edge

dislocations

(for

^a

^fully

^hydrated

bilayer

spacing

^{of 64 A}

[15])

^{is 80 J.1m, in close agree-}

ment with the observed

periodicity

^{of 90} J.1m. The total fluorescence

intensity

^{at a} place ^{where the} fringes

were clearly ^visible gave ^an

intensity

per

bilayer

ⁱⁿ

good

agreement with the observed step ^{size. Parallel}

1010

Fig. ^{3. -} Scan of well-annealed sample ^{in the} La phase showing

the regular steps ^{due to the} edge dislocations. Each step corresponds

to the termination of a single bilayer. The shot noise level is approxi- mately 10 % ^{of the} step ^size.

scans gave the same

periodic

steps,

indicating

^the

defects are

parallel

^{lines as}

expected.

^{We are unable}

to determine the location of the dislocations within the thickness of the sample, that is whether they ^are

near the quartz surfaces, ^{or near the centre as one} would expect ^from their elastic interactions

[2].

^This

central location is subsequently ^assumed.

It is important ^to realize that the steps in the inten-

sity

^{did not} appear immediately after the formation of the

specimens.

^{In fresh} samples, ^{there were} many

irregular

steps,

giving

^the appearance of

peaks

(Fig. 4).

For samples

kept

^{at 35 OC, 12} OC above the

gel

^tem- perature, ^{2-4 weeks of}

annealing

^was

required

^{for the}

regular

steps ^to appear. Once the steps appear, they

were easy ^to

reproduce

^{and to detect over most of the}

sample.

Furthermore, ^the regular steps ^{did not} appear in all of the samples

prepared though

^{there were}

Fig. ^{4. - Scan of a fresh} sample ^{in the} La phase ^{where the} regular steps ^{are not} visible. The vertical scale is the ^{same as} in figure ^{3 so}

that / f= ^{1 in} equation (5).

definite

improvements

after several weeks ; ^the pre-

sence of extra defects in these samples may have been due to, for

example, pits

and scratches in those

parti-

cular quartz plates ^{and need not be an inherent} pro- perty ^{of the}

lipid preparation.

^The extremely long

times involved made it difficult to obtain

good

^statis-

tics and to look for systematic ^trends.

A sample ^which showed the

uniformly spaced

steps in the Lex

phase

lost them when taken below the tran-

sition ; ^the irregular

peaks

characteristic of a fresh

sample appeared

^{in scans} taken minutes after the temperature

equilibrated.

^If

kept

^{at 17 OC for} only ^a

few hours, ^the specimen recovered its regular steps within two

days

^after rewarming ^{into the} Lex

phase;

however, ^if

kept

^{in the}

P.8, ^phase

^{at 17 OC for more}

than a

day,

^the specimen

required

^{the full two to four}

weeks to recover the

regular

steps again ^{after rewarm-}

ing

^{to 35 OC. This} suggests that the transition occurs

in two steps ^with the slower step

taking

^several hours, consistent with the water diffusion time [16]. ^This point will be discussed in more detail in a later

publi-

cation

[17].

^A sample left below the transition at about 17 OC did not anneal enough ^{to show the} uniformly spaced steps ^even after twelve weeks; ^{a fresh} sample

never taken above the transition also did not show these steps ^{after the same} length ^{of time.}

4. Discussion. ^- 4.1 ESTIMATION OF DISLOCATION DENSITY. ^- The size and frequency ^{of the} peaks ⁱⁿ

figure

⁴ contain information on the

density

^of edge

dislocations. For the sake of definiteness, ^{we assume}

a model with all the

edge

dislocations in excess of those needed for the step array ^to conform to the

wedge

angle ^{are in} the form of circular dislocation

loops ^{of radius} RL ^{which has a}

probability density

P(RL). ^{This is a} reasonable

assumption

^since

(1)

^a single ^finite bilayer fragment would take on a nearly

circular

shape

because of the dislocation line tension, and

(2)

dislocation loops intercalated between the

same extended bilayers should coalesce to form larger

loops increasing

^{the mean value of} RL. Negative loops,

that is a

bilayer

^{with a circular area}

missing,

^{can anneal}

away

quickly

by dislocation climb since intralayer

molecular transport ^{is fast.}

Consider a random

spatial

distribution of uncorre-

lated dislocation

loops,

with the random variable RL

at a loop

density

PL (loops per

cm’).

(The uncorrelated

loop approximation

fails when the

loop density

^is

high enough

^that

loops begin

^to

overlap,

^{i.e. when}

rcRL2

^apL > 1.) ^The

corresponding

dislocation density (cm of dislocation _per

cm’)

^{is then} p ⁼ 2

7rkL

^PL,

where

Neglecting edge

effects, ^the average number of loops

in a resolution volume nR2 H/4 ^is

where H is the thickness of the

sample

and R is the diameter of the resolution area

specified by

equa- tion

(2). Neglecting

^the interaction between

loops

^to

assume a random

spatial

distribution, the variance of N is

The corresponding variance in the fluorescence inten-

sity f,

measured relative to the

intensity

^{of one}

bilayer,

is

where

Using equation (4) and the relation between _p and _PL

where

oi2rl

⁼

RL - RL

is the variance of RL. ^There

is

only

^{a factor of 4} difference in _p between a very

sharply peaked distribution

^for

RL( (J RL

⁼

0) and a very

broad one

(aRL

^{= RL) ; for} definiteness, ^we

take

^an

intermediate case ^with

a 2 L

⁼

(J2 - ^{1) RL}

^or

p

=f2R 2/4 H7RL3.

The observed distribution of peaks ⁱⁿ

figure

^{4 is}

characterized

by f ~

1. Since the peak ^{widths are}

close to the

spatial resolution,

^{RL R. Other experi-}

mental results suggest

RL = R

^{10 J.1m.} First, ^the

size of

bilayer

vesicles formed from multilamellar

liquid

crystals ^on

adding

^excess pure ^water should be limited

by

^the dislocation

density

^{in the}

multilayers;

since we commonly

observe

DMPC vesicles _{10 J.1m} diameter and

larger, RL

^cannot be much smaller than 10 J.1m. Second,

light

scattering experiments ^have

indicated the _presence of _{10 J.1m} size objects ⁱⁿ

aligned multilayers

[16]. Thus, p ~ ^{1 x} 10’ cm - 2 when we

take H ⁼ 0.2 _J.1m.

4.2 ANNEALING ^{DYNAMICS. - Molecules in these}

10 um dislocation loops ^are effectively isolated;

therefore, ^further annealing ^must

proceed

^{via the} interlayer transport of molecules which is described

by

^a transport coefficient that ^we shall associate with the

permeation

coefficient

Ap

^[18]. A dislocation loop

contributes to the _energy of the system

through

^its

edge

^or line tension _energy and

through

^the compres- sional energy in

squeezing

^{an extra} bilayer ^between

the quartz plates [2]. The first contributes

while the second contributes

Using

the relation Ba2’ -= y

[2],

^{we find the} compres- sional _energy is about ten times greater ^when RL ⁼ 10 um, ^so is the dominant contribution. Thus,

a

rough description

^of interlayer transport ^{has the}

lipid

molecules in the extra loop

squeezed by

^{the other} bilayers ^{from the} loop ^{into the}

bilayer

^{above or below}

it.

Let us consider the visco-elastic

equations

^{of a}

smectic

liquid crystal [19,

20, 21]. ^In

particular,

^we

wish to find the characteristic time for the

disappea-

rance of a

bilayer

^{when there is a zz} strain where the z-direction is normal ^to the

bilayers.

The relevant

equations

^are

(using

the notation of

[19])

where u is the bilayer

displacement, vz

^{is the macro-} scopic velocity ^{in the} z-direction, 0 ^{is the volume} contraction,

Ap

^{is the}

permeation

coefficient, ^{and B}

and C are elastic constants. When the

bilayers

^{are fixed}

in _space

(ü = 0),

^the permeation coefficient relates vz

to the z-component ^{of the} force;

Ap-1 ’

^{resembles a}

^drag

coefficient, ^{for a case where the}

phospholipid

^mole-

cules are both the fluid and the moving

objects.

^Eli-

minating Vz

^{from the} equations,

For a sudden compression ^{at t =} 0, ^{we have 0 = -}

0o h(t)

where

h(t)

^{is the} step ^{function or}

where b(t) ^is the delta function.

Solving

this initial value

problem yields

^a relaxation time z for u(t) ⁱⁿ

terms of a characteristic

length

L, ^viz.

For a defect-free sample ^of infinite lateral extent, ^the characteristic

length

^{is the}

sample

thickness. Our situation is _very different, however, ^{in that we have} isolated dislocation loops embedded in a multilayer

matrix. The

bilayers

ⁱⁿ this matrix are not isolated from each other, ^{but are connected}

by

large Burgers

vector

edge

dislocations at the

edge

^{of the} sample

and

by

^screw dislocations. Once a molecule

jumps

from the isolated dislocation loop ^{to an}

adjacent

bilayer, ^the bilayer ^{with the extra molecule can} very

quickly

^{come into}

equilibrium

with the other bilayers by lateral diffusion through ^the short circuits

provided

by

^the

edge

^{and screw} dislocations. Thus the charac- teristic length ^{is the} bilayer

spacing

a, and we have for an annealing ^time T. ^{of 2 x 106 s}