Dynamics of Rayleigh-like Instability Induced by Laser Tweezers in Tubular Vesicles of Self-Assembled Membranes

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Dynamics of Rayleigh-like Instability Induced by Laser Tweezers in Tubular Vesicles of Self-Assembled

Membranes

R. Granek, Z. Olami

To cite this version:

R. Granek, Z. Olami. Dynamics of Rayleigh-like Instability Induced by Laser Tweezers in Tubular

Vesicles of Self-Assembled Membranes. Journal de Physique II, EDP Sciences, 1995, 5 (9), pp.1349-

1370. �10.1051/jp2:1995187�. �jpa-00248238�

Classification Physics Abstracts

68 10-m 47.20Dr 82 70Kj

Dynamics of Rayleigh-like Instability Induced by Laser Tweezers in Tabular Vesicles of Self-Assembled Membranes

R. Granek (~) ^{and Z. Olami} (2)

Department of Materials and Interfaces (~) ^and Department of Chemical Physics (~), Weizmann Institute of Science~ Rehovot 76100, Israel

(Recei;ed

3i January 1995, re;Jsed 12 May 1995, accepted I June 1995)

Abstract. We present a theoretical study of the Rayleigh-like, "pearling", instability _re- cently observed _in tubular vesicles of bilayer membranes under the action of optical tweezers.

The _tweezers

are argued to pull the membrane into their operation _zone, thereby inducing _a

tension _m it. Consequently, the membrane _can respond in _a Rayleigh-like instability, _m which its _area decreases. Our approach is based

on a linear stability analysis of the deformations. How-

ever, because of the non-linear coupling existing between the suction process and the growth of

the unstable modes, the instability develops different characteristics At long times the growth

of the instability is controlled by the suction rate of the tweezers, assumed constant _m time The wavelength ^J* of the most unstable mode is time-dependent at short times it decreases

as -~ t~~/~~ at long times it is either constant, of the order of the tube radius R, _or increases

as -~

t~& _The amplitude of the most unstable mode _increases

(at

long times) _as

-~

t~/~ _We

delineate the different _regimes under which these behaviours

are predicted~ and discuss their

possible implications _on experiment. We also briefly discuss the possibility for other non-linear effects, such _{as an} induced spontaneous curvature m the membrane~ which _may be responsible

for the _appearance of _{very narrow} tubes _in the late stages

1. Introduction

The

study

of the _structure and

dynamics

of

self-assembly

_systems is _a field of

growing

interest

[1-3].

In

particular, shape

fluctuations and instabilities of

bilayer

vesicles, under

equilibrium

or

metastability conditions,

have been the

subject

of several recent studies

[4-12], particularly

with respect to the question of

budding is-ii

Much less attention has been given ^to the

question

^of the response of vesicles to a strong ^external

perturbation

which drives them far from the

equilibrium

situation.

Nevertheless,

these questions are of primary importance in

trying

to understand the behaviour of

biological

membranes _in

living

tissue.

In this paper, we focus _on the _response of

cylindrical (tubular)

vesicles to a local

tweezing.

Metastable

cylindrical

vesicles of surfactant

bilayers

_can be formed in the

laboratory

under flow [13]. ^{Their life-time is}

sufficiently long

to allow for

studying

their response ^to an external perturbation.

Recently,

the effect of

using optical (laser)

tweezers on such vesicles has been

@ Les Editions de Physique 1995

reported

_[13]. A durable action of the tweezers _on the membrane at _a local

region

defined

by

the size of the laser beam _causes the

cylinder

to deform with _a well-defined

wavelength

of the order of the tube radius R. This reaction, which _{appears even} far _away from the

place

of action, is _a reminiscent of the well known

Rayleigh instability

of

cylindrical jets

and of

cylindrical

columns in systems ^of two imiscible fluids

[14-16].

In the latter systems the

instability

is

easily explained by

the introduction of surface tension alone: since the interfacial _area decreases in _a

volume

conserving deformation,

the surface _energy is also decreased.

The action of the _tweezers has been

recently

addressed [13,

iii (independently

of _us

[18])

in terms of _an induced surface tension. This is _a reasonable

assumption,

since surfactant

molecules have _a

dipole

_{moment so} that their _energy is reduced in the _presence of _a

gradient

in the intensity ^of an

electromagnetic

field. As _a result, the tweezers will tend to pull surfactant

molecules into their

operation

_zone. The initial action of the tweezers is thus

expected

to be

a

simple pulling

^of the membrane, which will

obviously

induce _a tension in it. As the tension is raised above _a certain "critical"

value,

it _can dominate _any

bending

forces that resist _a

deformation.

However,

just as in the _case of the

simple Rayleigh instability,

the _area of the

bilayer

must decrease in this volume

conserving

deformation.

Hence,

if the _area is

fixed by

the membrane

itself,

the

instability

cannot

develop.

For the

instability

to

develop significantly,

transport of membrane _area out of the unstable

region

must occur, I.e., molecules must be

pulled

out from the

cylindrical

part. The action of the tweezers cannot be therefore

explained only

in

terms of _an induced surface tension, _as

previously suggested [13,17].

The tweezers must also

continuously

suck molecules into their operation zone.

The way these two effects of induced tension and continuous suction act

together,

_can be

explained

_as follows. As the laser is turned _on, there _{appears a} force

(per

unit

length)

_a~xt

towards the center of the

operation

_zone

acting

_on the

bilayer.

The

strength

of this force is fixed

by

the laser _power.

If, hypothetically,

the rest of the

cylinder

outside the

operation

_zone

would not

respond

to the external

perturbation,

_a mechanical

equilibrium

could be reached when this force is balanced

by

the

(mostly stretching)

elastic forces. In such _an

equilibrium

situation, _a fixed tension

profile

_a~xt would be induced in the membrane

(the boundary

conditions _{are zero current} at the

ends.) However,

if this tension is

large enough

to induce the

Rayleigh instability,

the membrane will

begin

to react The

resulting

reduction in _area will reduce the tension and constrain the

instability.

This

negative

feedback mechanism _{appears as}

a nonlinear

coupling

between the induced _current in the system ^and the tension. As _a result,

though

the _area

instability

still exists, it

develops

different characteristics which _we discuss below

[19].

To be _more

explicit,

_{we may} estimate the maximum surface tension that _can be induced

by

the _tweezers [17] as aext +~

Ebed,

where E is the _energy

density

of the laser beam in

vacuum, be is the difference _in dielectric _constant of _water and surfactant

tail,

and d is the

bilayer

half-thickness.

Typical

values

(E

_m 3 _x 10~ _erg

cm~~,

be _m

0.2,

d _m 20

I)

_{lead to}

a~xt ^Cf 10~~ _erg cm~2. For R

-~ I _pm and _~

-~ 10kBT

rw

10~~2 _erg,

a~xt is indeed well above the critical surface tension _{ac m}

~/R2

rw

10~~ _erg cm~2 needed to initiate the

Rayleigh instability (see below).

Of

particular

note is the fact that the

cylinder

is not the minimal

bending

_energy

shape.

Consider _a

cylinder

of

length

L and radius Ro which is divided into N~

spheres

of radius R~

with the _same total volume

(but

not the _same

area).

The latter

implies

N~ ₌

3R(L/(4R)).

The

bending

_energy ^{of the}

^cylinder (dropping

the Gaussian

curvature)

is Ec

=

7r~L/Ro

and that of the

spheres

E~ ₌

67r~R(L/R].

It _can be _seen that for Rs >

6~/~Ro

_m

1.82Ro

_one

obtains E~ _< Ec and the

cylinder

will be metastable

against

its

breakage

into

spheres.

We may

apply

the _same argument to a "necklace of

pearls", I.e., spheres

connected

by sharp saddle-type

"necks",

_so

long

_as the local radii of _curvatures at the saddles

obey

Ri _m

-R2, Providing

a

negligible bending

_energy contribution. This

"pearling" instability

does not show _up in the linear

analysis (discussed

in Section

II)

since it is _a first-order type, which

requires

to _overcome a

bending

_energy barrier. It is

apparently

_never observed in experiment without the action of the tweezers [13], ^{consistent with the fact that} it

requires

_a

significant

decrease

of

the _area

(Otherwise,

_one could expect ^{that it would be} initiated

by

thermal

fluctuations).

Our

approach

is formulated in terms of _a suction rate _u which controls the decrease of the surface

"coverage" # (or

increase of _{area per}

molecule),

with 0 <

#

< 1 and

#

= I, taken

by

definition to be the

equilibrium

_coverage. For

simplicity,

it is assumed

(somewhat incorrectly)

that the

suction,

and hence the induced tension, are uniform

throughout

the

cylinder.

The rate u may be estimated _as follows. With the assumption ^{that the} tweezers act _{on a} piece of

membrane with _a force _per unit

length

_{aext, we can} estimate the

velocity

_v in which that

piece

of membrane _moves under

steady

state conditions. The

(membrane) envelope

of the

cylinder

has to flow with the _same

velocity

_v towards the trap, ^{which is}

equivalent

to the flow of water in the

opposite

^direction with the

envelope

held fixed. Hence, ⁱⁿ the Poiseuille

approximation

this

velocity

is

v m

~~vP

m

I _jl)

il

ilL

where VP _m

aext/(RL)

is the pressure

gradient along

the

cylinder.

The amount of

stretching

of the membrane after _a time it is bL _cf

vbt,

and the surfactant _coverage

#

becomes

roughly

L/(L

+

bL).

Since bL _< L _we have

ii

_m -bL

IL

e -ubt and _we find the desired

expression

for _u [20],

u rw

~~~j~

~L (2)

Together

with the estimate for _aext

given

above this

yields

_a

rough

estimate for

u. It shows that for _very

long

tubes where L _»

R,

^u~~ becomes

extremely long compared

to other relevant

timescales in the

problem.

Of course, the suction rate is in fact position

dependent

_so that the above estimate _may be

regarded

_{as an}

averaged

rate.

2. Free

Energy

Model

Let _{us assume} that the membrane forms _a

cylindrical shape

vesicle of radius Ro and

length

L » Ro

suspended

in the solution. The

bending

free _energy associated with deformations of the

cylinder

_can be described

by

the Helfrich Hamiltonian [21]. ^Since we

ignore topological

transformations, _{we may}

drop

the Gaussian curvature and the Helfrich Hamiltonian is

H~ ₌

Ids ^~~

+

~

l, ₍₃₎

2 Ri R2

~

where Ri and R2 _are local radii of _curvature and _~ is the

bending

modulus.

(It

is assumed that the membrane has _no spontaneous

curvature.)

In

equilibrium

the membrane has _no tension,

but the tweezers induce _a surface tension _a. In this _case the total free energy of the surface is

written _as

F = H~ + H~

(4)

where

H~ = ds _a

(5)

We consider small

axisymmetric

deformations of the vesicle from its

cylindrical shape

_conserv-

ing

the total enclosed volume [15]. We therefore _assume that _{we can} describe the

djviation

^from

the

cylindrical shape by

_a

single-valued

function

u(z)

which has the property

u(z)dz

= 0.

The local circular radius

W(z)

is written as

Wlz)

= R +

Ulz)

₁₆₎

L

so that

/ _W(z)dz

= RL. Conservation of volume allows _us to find R. We have

o

R(L

=

R~L

₊

/ ^~ _{u(z)~dz (7)}

Making

use of the Fourier transform

u(z)

=

£

_Uqe~~~

₍₈₎

q

we obtain

R~

=

R( £

_Uq

_{U-q (9)}

q

This result is exact _even for

large

_{u so}

long

_as it does not exceed R

(in

which _case the membrane cuts

itself).

It will be desirable _to calculate the _area of the

cylinder

at _a

given

deformation. For small deformations where

(0u/0=(

< 1, the _area may be

expressed

_as

S _m 2~RL + ~RL

£ _{q~UqU-q (10)}

q

Making

_use of the volume conservation _constraint

equation (9)

we get

~

=l-i

ill)

so j ~

l

2R( ~j(1 (QRO)~]UqU-~

~

(12)

Here So

" 2~RoL is the _area of the undeformed

cylinder,

and _so § _is the relative decrease in

area.

We _{can now}

proceed

to calculate the total

free-energy

for small deformations _where au

/0z(

< 1 Consider first the

bending

_energy, equation

(3).

In the

Appendix

_we

obtain,

to order

U~,

H~ = H~o +

£ ^~ _(qRo

_{)~ +}

_(qRo ^~j _{UqU-q (13)}

o

~

2 2

where H~o

=

~~~

is the

bending

_energy of the undeformed

cylinder

The energy associated Ro

with surface tension involves

only

the

change

in _area

(S)

and _we obtain the classical

(Rayleigh instability)

result [15]

H~ = H~o

~ _{~ Ii} (qRo)~j U~U-q (14)

o

~

(with

H,o

=

Soa).

We define the difference in

free-energy

densities A

f

_as SA

f

=

(H~ H~o)

₊

(H~ H~o) (15)

so

that,

to order

U~,

_we

finally

obtain

Al

₌

fl ) ₍₍

)iqRo)~

⁺

qRo)4j j _{[i iqRo)~] uqu-~ j16)}

q o o

The

free-energy (16) implies

that fluctuations _at all

wavelengths

_are stable if the surface tension is below the critical value

~~

°~

2R(

^~~~~

If _a is raised above _ac,

long wavelength

deformations become unstable For this

regime

^of _a it is convenient to introduce the surface tension difference d

= a ac. In _terms of d, equation

(16)

becomes

~/ ~ )

~~~~°~~ ^~

~~~0)~j )

~

(~~0

_)~j

~q~-q (18)

q 0 0

so that small deformations of wavenumber q ^< qc, where

q)

₌

~

(-(

+

(~

+

4dR( /~) ⁽¹⁹⁾

2Ro

(

= +

aR( _/~

,

(20)

will _grow in time rather than

decay (A

model for this _time

dependent growth

is discussed in the

following sections).

For d <

~/R(

_we have

qcRo

_Ci

@

^~ < l

(21)

while for _{a »}

~/R(,

_qc saturates at

qcRo

C~ I. For

brevity,

from _{now on we} do not

distinguish

between R and

Ro,

and all

equations

will be

expressed

in terms of R.

So far _we have been concerned

solely

with the energetic stability of the

cylinder.

These considerations suggest ^{that the}

longest wavelength

mode (rw

L)

is the most unstable _one However, _a different mode _may be selected

throughout

the

dynamics,

which _we consider next.

3.

Dynamics

3.I. HYDRODYNAMIC MODES _AT CONSTANT SURFACE TENSION Consider the _pressure

difference between the inside and the outside of the tube. As usual this _pressure difference is induced

by

the elastic forces of the membrane _so that

[9,15]

~~ ~"

~"~ ~~ 2~~7L ~~

~~~~

For the undeformed

cylinder

this

equation just

leads to the

generalized Laplace

formula

APO ₌

I ( ₍₂₃₎

(Note

that _~ enters with _a

negative sign.)

We denote the pressure difference of the deformed tube _as

AP ₌ APO + bP

(24)

so

that, according

to equation

(22),

bP

(to

linear order in

u)

_can be obtained from the free- energy

density Al by taking

its functional derivative with respect to _u,

ipjzj

m

) _j25j

In Fourier space this functional derivative

simplifies

to

lip)~

₌

) _j26)

Applying

this formula _{to our}

free-energy density

equation

(16)

_we obtain

(bP)~

=

~

j)~

_Uq,

(27)

where

~

3

~ 4

(28)

T(T)

# a

[1~

_T~j

_@

2

2~

^{~ ~}

This result allows _us to _use previous

hydrodynamic

calculations for the

Rayleigh

instabil-

ity.

This involves the

following

_two assumptions.

First,

_we allow for the membrane to be

compressible,

_as done in theories for

hydrodynamic

relaxation of

long wavelength

fluctuations in membrane lamellar

phases

[22]. ^{It is} true that the membrane is

nearly incompressible

_on

lengthscales

^smaller than its

persistence length. However,

the

instability

_we discuss _{occurs on}

wavelengths

^of the order of microns, and the membrane still

thermally

undulates _on smaller

lengthscales (which

_are not observable in

optical microscopy),

I-e-, at wavenumbers _q

larger

than qc.

Thus,

for the small tensions _we consider

here,

the membrane _is

effectively quite

_com-

pressible

_on the micron scale

(see

also the discussion

following Eq. (43) ). Second,

_{we assume} that the

viscosity

associated with the 2D flow of surfactant in the membrane is not too much

different from the

viscosity

of the

surrounding fluid,

which is also consistent with previous studies [22].

We _can thus _use Tomotika's result [16], ^{who considered} a

cylindrical

thread of _one viscous fluid

(viscosity ~i)

surrounded

by

another viscous fluid

(viscosity _~2)

with

no-slip boundary

conditions

(i.e.,

the

velocity

field is continuous at the

surface).

In this calculation _{we may}

safely

assume that inertia _can be

neglected

since the

wavelengths

of interest _are much smaller than

the

macroscopic length (p~~R la)

^~~~

rw 10 _cm, with _p the fluid

density. (More accurately [15],

it is

self-consistently

assumed that _q »

w(q)p/~).

The rate _w associated with the

decay

_or

growth

of

fluctuations,

_i-e-,

Uq cf

U(exp[w(q )t] (29)

can be obtained from Tomotica's work [16] as a special case

(~i

= ~2 =

~).

The result _can be put ⁱⁿ the form

"~~~ ~~~~~~~~

~~~~

where 16(x) ^is a

monotonically increasing

^function in the

regime

0 _{< x} < I. As _a

result,

the function

(I x~)16(x)

has

a clear maximum at _{x =} x* _ci o.562. This

implies

that the mode

with q = q* = x*

/R

will grow faster than _any other mode and will dominate the

shape

at

long

times. More

explicitly

_we find

~ ^x

(ii (x)Kojx) lo (x)Ki jx))

+

211jx)Ki (x)

~ "

T

(Ii lT)Ko _IT)

_{+ lo}

lT)Ki (x))

^~~~~

For small _x, 16(x)

adopts

the asymptotic _expansion

ibjxj

_m

-(Cl

₊

)In ^x)x~

⁺

^{O(x~) j32)}

where

Cl

is _a numerical constant, Cl _Cf 0.067.

Roughly speaking,

the

hydrodynamics

leads to w _r- q~ ^for small _q,

making

the

long wavelength

modes _{grow very}

slowly, although, according

to energy considerations

(as

described in the previous section

),

these should be the most unstable modes. This result differs

strongly

from the _case considered

by Rayleigh

where _~2

" 0, in

which the

long wavelength

modes remain the _most unstable. The _same is true for the _case of

a "hollow"

cylinder

where _~i

= 0. The difference between the _two results _may be understood in terms of _energy

dissipation

considerations. Consider the

buildup

of the

longest wavelength

mode

(q

₌

~/L,

_u

r-

sin

(~z IL))

When _~2

= 0

(Rayleigh's case)

the

tangential velocity

at the surface _near the tube ends _can be finite and close to the central

(z-axis) velocity.

Since

the outer fluid has to _move in the opposite direction _m order _to fill _{up space,} this _means that

there _are indeed

infinitely large velocity gradients

in the outer

fluid,

which _can be however tolerated since ~2 " 0. As _a

result,

there _are

only

small

velocity gradients

in the inner fluid and _energy

dissipation

in total is small. In the _case ~i " ~2, on the other

hand,

the

tangential velocity

is

essentially

_zero at the surface and _one has

significant velocity gradients

in both

regions

outside and inside the

cylinder leading

to a

large

_energy

dissipation

and therefore

a

suppressed growth

rate.

(We

note that the Poiseuille

approximation correctly

describes the

same effect and also leads to [9] ^cJ rw q~ for small

q.)

3.2. TIME-DEPENDENT SURFACE TENSION. So far _we have assumed in the discussion that

the surface tension does _not

depend

_on time. This however cannot be true if the tension _is induced

by

the tweezers, _as

explained

in Section I. Let _us first

generalize

our results for _an

arbitrary time-dependence

of the surface tension. In this _case the

growth

rate

cJ(q, t)

is also

time-dependent, however,

in the

high

viscous

regime (neglect

of

inertia)

the

following

kinetic

equation

~~ ~~~'

_~~~~ ~~~~

still holds. Its solution is

Uq(t)

₌

U(exp ~ dt'cJ(q, t') (34)

or

Uq(t)

=

U(exp [Q(x, t)] (35)

Wh~~~

njx,

~~ ⁼

jj( ^zi~) _ii

_T~)

i~ _{(I l~~}

_~

^~i

_~~~~

with _x

=

qR

and

L(t)

=

~ dt'a(t') (37)

Suppose

that there is _a

buildup

of the tension with time,

starting

from _zero at t = 0. There

is therefore _a critical time _Tc for which the

instability

starts to

develop;

I e., for t _> Tc,

a(t)

is

larger

than _ac

(= 3/2(~/R~))

and the

amplitudes

start to grow in time rather than

decay.

At

slightly

later times, of order _Tc, the

integral

_over the surface tension

L(t)

exceeds act. For clarity we shall _now focus

again only

_on ^the

instability regime.

_we shall _measure the surface

tension relative to its critical

value,

I-e-,

using

d ₌ a ac, and the time will be reset to _zero at t ₌ Tc, without

change of

notations however. With these definitions, equation

(36)

transforms

to

~~~'~~ ~$ ^~~~~

_{~~ ~~~}

_i~

_~~~ ~

~~~

~°~~~

_~~~~

where

fit)

is _an

integral

in time _over the surface tension difference a, and

no lx)

_is

independent

of time and will absorbed into

U(. ^(Later

on this contribution to

U(

^{will be}

^neglected

^{since it}

is important

only

at very

early

times of order T~.)

From equation

(38)

we observe that the critical wavenumber

describing

the range of the unstable band is

time-dependent

and is

given by (neglecting Qo(x))

q~(t)~

=

j _(-((t)

+

/((~)~

+

4tit)R~/l~t))

¹³⁹⁾

with

(it)

m i ₊

zjt)R2 /(~t) (40)

reducing

to the two limits

qc ~

/l

«

i/R

₁₄₁₎

for

f(t)

<

~t/R~,

and _{qc t}

I/R

for

f(t)

_»

~t/R~

The wavenumber of the fastest

growing

mode, q*

(t),

_is also, in

principle, time-dependent.

Ob-

viously,

_a definition of q* ^is

primarily

relevant for the _case where

Q(q,t)

_» I

(for

q rw

q*),

where mode selection is

sharp. Indeed,

_m this _case q*

it)

_may be obtained

correctly by maximizing Q(q, t)

instead of the _more exact maximization of the

product w(q, t)Uq(t).

For

fit

_<

~~/R~

_{we may assume}

q*R

< I and _use 16(x) to order x~

only.

The wavenumber q* ^is then obtained _{as a} solution of the

equation

which _in the limit

q*R

_-+ 0 leads to

~vhen

fit)

_»

K/R~

_we obtain _again

q*R10.56, independent

of time.

3.3 A SPECIFIC MODEL. We _now wish to deal with _a

specific

model for the effect of laser

tweezers on

cylindrical

vesicles. We introduce the

assumption

that the tweezers

only

suck out

surfactant molecules from the membrane

Removing

surfactant molecules from the membrane does not

correspond exactly

to removing part ^{of the membrane}

itself,

since there should be

a time

delay

for the _area to

respond

to a

change

in the 2D surfactant concentration. It is the osmotic

compressibility, namely

the

dependence

of surface tension _on concentration, which

controls the _nature of this _response [19].

Given _{a certain} concentration _c~ of surfactant molecules in _a

single monolayer

film

(of

the two films composing ^the

bilayer),

_{we now} define the surface

"coverage"

_as

#

_{= c~ao} where

ao is the

equilibrium

_{area per} molecule. When the membrane is in

equilibrium,

its

(true)

surface tension _a vanishes and the

equilibrium

_coverage

is, by definition, #

= I. For

#

_< I

the surface tension is _non-zero

(positive).

The _coverage

corresponding

to the critical surface tension _ac

=

(3/2)~/R~

is denoted _as

#c. Assuming (self-consistently,

_as will be shown

later)

that # remains close to

#c

at all times, we may linearize the surface tension _near

#c:

d C~

ao14c 4)

₁₄₃₎

Here _ao is

essentially

the compression modulus at coverage

#c: denoting

_a the _{area per}

molecule,

and II the surface _pressure, the

compression

modulus is B

= -a0II

Ida

₌

<off /0#

₌

-#0a/0#,

_so that

ao4c

=

B(#c).

We note in

passing

that the small surface tensions assumed here

imply

that

they correspond

to the thermal undulations

regime [3,13, 23].

^{When the membrane is}

pulled

out at surface tensions below _ac, the thermal

"ripples"

_are

being

stretched

corresponding

to _an increase of the

projected

_{area per} molecule which decreases the surface entropy ^{and therefore} increases the effective tension. For _a > ac the situation is _more

complicated

because the

long wavelength

axisymmetric

modes _are

unstable,

whereas all other modes still experience thermal undulations.

We _can take this effect into account

by redefining #

_as the

projected

_coverage, rather than the real coverage [24].

For

simplicity

_{we assume} that the surfactant is sucked

uniformly

from the membrane, and

at _a constant rate. If J is the suction rate of

molecules, then, by definition,

the

change

in

the number N of molecules in the membrane per unit time is

dN/dt

=

-J,

which _may be transformed to

( _#(t) ^~~

= -u

(44)

t o

where the

(coverage)

suction rate _u is _u

=

Jao/So. (Note

that in Section I _we have made _use of

a different _route and estimated _u

directly.) Integrating equation (44)

and

using

the definition

(12)

for

§(t)

_we obtain

<it)

m

(c =jj ^j45)

On the other

hand,

_{we can} evaluate

§(t)

in terms of the surface tension

d(t)

=

a[#(t)]

@(t) ci

j _£[I (qR)~](U(Ui~)exp 2

/~ ^cJ(q, _t')dt'j ⁽⁴⁶⁾

It is _a

good approximation

to _assume

that,

within the time

period

m which the

(true)

surface tension increases from _zero to its

(vanishingly small)

critical

value,

the undulations correlation function does not

change

from its value at

equilibrium.

Hence

(~/0~/0 j

_{~ (~/} ~/_ _~

~~ ~B~

~~)

~ ~~ ~ ~ ~~ 2~L

~i(qR)

1(x)

=

~

~x~

_{+ x~}

(48)

This leads to

Sit)

= Wo

£" _dx~il~exP i2Qix

_t)1

149)

Wo "

kBT/(8~~~) (50)

Equations (45)

and

(49)

for

#(t)

and

§(t)

_are

coupled

via

equations (43)

for _a.

Obviously,

this should lead to _a

highly

non~hnear

behaviour,

which is quite far from the

simple

linear

analysis given

in Section 3.1 [25]. ^{Exact solutions} to these

coupled equations

can be therefore obtained

only numerically (Section 3.5).

In the next section however _we will present a few

asymptotic

solutions which

already

bear most of the

important physics.

3.4. ASYMPTOTIC SOLUTIONS

3.4.1. At short times ut <

#c,

_{we may use}

§(t)

< I in

equation (45)

to obtain

#

ci

#c

ut +

#c§(t) (51)

Consider _{now even} shorter times where

#c§(t)

< ut, ^{for which}

#

is

decreasing linearly

with time:

4

_~

4c

Ut

152)

However, by looking

at

(51),

_{we can} expect _a crossover behaviour at a time t ₌ tc

obeying

@ltc) Cf

Utc/4c ₁₅₃₎

At these times the decrease in _area

resulting

from the deformations starts to dominate and

#

will increase towards

#c

at later times,

causing

the

instability

to slow down. Since the

instability

stops when

#

₌

#c,

it is clear that at all times t > 0 the

inequality #

<

#c

must hold.

Together

with equation

(45),

this determines _a definite _upper bound for

§(t).

ijt)

<

ut/ic j54)

This _means that

§(t)

must saturate at its upper

bound,

I-e-,

bit)

_Cf

Ut/4c

155)

at

long

times t » tc.

On the other

hand,

equation

(49)

^{still has} to be

obeyed.

Let _us consider this equation ^when

Q(x, t)

is much

larger

than unity for

(normalized)

wavenumbers _x around x*, which is the situ~

ation

expected

at

long

times t » tc. The

integral

in equation

(49)

is then dominated

by

these

wavenumbers.

(Since

the condition

Q(x*)

_» I corresponds also to

Q(x*)

» In

[(I (x* )~)/i(x*)],

_{we may}

safely neglect

the effect of the

weight (I x~)/i(x)

in

determining

the dominant value of

x.) Equation (49)

_can therefore be

approximated according

to the method of steepest descents to

give

~ ~

~fi~dT~

(~*

(~~~~

~~~ ~~~~~~'~~~ ~~~~

Equating

the latter

equation

with

ut/#c defines

_an equation

for L(t)

_m the

long

time regime t _» tc. It should be

noted, however,

that the steepest descents approximation ^is

only marginally

valid here because since

§(t)

must increase

linearly

with time in this

long

time

regime Q(x*

cannot get

typically

_more than _an order of

magnitude larger

than unity.

(This

_may also

give

somewhat _more

significance

to the

weight

in

equation (49),

which _may be however

easily

taken into account in the steepest descents

approximation.)

To

verify

these

approximations,

_a full

numerical

integration

will be used in _our numerical

analysis

in Section 3.5.

Consider _now the two

following

situations

according

to whether the value of x* in the relevant time

regime

is of order

unity

_{or not:}

I) ^x* _rw

@

_{< I. Here}

we

find,

to

leading order,

it)

x* _cf I. Here _we

have,

_more

precisely,

x* _ci 0 56 and _we find

S _~ CWO

l~

exP

Ill

158)

where C

=

1.37,

_{a =} 0.071

We _now equate

equations (57)

and

(58)

with

ut/#c

and solve

approximately

for

f(t) _(using,

for

simplicity, only

_a

single

iteration of the

solution):

I) ^{For x*} rw

h

_{< I}

we get

L(t)

rw

fi@

_{or, more}

_precisely,

~

^~~t

In ut/(#cA(t))j

^~/~

~~~~ ^~

~~

ln

(~t/(4~R3))

^~~~~

with

Ait)

=

@Wo ^~l~l~~~ _In lsll

^~~~

Evidently,

the

self-consistency requirement f(t)

_<

~t/R~ implies

that _we deal here with times much

longer

than the basic

hydrodynamic

relaxation time of the tube _Tr

=

~R~/~.

More

accurately, including

the numerical

prefactors

and

logarithmic dependences,

this

self-consistency

requires t » 100Tr for

typical

situations

(Wo

_rw ^10~~

-10~~).

ii)

For x* ₌ 0.56 _we get

~~~~ ~ °

~~~~~ ~li~oj ^l~°~

where _{p =} 2.74.

Accordingly,

the

self-consistency

requirement

fit)

_»

~t/R~ implies

_now that t < 100Tr.

We _see that for t » tc,

f(t)

increases with time much _more

slowly

^than

rw t, either _{as a} power law

rw

t~/~,

_or

logarithmically.

This slow increase

corresponds

to _an

algebraic decay

of the surface tension difference

a(t)

towards _zero

(or

of # towards

#c),

_as obtained from

a(t)

= df

/dt.

This

yields (to leading order)

d

rw

4(61)

t

in _case

(I)

where x* <

I,

^and

~-i~~

d _m

(62)

in _case

(it)

where x*

rw I. We _may conclude that for

sufficiently

small suction rates where 100UTr ~ l, the

(trUe)

SUrface _{tension a IS}

eSSenil3ily eqUal,

In the

long

time lImIi t $ 100Tr

(with

_Tr ci

~R~/~),

to the critical value _ac

rw

~/R~. However,

for

larger

rates, 100uTr » 1, ^the surface tension _a at the

longest

relevant time t

rw

u-~ _is still much

larger

than _ac.

Another quantity of interest to _us is _<

(Vu(t))~

_>, which is

expressed

_as

<

(VU(t))~

_>

=

L

_q~

(Uq(t)U-q(t)1 (63)

It _measures the _extent of surface

corrugation

and therefore _can be related

directly

to

experi-

ment.

Using

the _same

procedure

_as for @(t),

equation (63)

becomes

m ~2

<

(v"(~),)~

> " 2w0

/

dTj~XP ^l~Q(~,

^t)1

⁽⁶⁴⁾

0

For

large

values of the argument ^{of the} exponent,

corresponding

to t » tc, we may

again

_use

the method of steepest ^descents

leading

to

<

(vU(t))~

_{> ~} 2W0

2n)dx21~~

$exP12i~(x*,

^t)1

⁽⁶⁵⁾

~° ~~~~

<

(vu)2

_>

m

2~

~)jl~~

S(t)

_~~~~

Again

_we deal with this result

separately

for values of x* < I, and for x*

rw 1:

I) ^{For x*} rw

fR2 /(~t)

< I _we find

<

(VU)~

> t

2(T*)~§(t)

_t

~(~)~~

(67)

or,

using

equation

(59)

for

f(t)

in this

regime (ignoring

all numerical factors and

loga-

rithmic

terms),

<

(Vu)~

_>

+~ u

(~~~t)

~~~

rw

ufi ₍₆₈₎

~

it)

^For x*

rw I _we find

Let _{us now} summarize the

implications

of these results. The discussion henceforth will be divided into _{two cases,}

depending

_on the value of the _maximum

surface

tension dmax =

ac + amax that is induced in the membrane

during

the laser

operation,

where

°max ^~i °0~tc, ~~~)

as

compared

to the

bending

_energy

~/R~.

This _can be translated into _a condition _on the suction rate u, as shown below.

3.4.2. Case 1: dmax <

~/R~.

This is the less relevant _case for _us where the smallest value of the

wavelength

I* that is reached is still much

larger

than the tube radius R. This _case corresponds to very small suction rates. To be more

specific,

_we need to find the _crossover time tc. For t < tc we have

L

(t

_{nf 00}Ut~

71)

2

The time tc ^is obtained from

equating

this result with the result at

long

times,

equation (59), leading

to the

scaling

~

l/3

~c '~

~~)

~~2 ~~~)

0

(ignoring

here all

logarithmic corrections).

Hence the above condition _on dmax C~ acute translates into _a condition _on the suction rate _u,

~2

~ ~

ao~R5

or

UTr ~

~

_{~ l}

₍₇₃₎

R _ao

These _are indeed

extremely

small suction rates relative to the tube relaxation _rate

Tj~.

Note that this _case

corresponds

to Tr < tc, _so that the time

regime

t > tc

corresponds already

to t » Tr.

As

explained,

in the

regime

t < tc all

amplitudes

remain small

(Q(x)

<

I)

and mode selection is not

sharp. Nevertheless,

it is of interest to obtain the wavenumber of the fastest

growing mode, q*,

which in this regime is the _one which maximizes

cJ(q, t)U).

It increases from _zero

(actually,

from

rw I

IL)

_{as q*}

rw

t~/2,

_{or, more}

explicitly,

_as q* cr

I(74)

~

until it reaches _a maximum value

(q*)max.

^This value is

given by (x*

=

q*R)

~~« ~

fi~ _l~ao

uR~ ^~/~

~

(~

^~

R~ao

^~/~

j~~~

~~~ ~i~ ^{~2 ~} ~

In the

regime

t » tc _we have

§(t)

_ci vi

/#c

and the most unstable wavenumber

decays

back to zero as

j~3 ^1/4 x*

rw

~

(76)

~t

(ignoring logarithmic dependencies) showing

_{a very} slow _increase of the

wavelength

as

I*

rw

PM.

Since the

wavelength

I* of the most unstable mode is much

larger

than the radius R _at all times, the

instability

is

likely

to stay ^invisible. We may confirm this

by looking

at the time

dependence

of _<

(Vu)~

_>; after the

"exponential"

_{increase up} to the time tc, ^it increases _as

rw

t~/~ (following Eq. (68)),

indeed _{a very} slow increase.

Moreover,

since

by equation (73)

we consider here very small suction rates

corresponding

to 100uTr < 1, it follows from

equation (68)

that _<

(Vu)~

> is also

vanishingly

small _even at the

longest

relevant times vi _+~

I,

where almost all the membrane of the initial tube has been sucked _out.