Mechanics of interfaces : the stability of a spherical shape

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Mechanics of interfaces : the stability of a spherical

shape

M. M. Kozlov, V.S. Markin

To cite this version:

Mechanics of interfaces :

the

_stability

of a

_{spherical shape}

M. M. Kozlov and V. S. Markin

A. N. Frumkin Institute of

Electrochemistry,

The U.S.S.R.

_Academy

of Sciences, 31

_Leninsky

Prospect,

Moscow, U.S.S.R.

(Reçu

le 20

_février

1989, révisé le 17 novembre 1989,

accepté

le 23 novembre

₁₉₈₉₎

Résumé. 2014 On considère les conditions de

perte de stability d’une interface

_sphérique.

Une interface arbitraire est _{caractérisé par des facteurs tels que tension de} surface, le

_premier

et le deuxième moments, les modules d’élasticité. Dans un état _déformé, on _{suppose que les conditions}

d’équilibre à

l’interface interne sont

_respectées.

On obtient des

_{expressions générales}

_{pour les} critères de stabilité

_qui

mettent en _rapport les

paramètres

du

_système

et la différence de

_pression

entre

_phases

_{homogènes séparées}

_{par l’interface. On considère les} cas

_particuliers

de membranes

qui

ne se _{dilatent pas} et d’une interface ayant un

_petit

module de dilatation. Abstract. 2014 Conditions _are considered for the loss of

stability

of a

_spherical

interface. An

arbitrary

interface is characterized

_by

the force _factors, such as surface _tension, first and second

moments, moduli of

elasticity

and

cutting

forces. In a deformed state the conditions of internal interface

_equilibrium

are assumed to be satisfied. General

_expressions

are obtained for the

stability

criterion which relate the _{system parameters} and the pressure

drop

between homo-geneous

phases separated by

the interface. Particular cases are considered of a

_{non-expansible}

membrane and an interface with a small

_expansion

modulus of

_elasticity.

Classification

Physics

Abstracts 82.65D - 87.20C

Introduction.

In recent _{years researchers have become}

_quite

active in a

_{traditionally explored}

field of

chemistry, thermodynamics

of interfaces. This concerns the

_study

of

_systems

such as

microemulsions,

water solutions of

_amphiphilic

_substances,

and

_biological

membranes. The classical

_approach

to the

_description

of interfaces

_{developed by}

Gibbs

_[1]

and followed

_by

others

_[2-8]

involves the

_concept

of force

factors,

namely

the Gibbs surface tension and two moments,

Ci

and

_C2.

The interface

_shape

is described in terms of the Gibbs

_dividing

surface,

a normal to _{which in every}

_point

coincides with the

_{density gradients}

of the

_system

components

(as

well as other

_{thermodynamics}

_quantities).

_According

to this

_approach,

the variations of the Helmholtz free energy of a

_piece

of an interface _{may be described} as

where T is the absolute

_{temperature ;}

_g

_is

the chemical

_potential

of the i-th

_{component ;}

SS and

_nsi

are the excesses of

entropy

and the amount of the i-th

_component,

determined with

respect

to the Gibbs

_dividing

_{surface ;}

A is the area of the considered

_piece

of

interface ;

J is the sum of

_principal

curvatures hereafter referred to as the mean curvature ;

K is the

_product

of

_principal

curvatures to be referred to as the Gaussian curvature. Let us

emphasize

that extensive

_quantities

in

_expression

₍₁₎

have the

_meaning

of differential

quantities

describing

a

_piece

of interface.

The force factors were

_{postulated by}

Gibbs.

However,

within an

_{approach involving}

assumptions

of local

_{thermodynamics,}

the force _{factors may be}

_expressed

in terms of the

components

of a

_microscopic

_pressure tensor

_[2,

_3,

_6-11].

In this case the interface is

_regarded

as a continuous

_layer,

_{with the pressure} tensor

_{being specified}

_{in every}

_point.

In

_describing

classical

_capillary

systems

in which the surface tension yG is

high,

the

dependence

of the force factors on deformation _{may be}

_neglected.

The

_systems

_belonging

to another class

_(interfaces

with the

_{surfactant-containing}

microemulsions ;

films of

_amphiphilic

substances

_making

_up such structures as _e.g.

_{lipid mesophases}

and

biological

membranes)

have a _{very low} surface tension

[12].

For the

description

of such

_systems

in terms of

thermodynamics

the

_dependence

of the force factors of an interface on its

_geometrical

characteristics is

_significant.

This means that an interface should be

_regarded

as an elastic

system

characterized

_by

the moduli of

_elasticity

_{[10] :}

extension-compression

modulus of

_{elasticity :}

first modulus of

_{bending elasticity :}

second modulus of

_{bending elasticity :}

moduli of

_elasticity

for the mixed deformations :

Moduli of

_elasticity

_{may be}

_expressed

_through

the force factors as

On the basis of the

_{relationships}

between the force factors and

_geometrical

characteristics

the

_spontaneous

_geometrical

characteristics of an interface _{may be}

determined,

viz. the

The

_description

in terms of

_{thermodynamics}

and mechanics

_presented

in the above-mentioned references makes it

_possible

to

_study

the

_shape

and

_stability

of structures

_produced

within the

_multi-phase

_systems.

_{Experimental findings}

revealed a

_variety

of such structures

and confirmed the

_feasibility

of their transformations into each other. The most illustrative

examples

are

_lipid-water

mixtures and

_biological

membranes.

_Lipids

form

_bilayers

in water,

with the

_bilayer

surfaces

_being

formed

_{by polar}

heads of

_lipid

_molécules,

and the inner volume

occupied

by hydrocarbon

tails. Unilamellar

_spherical

vesicles,

multi-lamellar

_spherical

liposomes,

and flat lamellar

_systems

are the most

_wide-spread

structures formed

_by

the

_lipid

bilayers.

In

_systems

with a low water content,

lipids

also form normal and inverted

micelles,

and normal and inverted

_cylinders

within a

_{hexagonal lattice}

(HII-phase).

When

_temperature,

water content, or other external factors

_change,

a

_phase

transition occurs, followed

by

changes

in the

_shape

of

_lipid

structures

_[13].

A

_biological

membrane whose bulk is the

_lipid

bilayer

forms the cell

_envelope

and

_largely

dictates the cell

_shape

_[14].

An

_erythrocyte

is

mechanically

the most

_interesting

cell.

_Depending

on the

conditions,

an

_erythrocyte

is

_capable

of

_assuming

the

_shape

of a biconcave discs

_(dyscocyte),

a

_sphere

_{(spherocyte),}

a _cup

(stomatocyte),

a

_sphere

with

_spicules

_{(echinocyte),}

etc.

_[14].

Characteristic

_shapes

of the structures formed

_by

interfaces

_correspond

to the

_system’s

equilibrium

state. When

_equilibrium

is no

_longer

_stable,

the

_{shape changes.}

There are

_reports

[15, 16]

of the studies of interface

_stability.

One _paper

_[15]

discusses the

_stability

of an

interface with force factors and no

_elasticity,

i.e. the moduli of

_elasticity

are

_equal

to zero.

Reference

_[16]

is concemed with an

_opposite

case where variations in free energy are

_wholly

associated with the

_bending

_{elasticity Ejj}

which

_responds

to the variations of the total

curvature.

This paper considers the

stability

of a

_spherical

interface

_shape.

The contributions of all the force factors and moduli of

_elasticity

to the

_stability

criteria are taken into account.

Statement of the

problem.

Let the interface be

_spherical

_{with the radius ro. There is} a

_drop

of

_hydrostatic

_pressure at the

interface between the bulk

_phase

and the interior volume of the vesicle. The

_{thermodynamic}

properties

of the interface are determined

_by

the force

_factors,

_namely

the surface tension

Vo and the first and the second moments

Cl

and

C2.

Variations _{of the free energy of} a

_piece

of interface of an area A in the process of isothermal

deformation with a constant amount of

_components

_{may be described} as

where the functions

_{Î’G(A,J, K), C1(A,J, K)}

and

_C2(A,J,K)

allow for a

potential

redistribution of the

_{components ;}

all the

_geometrical

characteristics describe the

_dividing

surface. Variations of the

force

factors

_during

deformation are,

by

definition,

expressed

in

terms of the interface moduli of

_elasticity

_[10].

Let us consider the interface in the state of internal mechanical

_equilibrium.

The condition of

_equilibrium

force a

_piece

of interface with

respect

to lateral

_{displacements}

_[9]

is

_expressed

as

In

_{equations (6)}

and

₍₇₎

the

_operators

_81ax

and

_a/ay

denote differentiation in a

_system

of

coordinates whose axes

(x, y )

are in a

_plane

which is

tangential

to the considered interface and coincide with the direction of the

_principal

curvatures ; cx and

_cy

are the

_principal

curvatures,

Qx

and

_Qy

are the values of the

_cutting

forces which are

_applied

to the

_edges

of the surface element and are

_{perpendicular}

to the axes x and y.

_{Equilibrium equations describing}

rotation may be written as

The

_equilibrium

_equations

_(6)-(9)

were obtained on the basis of

[9]

assuming

the membrane

thickness h to be much less than the radius of curvature _{ro. In}

_{equations (6)-(9)}

the terms of a

higher

order than

_(h/ro)2

were

_disregarded.

Let us consider a minor deformation of a

_spherical

interface. This deformation may be

assumed to be

_{axially symmetrical,}

and the interface

_shape

_{may be described}

_by

the function

(Fig. 1)

as

Fig.

1. - Axial

where r is the

_length

of the radius vector drawn from the center of the

_sphere

to a

_point

on the

dividing

surface in the deformed _{state ;} cp is the meridional

_angles

is the

_{arbitrary angle}

function and À is the deformation

_amplitude.

The curvatures and the element of the area of the

_dividing

surface in the state of deformation may be

presented,

to an _accuracy of the second-order terms with

_respect

to

À,

as

Let us assume that the deformed interface maintains an

_equilibrium

_and,

in

_particular,

the

equation

of lateral

_equilibrium

₍₆₎

is satisfied. The

_objective

of the

_{investigation}

includes the

study

of the

_stability

of a

_spherical

interface to such deformations.

Stability

criteria.

The

_{thermodynamic stability}

of a

_{spherically shaped}

interface is determined

_by

the amount of work to be

_applied

to the interface and its environment at a minor deformation. Zero work

estimated in the first order of

_magnitude

of deformation

_yields

the condition of

_equilibrium

between the interface and the bulk

_phases.

The contribution made

_by

the deformation of the second order of

_magnitude

determines the

_stability

of the

_equilibrium

state. If the second order contribution is

_{negative, equilibrium}

is unstable. _

To determine the

_stability

_criterion,

work must be calculated to an _{accuracy of the second}

order terms of deformation

_{magnitude. By}

virtue of

_{equations (1), (3)-(5)}

an

_elementary

work

applied

to the deformed

_part

of an interface and

_required

for its further deformation is

_equal

to

The last term in

_equation

₍₁₄₎

is the work

_applied

to the bulk

_phases,

whereas AP is

_equal

to _{the difference between pressure values for the}

_sphere

interior and environment.

On the basis of the

_equation

of lateral

_equilibrium

₍₆₎

as well as

_equations

_{(7)-(9), (3)-(5),}

(11)-(13),

the relative

_expansion

of an interface may be found

_{(Appendix A)}

with an

The

_expression

for the deformation work obtained

_by

_{integrating (14)}

over the deformation

value and interface area _{may be}

_presented,

with an account for

_{(11)-(13), (15),}

in the first order with

_respect

to À as

When

_W1

is zero at an

_{arbitrary e}

an

_equilibrium

_equation

_{may be written :}

which is a

_{generalized Laplace}

_equation

for a

_spherical

surface

[9].

The second order work with

_respect

to À determines the

_stability

of the

_spherical

interface

equilibrium

and may be described

(Appendix B)

as

The work

_WII

to be

_applied

to the environment and the

_spherical

interface to deform the latter is determined

_by

the

_type

of the

_{deformation 03BE (~ )}

and

_by

the interface

_parameters,

such as the force factors associated with the initial state

_{yl , Co}

and

_Co,

the moduli of

_elasticity

EJJ, EA-41

EKK, EAJ, EAK

and

_EjK,

and the pressure

drop

between the internal and extemal

It is

_convenient,

and in most cases

_possible,

to estimate the

deformation e

*

(p )

at which the

system

loses

_stability

« for the first time ». Estimation

of § *

is based on an

_assumption

that there are values of the interface

_parameters

for which the amount of work

_{WH required}

to

implement

the

_{deformation e *}

_{may be}

_negative.

_{For any other}

_type

of deformation

e :0 e *,

however,

the same values of the interface

_parameters

result in a

_positive

WII ;

in

effect,

the

_system

is unstable

_only

with

_respect

to § *.

The criterion for a loss of

_stability

introduces a link between the force

factors,

the moduli of

elasticity,

and the pressure

drop

OP ,

at which the

_spherical

interface loses

_stability

with

respect

to § * .

Equation

(18)

appears to be too cumbersome for

_{analysis ;}

let us consider extreme cases.

NON-EXPANSIBLE INTERFACE. - Let

us consider an interface with a

_large

extension-compression

modulus of

_elasticity

_{(EAA --+ (0).}

Such

_systems

are

_{represented by}

_{e.g. thin}

insoluble films and

_{bilayer lipid}

membranes which do not

_exchange

_{material with any}

reservoir

_during

deformation. In this case the last term in

_equation

₍₁₈₎

_{substantially}

exceeds all the other terms for an

_arbitrary

form of

_{deformation e}

and is

_positive.

Therefore the

deformation g

*

at which

_stability

is lost « for the first time » should

comply

with the zero

value of this term

This class of deformations will be

_analysed

in this section. In this case

_equation

₍₁₈₎

_{may be}

reduced to a

_simplified

form where the surface tension

y 0

is eliminated

_{by using}

a

_generalized

Laplace equation

(17)

The

_analysis

of

_{equation (20)}

reveals a

_relationship

between the force factors and the interface

moduli of

_elasticity

for which the

_{spherical shape}

becomes unstable. The deformation at

which

_stability

is lost « for the first time » is associated with a maximum characteristic

wavelength

of the

_{perturbation imposed}

on the

_spherical

interface. The

_longest-wave

perturbation

which satisfies

_equation

₍₁₉₎

_{may be written} as

The criterion of

_stability

loss to deformation

₍₂₁₎

can be

_expressed

in terms of a _pressure

drop

OP

_equal

to _{the difference between the pressure values for the bulk}

_phase

inside the

sphere P in

and for the exterior bulk

_phase

_{P out,}

OP =

Pin - P out.

The

_{spherical shape}

loses

where

The value of the critical pressure

drop APcr

which

_designates

the

_boundary

between the

states of stable and unstable

_equilibrium

of a

_spherical

interface

_depends,

in

_compliance

with

equation

(22),

on the moments

Co

and

_Co.

As defined in references

_{[8, 9],}

a

_positive

value of the first moment

Co

is related to a

_tendency

of a

_part

of the interface to reduce the total

curvature or to cave in the

_{sphere. Similarly,}

at a

_positive

value of the second moment

Co 2

that

_part

of the interface tends to reduce the Gaussian curvature. The

_relationship

between the critical value of the pressure

drop

âP cr

and the moments is shown in

_figure

2. The unshaded area above the

_straight

line

_represents

a stable

_equilibrium

of the

_spherical

interface,

whereas the shaded area below

_represents

an unstable

_equilibrium

of the

_system.

Stability

is lost in the

_points

on the

_straight

line.

Qualitatively, figure

2

_suggests

that for the fixed moments

Co

and

_Co,

an increase in

pressure inside the interface-formed

sphere

can

_bring

the

_system

to a stable

_equilibrium.

Conversely,

a decrease in the ambient pressure results in the destabilization of the

_spherical

shape.

With the value of AP

_specified,

at a

_greater

Co

(the

interface tends to cave

in)

and a

greater

Co 2

(a

tendency

to reduce the Gaussian

_curvature)

the

_{spherical shape}

_{may lose}

stability.

The lesser and

_negative

values of the moments stabilize the

_system.

Let us comment on the

_{interceptions}

of the

_straight

line and the axes

_{(Fig. 2).}

If a

_sphere

with radius ro

yields

a

_spontaneous

interface

_shape

at

Co 1

= 0 and

Co 2

=

0,

the

shape

becomes

unstable when there is a

_negative

_pressure

_{drop equal}

to

If there is no _pressure

_drop

between two

_{homogeneous phases,}

AP

= 0,

the

_spherical

interface is found to be unstable when the moments are

_{large enough}

Let us consider an even more

_specific

case where the moments of a

_spherical

_{interface may}

be

_linearly

related to

_spontaneous

curvatures

_[10]

In this case the criterion of the loss of

_stability

_{relates the pressure}

_drop,

moduli of

_elasticity,

and

spontaneous

curvatures as

The

_relationship

_{between the critical pressure}

_{drop âP cr}

and the

_spontaneous

curvatures

Js

and Ks

is shown in

_figure

3. The unshaded area stands for a stable state, and the shaded

Fig.

2. -

Stability

criterion for a

_{spherical non-expansible}

interface

_expressed

in terms _{of the pressure}

drop

and moments : _{AP is the pressure}

_drop

between the bulk

phase

inside the

_sphère

and environment ;

C01

and

_{Co 2}

are the first and second moments

_{respectively ;}

the shaded area

_designates

the absence of a

_{spherical shape.}

Fig.

3. -

Stability

criterion for a

_{spherical non-expansible}

interface :

_relationship

between the pressure

drop

OP , spontaneous total curvature

_Js,

and _spontaneous Gaussian curvature

_K.,.

The

factors

a

_{and Q}

are

_expressed

on the basis of the interface moduli of

elasticity,

as

a =-2Ejj

+

2JK

;

1 ... ro

(

ro

/

of

_spontaneous

curvatures destabilizes the

_system.

If the

_spontaneous

state of the interface is

flat,

Js

=

0,

KS

=

0,

the critical _{pressure value below which}

stability

is lost is

According

to

_equation

_(27),

for a

_spherical

interface with a zero

_spontaneous

curvature of a

sufficiently large

radius

_(when

the last two terms in the

_right-hand

_{side may be}

_neglected)

the critical pressure

drop

is

_positive,

i.e. the

_{spherical shape}

_{may lose}

_{stability despite}

the pressure

being higher

inside than outside the

sphere.

The

_stability

criterion for a

_spherical

interface _{may be}

_expressed

in terms of the Gibbs surface tension and the moments. A

_{spherical shape}

_corresponds

to an

_equilibrium

state of the

_system,

thus the

_{generalized Laplace equation}

₍₁₇₎

holds.

_Using

this

_equation

as a

relationship

between AP and _{yG, we} obtain a

_{spherical shape}

which loses

stability

if the Gibbs

surface tension is less than the critical value

The criterion of the loss of

_stability

of a

spherical

_shape

of a

_{non-expansible}

interface,

expressed

in terms of the Gibbs surface

tension,

is shown in

_figure

4. The shaded area stands

for the absence of a stable

_equilibrium.

_Figure

4

_shows,

in

_particular,

that at low and

_negative

values of the moments

the

_{spherical shape}

is unstable even at some

_negative

values of the surface tension y G. The

spherical

interface for which the moments are

_{fairly large,}

.

loses

_stability

even for

_positive

values of the surface tension.

One case of an interface with a

_{large elongation}

modulus of

_elasticity

is a

_{bilayer lipid}

membrane,

which consists of two

_monolayers

of

_lipid

molecules. A

_spherical

vesicle is a most

often met membrane structure. A membrane is

_{selectively permeable}

for water. Due to this the _{transmembrane pressure}

_drop

in the vesicles is osmotic. The force factors in the vesicle

membrane,

in

_particular

the moments

Co

and

_Co,

are related to both the structure of the

_lipid

molecules

_forming

_monolayers

and the interactions of

_monolayers

_(for

more

details,

see

_[11]).

The most

_frequent

cause of the occurrence of moments in a

bilayer

lipid

membrane is a

non-optimum

ratio of the numbers of molecules in the external and intemal

_monolayers.

For the

moments to be absent in the membrane of a

_spherical

_vesicle,

there must be fewer molecules in the internal

_monolayer

than in the external one

_[11].

In this case,

according

to

_equation

(23),

a

_spherical

vesicle remains stable even _{if the osmotic pressure inside is lower than}

outside

_(however,

still

_being

within the limits defined

_{by Eq.}

_(23)).

If the number of molecules in the internal

_monolayer

is

_equal

to or

_greater

than the

respective

number in the

external

_monolayer,

the

_positive

values of

_C°

and

_C°

occur in the vesicle membrane

[11].

In

this case, as shown

above,

the

_{spherical shape}

loses

_stability

even for

_positive

values of the

transmembrane osmotic pressure

drop.

The loss of

_stability

should

_obviously

result in the membrane

_caving

in

and,

possibly,

in the

_{phenomena resembling}

the

_endocytosis

of live cells. To maintain a stable

_{spherical shape}

in this

situation,

the concentration of an

_osmotically

active substance inside the vesicle must be

_{high enough}

to _{enable the pressure}

_drop

to exceed

the critical value

_(27).

Fig.

4. -

An interface with a small

_{extension-compression}

modulus of

elasticity.

Let us consider an interface whose

_{extension-compression}

modulus of

_elasticity

is small

(EAA --> 0).

This is the case of interfaces which are

_capable

of

exchanging

_components

with the environment in the process of deformation

(an

extreme case of such a

_system

is an

interface of two _pure

_liquids).

In this case there is a weak

_dependence

of the values of all force

factors on the area A of a

_part

of the interface.

_Using

_{equation (2)}

which relates the moduli of

elasticity

with the force factors and

_neglecting

the area derivatives _{of yG,}

_Ci

and

C2, equation (18)

may be rewritten as

As in the case of a

_{non-expansible}

interface,

the

stability

of a

_spherical

_shape

is « for the first

time » lost to a deformation which has a maximum

wavelength.

In the case of the deformation

specified

as § = 03BC

+

(1/3

+ cos 2

cp)

the constant term determines the variation of the

interface area ; the contribution of 1/3 + cos 2 cp is for the

bending

deformation at a constant

area. In the case under consideration the work of an exterior source

_applied

to a

_spherical

interface in

_equilibrium

and to the environment

_during

deformation is

_equal

to

Equation

(29)

shows that there are more chances for the loss of

_stability

of a

_{spherical shape}

of an

_expansible

interface

_{(EAA -+ 0)}

than in the above case of a

_{non-expansible}

interface

(EAA -+ 00 ).

An

_expansible

interface may lose

stability

in two _{ways. The first is similar} to the above case

of a

_{non-expansible}

interface. In the case of deformation at g = 0 for the pressure

drop

àP

_(between

the bulk

_phase

inside the

_sphere

and its

_environment)

whose value is less than the critical one

0394Pcr(1)

the

_{spherical shape}

of an

_expansible

interface loses

_stability.

The deformation for which

stability

is lost in this case amounts, as in the case of a

_{non-expansible}

_interface,

to a flexure

For an

_expansible

_{interface there is another way} to lose

_{stability, by}

virtue of the last term

in the

_right-hand

side of

_equation

_(29).

If the coefficient

_by

_{IL 2 is}

_positive,

then

deformation,

with the _{constant IL}

_{being large enough,}

_requires

_negative

work to be

_{done ;}

thus the

system

proves to be unstable to this deformation.

_{Consequently,}

in the case of an

_expansible

interface the

_system

is characterized

_by

one more critical _{pressure value}

Ap cr(2)

_equal

to

If _{the pressure inside the}

_sphere

is

_{high enough,}

the

_sphere

loses

_stability

for a deformation whose constant

_{component 03BC}

satisfies the

inequality

provided

that the numerator in

₍₃₁₎

is

_greater

than zero. If the numerator is

_{negative, stability}

is lost for deformations with any values of 03BC.

The deformation for which the

_system

loses

_stability

entails an increase in the interface area, which

actually

amounts to a

_{sphere bulging.}

The criterion

₍₃₀₎

of the loss of

_stability

of an

_{expansible spherical}

interface is illustrated in

figure

5. The shaded area stands for the absence of a stable

_equilibrium

of a

_{spherical shape.}

Both ways in which the

_stability

of an

_expansible

_spherical

interface _{may be lost} are

illustrated in

_figure

6. The unstable

_equilibrium

of the

_system

is

_{represented by}

the unshaded

area. The

_{bulging-destabilized}

states are shown

_{by vertically}

shaded area. The

_unstability

to a

Fig.

5.

_{- Stability}

criterion for a

_bulging

deformation of a

_spherical

interface. For

notation,

see

_figure

2.

Shaded area

_designates

the absence of a stable

_equilibrium.

Fig.

6.

_{- Complete}

_stability

criterion for a

_{spherical expansible}

interface. For _notation, see

_figure

2. Unshaded area

_designates

stable

_{equilibrium. Shading designates}

the absence of

_{stability :}

vertical, to

flexural deformation with no

_changes

in the area is

_{represented by}

horizontal

shading. Finally,

the area shaded both

_vertically

and

_horizontally

denotes the states in which the

_expansible

spherical

interface is unstable to both

_types

of deformation.

An

_example

of the

_systems

discussed in this section is a

_drop

of oil

_(or

another

_liquid

which

does not mix with

_water)

in water. For all the above effects to be

_possible,

the internal volume of the

_drop

must be able to

_exchange

substances with the reservoir

_maintaining

a constant

pressure within the

drop.

Discussion.

The paper discusses the conditions of

thermodynamic stability

of a

_spherical

interface. The

problem

is solved

_{by using}

the Gibbs

_approach

to

_{thermodynamics}

of interfaces extended to

the case of elastic interfaces.

_Stability

was studied for deformations that do not disturb the

conditions of the internal interface

_equilibrium.

Equilibrium

and

_stability

criteria are defined with

_respect

to

_{axially symmetric}

pertur-bations. Such a loss of

_generality

is

_{compensated by}

the

_opportunity

of

analytical

solution of

the

_problem.

The

_{investigation}

_generalizes

the available

_approaches

to the

_study

of the

_stability

of

structures formed

_by

interfaces

_{[15, 16].}

In reference

_[15]

the

_problem

was solved within the Gibbs

_theory

of

_capillarity

for an

_{elasticity-free}

interface of an

_{arbitrary shape. Analysis}

reveals that for an interface whose moduli of

_elasticity

and

_cutting

components

of the

microscopic

pressure tensor are _zero, the

_equilibrium

_equation

for the lateral direction holds

only

for the deformations

_involving

a shift of every interface

_piece

without an additional

flexure. For this reason, the

_stability

criteria were found

_only

for that class of deformations

[15].

The results of the

_present

_{investigation}

include an

_analogue

of the

_stability

criterion for a

spherical

interface

_[15],

viz. an

_inequality

obtained for a membrane with a small

_elongation

modulus

_{(EAA ---> 0)}

at which the

_{spherical shape}

is unstable if AP >

_{Ap (2)}

_(30).

Reference

_[16]

discussed another extreme case, that of a

_{non-expansible}

interface whose

mechanical

_properties

are described

_{by using only}

the

_{first-bending}

modulus of

_elasticity

Ejj.

The authors

_[16]

did not use the Gibbs

_{approach ; they}

_proceeded

from a

phenomenologi-cal formula for elastic

_bending

_{energy which} was

_originally

derived and discussed

_by

Helfrich

[17].

The interface was assumed

_[16]

to be

_{completely non-expansible}

and the conditions of a

lateral

_{equilibrium during}

deformation was not discussed.

Besides,

the authors

_[16]

did not

study

the response of the

equilibrium

condition to the

_{elasticity-related}

factors for the

deformations

_{including changes}

in the Gaussian curvature, and to the factors related to the

cutting

components

of a

_microscopic

_pressure tensor. The

_findings

of the

_present

_{investigation}

suggest

that the

_{spherical shape}

of a

_{non-expansible}

interface

_{(EAA --> 00 )}

_{may lose}

_stability

if the pressure

drop

between the

_{homogeneous phases}

is less than the critical value

dP cr

obtained

_{by equation}

_(22).

A

_particular

case of

_expression

₍₂₂₎

is

where Ci

1 was assumed to be related

by equation

(25)

to the

_spontaneous

total curvature

Js ;

besides,

all the factors in the Gaussian curvature and

_cutting

_components

of a

_microscopic

pressure tensor were

_neglected.

In the

_present

_{investigation}

we took into account the full set of

_elasticity

moduli and

spontaneous

geometrical

characteristics of the membrane. The results of such a consideration

give

additional information about the

_dependence

of

_instability

criteria on the vesicle radius.

Appendix

A.

Calculation of a relative

_expansion

of a

_part

of interface.

An

_equilibrium

_equation

for the lateral interface

_displacement

_{may be}

_presented

in the form

of

_equation

_(6).

_Substituting

in

_{equation (6)}

an

_expression

for the force factors where

_only

the first order terms of the

_magnitude

of deformation

remain,

it is

_possible,

with an account for

equations

(3)-(5), (8)-(9)

and

_(11)-(13),

to

_integrate

this

_{expression, resulting}

in

- - - 1

To find the constant in the

_right-hand

side of

_equation

_(Al),

let us

_integrate

(Al)

over the area of a non-deformed

_sphere,

which

_yields

const.

The substitution of

_(A2)

into

_(A1)

and solution of the

_{resulting equation}

for

dA

_yields

A

equation

(15).

Appendix

B.

Calculation of the work of interface deformation.

The work of interface deformation found

_{by integrating}

_{equation (14)}

can be

_conveniently

calculated

_by

parts.

The

_elongation

deformation work is

The use of

_{equations (11)-(13), (15)}

results in

where V and m are defined in the text.

The

_integration

of

_{equation (Bl)}

over the

_angle

dcp

from 0 to 7r and over the deformation

The work of the

_{first-bending}

deformation is calculated in a _{similar way} as

Integration

of

_(B3)

results in the

_expression

And

_finally

the work of the second flexure is

Integration yields

The total of

_equations

_{(B2), (B4)}

and

_(B6)

_presents

a

_{complet expression}

for the work of

References

[1]

GIBBS G. W., The collected works

_{(Longmans-Green,}

New

_York)

1928, Vol. 2.

[2]

RUSANOV A. I.,

Phasengleigewichte

und Grenzflachemer

_{Scheinungen (Academia}

_{Verlag, Berlin)}

1978.

[3]

BUFF F. P., J. Chem.

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₍₁₉₅₆₎

146.

[4]

MELROSE J. C., Ind.

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Chem. 60

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53.

[5]

BORUVKA L., NEUMANN A. W., J. Chem.

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KOZLOV M. M., MARKIN V. S.,

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GRUEN D. W. R., MARCELJA S., PARSEGIAN V., Cell Surface

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Delisi,

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