Microphase separation in binary polymeric micelles

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Microphase separation in binary polymeric micelles

A. Halperin

To cite this version:

A. Halperin. Microphase separation in binary polymeric micelles. Journal de Physique, 1988, 49 (1),

pp.131-137. �10.1051/jphys:01988004901013100�. �jpa-00210667�

Microphase separation ⁱⁿ binary polymeric ^micelles

A. Halperin

The Fritz Haber Research Center for Molecular Dynamics, ^{The Hebrew} University, ^Jerusalem 91904, ^Israel (Requ ^{le 29} juin 1987, accept6 ^{le 18} septembre 1987)

Résumé.

On présente ^une analyse théorique ^{de la} séparation ^en microphases de micelles binaires formées par des copolymères ^en blocs. On considère le cas où les copolymères ^{sont de deux} types différents, ^{de telle}

sorte que la ^couronne micellaire contient deux types ^de blocs, ^{A et} B. Les processus de séparation peuvent ^être soit intramicellaires, soit intermicellaires. On étudie particulièrement le processus de ségrégation intermicel- laire qui ^se développe après ^une trempe ayant produit ^une séparation intramicellaire complète. ^La séparation

intermicellaire est rendue possible par les processus d’association qui forment les micelles, ^{et elle est favorisée} par la tension de la ligne ^de séparation intramicellaire. Cette force inhabituelle doit son importance ^{à la} longueur énorme de la ligne ^de séparation ^de phase. Pour des micelles dont les couronnes sont minces, ^la distribution ultime des compositions des micelles est proportionnelle à exp [2014 0393x 2014 03BAN1/3N14/27cx½ (1- x )½] où ^{x est} la fraction de blocs A dans la couronne, N et Nc ^{sont les} degrés ^de polymérisation ^{des blocs}

de la couronne et du c0153ur tandis que 0393 ^{et 03BA sont} des constantes numériques.

Abstract.

A theoretical analysis ^of microphase separation ⁱⁿ binary micelles is presented. ^{We consider}

micelles consisting ^{of two} kinds of diblock copolymers such that the micellar corona contains two chemically

distinct block types (A ^and B). Both intramicellar and intermicellar processes may ^occur. We focus on

intermicellar segregation occurring ^{after a} quench yielding complete intramicellar microphase separation. ^The

intermicellar process, made possible by the associative nature of the micelles, is driven by line tension. This rather special driving ^{force owes its} importance ^{to the} enormously long phase boundary ^{in the} system.

For micelles with relatively ^{thin coronas, the} final distribution of micellar composition ^is proportional ^to exp [- 0393x 2014 03BAN1/3N14/27cx½(12014 ^x )½] where ^x is the fraction of A blocks in the _corona, N and Nc ^{are the} degrees ^of polymerization of the coronal and the core blocks, 0393 and 03BA are numerical constants.

Classification

Physics ^Abstracts

36.20E

61.25H

64.75

1. Introduction.

Polymeric surfactants, typically ^diblock copolymers,

are known to form monolayers [1], ^micelles [2],

emulsions [3] ^and microemulsions [4]. The interest in such macromolecules is due, apart from their

practical importance, ^{to the} similarity ^{of their be-}

haviour to that of low molecular weight surfactants

[5a]. ^{In turn,} this last group is subject ^{to intensive}

current research [5]. ^{While the two} surfactant classes are similar they ^{are not} quite identical. For example, monolayers ^formed by polymeric surfactants do not

undergo configurational phase transitions [6, 7]

characteristic of ^« small » surfactant monolayers [5].

However, ^{some of the} distinguishing properties ^of polymeric surfactants render them particularly ^at-

tractive for study [7a]. Because of the chains’ length

one may ^use their asymptotic properties, ^thus gaining

considerable simplification ^{of the} theory. ^{It is also} possible ^to vary the number of ^monomers in a chain

over a wide range and study ^the ensuing ^trends.

Most studies of polymeric surfactants focus on single component systems. ^{In the} following ^{we will discuss}

theoretical aspects ^{of a} multicomponent system.

Specifically, ^{we consider} microphase separation ⁱⁿ binary polymeric ^{micelles i. e. , micelles} composed ^of

two chemically ^distinct polymeric surfactants. We shall consider micelles formed by ^a mixture of A-C and B-C diblock copolymers ^{in a} highly ^selective

solvent S, such that the C blocks are immiscible in S. S is a good, low molecular weight solvent for both A and B blocks. For simplicity ^{we take S to be an} equally good ^solvent for A and B. In the micelles thus formed we distinguish ^{between two concentric}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01988004901013100

132

spherical regions : ^{a central} core, consisting solely ^of

immiscible C blocks in a melt-like state, and ^an outer corona made up of A and B blocks swollen by ^the

solvent (Fig. 1). ^{In the} high temperature limit, ^{the A}

and B blocks are uniformly distributed in the corona

(Fig. 2a). Upon lowering ^the temperature, microph-

ase separation may result. For ^a micellar system ^this may ^{assume two} forms : (1) Intramicellar microph-

Fig. ^1.

^{A schematic cross} section of a single binary

micelle. The inner core consists of C blocks in a melt-like state. The external corona consists of A and B block swollen by the solvent.

ase separation involving ^AB segregation ^{within a} single ^micelle (Fig. 2b). ^{This is a} microscopic analog

of the phase transition associated with AB separation

in mixed monolayers ^of polymeric surfactants [8]. ^A

similar phenomenon may ^occur in binary ^{stars with}

both A and B arms [9]. (2) Intermicellar microphase separation. ^{In this case the AB} segregation ^involves

two or more micelles and results in the formation of A-rich and B-rich micelles (Fig. 2c). ^{Because this}

process depends ^{on the} associative nature of the micelles it has no molecular counterpart. ^Further-

more, this type ^of microphase separation ^{seems to}

have no macroscopic analog either. This is because the intermicellar process, when following ^{the in-}

tramicellar one, is driven by line tension. This factor is of negligible importance ^when considering ^the stability ^of macroscopic phases. However, ^the geometric constraints of the micellar solution impose

an enormously long phase boundary ^{on the} system

(it’s length ^is proportional ^to the total number of micelles in the solution). ^The product of the line tension and boundary’s length yields ^the boundary’s

free energy. This term, which owes it’s importance

to the magnitude ^{of the} boundary’s ^total length,

causes the intermicellar microphase separation ^under

these conditions. We will consider the two types ^of micellar microphase separation ^while focusing mainly ^{on the} intermicellar type. Our interest in the

Fig. ^2.

^{Schematic cross sections} depicting ^{three extreme}

micellar states. (a) ^The high temperature ^{state in which A} and B blocks are uniformly mixed in the corona ; (b) ^an

intramicellar microphase separation ^{leads to AB} segrega- tion within the _{corona ;} (c) ^{two or more} micelles may

exchange ^{chains to achieve} intermicellar microphase separation. ^{In the text we consider} the sequence (a)-(c)

where intramicellar microphase separation ^{is followed} by

the intermicellar process.

problem ^is two-fold : Experience with « small » surfactants suggests the control of micellar properties by using ^a mixture of polymeric surfactants. Under-

standing ^{of the} possible microphase separations ⁱⁿ

the system is then crucial. From ^a more fundamental

point ^{of view, the} intermicellar process is of interest because of the special ^role played by ^{the line tension}

in driving ^this type ^of microphase separation.

For simplicity ^{we assume} A-C and B-C are

monodispersed. ^The degrees ^of polymerization (DP)

of the A block (NA ) and of the B block (NB ) ^are

taken to be equal, ^i.e. NA

NB

N. To avoid considerations of polyelectrolyte behaviour, ^{we as-}

sume the chain to be uncharged. ^{We confine our} study ^{to micellar} systems ^well past ^{the critical} micelle concentration (cmc ). ^{In this} regime ^we may

assume the micelles are monodispersed consisting

each of f ^diblock copolymer chains. We further limit ourselves to dilute micellar solutions thus eliminating

micelle-micelle interactions and possible ^macro- scopic phase transitions. In general, the intramicellar and intermicellar microphase separations may ^occur

simultaneously. However, following ^a quench ^{of a}

dilute micellar solution the two processes ^{are ex-}

pected ^{to occur} separately, and the intramicellar

microphase separation ^to precede. This is because

intermicellar exchange is slowed down by ^{the in-}

creased micelle-micelle distances, while the in- tramicellar transport is unaffected. A similar effect

might be obtained by ^{use of} highly viscous solvents.

The emergence, under this conditions, ^{of two time}

scales allows us to choose a complete intramicellar

microphase separation ^{as an} initial condition. Be-

cause our main interest is in the role of the line tension in driving ^the intermicellar _process, we

confine ourselves to such cases. The equilibrium

distribution of the intermicellar microphase separation, following ^the intramicellar stage, ^{is de-} termined primarily by the free energy associated with the phase boundaries. This in turn is given by

the product ^{of a line} tension, T, and the length ^{of the} phase boundary. ^{In order to} facilitate the estimate of

T we only consider micelles with coronas which are

thin in comparison ^{with the core.} The coronal blocks in such micelles may be considered ^as grafted (attached by ^{a head} group) ^{to a} flat surface. We thus avoid complications ^{due to the star like structure of}

micelles with extended coronas [10]. ^A brief review of the relevant properties ^{of chains} grafted ^{to flat}

surfaces is given in section 2. The intermicellar

microphase separation and the line tension causing ^it

are discussed in section 3.

2. Grafted layers.

We refer to polymers ^{attached to an interface} by ^a head-group ^as grafted chains. A common approach

to grafting employs amphiphilic ^diblock copolymers

anchored to a liquid interface. Each block is chosen

so as to be miscible only ^{in one of the} liquid phases,

and the junction ^{of the} blocks is thus constrained to the interface. From this point ^of view, ^the resulting

structure consists of two grafted layers corresponding

to the two blocks, with the added constraint that any chain in a grafted layer ^{is bonded at the} grafting ^site

to a chain belonging ^{to the second} layer. In par-

ticular, ^a micelle is obtained when amphiphilic

diblock copolymers ^{are anchored to a} spherical

interface. The core chains are grafted ^{to the inner}

surface of the sphere, while the coronal chains are

grafted ^{to the} exterior. Grafted chains which are

anchored to a liquid ^interface by ^{virtue of their} amphiphilic ^nature preserve their lateral freedom of

motion i.e., the diblock copolymer junction ^{is free to}

move along the interface. This is not true for all

grafting methods. For example, chains which ^are

grafted by irreversible chemical bonding ^{to a solid}

surface ^are completely immobilized. This distinction is important ^because segregation ^of chemically ^diffe-

rent chains is only possible ^{for the} laterally ^mobile

chains.

Consider first chemically identical chains grafted

to a flat interface [6, 7]. ^{For low} grafting densities,

such that different chains do not overlap, ^we may

picture ^each grafted ^{chain as a « mushroom » oc-} cupying ^a hemisphere ^{of radius} RF, ^the Flory ^radius

of a coil in a good ^solvent

where N is the degree ^of polymerization ^{and a the}

monomer size. On increasing ^the grafting density

past ^the overlap threshold, ^the polymers ^stretch along the normal ^to the surface assuming ^« cigar »

like shapes. The associated concentration profile ^is

characterized by ^{a wide} plateau. ^{In this} regime ^the layer thickness is given by

where D is the average distance between adjacent grafting sites. Within mean field theory ^the layer’s

state is determined by the balance of two contribu- tions to the free energy of ^a grafted chain : the

mixing free energy Fm;x ^and the elastic free energy

F cl . Fni,,, ^{corrected to} allow for the grafting, ^ac-

counts for the solvent-polymer mixing free energy.

The effect of excluded volume interactions is in- cluded in F mix. ^{This term} favors the layer’s swelling

or chain stretching. Fel, ^{a term} wich is due to the coil

configurational entropy, ^{tends to control the} swelling i. e . , minimize the deformation of the coil [7]. Scaling analysis ^{is based on the same} general approach.

However, the basic unit is now the « blob » : a

volume element of diameter §, ^the correlation length given by 6 - a-0 -314 ^{where 0 is the monomer volume}

fraction. Within the blob, for distances r g, ^corre- lations are dominated by excluded volume interac- tions. On larger length scales, r >- g, ^{the chain} behaviour is ideal. We now picture each chain as a

string of blobs of size 6. ^{To account for the} repulsive

interactions we assign ^{kT to} each blob in the chain.

This term, ^as Fnix ^{in the mean field} theory, ^{tends to}

swell the layer. ^The swelling ^is again ^checked by ^the

elastic free energy. Now, however, the elastic free energy is calculated for ^a string of blobs. D is found

to be the fundamental length ^{in the} problem, ^{that is} 6

D. Different blobs pack ^as space-filling ^hard spheres. ^In essence, each chain behaves as if confined

to a narrow cylinder of radius D (the ^« cigar »). ^{It is}

however important ^{to note that each chain has a}

random walk component parallel ^{to the interface} plane. This last remark is relevant to binary ^mixtures

where interactions between chemically ^different

blobs should be taken into account.

A multicomponent grafted monolayer differs from the neat layer ^{in two} respects : (1) ^the layer ^thick-

ness, L, depends ^{on the} layer composition. (2) ^For

the case of laterally ^mobile grafted ^{chains, a} phase

transition associated with the segregation ^{of the}

different chemical species ^is possible. Consider then

a two-component system ^{such that out of a total of}

134

nP grafted ^chains _xnp ^{are of one chemical} species, A,

and (1- x ) np ^{of the other, B.} We denote the interaction parameter between the two kinds of

monomers by _Xpp. ^{L is found to be} weakly dependent

on x. This is an enthalpic effect related to the random walk component of the chain configuration,

which is parallel ^to the interface. Repulsive ^interac-

tions between the different monomers (X pp > 0) partly suppress this component, ^thus increasing ^the

chain stretching and L. For attractive interactions

(Xpp.O) ^{this component is amplified thus coun-} teracting ^the stretching ^{of the} polymers. ^{In the case}

of chemically grafted chains, ^{which are} completely immobile, ^we expect ^no other effect. For binary layers composed ^of laterally mobile chains the phase

transition associated with the AB segregation ^is by

far the more significant ^effect. Momentarily neglect- ing ^{the x} dependence ^{of L} which is rather weak, ^we

may write the free energy of such ^a binary grafted layer ^as

where Fo is the free energy of the single component grafted monolayer and X the interaction parameter between A and B chains. This free energy suggests ^a

two dimensional regular binary solution. The mixing entropy ^is essentially ^{due to the two} dimensional gas of grafted head groups. The three dimensional, polymeric, ^{nature of the} grafted chains determines the form of _X. In particular, ^{an effect} special ^{to this} system arises because of the coupling ^{of chain} configurations ^{and the} grafting density. ^{As a result}

one may distinguish ^two concentration regimes

which differ in the form assumed by X (RFI D). ^Past

the overlap threshold X (RFID) ^{increases at a slower}

rate with the grafting density. This is due to the chain stretching, which tends to lower the growth

rate of the monomer volume fraction in the layer.

We however are going ^to deal with a system characterized by ^a high ^{and constant} grafting density.

Accordingly ^the detailed behaviour of X (RFID) ^is

not relevant to our needs. It is sufficient for us to note that to coexistence curve of the grafted layer

with respect ^{to AB} segregation is determined by aF/ax

⁰ [11]. ^{One then} expects ^the phase ^be-

haviour of the corresponding bulk solution [11, 12]

which is essentially ^{that of a} regular ^solution.

3. Microphase separation ⁱⁿ binary polymeric ^micel-

les.

Micellar coronas may be viewed ^as spherical grafted layers [10]. ^{Other similar} systems ^{include star} polym-

ers [13, 14] and colloidal particles ^coated by grafted

chains [14, 15]. ^In general, certain modifications ^are

called for when spherical grafted layers ^{are con-}

sidered. Primarily, ^{one must} allow for the larger

volume available to the chains the farther they ^are

from the centre. As a result 0 ^{must be a} decreasing

function of the distance from the origin. ^{In the} special ^case of micelles with relatively ^{thin coronas a}

much simpler ^{model is} adequate. ^{In this} model, ^due

to de Gennes [7a, 16, 17], ^{the monomer volume}

fraction in both the core and the corona is assumed to be constant. For highly ^{selective solvents a} sharp

core-corona interface is expected. Micelle formation is then driven by ^the system’s tendency ^{to lower the}

surface free energy per chain. Because the diblock

copolymer junctions ^{are localized at the} core-corona

interface, ^micellar growth ^causes stretching ^{of the}

core blocks. The resulting increase in free energy controls the growth of the micelles. The free energy term associated with the coronal blocks may be

ignored in this limit. For micelles well past ^the

cmc, where entropy ^{effects are} negligible ^{we have}

where N c denotes the DP of the ^core blocks. This model is particularly suitable for our ends. As was

mentioned already, the coronal blocks may be

regarded ^as grafted ^on the core’s surface. When the thickness of the corona is small compared ^{with the}

radius of curvature (Rcore), its behaviour is essential-

ly ^{that of a flat} grafted layer. We may then ^use results obtained for flat, mixed, grafted layers ^{as a}

basis for ^our discussion of binary ^micelles. Complica-

tions associated with the estimation of X ^{in a star-like}

structure are thus avoided. Furthermore, in this limit the free energy variations resulting ^from changes ⁱⁿ

coronal states are small enough ^{so as not to affect}

micellar structure. In particular, f and ^{R are} given by (3.1) ^and (3.2) respectively for all the possible

micellar species involved in our discussion.

Consider the following situation : a solution of

binary micelles with uniformly ^{mixed coronas is} deep quenched ^{so as to} bring ^{about a} complete

intramicellar microphase separation, ^i.e., AB sep- aration within a single ^{micellar corona.} This is the micellar analog ^{of the} phase transition associated with the AB segregation ^{in flat} grafted layers. ^A micelle, ^{as a finite} system, may ^not undergo ^a proper

phase transition. Yet, ^{if one} avoids the vicinity ^{of the}

critical point, the micelle is big enough ⁱⁿ comparison

with the relevant correlation length. Under these conditions a sharp intramicellar transition may be

expected. The initial composition of the micelles is such that _xo f of the coronal blocks are A blocks while the remaining (1 2013 xo) f ^{are B} blocks. The

quench yields ^{demixed micelles with} regions,

« phases ^», consisting solely ^{of A or B} blocks. Out of

the total coronal area, Stot

⁴ 7T R;orc, ^{the two} single

component phases occupy spherical caps of ^areas xo Stot ^and (1 - xo) Stoto ^The length, Lo, ^{of the} boundary between the phases ^is given by (Appen-

dix A).

This boundary is associated with a free energy per unit length, ^{T, to which we} refer somewhat loosely ^as

line tension (*). ^The free energy ^term originating

from the phase boundary ^{is then}

where ntot is the total number of micelles in the system, ^{which is a constant in our case. An inter-} micellar microphase separation ^takes place ^because

the system free energy is lowered by redistribution of the chain species amongst ^the ntot micelles. Such redistribution lowers the total length ^{of the} phase boundary and the free energy associated with it. To determine the final state of the system ^{one must}

account for the concurrent increase in mixing ^en- tropy. This is due to the fact that micelles of different composition ^{should be} considered as diffe- rent species. ^{The final state is thus} expected ^to

consist of a mixture of demixed micelles of continu-

ously varying ^overall compositions. The mixture is characterized by ^a distribution function p (x ) ^defined

as the fraction of micelles having x f ^A blocks in their

corona. The length ^{of the} phase boundary ^{in micelles}

with x f ^A blocks, L (x), ^is given by (3.3) ^{with x} taking ^the place ^of xo. We thus seek p (x ) ^that

minimizes the system’s free energy

subject ^to the constraints

where (3.6) ^assures proper normalization while (3.7)

insures conservation of the overall composition ^of

the system. The variational calculation yields

(*) Properly speaking ^{the term} line tension refers to the free energy per unit length associated with the three phase boundary [18]. ^« True » line tension, ^as opposed ^{to surface} tension, ^{can be} negative. We however use the term in reference to the two dimensional analog ^of surface tension which _may not assume negative ^values.

here j3 = 11 kT, ^{r is the} Lagrange multiplier originating ^with (3.7) and Z is the partition ^function

To complete ^our argument ^{we need an estimate} for the line tension, ^{T. We seek T in the} sharp

interface limit which is appropriate ^{far from the}

critical point. ^{The two} coexisting phases consist then almost entirely ^{of A or B} chains. The two phases ^are separated by ^{an interface zone} of thickness 2A. We take d as a measure of the lateral dimensions of chains in the interfacial region. ^{The relative abund-}

ance of A and B monomers varies continuously along the normal to the interface. We now argue that the free energy per chain consists of two terms :

(1) ^an elastic free energy, F,I, accounting ^{for chain}

deformations. This term is due to the tendency ^{of A}

and B monomers to segregate within the intrafacial

region. ^{this in turn leads to lateral} compression resulting in loss of configurational entropy. ^We estimate Fel by : Fel = R2/ L12 ^{where R is the} inplane

radius of a chain in one of the bulk phases. ^Blob analysis yields [7a] R 2 _ (Nlg) D 2 ^where

g = (D/a)5/3 ^is the number of monomers per blob,

thus leading ^to R = N 1/2 a5/6 D1/6. (2) Local devia- tions from the equilibrium composition give ^{rise to}

free energy density ^{excess. This is} usually ^assumed

to depend ^on the local variables as does its bulk counterpart. As chains span the whole interfacial

region they experience ^an average ^excess free energy

density eA:T. The excess free energy per chain may then be written ^{as :} FclkT , A E where E has units of

l ength -1. F, ^{tends to} produce ^a sharp interface while

Fel ^{tends to} widen it. All together, ^the free energy per interfacial chain is given by

Upon minimizing (3.10) ^with respect ^{to A we obtain}

and

As the average distance between grafting ^{sites is D, T}

is given by

Equation (3.13) ^was obtained for a flat grafted layer, ^{where D} may be ^set arbitrarily. ^{In a micelle D}

is uniquely determined by Rcore ^and f :

Furthermore, ^the quantity of interest in our case is

136

f3 TL (x ) rather than r itself. This in ^turn is given by

The final form for p(x) ^{is thus}

where K is a numerical constant which may ^not be determined by ^this type ^of analysis. ^Because L (x) ^is symmetrical ^with respect ^{to a maximum attained at}

x = 1/2, p(x) ^{has a U like} shape. ^For xo = 1/2 F vanishes and p (x ) ^{too is} symmetrical ^around

x = 1/2. In cases where xo =A ^{1/2 the} p (x ) ^{curve is}

deformed so as to conserve the overall AB compo- sition of the system. ^The precise ^{value of K not-} withstanding, ^for polymeric ^micelles K N 1/3 N ;4/27

may be varied ^{over a} wide range. By properly choosing ^{N and} Nc ^{it should be} possible ^{to obtain} KN’I’N c 14/27 _ 10. ^{In such cases a} clear intermicellar

microphase separation ^is expected (Fig. 3) : ^{most of}

the micelle population ^is concentrated around

x 0 and x , 1 with negligible, though ^{non zero,} p (x) values for 0.2 $; ^x s 0.8. The main effect of xo =A 1/2 is then to produce ^a distorted U like

p (x ) ^{curve, with one arm} higher then the other.

Fig. ^{3. - A} plot of p (x ) ^{versus x for KN 1/3} N14111 ^{= 10. The}

continuous curve corresponds ^to xo

0.5 while the dashed

curve is obtained for _xo

0.25.

4. Discussion.

In binary ^star polymers ^{i.e., star} polymers ^{with arms}

of two chemically ^distinct types, ^one may expect

intramolecular microphase separation ^{to occur. This}

is ^an analogous process ^to the phase ^transition expected ^{in flat} binary grafted layers. ^In binary

polymeric ^{micelles one} may expect ^a similar in- tramicellar process ^{to occur as} well as the occurrence

of intermicellar microphase separation. This second type of process is made possible by the associative nature of the micelles which allows intermicellar

exchange. Accordingly, this process has ^{no counter-} part ^{in star} polymers. Furthermore, this process has

no macroscopic analog ^{because it} is driven by ^line

tension which is of negligible importance ^{for macro-} scopic systems. In micellar solutions, ^because they

are highly dispersed, ^the phase boundary ^{is enorm-} ously long and the line tension contribution is

important.

Our analysis ^{of the} system avoided, ^on purpose,

complications ^{due to the star like structure of the}

micelles. We have only considered micelles with thin

coronas which may be considered ^as flat grafted layers. ^Our conclusions, however, ^{should be} qualitat- ively valid for micelles with extended coronas. A different expression ^{for T} should obtain in such

cases : T must certainly increase with N but ^at a

lower rate because 0 decreases with the distance from the micelle’s centre. Yet ^T can reach higher

values in micelles with extended coronas, thus

making ^{them the} system of choice from an exper- imental point ^{of view.} Also, ^one may wish ^to follow these processes using ^neutron scattering, ^{and micel-}

les with thick coronas should be better suited for such experiments.

Appendix ^A.

The two phases occupy spherical caps having ^areas

of xStot ^and (1 2013 x ) Stot, ^where Stot ^{= 4} 7T R;orc. ^The

area of ^a spherical cap is given by [19] :

Fig. ^4.

^{A cross section of the core of a} segregated binary

micelle. A and A’ denote the position ^{of the} phase

boundary and r is the radius at the phase boundary.

where h is the height of the cap (Fig. 4). Accordingly

We seek the length, L (x ), ^{of the} caps’ boundary. ^To

obtain L (x ) ^we first find an expression ^{for r, the}

radius of the boundary. ^{We denote} by a ^the angle

subtended by ^{the arc} formed between the cap’s

centre and it’s boundary. We then have

On substituting (A.2) ^into (A.3) ^{we obtain}

which leads to the desired expression ^{for L} (x).

Acknowledgments.

It is a pleasure ^to acknowledge instructive discussions with Prof. S. Alexander. This work was supported by ^the U.S.-Israel Binational Foundation (BSF) Jerusalem, Israel. The Fritz Haber Research Center is supported by the Minerva Gesellschaft fiir die

Forschung, mbH, Munchen, ^BRD.

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