Nematic stability and the alignment-induced growth of anisotropic micelles

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Nematic stability and the alignment-induced growth of anisotropic micelles

W.M. Gelbart, W.E. Mcmullen, A. Ben-Shaul

To cite this version:

W.M. Gelbart, W.E. Mcmullen, A. Ben-Shaul. Nematic stability and the alignment- induced growth of anisotropic micelles. Journal de Physique, 1985, 46 (7), pp.1137-1144.

�10.1051/jphys:019850046070113700�. �jpa-00210056�

Nematic stability ^{and the} alignment-induced growth

of anisotropic ^{micelles +}

W. M. Gelbart (*), ^{W. E. McMullen} (*), ^and A. Ben-Shaul (**)

(*) Department ^of Chemistry ^and Biochemistry, University ^of California, ^Los Angeles ^CA 90024, ^U.S.A.

(**) Department ^of Physical Chemistry and The Fritz Haber Research Center for Molecular Dynamics,

The Hebrew University ^of Jerusalem, ^Jerusalem 91904, ^Israel.

(Reçu ^le 9 juillet ^1984, accepté ^{le 7 mars} 1985)

Résumé. 2014 Dans cet article, ^nous examinons pour la première ^{fois un} phénomène qui ^se produit dans les solutions micellaires de _savon, uniquement ^en raison de leur nature de suspensions colloidales formées de particules qui ^ne

conservent pas leur intégrité. ^En particulier, ^{nous nous} intéressons à la croissance d’agrégats ^{en relation avec leur}

ordre à grande distance. Nous considérons une forme analytique simple pour l’énergie ^libre par molécule et nous

comparons explicitement ^{la taille des micelles en} bâtonnets dans des systèmes où coexistent des phases isotropes (I)

et nématiques (N). ^{Nous montrons} que le couplage ^entre croissance et alignement ^{limite la stabilité des} agrégats partiellement ^{ordonnés et de taille finie. La} phase nématique ^{est limitée à une} gamme de concentrations très res-

treinte car ses micelles ne peuvent ^survivre (c’est-à-dire ^rester finies) que si elles ^sont petites. Le rôle des degrés ^de

liberté de translation et de rotation ainsi _{que les} effets de cosurfactant sont également pris ^en considération : tous

deux augmentent ^{le domaine} de stabilité de la phase nématique. Dans le cadre de la théorie d’Onsager ^de l’aligne-

ment à longue portée ^de particules ^en bâtonnets, ^{nous en} arrivons à la conclusion (1) que la croissance des micelles à la transition I ^~ N est contrôlée par l’« entropie ^de melange » orientationnel, (2) que le paramètre d’ordre néma-

tique ^{et le} rapport ^{entre tailles} qui coexistent est essentiellement universel, la fraction moyenne ^en volume à la transition se mesurant en termes de l’inverse du nombre d’agrégation moyen.

Abstract.

In this paper ^{we treat} for the first time a phenomenon ⁱⁿ micellized soap solutions which arises uniquely

from their being ^colloidal suspensions ^{whose «} particles » ^{do not maintain their} integrity. ^In particular ^{we focus on}

the growth ^of anisotropic aggregates ^{which is attendant} upon their long-range orientational ordering. ^{We consider}

a simple analytical form for the free energy per molecule and _compare explicitly ^{the sizes of} rod-like micelles in

coexisting isotropic (I) and nematic (N) phases. ^The coupling ^between growth ^and alignment ^{is shown to limit the} stability of finite-size, partially-ordered aggregates : the nematic phase ^{is confined to a} highly restricted concentra- tion range, because its micelles ^can only ^survive (i.e. ^remain finite) ^if they ^are small. The roles of translational and rotational degrees ^of freedom, ^and of cosurfactant effects, ^are also considered : both are shown to enhance the

stability range of the nematic. Within Onsager’s theory ^{for the} long-range alignment ^{of rod-like} particles, ^we conclude further that : (i) ^the growth ^{of micelles at the I ~ N} transition is driven largely by the orientational

« entropy ^of mixing » 2014 bigger rods allow this entropy ^{loss to be} minimized; ^and (ii) ^{the nematic order} parameter and the ratio of coexisting ^{sizes are} essentially universal, with the average volume fraction ^at the transition scaling

as the reciprocal ^of the average aggregation ^number.

Classification

Physics ^Abstracts

61.30C - 82.70D

1. Introduction.

« Ordinary » ^colloidal suspensions ^have long ^chal- lenged ^both chemists and physicists [1]. ^{On a} statistical

thermodynamics level, ^the problem ^{reduces to the}

familiar one of accounting ^{for bulk} properties ^{in terms}

of interparticle interactions. A classic example ^{is that}

of osmotic _pressure and compressibility : ^a large

number of experimental behaviours have been des- cribed via approximate treatments of hard-core

(« ^steric »), electrostatic (o screened-Coulomb »)

and dispersion (« ^Hamaker ») ^forces between the colloidal ^« particles ^» [2]. ^{At the same time, much}

attention has been directed towards understanding

the nature of phase transitions in these macro-particle suspensions. Important examples here include the work by Derjaguin, Landau, Verwey ^{and Overbeek}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019850046070113700

1138

on flocculation and lyophobic stability [3], ^and by Onsager ^on long-range orientational alignment ⁱⁿ hydrophilic systems [4]. Rheological properties, ^sedi-

mentation characteristics, ^and sorption ^behaviours

of colloidal solutions have also been thoroughly investigated [1].

Micellized _soap solutions, e.g. surfactant aggregates in water, ^{have also} been treated within the context of colloid science since the dispersed ^« particles »

here commonly range from tens to thousands of

Angstroms in size. From a fundamental research

point-of-view, much effort has been devoted to

understanding the size and shape of micelles in dilute solution [5] ^{and the} phase transitions between lamel- lar, hexagonal ^{and other} highly-organized ^« crystal- like » states at very high concentrations [6, 7]. ^{As we}

have recently emphasized [8], ^{however, there is a}

crucial conceptual difference between ^« association )) and « ordinary » ^{colloids which} requires ^essential

differences in their statistical thermodynamic ^treat-

ments. More explicitly, ^{the «} particles » in the former

case do not maintain their integrity. ^{Instead of} being

macromolecules or micrograins, they ^are aggregates of small molecules in exchange equilibrium ^{with one}

another. Accordingly, ^even in the absence of inter-

actions, ^a change ⁱⁿ temperature ^or concentration results in a reorganization ^{of the} system ^{into a new} size and shape distribution of« particles » (aggregates).

The fact that the colloidal particles ^themselves change ^with thermodynamic ^{state means of course}

that the interparticle forces also change. ^{Thus one}

must expect ^a coupling between the self-assembly behaviour, ^the interaction between aggregates, ^and the thermodynamic stability. ^We have shown [9],

for example, that when a micellized _soap solution becomes sufficiently concentrated, ^the effect of exclud- ed volume (o ^steric ») ^{forces will be to enhance the size}

of rod- and disk-like aggregates. ^That is, ^{the inter-} action free energy of the suspension ^{is minimized} by ^a re-assembly ^{of the} system ^into larger micelles, thereby stabilizing further the isotropic phase.

The most dramatic effect of the intermicellar forces

on particle ^{size is} manifested at still higher ^concentra- tions, ^where long-range orientational ordering ap- pears. In several quasi-binary (surfactant ⁱⁿ water, plus salt) systems [10], ^for example, ^a phase ^transition

occurs (at volume fractions of 10-30 %) ^{from an iso-} tropic suspension (of ^rods, say) ^{to a nematic} (o magne-

tically orientable ») ^{state. At} slightly higher ^concentra-

tions the partially-aligned, ^finite aggregates give way to perfectly-ordered, ^infinite cylinders (whose ^axes,

moreover, ^are hexagonally packed). The sequence of

isotropic (I) ^{- nematic} (N) -+ hexagonal (H) phase

transitions is observed in _many other systems, upon the addition of a « cosurfactant » (e.g. alkanols) [11].

Nematic states involving ^the partial alignment ^{of the}

short-axes of finite disk-like aggregates ^{are also observ-} ed under these circumstances; ^{an increase in concen-}

tration here leads to a lamellar (i.e. infinite-disk)

phase. ^In some cases the isotropic solution of rod-

or disk-like micelles is found to transform directly

into the hexagonal ^{or lamellar state.}

In view of the large ^{amount of} experimental interest, theoretical treatment of the isotropic-to-nematic phase

transition in simple soap solutions is notably lacking.

This phenomenon ^{must be} expected ^{to be} qualitatively

different from its counterpart ^{in «} ordinary » ^colloidal suspension ^{for at least two reasons :} (i) anisotropic

micelles of large axial ratio are almost certainly very

polydisperse ^{in size} _{[12] ;} ^and (ii) alignment ^{of the}

aggregates changes the effective forces between them and hence their self-assembly characteristics. The effect of polydispersity in rod sizes on the isotropic

nematic transition has been treated recently [13, 14]

for the ^« ordinary » ^{case where the} particles ^maintain

their integrity. ^The long ^{rods are found to} partition preferentially ^{into the} aligned phase, ^where they ^{can be}

more easily accommodated by ^their neighbours. ^The

average size (axial ratio) in the nematic state is thus

larger than that in the isotropic solution, correspond- ing ^{to an} apparent growth of the rods. This fractiona- tion effect will of course occur as well in the micellar case, but it will be magnified by ^{the real} growth ^of

rods which is attendant _upon their long-range ^orien-

tational ordering [10b].

In the present communication we shall feature the

coupling between rod growth ^and alignment by sup-

pressing ^the polydispersity in size. The micellar size difference between isotropic and nematic phases ^is

then due exclusively ^{to real} growth, rather than to

apparent effects associated with the partitioning

mentioned above. The suppression ^of polydispersity

also simplifies enormously the statistical thermo-

dynamics. ^More explicitly, ^the micellar chemical

potential ^can be written as an elementary ^function

of size (aggregation ^number s) ^and alignment (orien-

tational order parameter a), parameterized by ^the

overall _soap concentration (volume ^fraction v). ^Mini- mizing this free energy with respect ^{to s and a} provides directly ^a simple analytic ^{treatment of} the relevant

phase transition behaviours. In this _way we are able to show that :

(i) previously neglected translational and rotational contributions to the ^« ideal-gas » ^chemical potential

are necessary ^to stabilize the nematic as a phase ^of

finite aggregates;

(ii) ^the stability range of the nematic is enhanced

by ^addition of cosurfactant whose effect is to introduce

« negative feedback » into the coupling between rod

growth ^and alignment; ^and

(iii) upon aligning, ^{the micellar rods} undergo ^a

« growth spurt » which is driven largely by ^{the orien-}

tational entropy, ^i.e. bigger rods allow this entropy loss to be minimized.

In section 2 we consider the relevant contributions

to the chemical potential ^{of a} soap molecule in a rod-

like aggregate ^{of size s.} These include the « usual »

standard free energy term ji associated with the

reversible work of moving ^a micelle from vacuum into solvent, plus ^{the «} corrections )) due to translational and rotational degrees of freedom. These latter are

evaluated explicitly ^{and shown to « add » to the} 1 _term In,us ^{and to} t e - n s mono-dispersity repre-

s s p Y p

sentation of the entropy ^of mixing. ^{In this same} single-

size model, the orientational entropy and intermicellar

packing free energy reduce for long ^{rods to} particularly suggestive forms. The free energy per molecule is

thereby ^{obtained as an} elementary function of the sin-

gle size s and a single orientational order parameter ^a

(see Eq. (7)). Stability ^{of the} finite-rod, partially aligned

nematic is treated in section 3, ^without (i) ^{and with} (ii) added cosurfactant. In section 4 we examine the

alignment-induced growth ^{of rods} by comparing

the coexisting isotropic and nematic phases ⁱⁿ binary (soap/water) (i) ^and ternary (soap/cosurfactant/water) (ii) systems. ^Further interpretation and discussion of these results are given ^{in the} concluding ^{section 5.}

2. The free _energy.

As discussed elsewhere [8, 15], ^the average chemical

potential ^{of a} molecule in an s-micelle can be written

as (in ^{units of} kT)

Here sjifl is the reversible work associated with taking

an s-micelle ^{from vacuum and} placing ^{it in} aqueous

solution at a particular position ^{and with a} particular

orientation. It is conveniently expressed ^as

.where jiO(ai) ^{is the} free energy of ^a molecule in ^« envi- ronment » i (spherical ^cap, cylindrical body, etc.) having ^head group ^area ai. ^In the case of rod-like aggregates, ^for example

^{where the «} growing »

dimension scales with s

it is _easy to show that

with 6 proportional ^to the chemical potential ^diffe-

rence between « cap ^» fi1£) ^{and «} body ^» (j1) ^molecules.

For disks, whose diameters _grow as the _square root

of s, ,us ^varies asymptotically as ,uoo,d ⁺ s 1/2. ^To

keep ^our analysis ^of growth/alignment coupling ^as simple ^as possible, ^we shall restrict all further discus- sion to rods.

jit describes the contributions to ¡is ^{from the trans-}

lational and rotational degrees ^of freedom which have

commonly been overlooked in recent treatments of micellar self-assembly. ^{As we} have shown [15], they

comprise nothing ^{more or} less than the ^« ideal-gas »

terms for a particle (here ^the s-micelle) without inter- nal structure :

Here As ^{is the de Broglie} wavelength ^{of the} s-aggregate, qrot,s ^{is its} partition function for overall rotation, ^and

p is the total number density of solution molecules.

Since the micellar ^mass is s times the monomer’s

(ml) ^{we have}

Absorbing ^In Ai ^p into the 6 »-coefficient from

equation (2B), ^{we have}

with the s-dependence of qrot following ^{from the stan-}

dard relations between qrot and rod-lengths (moments

of inertia) [15]. ^More explicitly

where I is the length ^{of a} single soap molecule and

for s greatly exceeding ^{the « minimum »} aggregation

number m -= 4 n/3/3 vi. Taking the molecular length

and volume to be 1 ^- 10 A and vi - 360 A3, M1 250 _amu, T - 300 K and _p

0 .033/A3 ^{we find}

with 6* = 10 In 10. Note that the ^« 3/2 » ^{and «} 7/2 ^»

coefficients arise from the translational and rotational

« corrections » respectively.

All of the above contributions to ji,: ^{are « small »,}

cordingly, ^{one does not} expect ’them, ^for example, ^to

influence significantly the critical micelle concentration

(a ^{property which is} determined primarily by ^the

« large », s-independent ^chemical potential ^difference

,u° - ;u ⁼ 0(10)). ^{We note in} this connection that

Nagarajan ^and Ruckenstein [16] ^{have examined the}

translational and rotational contributions to aggrega- tion free energies, analysing ^{them via a} very different model for the micellar interiors. They suggest partial

concellations amongst ^{these terms} and conclude that

1140

the critical micelle concentration ^is relatively ^unaffect-

ed. In our present work, however, ^{we are concerned} with phenomena ^{such as} aggregate growth ^and align-

ment which are determined chiefly by ^{« small} », ^s-

dependent contributions. We shall show in particular

that terms of order In sls ^are indeed crucial in stabi-

lizing finite micelles in orientationally-ordered phases.

The third term in equation (1) is the usual« entropy of mixing » contribution. Upon suppression ^of poly- dispersity ^{it can be} approximated by (here ^{v -} v, pX

is the volume fraction ^of soap)

where Xs - ^the molefraction of _soap molecules

incorporated ^into s-aggregation

^{has been} replaced by ^{X, the total mole fraction.} (s is henceforth referred to as the« micellar size ».) Note that s > 1 and X 1, implying ^{that the} ln X _s coefficient is always negative.

Thus the free energy of mixing ^increases with s, consistent ^{with our} having organized ^the system ^into fewer (larger) aggregates. ^This driving ^{force for} keeping

the micelles small is opposed by ^the A,,o ^{term of equa-}

tion (2B) ^which decreases with s. The equilibrium ^size

of rods is determined by ^the play-off between these two competing effects, their relative magnitudes being

controlled by ^{the total} soap concentration X. More

explicitly, minimization of 1 6 - I ^{s s In} s ^{X with respect}

to s leads to seq ⁼ X e" + 1. That is, ^the equilibrium

size (in ^the monodisperse limit) ^increases exponen-

tially ^{with 6, the chemical} potential difference between molecules in the ^« _{cap »} and « body », ^and linearly

with molefraction. At high concentration the In s X coefficient is small and the entropy ^of mixing ^{is domi-}

nated by ^the single ^micelle free energy #,,o

^large

aggregates become favoured. (This zero-order picture

is preserved ^{in essence when} polydispersity ^{and the}

other chemical potential contributions from _equa- tion (1) ^are included.)

The ! Xs ^{term in (1) arises} from interactions bet-

s Xs

ween micelles, ^{i.e. from non-ideal} solution corrections.

Elsewhere we have shown how these contributions

can be summed to all orders in the density [9b] ^and

also how they ^{can be} generalized ^to include the _pos-

sibility ^of long range orientational ordering [8].

At the second virial level of approximation, ^{for exam-} ple, ^{we find}

where fs(Q) ^{is the} fraction of s-micelles having ^orien-

tation Q, ^and vss,(Q, Q’) ^{is the} pair excluded volume associated with ^s- and s’-rods having orientations Q and Q’. Using ^the well-known Onsager ^results [4]

for vss,(Q, Q’), ^{and his} single parameter choice for

fs(Q) [ - cosh (a, ^cos 0)], ^{it is easy to show that} (dropping numerical factors 0(1))

Here we have again suppressed polydispersity (XS, ^{-+ X} £5ss’)’ and assumed the rods to be large (s >> ^{1 and a} >> 1). ^{In the} isotropic phase,

As alluded to parenthetically just ^before equa- tion (5B), ^we formulate the present theory ^without regard ^for numerical factors of order unity. ^This

is consistent with the phenomenological spirit ^{of our} analysis, e.g. with the use of Onsager theory ^{and the} neglect ^of polydispersity, attractive energies, ^etc.

For example, ^we drop ^{a factor of two in} writing ^the asymptotic (a >> 1) second-virial (v 1) ^result

X o -- I v - ^{since the} effects of smaller alignment ^and

met

larger density ^are of this order. The only important

fact is that _xo increases with concentration (~ v) ^and

decreases with long-range orientational order

as 1 Similarly, _Y’ ^the 6S _term decreases with _aggre-

s

gation ^number according ^to ln a N ln s ^{s s as we}

show below. It follows from these physical arguments that the _average size (s) necessarily ^{increases at the} isotropic-nematic transition, independent ^{of numerical}

factors 0(l) and of whether Helmholtz or Gibbs free

energies ^are minimized, ^etc.

Finally, Y ⁱⁿ (1) corresponds ^{to the} entropy ^loss

s ^S ( ) p pY

due to long-range orientational-order :

This term vanishes identically ^{in the} isotropic phase;

for nematic states, ^{with the} Onsager ^choice for f,

Here, ^{as above we have taken} a(=- aseQ) ^{to be large}

compared ^to unity. ^{Note that a is related to the usual} (« P2 » orientational order parameter via q = I - 3*

( 2 ) ^p _cx

Collecting equations (1), (2B), (3E), (4), (5B-D) ^and (6B), ^we obtain the following, simple ^result for micellar

free energy (per molecule) ^{as an} explicit ^{function of}

size (s) ^and alignment (a) :

Here « N » and ^« I » refer to the nematic and isotropic phases, respectively, ^and

with 6* ⁼ 10 In 10 the correction mentioned earlier (cf. Eq. (3E)) ; ^{« -} In 10 » is the coefficient In 20132013 V1 p ^arising from the - ^{term in the entropy of mixing (see Eq. (4)). Note that the factor « - 6 » in} the - ^{In s} term of(7) ^is comprised primarily of contributions from the translational and rotational degrees of freedom :

As we show in the next section, these latter two effects

provide ^{for the} stability of the nematic phase. ^Fur- thermore, ^for isotropic states, they help ^the entropy of mixing ⁱⁿ favoring ^small aggregates : this follows from the « braking ^{» mechanism}

being ^enhanced [ - ^{(1) -+ -} (6)] and from the single-

micelle ^« growth ush » + 6 being ^diminished

s g

(6 - 6’ = 6 - 1 1 In 10).

3. Nematic stability

3.1 ROLE OF TRANSLATIONAL/ROTATIONAL ^{DEGREES OF}

FREEDOM. - In order for the partially-ordered, ^finite-

rod phase ^{to be} stable it is of ^course _necessary that

AN(S’ a) ^{show a} minimum for some pair of finite values of s and a. This in turn requires

straightforward ^{to show from} equation (7) ^for AN(S’ a)

that conditions (8) ^{are fulfilled} if ^and only if ^the

-

3 7 1 In s _{2-’2 s} contributions in (7B) ( ^{are included.}

Otherwise, i.e. with only ^the entropy ^of mixing ^term

-

(1) 1In s, _s ^the determinant in (8) () ^is negative g ^{for all} volume fraction v. It is possible, ^but unlikely, th£t ^the finite, a)-nematic might ^be stabilized by polydis- persity, ^i.e. by ^the {X s’ }-distribution ^neglected [{ Xs’ -+ Xass, }] ^{in the present} analytical ^treatment.

Preliminary numerical calculations [17] ^which expli- citly include the full size distribution demanded by

micellar equilibrium (i1s = ,ul’ all s) indicate that the free _energy indeed has no minimum for any finite (s, a)

when translational and rotational degrees ^{of freedom}

are left out.

3.2 ROLE OF COSURFACTANT.

As discussed else- where [8, 18] the effect of adding cosurfactant to a

soap/water system ^{can be shown to be a} replacement

of cap/body ^chemical potential difference 6 by ^a decreasing ^{function of} aggregation ^{number s :}

This comes about because the cosurfactant molecules

(alkanols, say) ^are preferentially partitioned ^{into the}

lower-curvature micellar region ^where they ^{can be}

1142

more effective in relieving the surface charge density

associated with small head _group area. This « relief of electrostatic strain » becomes saturated at large aggregation number, for which the Nalcohol

ratio

gg g ^’

Nsoap

approaches its value in the overall solution

i.e. end

(cap) contributions become negligible. ^{It can be}

shown specifically ^{that the} effective 6 will decrease with s according ^to equation (9), ^with 61 ^and 62

both positive.

As far as micellar equilibrium ^is concerned, equa- tion (9) implies ^{that as the} rod-like aggregates get bigger (s increases) ^{the «} push » (effective) ^{for them}

to grow further is decreased This provides ^{a «} nega- tive feedback » mechanism in the coupling ^between

rod alignment ^and growth. Indeed, substituting equa- tion (9) ^{for 6 into} equations (7) ^and (7A) ^for i1N(S, a)

it is trivial to recompute ^the second-derivative matrix :

we find again that conditions (8) ^are satisfied for ^all

volume fractions. Furthermore, ^a minimum in the free energy is obtained for finite (s, oc) even in the absence of rotational and translational contributions.

In this latter _case, however, ^we need to impose ^the

extra constraint that s 2 62. ^This suggests ^that micellar sizes must remain small in order for the nematic to be stable : for larger ^{rods the} growth/align-

ment coupling ^{becomes too} strong ^{for finite} aggregates

to survive. We explore ^this point ^{below in our dis-}

cussion of isotropic nematic coexistences.

4. Alignment ^Induced growth ^{of rods.}

4.1 I ^- N TRANSITION IN SOAP/WATER ^SYSTEMS.

We return to equations (7) and minimize J1N and J1I

with respect ^{to a} and s and s respectively. ^{For the} isotropic phase, this leads to

For the nematic phase ^we have, similarly,

with and

Imposing fiI = ;uN ^{at the} transition, equations (lOB)

and (11C) give

Using a = s’N vllm’ and combining (I OA) and (I I A) to

express sI ^{in terms} of sN, ^we find

The physically reasonable root of equation (13) ^is

giving

Note, that, independent of molecular ^constants (e.g. 61), equations (15), (16) characterize the alignment ^and

size changes ^at the transition. These results are also consistent with our earlier assumption ^{that a} >> 1.

They demonstrate qualitatively ^the growth ^{of rods}

-

see equation (16)

^attendant upon their orien- tational ordering. Inspection ^of equation (7), ^or comparing equations (10A) ^and (11A), suggests ^that this growth ^{is due} primarily ^to the orientational

« entropy ^of mixing » term S Q N ^{s In (X - 1 s}

More explicitly,

decreases with aggregation ^number (s). ^That is, ^the

orientational entropy ^loss (free energy gain) ^attendant

upon alignment ^{is minimized} by reorganization ^of

the system ^into larger ^{rods. In a more} thorough

treatment of the I/N coexistence, ^{i.e. one which} allowed properly ^{for an} increase in concentration

(v) ^{for the} nematic, ^{the terms in the} 1/s-coefficient (see ^Eq. (7)) would further enhance sN over SI ^since the dominant contribution scales ^{as v.} (Recall ^that

this - « v » _term arises from the « packing » ^free

s

energy effects comprising x.)

Consider now equations (10A), (11A), (11B) ^and (12), ^{for the} particular ^{choice 6’ = 28} [19]. ^{We have}

and thus

Both sN+ and sI ^increase exponentially ^{with 6’. From} equations (11B) ^and (15), (16) ^{we have} furthermore that v + ’" 4- (or 4-). Since for large aggregates

;7 ^N B ^{SI 1 g gg g}

the rod length ^{L is} roughly proportional ^{to s, this}

means that v + - 1 , ^{in agreement with the Onsager}

result according ^{to which} v i5 - ^constant.

4. 2 I ^- N TRANSITION IN SOAP/ALCOHOL/WATER ^SYS-

TEMS.

With 6 - 61 ⁺ 62 ^{s in equation (7), the}

a- a-

as ^{= 0} and aS ^{= 0 equations become} transcenden- tal and must be solved iteratively for sI and sN. Using

a-

(X s2v2/M2 - the agN _aa 0 equation is unaffected by

addition of cosurfactant

we again obtain the I/N

coexistence by finding ^{v = v + such that}

Unlike in the binary (soap/water) ^{case, a + and}

SN

+

are not quite universal. Rather they depend

SI

weakly ^on £52.

Taking ^{the same} self-assembly parameters ^{as in} part (i) immediately above, ^{and with a reasonable}

estimate of the cosurfactant partitioning contribution

Comparing equations ^{(20) with (18), we see that the} qualitative consequence of cosurfactant is ^to push

the I ^- N transition to lower concentrations because of enhanced rod growth. ^As pointed ^{out elsewhere} [17], however, ^the present ^treatment of cosurfactant

partitioning ^includes only its effect on the intra- aggregate free energy jif. ^Lowering the surface charge density ^{will also} modify ^{the inter} micelle contribution

’ Zs _s ^xs

^see equation ^q (1)

^and

^{hence is} expected pe ^to affect both _average sizes in the I and N phases ^as

well as characteristics of the coexistence.

Finally ^{we note that the} presence of cosurfactant

changes only slightly ^{a+ and the} scaling ^result

5. Concluding discussion.

In the above we have argued ^{that the} coupling ^between

micellar growth ^and alignment gives ^{rise to} signifi- cantly larger aggregates in the nematic (vs. isotropic) phase. Enhancement of nematic stability by ^cosur-

factant was also demonstrated. The theory ^{on which}

these conclusions were based is simple ^and analytic,

because of several drastic approximations ^{used to}

formulate the free energy. ^Most significant ^{is our} suppression ^of polydispersity. Nevertheless, preli- minary numerical calculations [17] indicate that

alignment-induced growth ^of anisotropic aggregates

persists ^even upon including all molecular exchanges

relevant to the full equilibrium distribution of rod sizes.

Because the coupling ^between self-assembly ^and long-range orientational order becomes stronger ^with

increasing rod-size, ^it appears that nematic phases

of amphiphilic ^{solutions are} only marginally ^stable against « explosion » ^into hexagonal ^states (and similarly for discotics ^-+ lamellae). ^{First of} all, they

will only ^{arise in} soap/alcohol systems ^{whose trans-} lational, rotational and cosurfactant features

see

(i) ^and (ii) in section 3

allow a significantly deep finite-(s, a) minimum in the free energy per molecule.

And, ^even then, they ^will quickly destabilize _upon ^an increase in concentration, because of the « snow-

balling » growth (s)-alignment (a) coupling. ^Accor- dingly, ^we expect ^{that nematic} phases ^will always

occupy ^a small region of overall ternary phase ^dia-

grams and that their aligned aggregates ^{will necessa-} rily ^{be small} (in comparison with those of more

dilute regions, say, of the isotropic state, ^far away from the coexistence region).

References

[1] See, ^for example, ^the collection _{of papers} recently presented ^{at a review} symposium organized by

the Royal Society ^of Chemistry : Colloidal Dis-

persions, ^edited by J. W. Goodwin (Special ^Publi-

cation No. 43, ^The Royal Society ^of Chemistry,

Dorset Press) ^1982.

[2] (a) VRIJ, A., NIEUWENHUIS, ^E. A., FIJNAUT, ^{H. M.}

and AGTEROF, ^{W. G.} M., Disc. Faraday ^{Soc. 65} (1978) ^101.

(b) OBER, ^{R. and} TAUPIN, C., ^J. Phys. ^{Chem. 84} (1980)

2418.

[3] VERWEY, E. J. W. and OVERBEEK, ^{J. Th.} G., Theory of

the Stability of Lyophobic ^Colloids (Elsevier, Amsterdam) ^1948.

[4] ONSAGER, L., Ann. N. Y. Acad. Sci. 51 (1949) ^627.

[5] (a) ISRAELACHVILI, ^J. N., MITCHELL, ^{D. J. and NIN-}

HAM, B. W., ^J. Chem. Soc. Faraday ^{Trans. II} 72 (1976) 1525;

(b) TANFORD, C., ^The Hydrophobic Effect, ^{2nd edi-}

tion (Wiley, ^New York) 1980 ;

(c) MCMULLEN, ^W. E., BEN-SHAUL, ^{A. and} GELBART,

W. M., ^{J. Coll.} Interface ^{Sci. 98} (1984) ^523.

1144

[6] Transitions between smectic-like liquid-crystalline ^sta-

tes in simple soap solutions have been discussed recently by ^P. Pershan, ^for example, ⁱⁿ Phys.

Today ³⁵ (5) (1982) ^34.

[7] SAFRAN, S., TURKEVICH, ^{L. and} PINCUS, P., ^J. Physique

Lett. 45 (1984) ^{L-69 treat the} possibility ^of cylin-

drical phases ^{in model} microemulsions. See also the discussion by ^G. Porte, ^J. Phys. ^{Chem. 87} (1983) ³⁵⁴¹ for the pure micellar case.

[8] GELBART, ^W. M., BEN-SHAUL, A., MASTERS, ^{A. and} MCMULLEN, W. E., in Physics of Amphiphiles : Micelles, ^{Vesicles and} Microemulsions, ^edited by

V. Degiorgio ^{and M. Corti} (North Holland) ⁱⁿ

press.

[9] (a) BEN-SHAUL, ^{A. and} GELBART, ^W. M., ^J. Phys. ^Chem.

86 (1982) 316;

(b) GELBART, ^W. M., BEN-SHAUL, A., MCMULLEN, W. E. and MASTERS, A., ibid ⁸⁸ (1984) ^861.

[10] (a) LINDMAN, B., private communication;

(b) CHARVOLIN, J., private communication.

[11] TIDDY, ^{G. J.} T., Phys. Rep. ⁵⁷ (1980) 1 ;

FONTELL, K., ^Mol. Cryst. Liq. Cryst. ⁶³ (1981) ^59.

[12] ^{As discussed in} references 5 (a)-(c), the variance of the size distribution scales as the average size, ^i.e.

the fractional polydispersity ^{is of order} unity.

[13] FROST, R. S. and FLORY, ^P. J., Macromol. 11 (1978)

1134.

[14] ^{DEBLIECK, R. and} LEKKERKERKER, ^{H. N.} W., J. Physi-

que Lett. 41 (1980) ^L-351.

[15] MCMULLEN, ^W. E., GELBART, ^{W. M. and} BEN-SHAUL, A., ^J. Phys. ^{Chem. 88} (1984) ^6649.

[16] NAGARAJAN, ^{R. and} RUCKENSTEIN, E., ^{J. Colloid} Interface ^{Sci. 60} (1977) 221 and 71 (1979) ^580.

[17] MCMULLEN, ^W. E., GELBART, ^{W. M. and} BEN-SHAUL, A., Isotropic-Nematic Transition in Micellized Solutions, J. Chem. Phys., in press.

[18] GELBART, ^W. M., MCMULLEN, ^W. E., MASTERS, ^A.

and BEN-SHAUL, A., Langmuir ¹ (1985) ^101.

[19] This choice corresponds ^to amphiphilic length ¹² A,

volume 360 Å3, interfacial tension 60 erg/cm2

and optimum head group ^area 45 Å2.

[20] ^{Here we use the same} self-assembly parameters ^{as in}

18, ^{with a} alcohol-to-soap ^{ratio of 0.9} implying

03B41 ~ ^{60 and} 03B42 ~ ^140.