Bending moduli of polymeric surfactant interfaces

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Bending moduli of polymeric surfactant interfaces

S.T. Milner, T.A. Witten

To cite this version:

S.T. Milner, T.A. Witten. Bending moduli of polymeric surfactant interfaces. Journal de Physique,

1988, 49 (11), pp.1951-1962. �10.1051/jphys:0198800490110195100�. �jpa-00210874�

Bending ^{moduli of} polymeric surfactant interfaces

S. T. Milner and T. A. Witten

Exxon Research and Engineering, Corporate Research Science Laboratories, Annandale, ^NJ 08801, ^U.S.A.

(Reçu ^{le 24 mai 1988,} accept6 ^sous forme définitive ^{le 26} juillet 1988)

Résumé.

Nous étendons notre théorie récente des ^« brosses ^» polymériques greffées ^{en bout} de chaînes à des polymères ^{attachés sur} des interfaces courbées. Plusieurs systèmes importants, par exemple les surfactants

polymériques ^{ou les} copolymères formés de deux blocs fortement ségrégués, peuvent être décrits comme des brosses. Par un développement ^de l’énergie libre d’une brosse sur une surface courbée en puissances ^{de la} courbure, ^on obtient des expressions analytiques pour les modules de courbure moyenne ^et de courbure

gaussienne. ^{Les valeurs K et K} pour des brosses monodisperses ^{sont en accord avec les} arguments ^d’échelle qui donnent, K, K ~ ^N303C35 pour les conditions de ^« fondu ^» et ~ N 3 03C37/3 pour des brosses de densité modérée. On traite également de manière analytique ^{le cas} important d’une brosse formée d’un mélange de chaînes longues

et courtes. On montre aussi qu’en remplaçant ^une faible fraction des chaînes longues par des chaînes courtes,

on réduit de manière spectaculaire les modules de courbure.

Abstract.

Our recent theory ^{of the} free energy and conformations of end-grafted polymer ^{« brushes » is}

extended to polymers ^{attached to} curved surfaces. Several important systems, e.g., layers ^of polymeric

surfactants or of strongly segregated ^diblock copolymers, ^can be well described as brushes. By expanding ⁱⁿ

powers of the ^curvature the free energy of ^a brush on a curved surface, ^{the mean} and Gaussian bending ^moduli

may be obtained analytically. Results for K and K of monodisperse ^{brushes are} consistent with scaling arguments, ^which imply K, K ~ ^{N3 03C3 5} for melt conditions and ~ N3 03C3 7/3 for moderate-density brushes with solvent. The important ^{case of a brush} composed ^{of a} mixture of long- and short-chain molecules is also treated

analytically. ^The replacement ^{of a} small fraction of long-chain molecules in a brush by short chains is shown to

dramatically reduce the bending ^moduli.

Classification

Physics ^Abstracts

36.20E

81.60J

87.15D - 82.70D

Introduction.

Among ^{the most} basic and important physical properties ^{of an} adsorbed surfactant layer ^{at a} liquid-liquid interface, ^{or of a} bilayer composed ^of amphiphilic molecules, ^{are the mean and Gaussian} bending moduli. Much progress has been made in

predicting ^the phase ^behavior [1-5] ^and fluctuations

[6-9] ⁱⁿ systems ^of self-associating amphiphiles, by parameterizing ^the monolayers ^or bilayers ^{in these} systems ^{in terms of} bending ^{moduli and} spontaneous

curvature. However, progress ⁱⁿ understanding ^the origins [10-14] ^{of these} fundamental parameters ^has been somewhat less complete.

One origin of the elastic constants of amphiphilic layers is the interaction of the long hydrocarbon

« tails » of the molecules. These tails may be de- scribed with the language ^of polymer physics, ^a description which becomes increasingly good ^{as the}

molecular weight ^{of the tails} increases, i. e. , ^{in the}

limite of a « polymeric surfactant ^».

Recently, progress has been made both exper-

imentally [15, 16] ^and theoretically [17, 18] ^is understanding the conformations and elastic proper- ties of a layer ^of polymer ^{molecules attached} by ^one

end to a surface at relatively high ^surface coverage.

The resulting structure, called a « brush ^» because the polymer chains stretch away from the grafting

surface to avoid high ^monomer concentrations, ^has

been studied both for the monodisperse case, [17, 19, 20] i. e. , when all the chains are of the same

molecular weight N, and for the case of arbitrary

molecular weight distributions [18].

A strong analogy may be drawn between the system ^of end-grafted polymer ^{chains, and a surfac-}

tant monolayer ; ^a bilayer may be regarded ^{as two}

such monolayers ^{back to back. In this} paper, ^we shall exploit ^this analogy, ^{and extend our results on}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198800490110195100

polymer ^{brushes to the case} of curved grafting ’ ^I surfaces, ^{in order to calculate} the elastic properties

of the amphiphilic layers. The paper is organized ^as

follows. In section 1, ^a brief review of methods and results for monodisperse brushes is presented. ^In section 2, ^the bending moduli for a monodisperse

brush are calculated by comparing ^{the free} energies

of the brush in flat, cylindrical, ^and spherical geomet- ries. In section 3, ^{we treat the} important ^{case of an} amphiphilic layer composed ^{of chains of two differ-}

ent molecular weights

^a model for the effects of a

short-chain « cosurfactant » on the properties ^{of a} layer ^of long-chain surfactant molecules. Recent exact enumerations of configurations of very short- chain molecules [14] ^{observe that a small admixture}

of shorter-chain molecules to a surfactant layer

reduces the bending ^stiffness with remarkable effi-

ciency. Using ^our analytic methods, ^{we calculate a} dramatic nonlinear dependence ^{of the} bending ^mod-

uli on the amount of cosurfactant added : a relatively

small amount of short chains serves to make the

layer nearly ^{as flexible as a} layer ^made only ^{of short}

chains. We conclude with a short discussion of

physical features left out of the present ^treatment, including interactions between the hydrophilic

« heads » of the amphiphiles, and the effects of the

(very) ^{finite tail molecular} weight. ^{Details of the}

mixed-brush calculations are relegated ^to Appendix A ; ^a prescription ^for converting ^our expressions ^for bending moduli from « theorists’ units » to physical

units is contained in Appendix ^B.

1. Review of ^« Brushes ».

Consider the problem ^{of a} polymer chains per unit

area attached by ^{one end to a} surface, ^and exposed

to a not-too-good ^solvent (such ^{that the monomers}

interact with one another through a mean monomer

density). ^A description of this brush would consist of the conformations of the chains, and thus such

quantities ^{as the monomer} density 0 (z ) ^{at a} height ^z

above the surface, ^{the local monomer chemical} potential V (z ), ^{and the} density ^{of free chain ends} (z).

Energy-balance [19] ^{and blob} [20] arguments ^have been used to show that, ^{as a} function of molecular

weight and surface coverage a, the thickness of this brush h scales as NQ ll3 and the free energy per unit

area F scales as No- 513

More recent work [17, 21] ^has shown that these

simple scaling arguments miss several important

features of the brush ; namely, 1) ^the conformations of different chains in the brush are not necessarily similar, ^{nor is a} particular ^chain uniformly stretched ; 2) ^the density profile, rather than being nearly ^a step-function ^{as was} suggested, [19, 20] ^{is instead} parabolic ; ^and 3) ^{because the} density profile goes

continuously ^{to zero at the outer} extremity ^{of the}

brush, the force to compress the brush slightly ^is

weaker (by ^one power of the strain) than calculated

using ^a step-function ^ansatz profile.

These results [17] ^{were obtained} by ^a solution of the one-dimensional self-consistent field (SCF) equations [22], ^which give ^a mean-field description [23] ^{of the system valid for chains at} high coverage

interacting through ^a sufficiently ^weak two-body repulsion. ^In practice ^this description should be valid for chains at moderate concentration in a not-too-

good ^solvent [24] ; ^for brevity this limit is referred to below as the ^« moderate density ^» regime. ^A closely analogous ^{treatment was} given ^earlier by ^Semenov [25] ^to describe the statistics of chains in a molten

block-copolymer system having ^{a lamellar} mesoph-

ase geometry. This situation corresponds formally ^to

a polymer brush in which no solvent whatever is present ; ^for brevity ^we below refer to this as the

« melt » regime. ^{In both} the moderate density ^and

the melt regimes, the solution of the SCF equations

makes use of the fact that the chains in a brush are

strongly stretched. This strong-stretching assumption

allows a « classical limit » to be taken, ^{in which} fluctuations of chain conformation about the most

probable path between its endpoints ^can properly ^be neglected [17, 25]. What remains is the many-chain problem ^of finding ^the self-consistent set of chain

paths, ^{and chain-end} positions ^with density E (z), ⁱⁿ

a monomer chemical potential V (z). ^This V(z) ^is proportional ^to the average ^monomer density 0 (z) ^{in the moderate} density regime. ^{In a melt,} V(z) ^{is fixed} by ^the requirement that 0 = 1. The self-consistent solution is one in which 1) ^{all chains}

are in equilibrium ^{in the} potential V (z ), ^with configurations ^{of the correct number of monomers ;} 2) ^{the monomer} density cP (z) ^is reproduced ^from

the sum over the end density E (z ) of the contribution to the density d 0 of each chain.

This many-chain problem ^{was solved} exactly, ^for

both the melt and moderate density cases, by employing ^an « equal-time » argument [17, 25],

which we now summarize.

First we assume that the polymer ^{chains are all of} equal ^molecular weight, and that their free ends are

distributed with nonzero density ^{at all distances from}

the grafting surface, up ^to the « brush height » h, beyond ^{which 0} (z > h ) ^{= 0.} (Detailed stability arguments ^are given ^{in reference} [17] ^to demonstrate that the chain end density is indeed nonzero

everywhere ^{in the} brush.)

Then the self-consistent chemical potential V (z)

must be such that a chain of molecular weight ^N may be in equilibrium

^{with no force} applied ^{to its free}

end

with that end located anywhere in the brush

(at zo, say). There is then a precise analogy [23]

between the most probable path of the chain be- tween z ⁼ zo and ^{z =} 0, ^{and the} trajectory ^{of a}

certain classical particle. Under this analogy, ^discus-

sed in detail in reference [17], ^the arc-length ^{of the}

chain (roughly, ^{the monomer} index) corresponds ^to

« time _», and the degree of local chain stretching ^to velocity. The fictitious particles ^{move with} equation

of motion

where the potential energy U(z) ^{of the fictitious} particle is related to the monomer chemical potential V (z ) by U (z ) = const - V (z ). (We ^define U(O) = 0, ^and assign V (h ) = 0 ^for equilibrium brushes.) Since there is no tension at the free end of

a chain, dz/dn ^{= 0} there ; i.e., ^the particle ^starts (at

« time » _zero, at point zo) ^{from rest.}

Then it is evident that for monodisperse chains,

the potential U (z ) ^{must be an} equal-time potential, i. e. , ^{it must} give ^{the same « time of} flight » (chain length) ^{for a} particle starting ^{from rest at} any distance away from the grafting surface. That is, ’

U(z ) ^{must be a} harmonic-oscillator potential, ^{so that} V (z ) ^{= A -} Bz. (See Fig. 1). ^The coefficient B then determines the period ^{of the} « oscillator » ; ^one quarter period ^{is the} length ^{N of the chain. The}

coefficient A may be determined e.g., by 1) insisting on a brush profile ^which accomodates the required

number of monomers per unit area ; and 2) requir- ing ^{the monomer chemical} potential ^{to be zero at the}

outer extremity ^{of the} brush, V (h ) ^{= 0.}

Finally, ^we may check ^our assumption ^{that a} positive ^end density c(z) exists which reproduces ^the

monomer density 0 according ^to the second self-

consistency condition. The contribution dO (z ; zo)

of a chain with free end at zo to 0 ^{at a} point ^{z is the}

inverse of its velocity [17]

The density is then the sum of contributions from all the chains, and may be written

This equation may be solved explicitly ^for e (z) ⁱⁿ

both the moderate density ^{and melt cases} (see Fig. 1) ; ⁱⁿ addition, general arguments may be

given ^{which establish that} E (z) > 0 in a brush grown

on a flat surface [17]. (This point is discussed further in Sect. 2).

In a subsequent paper, ^we explored ^{the ever-}

present effects of polydispersity in molecular weight

on the properties ^of end-grafted polymer ^brushes [18]. ^An understanding ^{of these effects} requires ^an

extension of the equal-time arguments of reference

[17] ^{to a case} where many molecular weights ^and

thus « transit times » coexist within a single ^brush.

Fig. ^1.

^The self-consistent potential V (z ) (solid curve),

and density of free chain ends s (z) ^{for the case of a melt}

brush (dashed curve) ^{and a moderate} density ^brush (dotted curve). ^The height ^{z is} given in units of the total brush height ho ; ^the y-axis ^{units are} arbitrary.

One characteristic feature of the polydisperse ^brush

is that the ^« equal-time ^» requirement forces the free ends of chains of different molecular weight ^to segregate in the z-direction : the self-consistent _po- tential V (z ) ^{has an} unique ^{time of} flight (molecular weight) associated with each height ^{z above the} grafting surface. All shorter chains will have their free ends closer to the grafting ^{surface than that of}

any longer chain. This segregation ^{is a} general

feature of the ^« classical » long-chain limit for the

polydisperse ^brush [26].

We were able in reference [18] ^{to construct a} complete extension of our description ^{of the mono-} disperse ^{brush to the case of} arbitrary polydispersi- ty ; analytical expressions ^{in terms} of the molecular

weight distribution were obtained for the potential V (z) (and ^{thus the} density, ^{via the} equation ^of state), ^{as well as the force} required ^to compress ^a

polydisperse brush. In particular, explicit ^formulas

for the potential V (z ) ^were obtained for a « bimod- al ^» brush, composed ^{of a} mixture of two molecular

weights ^M and N. These results, used in the calcu- lation of the bending ^{moduli of a bimodal} brush, ^are quoted in section 3.

2. Monodisperse ^case.

A formal expansion ^{of the} free energy per unit area

of a bent surface (brush, ^{membrane, or other thin} sheet) may be written ^as [27, 28]

Here K and K are the mean and Gaussian rigidity

moduli respectively, cl, C2 = I/Ri, 1/R2 ^{are the}

local curvatures (inverse ^{radii of} curvature), ^and

co is the preferred ^{or «} spontaneous ^{» curvature.}

If we can compute ^{the free} energy of ^a brush bent

into the inside or outside of a cylinder ^or sphere, ^we

can extract K, K, and co. We write F, (R) (Fc (R)) ^as

the free energy per unit ^area of a brush assembled on

the surface of a large sphere (cylinder). (Throughout

this section, subscripts ^{c ans s} will refer to cylindrical

and spherical ^bends respectively). We write R posi-

tive (negative) if the brush is on the outside (inside)

of the bend (see Fig. 2). Fo is the free energy per unit area of the corresponding flat brush. Then

equation (3) implies

We may ^now compute ^the bending rigidity ^moduli

of a « monodisperse ^» brush, i. e. , ^one consisting ^of

chains of a single ^molecular weight. ^{We consider a}

bend performed with the constraint of fixed cover-

age, i.e., ^{a fixed} number per unit ^area of chain

attachments, ^denoted by ^{cr. The} grafting surface will taken to be the « surface of constant area » of the bend (see Fig. 2). (Either ^{of these two} assumptions

may be later relaxed, by adjusting the coverage and/

or area of the brush prior ^{to the} bend, ^to satisfy

some other physical requirement than fixed coverage and a grafting surface of constant area during ^the bend).

Fig. ^2.

^A brush bent with « positive » ^{radius of}

curvature (in this paper, R > 0 means the chains are on the

« outside ^» of the bend).

To compute ^the quantities F, (R ) ^and Fc(R), ^we imagine assembling ^{the brush on a} slightly ^curved substrate, grafting ^{chains to this substrate at the}

same coverage (chains per unit area) a present ^{in the} flat brush. We assume for the moment (this ^{will be}

examined below) that in the bent configuration, ^the

local monomer chemical potential V (z ) is still para-

bolic, V (z, R) = A (R) - Bz 2. ^{This is} equivalent (because ^{of the} « equal ^{time »} arguments ^{of the} introduction and Ref. [17]) ^{to the} assumption ^that

some free ends of the grafted chains may be found

anywhere inside the brush. The coefficient B must still be given by ^{the «} equal-time ^» requirement ^on

the chain conformations, i. e. , B = ir 2/ (8 N 2) ^so

that the chains of molecular weight ^N may reside in

equilibrim ⁱⁿ V (z, R). The values of A (R) ^and h (R) ^must depend ^on the radius (and type) ^{of bend,}

as we will now show.

Consider first the case of a « melt brush », i. e. , ^on

in which the density ^{is forced to be uniform.} (This ^is perhaps ^a reasonable model for the hydrocarbon

tails of a surfactant layer). ^The height of the brush is

now given trivially by counting monomers ; in the flat geometry ^{we have h =} Na , ^{while in a} cylindrical geometry ^{we have}

with ho - ^{No- and E =} ho/R.

Because we require ^{the monomer chemical} poten- tial to vanish at the outer extremity ^{of the} brush, V (h (R), R ) ^{= 0, we have} A (R ) ⁼ Bh 2(R), ^{so V is}

now determined. _{We may} now compute ^{the free} energy ^to assemble the melt brush on a cylindrical

surface by ^{the same} technique used in the flat _case,

developed in reference [17]. ^The method, ^which calculates the free energy ^to assemble the brush by adding ^{chains to an} existing brush of lower coverage, may be briefly ^{described as} follows. The free energy increment to add a chain to a brush can be shown to be independent of where the free end of the new

chain is placed. ^{There are} chains with their free ends

arbitrarily ^{close to the} grafting surface, for which the free energy increment is purely ^{due to the} potential

V and is simply ^NV (o ) ^{= NA. Hence we} may write the free energy per unit ^{area as}

Application ^of equation (6) in the above case of a

melt brush grown ^{on a} cylinder gives

The analogous calculations in a spherical geometry give

Then equation (4) ^{can be} applied ^to extract K and

K for the brush as

A bilayer, composed ^{of two} such brushes back to

back, would have bending ^{moduli twice as} large, ^and

co ⁼ 0 by symmetry. (Appendix ^{B outlines the}

conversion of Eq. (10) ^into physical units).

We may also consider ^a moderate density brush, i. e. , grafted ^{chains at} moderately high coverage in the presence of ^a not-too-good solvent. As above,

we assume the monomer chemical potential ^{of the}

bent brush remains parabolic,

V (z, R) = A (R) - BZ2. ^{In the moderate} density

case, the potential ^{V is} proportional ^{to the} density,

V = 0 (we shall work in units such that the interac- tion parameter ^{U is} unity). ^The height h,,(R) ^{of such}

a brush when bent into a cylinder ^{of radius R is} again given by integrating ^the density profile in the bent

geometry,

With the explicit expression ^{for 0 above, this} implies

where in the moderate density ^{case, we have}

and E - ho/ R ^{as before.}

Now equation (6) may be used ^to give, ^{in the}

moderate density ^case,

The analogous calculations for the moderate den-

sity ^{case in a} spherical geometry give

Then equation (4) may ^once again ^be applied ^to

extract K and R for the moderate case as

Our results for the bending moduli of monodis- perse brushes ^are naturally consistent with a simple scaling argument ^{for their} magnitude ; namely, ^since

K and 9 have dimensions of energy, they ^should

scale as the free energy ^{of a} characteristic volume

ho of ^{the brush} [29], ^{or as} Fo h2 ^{Recalling the} scaling arguments given by ^{de Gennes} [19] (or examining equations (7, 9, 12, 14)) ^{we observe} F 0 ’" NO’ 5/3, ho - No-’13 ^{in the moderate} density ^{case and} Fo - Nu3, ho - No, in the melt case. These imply K and K scale as N 30,7/3 in the moderate density ^case

and N3 uS in the melt case.

Several previous authors, including references [1, 10, 14], ^{have obtained} estimates of the bending

stiffness of surfactant layers ^{which are} also consist- ent with these scaling ^results. Analogous scaling

results are contained in reference [1] ^{for the case of a} good solvent ; ^reference [14] presents ^a simple ^model

which results in K - N 3 cr 5for the melt case.

Other authors derive results which are equivalent

to the above scaling ^when combined with some

model of how the coverage ^a depends ^{on N. For} instance, in reference [10] the coverage cr is op- timized within a model in which a « bare surface tension » y is assigned ^to the surfactant interface, ^so

that a minimizes (F - y )/ o-. (This ^{is not} necessarily

a good ^{model for} determining a (N ) ; ^{see Ref.} [30]).

This implies ^{0’ --} N -315 for the moderate density

case. Together with the above scaling result for K,

K, ^this gives K, k - N 8/5 for the moderate density

case, which agrees with the scaling of the results

reported in reference [10].

The sign ^{of the Gaussian modulus 9} is such that

bending ^{into a «} saddle surface ^» (with Cl ^{= -} C2)

costs free energy. This feature may be anticipated.

quite generally ^{within a} polymer ^model [31] ^of bending energy, by ^the following argument. ^Con- sider the volume V of a thin shell of height h ^{above a}

surface of area A bent with local curvatures cl and C2. The ratio VI(AH) is the ratio of volume available for chains in a layer ^of height h ^and grafting

area A in the bent and unbent geometries, and may be expanded ^as [32]

Then for a saddle surface, ci = - c2 ⁼ h/R ^and VI(AH) -- ^/ 1 - 3 ^{3 (hlR )2 the bent thin shells have}

less room for monomers than the unbent shells, ^and

so the chains must stretch upon bending, ^{which must}

cost free _energy.

The spontaneous ^curvature co may also be ^ex- tracted from equations (4, 7, 12) for the moderate

density ^{and melt cases ;} co is of order 1 /ho ^{in each}

case, ^{as a} scaling argument ^would suggest. ^This is, ^of

course, ^a strong ^{hint that a flat} monolayer ^of grafted

chains would prefer ^to break up into micelles.

However, the calculated value of co ^cannot be

identified with the actual preferred curvature, be-

cause the value was obtained in an expansion ⁱⁿ

powers of hc about ^{c =} 0, ^{the flat surface. When a}

brush is bent a finite amount, ^two things ^{occur which}

cause the present calculation to break down.

First and most fundamentally, ^the strong-stretch- ing assumption fails when R - h ; this is because chains find so much free space far away from the

grafting surface, they ^no longer ^{stretch to avoid} high

monomer density. ^{To treat} properly the corrections to a scaling picture ^of polymeric ^{micelles or « stars »} [33] requires ^a major ^{extension of the} strong-stretch- ing ^methods.

Second, ^a difficult technical problem arises when

we consider a brush bent by any ^nonzero amount ;

namely, ^{it is no} longer ^{true that the} density ^{of free}

chain ends E(z) ^{is nonzero} everywhere inside the brush. A « dead zone » (region ^{of z where no free}

chain ends are found) opens up ^near the grafting

surface. We may estimate the size of this region by assuming ^{that the} potential ^{V is still} parabolic ^{in a}

bent brush, ^and computing in the bent geometry ^the

E (z, R ) required ^to reproduce ^{the monomer} density (0 ⁼ V for the moderate density case). ^This c (z, R ) ^{will be} negative ^{in a small} region ^near

z ⁼ 0, which will be approximately ^the region ^{of the}

dead zone.

For convenience, ^we define the end density e(z) in the bent (cylindrical) geometry ^{to be the} number of ends per unit (height ^x projected area).

Then, in the melt case, the requirement ^that E (z) reproduce ^the (uniform) density is, by analogy

to the discussion in section 1,

Assuming that V is a parabola amounts to writing U (z ) ⁼ BZ2; using this and A ⁼ Bh 2, equation (16)

may be expanded. It is convenient [18] ^to rewrite the

equations ⁱⁿ « U space », defining E (U) d U =

= e(z) dz, ^as

This may be solved perturbatively in powers of 5 -= h/R for e (U), yielding ^{the end} density ^{of a} monodisperse brush bent into a cylinder ^as

The second term in equation (18) diverges ^at U = 0, i.e., ^{at z = 0.} However, ^{if we} ask for the root of E (z), ^{we find z- h} exp(2013 1/6 ) ; ^{the dead zone is} exponentially ^{small in} 8, ^{and will not} contaminate an

expansion ⁱⁿ h/R. ^{We need} only contend with the

9

Fig. ^3.

^{From reference} [18], the self-consistent potential U(z ) (V ^{= A -} U ) ^{for a bimodal} brush, ^with M/N ^{= 3/2} (solid curve). ^Below h, ⁼ 1, ^the potential is that of a

monodisperse brush of N-chains (dashed curve) ; ^above hl, ^the potential approaches ^{that of a} brush of M-chains

(dotted curve).

complications ^{of a dead zone} (which e.g., makes it

impossible ^to guess the form of V) ^when considering

« large » (nonperturbative) curvatures.

3. Mixing ^{short and} long ^chains.

The methods of treating strongly ^stretched polymers developed in reference [17] ^were extended in refer-

ence [18] ^to brushes of arbitrary ^molecular weight

distribution. Using ^some of the results of reference

[18], ^{we can} compute ^the bending moduli K and

K for the interesting ^and important ^{case of a melt}

brush composed ^{of a mixture of} short and long polymer chains. This may be regarded ^{as a model for}

the effectiveness of short-chain cosurfactants in

reducing ^the bending ^{constant of a} layer composed

of long-chain surfactants.

We begin ^{with a} brief summary of results for the unbent mixed melt brush. In reference [18], ^{it was}

shown that in a brush composed ^{of a} mixture of chains of molecular weights N and M::-. N, ^{the free}

ends of chains segregate ^{in the} z-direction (normal ^to

the grafting surface). ^The segregation results from the « equal-time ^» requirement : ^{chains of two differ-}

ent molecular weights, experiencing ^{the same} poten- tial V, ^cannot both be in equilibrium with their free ends at the same distance from the grafting ^surface.

The free ends of the shorter (longer) chains will be found at distances z -- z 1 (z :-.. z ^{from the} grafting

surface.

The self-consistent potential V (z ) ^for this mixed

brush, ^{with a} partial coverage a 1 of short chains and

o- 2013 o-i 1 of long ^{chains, was} shown in reference 18 to

be

where Ul ⁼ U(zl) is determined below. This may be inverted to find U(z) ^{for z} > zl ^as

with a = M/N -1 ^and BN M lr2/(8 N 2). (We ^have U(z) ⁼ BN ^{Z 2for z :} zl, just ^{as for a brush made of}

pure N-chains). This bimodal brush potential ^is displayed ⁱⁿ figure ^3.

It was also shown in reference [17] ^{that the end} density, ^{converted to} «t/-space» by c(z)dz=

E ( U ) ^dC/ (as proved convenient in the calculation of

E in a bent geometry ⁱⁿ sect. 1), ^{could be determined} independently of polydispersity ^as

Using equation (21), the value of U(zl ), ^where

zl is the location of the boundary ^{between the} regions ^{where short and} long ^{chain ends are} found,

was determined by integrating ^{the end} density ^until

all the short chain ends are accomodated :

The value of A can be obtained in the same way

(replacing U1 by A and o-i by a ⁱⁿ Eq. (22)). ^The

results [18] ^are

The relation between cr 1 and Ul may be conveniently given ^{in terms of A as}

Solving ^for Ul, we ^obtain

The free _energy of the brush in a bent geometry

can be calculated in two equivalent ways : by directly assembling the brush on the bent surface, ^{as was}

done for the monodisperse ^{brush in section} 1 ; ^or by evaluating ^{the work} required ^{to bend a flat brush,}

discussed by ^Helfrich [34]. ^{To extract} bending ^con-

stants, ^{one must work to} 0 (c 2) ^{in the first} approach

and only ^to 0 (c ) ^{in the second.} The second method offers a welcome advantage ^{in the more tedious case}

of the mixed brush.

Consider a unit area of unbent brush. To deform it

so that it may become part ^of the inside of a

cylindrical ^{shell of radius} R, ^{we must reduce the}

cross-section C of the brush at a height ^{z from}

C = 1 to Cc (z, R) ^{= 1 -} z/R. ^{When we do this, the} height ^{of the} brush grows from the unbent value

ho ⁼ M((7- -al)+Nal ^to h (R ) given by

JOURNAL DE PHYSIQUE. - T. 49, N 11, NOVEMBRE ¹⁹⁸⁸

equation (5), ^{i.e., to} h (R ) ⁼ Ao(l - E/2 ⁺

E2/2 ... ), ^{with E =} ho/ (- R ), in order that the same

number of (incompressible) ^{monomers be accomo-}

dated in the reduced cross-sections. Work is done

during ^this process only [35] against ^{the osmotic}

pressure nc(z, R ), ^{which is a function both a} height

z above the grafting surface and the radius of the bend. Writing ^c = 1/7?, ^we may integrate ^with respect ^{to c to find the} change ^{in free} energy per unit

area in a brush bent into a cylinder ^as

The analogous construction for compressing ^a

brush into a section of a sphere ^{of a radius} R, ^for

which C s (z, R) = (1 - zIR)’ ^and hs (R ) ^is given by equation (13), yields

(In Eqs (26, 27), ^the signs ^have been chosen so that R > 0 corresponds ^{to a brush on} the outside of a

bend).

If we expand the osmotic pressure in equations (26, 27) ^to 0 (c ), and compare ^to the formal bending

energy expansion equation (3), ^we may obtain _gen- eral expressions ^for K, K, and co. What remains is to relate 8H/ac I c = 0 ^{in the} cylindrical ^and spherical geometries. ^The result,

may be understood in several _ways. Physically, ^the

factor of two arises from the fact that the cross-

section at a height ^{z in the} spherical bend is reduced twice as much (to 0 (c )) ^{as for a} cylindrical ^{bend of}

the same radius ; thus, ^the first-order change ^{of the}

osmotic _pressure II(z ) ^{from its} flat-geometry ^value

is doubled. More formally, ^we may write for ^a

general ^bend H = H(Z, Cl, C2) ; ^then Hc(z, c) = II (z, c, 0) ⁼ II (z, ^0, c ), ^and IIS (z, c ) ⁼ II (z, ^c, c ).

Differentiating ^with respect ^{to c,} equation (28) ^fol-

lows immediately.

Now we carry ^out the expansion ^of equation (27) (noting ^that H(h (c), c) always vanishes, ^{we need}

not expand ^the dependence ^{in the limit of} integration

of h(c)). ^{The final result is}

We see from equations (29-31) ^{that K and} Kco ^are properties ^of the unbent configuration ^only,

while K depends ^{on how the osmotic pressure} begins

to change ^when the brush is bent slightly. Noting ^the signs ^of equations (30, 31) (and assuming K > 0), ^we

see that brushes generically prefer ^{to bend so that}

the chains are on the outside (not surprisingly) ; also, ^{we see that the} sign ^{of k} generically opposes ^a

« saddle surface ^» bend. These conclusions depend only ^{on the} assumption ^{of a bend at fixed} coverage, and not on the molecular weight distribution. Huse and Leibler [9] have noted the importance ^{of the} sign ^{of K for} interesting phase ^{behavior in am-} phiphilic ^{systems. The present} polymeric ^{model for}

the origin ^of bending ^moduli suggests ^that positive .K, which may give ^{rise to «} plumber’s nightmare ^» phases [9], may be difficult to achieve.

We now proceed ^{to evaluate} equations (29, 30) ^for

the case of the bimodal melt brush. For a melt brush,

the osmotic pressure 77(z) ^{is the same as the}

monomer chemical potential V (z ), ^{since monomers}

are incompressible. ^Hence equation (30) for K may be evaluated, using ^{II = V = A - U, with A and} U(z) ^from equation (20) ^and equation (23). ^After

some algebra, the results are

with a = (MIN - 1) ^as given ^after equation (20).

Writing n =- N IM and 0 = o, 1 / o,, ^the quantities

h= (1-0)+no ^and hI = n (1 - (1 - p )2 )112 ^are

respectively the total height of the flat bimodal

brush, ^{and the} height ^of region where short chain ends are found in the flat bimodal brush (both ⁱⁿ

units of Me-). ^The quantity ⁱⁿ curly ^brackets equals unity for 0 ^{= 0} and n 3for 0 = 1, ^thus reproducing

the monodisperse result of equation (10) ^{in the} appropriate ^limits.

Before examining ^this (somewhat opaque) ^result,

we shall compute ^{the mean} bending modulus K. If

we write equation (29) ⁱⁿ U-space ^and integrate by parts, ^we get

The bimodal melt profile, equation (19), ^remains

valid in the bent geometry, ^with A(c) ^and Ul(c) replacing ^their flat-geometry values ; hence, z ( U, c)

is a function of ^c only through ^A (c ) and U1 (c). ^Thus

we may expand the innermost integral ⁱⁿ equation (33) ^to 0(c) ^as

where M and 5 U, ^{are the} 0 (c) changes ^{in A and} Ul ; ^and z(U), az/aUl ^{and A are} evaluated for an

unbent brush.

The changes ^{SA and 5} Ul ^{could be} calculated from

equation (22), ^{which counts the} free ends of the

chains, ^{if we} knew the end density E (z,.c) ^to 0 (c ) for the bimodal melt brush in the cylindrical geometry. ^{The details of this} procedure ^{are con-}

tained in Appendix A. The results are

with h again the total height of the flat bimodal

brush, ^as given ^after equation (32).

Combining equation (35, 36) ^with equation (34)

and inserting the result into equation (33) gives (after ^more algebra)

where we have defined n - N 1M, cp == 0’ II 0’, ^and

k - (1 - cp ) ^{+ n4,.} (Appendix ^{B describes the con-}

version of equations (32, 37) ^into physical units).

The somewhat formidable-looking expressions ^for

K is plotted ⁱⁿ figure ^{4 as a function} of .0 for several values of n. (It ^{turns out that the ratio} K/K ^is

constant to within a few percent for all values of 0

and a wide range of values of M/N ; ^{hence we have}

not included a plot ^of K(cp) for various MIN).

Notice that the dependence of the modulus on the fraction of short chains present is far from an

interpolation between the bending ^{moduli of a} pure short-chain brush and a pure long-chain ^brush (dashed line). We may understand this without

appealing ^{to the} complete ^formula by considering

two limits : a long-chain brush with a few short chains replacing long chains, ^{and the} opposite ^case

of a short-chain brush with a few long ^chains

substituted for short chains.

In the case of a few short chains, if M > N, ^{we can} crudely think of the effect of shortening ^{a few chains}

as removing ^them, since the space occupied ^{and free}

energy per chain ^are both proportional ^{to chain}

length. ^{Hence the} largest ^{effect of} shortening ^{a small}

Fig. ^4.

^The change ^{in mean} bending ^modulus AK(O’)

upon substituting by chains of length ^{M a fraction} 0’ of the chains in a brush of N-chains (At > lV ), ^{in units}

of AK (1 for several values of n = N/M. ^{Solid curve,}

n = 1/10 ; ^dotted curve, n = 1/2 ; ^dashed curve, n ⁼ 9/10.

(The analogous ^curves for the Gaussian bending ^modulus

look essentially identical).

fraction 0 ’ == 1 - cp of the chains is the same as

reducing the coverage ^{to o-} (1 - 0’) ⁱⁿ the monodis- perse long-chain brush. Recall that the monodisperse

melt brush bending ^{constants scaled} as o- 5 ; ^{then in}

the reduced units of figure 4, ^the slope ^{of the curve} near 0 = 1 should be five (rather ^than unity, ^{as the} interpolation ^would suggest). Indeed, ^{if we} expand equation (32, 37) ^around .0’ ⁼ 0, ^we obtain the first correction to the monodisperse M-chain value of

equation (10) ^as

For a small fraction 0 ^of long ^chains substituting

for short chains, ^we observe that the conformations of the long ^{chains are} very much like the ^most- stretched short chains (those extending ^{to the full} height ho), ^{with an extra} unstretched piece ^of length

M - N. These extra pieces ^are unstretched because

they ^{are in a} region ^of vanishingly ^{small chemical} potential ^{V. Thus} they ^make only ^a very small contribution to an integral of the osmotic pressure such as appear in equations (29, 30) ^{for K and} K. In fact, expanding equations (32, 37) ^around 0 ^{= 0} gives

Any ^{smooth curve} connecting ^the slower-than- linear behavior at small 0 ^{with the} steep slope ^at

0 ^{= 1 must deviate} strongly ^from the linear interpo- lation, ^{as does the exact result.}

Recent studies of mixed short and long ^{chains in a}

melt brush [13], though only very small (N -- 20)

molecular weights ^{are examined, show several of the}

same features as the above analytic calculation. In

particular, ^reference [13] ^{notes the} qualitative impor-

tance of an admixture of shorter chains in lowering

the stretching energy of ^a brush made of long ^chains (see especially their sketch in Fig. 1). ^{The less-}

stretched long chains interact less violently ^{when the}

brush is bent; the reduction of K and 9 which results is expressed ⁱⁿ equations (38, 39).

Conclusions.

We have employed ^an analogy between surfactant

monolayers ^and bilayers, ^{and the} well-characterized

polymer ^{« brush », to obtain an} analytic calculation of the bending ^{moduli of} amphiphilic bilayers. ^The

method becomes increasingly ^{valid in the limit of} long hydrocarbon ^{tails on} the surfactant molecules.

Our results for a monodisperse brush, correspond- ing ^{to an} amphiphilic layer ^{of a} single ^molecular weight, ^are consistent with simple scaling arguments for the dependence ^of bending ^{moduli on} coverage

(interfacial ^area per molecule) and tail molecular

weight.

In our polymeric model of the origin ^of bending stiffness, ^{the Gaussian} modulus in always ^{of a} sign (negative) which opposes bending ^{into « saddle}

surfaces ». This means that a special ^molecular interaction, perhaps ^{between the surfactant}

« heads » (see below), ^is required ^to produce ^a positive ^{Gaussian modulus,} which may give ^{rise to} interesting ^« plumber’s nightmare ^» phases [9].

Our methods break down when applied ^to sharply

curved interfaces, ^{such as occur} in wormlike or

spherical micelles. This results from the breakdown of the ^« strong stretching ^» approximation ^{when the}

bend radius is of the order of the layer ^thickness.

Thus, ^{we cannot} perform ^a reliable calculation of the spontaneous ^{radius of curvature} for surfactant mono-

layers.

The present ^{model can be} applied ^to the interest-

ing ^{case of mixed} long- ^and short-chain amphiphiles ;

the bending moduli for this system ^{are found to} depend nonlinearly ^{on the fraction of short chains.}

In particular, replacing ^a relatively ^{small fraction of} long ^{chains with shorter ones serves to make the} bilayer nearly ^{as flexible as} the pure short-chain

bilayer. ^This phenomenon is observed in preparation

of surfactant-cosurfactant mixtures used in the study

of fluctuating ^flexible bilayers [36]. ^{We have} given

no consideration in this work to interactions of

amphiphile ^« head groups ^» (which, by ^{their aversion}

to organic ^{solvents, bind the} amphiphile ^{to the}

interface). However, any specific model of head-

head interactions can be combined with the present work to give ^a complete description ^{of a flexible} amphiphilic layer. ^{If the} head-head interaction en-

tails a change ^{in the area} per molecule ^as the layer ^is bent, ^{this can be} incorporated by calculating ^{the free}

energy change upon bending ^{in two} stages : ^first change ^{the area} per molecule in the flat layer, ^and

then bend the layer ^at fixed coverage.

An important question (discussed ^{in Ref.} [17, 18])

is the size and nature of finite molecular weight

corrections to the fluctuation-free ^« classical limit »,

upon which our methods rely. Briefly, ^we may say that all quantities have corrections of relative order

Rglh - O(N-1/2); features of the local monomer

chemical potential V (z ), ^the positions ^{of chains, etc.}

will all be smeared out by ^{this amount.}

However, ^{a direct} investigation ^{of small-N} poly-

mer models of amphiphilic layers ^{seems to indicate}

that these fluctuation effects are not severe. Re-

cently, Szleifer et al. have performed ^{exact enumer-}

ations of configurations of short random walks attached at one end to a curved surface, ^and interacting ^{with the} mean monomer density. ^From

these calculations [14], ^the bending stiffness of

monolayers ^and bilayers ^were extracted. Even for such short random walks as five to 20 steps, ^the scaling dependence of the mooduli on molecular

weight, ^{and the} dependence ^{of the} bending ^moduli

in mixed bilayers ^on the fraction of short chains, ^are

in qualitative agreement ^{with our} analytic ^calcu-

lations. (Szleifer ^{et al.} observe several features, including ^a slight ^{minimum in the} bending ^{moduli of}

the mixed bilayer ^{as a} function of the fraction of

short chains, ^which they ^{attribute to} interdigitation

of the chains in the two monolayers forming ^the bilayer. Our work indicates [17] ^that interdigitation

effects should disappear ^as the molecular weight ^is increased).

Acknowledgments.

We thank S. Alexander, ^{M. E.} Cates, ^W. Gelbart,

P. Pincus, and S. A. Safran for helpful discussions,

and I. Szleifer for a stimulating discussion of his work prior ^to publication.

Appendix ^A.

Computing ^E (z, c).

To reproduce ^{the uniform} density ^{in a bent} (cylindri- cal) geometry, ^{we use} equation (16) together ^with

the bimodal potential profile ^of equation (19).

Writing ^the equation ⁱⁿ U-space ^and expanding ^{as in}

section 1, ^we obtain for C/> U1

where 6 ⁼ Nac. A set of solutions of this integral equation for various left-hand sides was obtained

[25] ^and give

We would like to obtain two equations ^{for the two}

unknowns A (c) ^and U, (c) by counting ^{chain free} ends, ^{as in} equation (22). ^That is, ^{we write}

Rather than compute ^E (U, c) ^{for U :} Ul, ^we may argue ^as follows. The first term in equation (43) ^for 8 (U, c) ^for U:--. U, ^satisfies equation (16) ^for

U : Ul, i.e., ^{with U =} BN ^z2. The contribution from the second term (proportional ^{to 8} (Af/W - 1) ^{of e}

in equation (43) spoils ^the equation, ^{and must be}

cancelled by ^{a small} negative contribution to E (U, c )

for U ,- Ui. The number of monomers cancelled out

by ^this negative contribution must equal the contri- bution of the second term in equation (43) ^{to the}

left-hand side of equation (16). ^{Each chain removed}

has N _monomers, which makes it possible ^{to count}

the free chain ends inside z, ^as

where _{81 I} and E2 ^{are the first} and second terms in

equation (43).