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Bending moduli of polymeric surfactant interfacesS.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�

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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. 2014 Nous étendons notre théorie récente des « brosses » polymériques greffées en bout de chaînes àdes polymères attachés sur des interfaces courbées. Plusieurs systèmes importants, par exemple les surfactantspolymériques ou les copolymères formés de deux blocs fortement ségrégués, peuvent être décrits comme desbrosses. Par un développement de l’énergie libre d’une brosse sur une surface courbée en puissances de lacourbure, 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 quidonnent, K, K ~ N303C35 pour les conditions de « fondu » et ~ N 3 03C37/3 pour des brosses de densité modérée. Ontraite également de manière analytique le cas important d’une brosse formée d’un mélange de chaînes longueset 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. 2014 Our recent theory of the free energy and conformations of end-grafted polymer « brushes » isextended to polymers attached to curved surfaces. Several important systems, e.g., layers of polymericsurfactants or of strongly segregated diblock copolymers, can be well described as brushes. By expanding inpowers 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 scalingarguments, which imply K, K ~ N3 03C3 5 for melt conditions and ~ N3 03C3 7/3 for moderate-density brushes withsolvent. The important case of a brush composed of a mixture of long- and short-chain molecules is also treatedanalytically. The replacement of a small fraction of long-chain molecules in a brush by short chains is shown todramatically reduce the bending moduli.

J. Phys. France 49 (1988) 1951-1962 NOVEMBRE 1988,

Classification

Physics Abstracts36.20E - 81.60J - 87.15D - 82.70D

Introduction.

Among the most basic and important physicalproperties of an adsorbed surfactant layer at a

liquid-liquid interface, or of a bilayer composed ofamphiphilic molecules, are the mean and Gaussianbending moduli. Much progress has been made inpredicting the phase behavior [1-5] and fluctuations[6-9] in systems of self-associating amphiphiles, byparameterizing the monolayers or bilayers in thesesystems in terms of bending moduli and spontaneouscurvature. However, progress in understanding theorigins [10-14] of these fundamental parameters hasbeen 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, adescription which becomes increasingly good as the

molecular weight of the tails increases, i. e. , in thelimite 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 oneend to a surface at relatively high surface coverage.The resulting structure, called a « brush » becausethe polymer chains stretch away from the graftingsurface to avoid high monomer concentrations, hasbeen studied both for the monodisperse case, [17,19, 20] i. e. , when all the chains are of the samemolecular weight N, and for the case of arbitrarymolecular 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 twosuch monolayers back to back. In this paper, weshall 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

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polymer brushes to the case of curved grafting ’ I

surfaces, in order to calculate the elastic propertiesof the amphiphilic layers. The paper is organized asfollows. In section 1, a brief review of methods andresults for monodisperse brushes is presented. Insection 2, the bending moduli for a monodispersebrush are calculated by comparing the free energiesof the brush in flat, cylindrical, and spherical geomet-ries. In section 3, we treat the important case of anamphiphilic layer composed of chains of two differ-ent molecular weights - a model for the effects of ashort-chain « cosurfactant » on the properties of alayer of long-chain surfactant molecules. Recent

exact enumerations of configurations of very short-chain molecules [14] observe that a small admixtureof shorter-chain molecules to a surfactant layerreduces the bending stiffness with remarkable effi-ciency. Using our analytic methods, we calculate adramatic nonlinear dependence of the bending mod-uli on the amount of cosurfactant added : a relativelysmall amount of short chains serves to make the

layer nearly as flexible as a layer made only of shortchains. 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 themixed-brush calculations are relegated to AppendixA ; a prescription for converting our expressions forbending moduli from « theorists’ units » to physicalunits is contained in Appendix B.

1. Review of « Brushes ».

Consider the problem of a polymer chains per unitarea attached by one end to a surface, and exposedto a not-too-good solvent (such that the monomersinteract with one another through a mean monomerdensity). A description of this brush would consist ofthe conformations of the chains, and thus such

quantities as the monomer density 0 (z ) at a height zabove 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 molecularweight and surface coverage a, the thickness of thisbrush h scales as NQ ll3 and the free energy per unitarea F scales as No- 513More recent work [17, 21] has shown that these

simple scaling arguments miss several importantfeatures of the brush ; namely, 1) the conformationsof different chains in the brush are not necessarilysimilar, nor is a particular chain uniformly stretched ;2) the density profile, rather than being nearly astep-function as was suggested, [19, 20] is instead

parabolic ; and 3) because the density profile goescontinuously to zero at the outer extremity of the

brush, the force to compress the brush slightly isweaker (by one power of the strain) than calculatedusing a step-function ansatz profile.

These results [17] were obtained by a solution ofthe one-dimensional self-consistent field (SCF)equations [22], which give a mean-field description[23] of the system valid for chains at high coverageinteracting through a sufficiently weak two-bodyrepulsion. In practice this description should be validfor chains at moderate concentration in a not-too-

good solvent [24] ; for brevity this limit is referred tobelow as the « moderate density » regime. A closelyanalogous treatment was given earlier by Semenov[25] to describe the statistics of chains in a moltenblock-copolymer system having a lamellar mesoph-ase geometry. This situation corresponds formally toa 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 andthe melt regimes, the solution of the SCF equationsmakes use of the fact that the chains in a brush are

strongly stretched. This strong-stretching assumptionallows a « classical limit » to be taken, in which

fluctuations of chain conformation about the most

probable path between its endpoints can properly beneglected [17, 25]. What remains is the many-chainproblem of finding the self-consistent set of chainpaths, and chain-end positions with density E (z), ina monomer chemical potential V (z). This V(z) isproportional to the average monomer density0 (z) in the moderate density regime. In a melt,V(z) is fixed by the requirement that 0 = 1. Theself-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 fromthe sum over the end density E (z ) of the contributionto the density d 0 of each chain.

This many-chain problem was solved exactly, forboth the melt and moderate density cases, byemploying an « equal-time » argument [17, 25],which we now summarize.

First we assume that the polymer chains are all ofequal molecular weight, and that their free ends aredistributed with nonzero density at all distances fromthe grafting surface, up to the « brush height » h,beyond which 0 (z > h ) = 0. (Detailed stabilityarguments are given in reference [17] to demonstratethat 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 maybe in equilibrium - with no force applied to its freeend - 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 acertain classical particle. Under this analogy, discus-

1953

sed in detail in reference [17], the arc-length of thechain (roughly, the monomer index) corresponds to« time », and the degree of local chain stretching tovelocity. The fictitious particles move with equationof motion

where the potential energy U(z) of the fictitious

particle is related to the monomer chemical potentialV (z ) by U (z ) = const - V (z ). (We define

U(O) = 0, and assign V (h ) = 0 for equilibriumbrushes.) Since there is no tension at the free end ofa 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 » (chainlength) for a particle starting from rest at anydistance away from the grafting surface. That is, ’

I

U(z ) must be a harmonic-oscillator potential, so thatV (z ) = A - Bz. (See Fig. 1). The coefficient B thendetermines the period of the « oscillator » ; onequarter period is the length N of the chain. Thecoefficient A may be determined e.g., by 1) insistingon a brush profile which accomodates the requirednumber of monomers per unit area ; and 2) requir-ing the monomer chemical potential to be zero at theouter extremity of the brush, V (h ) = 0.

Finally, we may check our assumption that a

positive end density c(z) exists which reproduces themonomer 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 theinverse of its velocity [17]

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

This equation may be solved explicitly for e (z) inboth the moderate density and melt cases (seeFig. 1) ; in addition, general arguments may begiven which establish that E (z) > 0 in a brush grownon a flat surface [17]. (This point is discussed furtherin Sect. 2).

In a subsequent paper, we explored the ever-

present effects of polydispersity in molecular weighton the properties of end-grafted polymer brushes[18]. An understanding of these effects requires anextension of the equal-time arguments of reference[17] to a case where many molecular weights andthus « 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 meltbrush (dashed curve) and a moderate density brush

(dotted curve). The height z is given in units of the totalbrush height ho ; the y-axis units are arbitrary.

One characteristic feature of the polydisperse brushis that the « equal-time » requirement forces the freeends 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 (molecularweight) associated with each height z above thegrafting surface. All shorter chains will have theirfree ends closer to the grafting surface than that ofany longer chain. This segregation is a generalfeature of the « classical » long-chain limit for thepolydisperse 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 molecularweight distribution were obtained for the potentialV (z) (and thus the density, via the equation ofstate), as well as the force required to compress apolydisperse brush. In particular, explicit formulasfor the potential V (z ) were obtained for a « bimod-al » brush, composed of a mixture of two molecularweights M and N. These results, used in the calcu-lation of the bending moduli of a bimodal brush, arequoted in section 3.

2. Monodisperse case.

A formal expansion of the free energy per unit areaof a bent surface (brush, membrane, or other thinsheet) may be written as [27, 28]

Here K and K are the mean and Gaussian rigiditymoduli respectively, cl, C2 = I/Ri, 1/R2 are thelocal curvatures (inverse radii of curvature), andco is the preferred or « spontaneous » curvature.

If we can compute the free energy of a brush bentinto the inside or outside of a cylinder or sphere, we

1954

can extract K, K, and co. We write F, (R) (Fc (R)) asthe free energy per unit area of a brush assembled onthe surface of a large sphere (cylinder). (Throughoutthis section, subscripts c ans s will refer to cylindricaland 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 perunit area of the corresponding flat brush. Then

equation (3) implies

We may now compute the bending rigidity moduliof a « monodisperse » brush, i. e. , one consisting ofchains of a single molecular weight. We consider abend 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 willtaken to be the « surface of constant area » of thebend (see Fig. 2). (Either of these two assumptionsmay be later relaxed, by adjusting the coverage and/or area of the brush prior to the bend, to satisfysome other physical requirement than fixed coverageand a grafting surface of constant area during thebend).

Fig. 2. - A brush bent with « positive » radius ofcurvature (in this paper, R > 0 means the chains are on the« outside » of the bend).

To compute the quantities F, (R ) and Fc(R), weimagine assembling the brush on a slightly curvedsubstrate, grafting chains to this substrate at thesame coverage (chains per unit area) a present in theflat brush. We assume for the moment (this will beexamined below) that in the bent configuration, thelocal 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 theintroduction and Ref. [17]) to the assumption thatsome free ends of the grafted chains may be foundanywhere inside the brush. The coefficient B muststill be given by the « equal-time » requirement onthe chain conformations, i. e. , B = ir 2/ (8 N 2) so

that the chains of molecular weight N may reside inequilibrim in V (z, R). The values of A (R) andh (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. , onin which the density is forced to be uniform. (This isperhaps a reasonable model for the hydrocarbontails of a surfactant layer). The height of the brush isnow given trivially by counting monomers ; in theflat geometry we have h = Na , while in a cylindricalgeometry 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 isnow determined. We may now compute the free

energy to assemble the melt brush on a cylindricalsurface by the same technique used in the flat case,developed in reference [17]. The method, whichcalculates the free energy to assemble the brush byadding chains to an existing brush of lower coverage,may be briefly described as follows. The free energyincrement to add a chain to a brush can be shown to

be independent of where the free end of the newchain is placed. There are chains with their free endsarbitrarily close to the grafting surface, for which thefree energy increment is purely due to the potentialV and is simply NV (o ) = NA. Hence we may writethe free energy per unit area as

Application of equation (6) in the above case of amelt brush grown on a cylinder gives

The analogous calculations in a spherical geometrygive

Then equation (4) can be applied to extract K andK for the brush as

1955

A bilayer, composed of two such brushes back toback, would have bending moduli twice as large, andco = 0 by symmetry. (Appendix B outlines theconversion of Eq. (10) into physical units).We may also consider a moderate density brush,

i. e. , grafted chains at moderately high coverage inthe presence of a not-too-good solvent. As above,we assume the monomer chemical potential of thebent brush remains parabolic,V (z, R) = A (R) - BZ2. In the moderate densitycase, 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 sucha brush when bent into a cylinder of radius R is againgiven by integrating the density profile in the bentgeometry,

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 toextract K and R for the moderate case as

Our results for the bending moduli of monodis-perse brushes are naturally consistent with a simplescaling argument for their magnitude ; namely, since

K and 9 have dimensions of energy, they shouldscale as the free energy of a characteristic volume

ho of the brush [29], or as Fo h2 Recalling the scalingarguments given by de Gennes [19] (or examiningequations (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 Kand K scale as N 30,7/3 in the moderate density caseand N3 uS in the melt case.

Several previous authors, including references [1,10, 14], have obtained estimates of the bendingstiffness of surfactant layers which are also consist-ent with these scaling results. Analogous scalingresults are contained in reference [1] for the case of agood solvent ; reference [14] presents a simple modelwhich results in K - N 3 cr 5for the melt case.

Other authors derive results which are equivalentto the above scaling when combined with somemodel of how the coverage a depends on N. Forinstance, in reference [10] the coverage cr is op-timized within a model in which a « bare surfacetension » y is assigned to the surfactant interface, sothat a minimizes (F - y )/ o-. (This is not necessarilya good model for determining a (N ) ; see Ref. [30]).This implies 0’ -- N -315 for the moderate densitycase. Together with the above scaling result for K,K, this gives K, k - N 8/5 for the moderate densitycase, which agrees with the scaling of the resultsreported 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 asurface of area A bent with local curvatures

cl and C2. The ratio VI(AH) is the ratio of volumeavailable for chains in a layer of height h and graftingarea A in the bent and unbent geometries, and maybe expanded as [32]

Then for a saddle surface, ci = - c2 = h/R and

VI(AH) -- 1 - 3 (hlR )2 the bent thin shells have/ 3 less room for monomers than the unbent shells, andso the chains must stretch upon bending, which mustcost free energy.The spontaneous curvature co may also be ex-

tracted from equations (4, 7, 12) for the moderatedensity and melt cases ; co is of order 1 /ho in eachcase, as a scaling argument would suggest. This is, ofcourse, a strong hint that a flat monolayer of graftedchains would prefer to break up into micelles.

However, the calculated value of co cannot beidentified with the actual preferred curvature, be-

1956

cause the value was obtained in an expansion inpowers of hc about c = 0, the flat surface. When abrush is bent a finite amount, two things occur whichcause the present calculation to break down.

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

grafting surface, they no longer stretch to avoid highmonomer density. To treat properly the correctionsto 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 whenwe consider a brush bent by any nonzero amount ;namely, it is no longer true that the density of freechain ends E(z) is nonzero everywhere inside thebrush. A « dead zone » (region of z where no freechain ends are found) opens up near the graftingsurface. We may estimate the size of this region byassuming that the potential V is still parabolic in abent brush, and computing in the bent geometry theE (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 nearz = 0, which will be approximately the region of thedead zone.For convenience, we define the end density

e(z) in the bent (cylindrical) geometry to be thenumber of ends per unit (height x projected area).Then, in the melt case, the requirement that

E (z) reproduce the (uniform) density is, by analogyto the discussion in section 1,

Assuming that V is a parabola amounts to writingU (z ) = BZ2; using this and A = Bh 2, equation (16)may be expanded. It is convenient [18] to rewrite theequations in « U space », defining E (U) d U == e(z) dz, as

This may be solved perturbatively in powers of5 -= h/R for e (U), yielding the end density of amonodisperse 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 rootof E (z), we find z- h exp(2013 1/6 ) ; the dead zone isexponentially small in 8, and will not contaminate anexpansion in h/R. We need only contend with the

9

Fig. 3. - From reference [18], the self-consistent potentialU(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) ; abovehl, the potential approaches that of a brush of M-chains(dotted curve).

complications of a dead zone (which e.g., makes itimpossible to guess the form of V) when considering« large » (nonperturbative) curvatures.

3. Mixing short and long chains.

The methods of treating strongly stretched polymersdeveloped in reference [17] were extended in refer-ence [18] to brushes of arbitrary molecular weightdistribution. Using some of the results of reference[18], we can compute the bending moduli K andK for the interesting and important case of a meltbrush composed of a mixture of short and longpolymer chains. This may be regarded as a model forthe effectiveness of short-chain cosurfactants in

reducing the bending constant of a layer composedof 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 ofchains of molecular weights N and M::-. N, the freeends of chains segregate in the z-direction (normal tothe grafting surface). The segregation results fromthe « equal-time » requirement : chains of two differ-ent molecular weights, experiencing the same poten-tial V, cannot both be in equilibrium with their freeends at the same distance from the grafting surface.The free ends of the shorter (longer) chains will befound at distances z -- z 1 (z :-.. z from the graftingsurface.The self-consistent potential V (z ) for this mixed

brush, with a partial coverage a 1 of short chains ando- 2013 o-i 1 of long chains, was shown in reference 18 tobe

1957

where Ul = U(zl) is determined below. This may beinverted to find U(z) for z > zl as

with a = M/N -1 and BN M lr2/(8 N 2). (We haveU(z) = BN Z 2for z : zl, just as for a brush made ofpure 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 ofE in a bent geometry in sect. 1), could be determinedindependently of polydispersity as

Using equation (21), the value of U(zl ), wherezl is the location of the boundary between theregions where short and long chain ends are found,was determined by integrating the end density untilall 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)). Theresults [18] are

The relation between cr 1 and Ul may be convenientlygiven in terms of A as

Solving for Ul, we obtain

The free energy of the brush in a bent geometrycan be calculated in two equivalent ways : by directlyassembling the brush on the bent surface, as wasdone for the monodisperse brush in section 1 ; or byevaluating 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 approachand only to 0 (c ) in the second. The second methodoffers a welcome advantage in the more tedious caseof the mixed brush.

Consider a unit area of unbent brush. To deform itso that it may become part of the inside of a

cylindrical shell of radius R, we must reduce thecross-section C of the brush at a height z fromC = 1 to Cc (z, R) = 1 - z/R. When we do this, theheight of the brush grows from the unbent valueho = 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 samenumber of (incompressible) monomers be accomo-dated in the reduced cross-sections. Work is done

during this process only [35] against the osmoticpressure nc(z, R ), which is a function both a heightz above the grafting surface and the radius of thebend. Writing c = 1/7?, we may integrate with

respect to c to find the change in free energy per unitarea 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, forwhich C s (z, R) = (1 - zIR)’ and hs (R ) is given byequation (13), yields

(In Eqs (26, 27), the signs have been chosen so thatR > 0 corresponds to a brush on the outside of abend).

If we expand the osmotic pressure in equations(26, 27) to 0 (c ), and compare to the formal bendingenergy expansion equation (3), we may obtain gen-eral expressions for K, K, and co. What remains is torelate 8H/ac I c = 0 in the cylindrical and sphericalgeometries. The result,

may be understood in several ways. Physically, thefactor of two arises from the fact that the cross-section at a height z in the spherical bend is reducedtwice as much (to 0 (c )) as for a cylindrical bend ofthe same radius ; thus, the first-order change of theosmotic pressure II(z ) from its flat-geometry valueis 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 neednot expand the dependence in the limit of integrationof h(c)). The final result is

1958

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 beginsto change when the brush is bent slightly. Noting thesigns of equations (30, 31) (and assuming K > 0), wesee that brushes generically prefer to bend so thatthe chains are on the outside (not surprisingly) ;also, we see that the sign of k generically opposes a« saddle surface » bend. These conclusions dependonly on the assumption of a bend at fixed coverage,and not on the molecular weight distribution. Huseand Leibler [9] have noted the importance of the

sign of K for interesting phase behavior in am-phiphilic systems. The present polymeric model forthe 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 monomersare incompressible. Hence equation (30) for K maybe evaluated, using II = V = A - U, with A and

U(z) from equation (20) and equation (23). Aftersome algebra, the results are

with a = (MIN - 1) as given after equation (20).Writing n =- N IM and 0 = o, 1 / o,, the quantitiesh= (1-0)+no and hI = n (1 - (1 - p )2 )112 arerespectively the total height of the flat bimodal

brush, and the height of region where short chainends are found in the flat bimodal brush (both inunits of Me-). The quantity in curly brackets equalsunity for 0 = 0 and n 3for 0 = 1, thus reproducingthe monodisperse result of equation (10) in the

appropriate limits.Before examining this (somewhat opaque) result,

we shall compute the mean bending modulus K. Ifwe write equation (29) in U-space and integrate byparts, we get

The bimodal melt profile, equation (19), remainsvalid 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 andUl ; and z(U), az/aUl and A are evaluated for anunbent brush.The changes SA and 5 Ul could be calculated from

equation (22), which counts the free ends of thechains, if we knew the end density E (z,.c) to

0 (c ) for the bimodal melt brush in the cylindricalgeometry. 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’, andk - (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 severalvalues of n. (It turns out that the ratio K/K isconstant to within a few percent for all values of 0and a wide range of values of M/N ; hence we havenot included a plot of K(cp) for various MIN).Notice that the dependence of the modulus on thefraction of short chains present is far from an

interpolation between the bending moduli of a pureshort-chain brush and a pure long-chain brush

(dashed line). We may understand this without

appealing to the complete formula by consideringtwo limits : a long-chain brush with a few shortchains replacing long chains, and the opposite caseof a short-chain brush with a few long chainssubstituted for short chains.

In the case of a few short chains, if M > N, we cancrudely think of the effect of shortening a few chainsas removing them, since the space occupied and freeenergy per chain are both proportional to chainlength. Hence the largest effect of shortening a small

1959

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 unitsof 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 moduluslook 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 monodispersemelt brush bending constants scaled as o- 5 ; then inthe reduced units of figure 4, the slope of the curvenear 0 = 1 should be five (rather than unity, as theinterpolation would suggest). Indeed, if we expandequation (32, 37) around .0’ = 0, we obtain the firstcorrection to the monodisperse M-chain value ofequation (10) as

For a small fraction 0 of long chains substitutingfor short chains, we observe that the conformationsof 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 lengthM - N. These extra pieces are unstretched becausethey are in a region of vanishingly small chemicalpotential V. Thus they make only a very smallcontribution to an integral of the osmotic pressuresuch as appear in equations (29, 30) for K andK. In fact, expanding equations (32, 37) around0 = 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 amelt brush [13], though only very small (N -- 20)molecular weights are examined, show several of thesame features as the above analytic calculation. Inparticular, reference [13] notes the qualitative impor-tance of an admixture of shorter chains in loweringthe 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 thebrush is bent; the reduction of K and 9 whichresults is expressed in equations (38, 39).

Conclusions.

We have employed an analogy between surfactantmonolayers and bilayers, and the well-characterizedpolymer « brush », to obtain an analytic calculationof the bending moduli of amphiphilic bilayers. Themethod becomes increasingly valid in the limit oflong hydrocarbon tails on the surfactant molecules.Our results for a monodisperse brush, correspond-

ing to an amphiphilic layer of a single molecularweight, are consistent with simple scaling argumentsfor 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 « saddlesurfaces ». 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 tointeresting « 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 breakdownof the « strong stretching » approximation when thebend radius is of the order of the layer thickness.Thus, we cannot perform a reliable calculation of thespontaneous 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 todepend nonlinearly on the fraction of short chains.In particular, replacing a relatively small fraction oflong chains with shorter ones serves to make thebilayer nearly as flexible as the pure short-chain

bilayer. This phenomenon is observed in preparationof surfactant-cosurfactant mixtures used in the studyof fluctuating flexible bilayers [36]. We have givenno consideration in this work to interactions of

amphiphile « head groups » (which, by their aversionto organic solvents, bind the amphiphile to the

interface). However, any specific model of head-

1960

head interactions can be combined with the presentwork to give a complete description of a flexibleamphiphilic layer. If the head-head interaction en-tails a change in the area per molecule as the layer isbent, this can be incorporated by calculating the freeenergy change upon bending in two stages : first

change the area per molecule in the flat layer, andthen bend the layer at fixed coverage.An important question (discussed in Ref. [17, 18])

is the size and nature of finite molecular weightcorrections to the fluctuation-free « classical limit »,upon which our methods rely. Briefly, we may saythat 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 indicatethat these fluctuation effects are not severe. Re-

cently, Szleifer et al. have performed exact enumer-ations of configurations of short random walksattached at one end to a curved surface, and

interacting with the mean monomer density. Fromthese calculations [14], the bending stiffness of

monolayers and bilayers were extracted. Even forsuch short random walks as five to 20 steps, the

scaling dependence of the mooduli on molecularweight, and the dependence of the bending moduliin mixed bilayers on the fraction of short chains, arein qualitative agreement with our analytic calcu-

lations. (Szleifer et al. observe several features,including a slight minimum in the bending moduli ofthe mixed bilayer as a function of the fraction of

short chains, which they attribute to interdigitationof the chains in the two monolayers forming thebilayer. Our work indicates [17] that interdigitationeffects should disappear as the molecular weight isincreased).

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 hiswork 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 withthe bimodal potential profile of equation (19).Writing the equation in U-space and expanding as insection 1, we obtain for C/> U1

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

[25] and give

We would like to obtain two equations for the twounknowns A (c) and U, (c) by counting chain freeends, as in equation (22). That is, we write

Rather than compute E (U, c) for U : Ul, we mayargue as follows. The first term in equation (43) for8 (U, c) for U:--. U, satisfies equation (16) forU : Ul, i.e., with U = BN z2. The contribution fromthe second term (proportional to 8 (Af/W - 1) of ein 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 outby this negative contribution must equal the contri-bution of the second term in equation (43) to theleft-hand side of equation (16). Each chain removedhas N monomers, which makes it possible to countthe free chain ends inside z, as

where 81 I and E2 are the first and second terms inequation (43).

1961

After considerable algebra, the resulting equationsare

where we hale kept only terms of first order in 8,using the unbent values of A and Ul from equations(23, 25) wherever possible. Expanding A(c) andU, (c) in equations (46, 47) to 0 (c ), and solving theresulting linear equations for 8A and 6 Ui, yieldsequations (35, 36).

Appendix B.

Physical Units. °

To convert the expressions for K and K of equations(10, 32, 37) into physical units, we first obtain

Fo and ho (the equilibrium height and free energyper unit area for the monodisperse melt brush) inphysical units. Then we may use equations (10, 32,37) directly, as the bending moduli are expressed asa dimensionless function times Fo ho.We begin by writing the system free energy of a

melt as

which leads to an effective single-chain free energynf

From this, and the melt constraint of cp = 0 0 (ratherthan 0 = 1), we obtain (following the Introductionand Ref. [17])

and finally

(Here for definiteness we shall express N in mono-mers, tPo in monomers per unit volume, energy inkT, coverage u in chains per unit area, and lengths incm). Now we use equation (48) to describe the end-end displacement Re of a melt chain, which yields

where the second two equalities express Re in moreconventional notation, in terms of the monomerbond length b and the characteristic ratio Coo.As an example, for ethylene chains, reported

values [37] of the parameters are b = 1.53 A ;p o = 0.806 glcm3 (so that 00 = (3.07 A )-3) ; andCoo = 6 at 273 K. Hence using equations (50, 51)and equation (10), we find K = 9 x

104 (1011 x u)5 N3 kT. If we are inclined to applythis expression to short-chain surfactants (for N = 6,say), for which a typical value of coverage iso- = (5 x 10-8)-2, we find K = 2 kT, which is notan unreasonable value.

References

[1] DE GENNES, P.-G., TAUPIN, C., J. Phys. Chem. 86(1982) 2294.

[2] SAFRAN, S. A., TURKEVICH, L. A., Phys. Rev. Lett.50 (1983) 1930.

[3] SAFRAN, S. A., TURKEVICH, L. A., PINCUS, P. A., J.Phys. Lett. France 45 (1984) L69.

[4] SAFRAN, S. A., ROUX, D., CATES, M. E., ANDEL-MAN, D., Phys. Rev. Lett. 57 (1986) 491.

[5] ANDELMAN, D., CATES, M. E., ROUX, D., SAFRAN,S. A., J. Chem. Phys., in press.

[6] HELFRICH, W., Z. Naturforschung 33 a (1978) 305.[7] HELFRICH, W., J. Phys. France 46 (1985) 1263.[8] PELITI, L., LEIBLER, S., Phys. Rev. Lett. 54 (1985)

1690.

[9] HUSE, D. A., LEIBLER, S., J. Phys. France 49 (1988)605.

[10] CANTOR, R., Macromolecules 14 (1981) 1186.[11] EVANS, E., Biophys. J. 14 (1974) 923.[12] PETROV, A. G., BIVAS, I., Prog. Surf. Sci. 16 (1984)

389, and references therein.[13] BEN-SHAUL, A., SZLEIFER, I., GELBART, W. M.,

Physics of Amphiphilic Layers, Ed. J. Meunieret al. (Springer-Verlag, New York) 1987, p. 2.

[14] SZLEIFER, I., KRAMER, D., BEN-SHAUL, A., ROUX,D., GELBART, W. M., Phys. Rev. Lett. 60

(1988) 1966.[15] HADZIIOANNOU, G., PATEL, S., GRANICK, S., TIR-

RELL, M., J. Am. Chem. Soc. 108 (1986) 2869,and references therein.

1962

[16] TAUNTON, H. J., TOPRAKCIOGLU, C., FETTERS,L. J., KLEIN, J., Nature 332 (1988) 712.

[17] MILNER, S. T., WITTEN, T. A., CATES, M. E.,Europhys. Lett. 5 (1988) 413 ;

MILNER, S. T., WITTEN, T. A., CATES, M. E.,Macromolecules 21 (1988) 2610.

[18] MILNER, S. T., WITTEN, T. A., CATES, M. E., to bepublished.

[19] DE GENNES, P.-G., J. Phys. France 37 (1976) 1443 ;Macromolecules 13 (1980) 1069 ; C. R. Acad.Sci. Paris 300 (1985) 839.

[20] ALEXANDER, S., J. Phys. France 38 (1977) 983.[21] HIRZ, S., unpublished Thesis, University of Minneso-

ta. For similar calculations, see also : Cos-

GROVE, T., HEATH, T., VAN LENT, B., LEER-MAKERS, F., SCHEUTJENS, J., Macromolecules20 (1987) 1692.

[22] EDWARDS, S. F., Proc. Phys. Soc. London 85 (1965)613 ; Proc. Phys. Soc. London 88 (1966) 265 ;

DE GENNES, P.-G., Rep. Prog. Phys. 32 (1969) 187.See also DE GENNES, P.-G. Scaling Concepts inPolymer Physics, Cornell, Ithaca (1979), Ch. IX.

[23] DOLAN, A. K., EDWARDS, S. F., Proc. R. Soc.

London A 337 (1974) 509 ; Proc. R. Soc. LondonA 343 (1975) 427.

[24] See, e.g., Flory P., Principles of Polymer Chemistry,Cornell, Ithaca (1971), Ch. 12.

[25] SEMENOV, A. N., Sov. Phys. JETP 61 (1985) 733,[Zh. Eksp. Teor. Fiz. 88 (1985) 1242].

[26] Of course, fluctuations will smear the « time zone »boundaries over a distance of order Rg ~ N 1/2,which is the characteristic scale of fluctuationcorrections to the classical limit. Nonetheless, inthe limit of large molecular weight at fixed

coverage, the time zones are sharply defined,insofar as Rg is small compared to the brushheight.

[27] LANDAU, L. D., LIFSHITZ, E. M., Theory of Elastici-ty (Pergamon) Oxford (1970), Ch. 2.

[28] HELFRICH, W., Z. Naturforschung 28c (1973) 693.[29] This is the relevant energy scale because a brush bent

on a radius R of order ho would have a bendingenergy for an area h20 of order K 2014 and, equiva-lently, a change of relative order unity in con-figurational free energy per unit volume.

[30] LEIBLER, L., Makromol. Chem. Macromol. Symp.16 (1988) 1-17.

[31] Of course, real long-chain surfactant interfaces mayhave many features which alter these con-

clusions, such as a tendency to pack with crystal-line order, or particular interactions between theadsorbing ends (« heads ») which are outside thescope of this calculation.

[32] This may be obtained by expanding V / (Ah) =1 + 03B1 (c1 + C2) + 03B2 (c21 + c22 ) + 03B3c1 C2 ... and fix-ing a, 03B2, 03B3 by comparison to the easily computedexpansions of V / (Ah ) for a cylinder and asphere.

[33] DAOUD, M., COTTON, J. P., J. Phys. France 43

(1982) 531.[34] HELFRICH, W., Physics of Defects, Les Houches

Session XXXV, Ed. R. Balian et al., (North-Holland) 1981.

[35] The chain backbones do not cross the surface normalto which the brush is compressed, hence thework is done against osmotic pressure only.

[36] DI MEGLO, J. M., DVOLAITZKY, M., TAUPIN, C., J.Phys. Chem. 89 (1985) 871.

[37] CHIANG, R. J., J. Phys. Chem. 69 (1965) 1645, J.Phys. Chem. 70 (1966) 2348 ;

STACY, C. J., ARNETT, R. L., J. Phys. Chem. 69(1965) 3109 ;

NAKAJIMA, A., HAMADA, F., HAYASHI, S., J.

Polym. Sci. C 15 (1966) 285.