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Variational theory of undulating multilayer systems

P. Pieruschka, S. Marcelja, M. Teubner

To cite this version:

P. Pieruschka, S. Marcelja, M. Teubner. Variational theory of undulating multilayer systems.Journal de Physique II, EDP Sciences, 1994, 4 (5), pp.763-772. <10.1051/jp2:1994163>. <jpa-00247999>

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J. Phys. II IYance 4 (1994) 763-772 MAY 1994, PAGE 763

Classification

Physics Abstracts

05.40 68.10 61.30

Variational theory of undulating multilayer systems

P. Pieruschka, S. Martelja and M. Teubner (*)

Department of Applied Mathematics, Australian National University, Canberra 0200, Australia

(Received 5 November 1993, accepted in final form 15 February1994)

Abstract. Weuse a

variational approach to determine the equilibrium properties of lamellar

surfactant phases. The variational theory yieldsa

general expression for the renormalization of

the bending constant of undulating sheet-like membranes. The method is then applied to lamel-

lar ensembles characterized by conserved surfactant film area and the full, non-linear bendingHamiltonian. In the limit of large bending modulus the theory converges towards Helfrich's

model. For realistic values of the bending constant wefind an increase in the equilibrium

crum-

pling and layer separation and characteristic changes in the structure factor and swelling law

due to film area conservation and non-linear terms in the Hamiltonian. The scaling of the free

energy density, however, appears to be largely unaffected by first order crumpling corrections.

Introduction.

Amphiphilic molecules when dissolved in water (and oil) often self-assemble in two-dimensional

bilayers (monolayers) which extend over scales much larger than molecular size [Ii. A commonlyobserved film geometry is lamellar, where the bulk material is separated by regularly stacked

sheets of surfactant film with a characteristic average layer spacing d which can be measured

in scattering experiments. In effective interface models [2, 3] the physical parameters which

specify the equilibrium state of a lamellar system are essentially reduced to the amphiphileconcentration is and the bending stiffness ~

of the elastic surfactant film. These model systemshave so far been described by a harmonic approximation of the bending Hamiltonian [4] which

allows for application of the equipartition theorem to determine the mode distributions of

thermally undulating layers and to estimate the repulsive entropic force which is due to the

excluded volume occupied by neighboring layers [5]. The theory operated in an ensemble

characterized by fluctuating film area and fixed area of the associated projected surface (open-framed ensemble in the classification of David and Leibler [2]). The crumpling of the system

was assumed negligible at all concentrations. This can be shown to be self-consistent at large ~.

Therefore, to zeroth order, the layer spacing d and surfactant concentration is were essentially

(*) Permanent address: Max-Planck-Institut fur Biophysikalische Chemie, G6ttingen, Germany.

764 JOURNAL DE PHYSIQUE II N°5

equivalent quantities; the steric force law for the free energy density which was expressed as

f ccd~~ in [5] is equivalent to the scaling relation

f ~j3 (~)~S

which can be directly derived from the scale invariance of the bending Hamiltonian and stan-

dard scaling arguments [6, 7].We investigate a different ensemble in which the film area is fixed, but the projected film

area can fluctuate (closed-unframed ensemble [2]). This ensemble appears suitable to model

the most commonly studied experimental situation where a given amount of surfactant is is

confined in a sealed container. The remaining free parameter in our theory is the bendingstiffness

~of the film. Other physical quantities can be derived from the minimization of the

free energy density. In particular, we consistently evaluate the scattering structure factor, the

layer density and crumpling, renormalization of the bending modulus (which finds a simpleformulation in the variational theory), and the steric repulsion force for an ensemble charac-

terized by the full, non-linear bending Hamiltonian. For large values of ~ the theory is shown

to be equivalent to Helfrich's theory. Away from this regime, however, we find characteristic

corrections to the structure factor, swelling law, and renormalization of the bending constant

which become significant for small ~ < 5kT. In contrast, the scaling of the steric force law is

found to hold even for low values of the bending constant.

Theory.

The thermodynamics of larnellar surfactant phases can be conveniently studied using a model

ensemble of essentially parallel, but thermally undulating interfaces as long as a single layer has

essentially two-dimensional character I.e. does not crumple too mucl~ [8]. The mean positionsof the undulating interfaces with surface area S are given by a set of flat, parallel surfaces

with projected or base area A. The surface position of an individual undulating layer can be

described by the displacement variable t~(r) normal to the projected surface

i~(r)"

~ji~(k) e'~~ (2)

Although commonly used, the Monge representation of states, equation (2), is only an ap-

proximation as it is single-valued and does not describe surfaces with overhanging parts or

topological defects, such as saddle structures which could connect neighbouring layers. How-

ever, even in the most swollen experimental samples [9], the ratio of real to projected surface

area or crumpling ratio is C=

S/A m 1.2. Usually the crumpling ratio is close to unity [10],and hence the single-valuedness should be a minor deficiency of the states equation (2). On

theoretical grounds it has been suggested that topological changes can be neglected as long as

the interlayer spacing is much smaller than the persistence length d < fk" Tm exp(47r~/3) [2]

I-e- for membranes with not too small bending modulus ~. We restrict our study therefore to

membranes with ~ > lkT. Below this value a more complete state representation has to be

chosen.

The undulations t~(r) are of thermal nature and we assume their Fourier amplitudes to be

uncorrelated and Gaussian with structure factor v(k) ill]. The structure factor is related to

the fluctuation of a mode by v(k)=

A ((t~(k)(~)o where ()o denotes statistical averaging over

the Gaussian ensemble.

The closed, unframed lamellar ensemble is essentially determined by the bending Hamilto-

njar~ and non-local interactions caused by the topological and surface area constraints. The

N°5 VARIATIONAL THEORY OF UNDULATING MULTILAYER SYSTEMS 765

local steric constraint due to adjacent layers is approximated by the usual global constraint [5]

d=

1 13)

where p is a numerical factor. We follow [5] and will set p =1/24 (in

our definition we

operate with walls at +d/2) later for numerical calculations. Due to the incompressibility of

the surfactant film [12], closed surfactant systems with surfactant volume fraction is have an

approximately constant surface to volume ratio

is cc

~=

~d~~

=Cte. (4)

I-e- in the closed-unframed ensemble the total surface area is kept constant (whereas the in-

dividual layer area and crumpling parameter may vary). The projected area to volume ratio

A/V is not a conserved quantity. Symmetric systems are characterized by a simple bendingHamiltonian without spontaneous and saddle-splay curvatures

7i=

2~/

dS H~ (5)s

where ~ is the bare bending modulus (in units of kT) and H is the mean curvature.

Minimization of the free energy density comprising the relevant interactions determines equi-librium structure factor, layer spacing and crumpling, and free energy at given values of ~ and

is. Using Gaussian model states, the free energy F (in units of kT) associated with the bendingHamiltonian 7i can be approximated [13-15]

F s F=

Fo + 17i 7io)o=

-TSO + (7i)o=

~ in AT+ (7i)o (6)

where the subscript 0 refers to Gaussian states characterized by the Hamiltonian 7io=

~j v(k)~~ t~(k)t~(-k). The entropic term TSO"

-Fo has been derived from the partition2

~

function of the Gaussian ensemble Zo"

e~'°=

f Dqe~~°~~) and the average of the bending

energy (7i)o can be calculated using the joint probability distribution p(t~~, t~y, t~~~, t~~y, t~yy) of

the first and second derivatives of the height field t~(r) which is given by a Gaussian distribution

[16] with the non-zero correlation matrix elements

l~l)0"

l~()0"

)(k~l' l~lz)0"

l~]Y)0"

)(k~l, (t~lV)0 "(UzzUyy)0

"~(k~). (7)

The moments of the structure factor are defined by

(kn)=

~~kn+i v(k) dk (8)

and the cut-off k~ is of the order of an inverse molecular size, k~=

~'~=

l. We will user~

(k°)=

(I) as a convenient notation for the 0th moment which is proportional to the mean

square average (t~~)o. Thus we can write

(7i)o"

(2~/

dA~~

H~)o"

2~A (~~ H~)o (9)A

dA dA


with

~~ ~~~

2~~~ fi ' dA

~~~~~

and we find for the ensemble average over the weighted mean square curvature

i$H~)= (ik~) iii + (vl~)~)~~/~ + (ivl~)~ (i + (vl~)~)~~/~)o

=)ik~) Giik~)) iii)

where

G(x)- T

II i+ 11 13 4T + 4T~) §(1 erffi)1 (12)

with x =27r/(k~). G((k~)) is bounded and monotonically decreasing

0 £ G((k~)) £ I, G'(ik~)) § 0 for (k~) / 0. (13)

For small (k~) it can be expanded into

G((k~))=

i )(k~) + $lk~)~ O((k~)~). (14)

The function G((k~)) contains the non-linear coupling between modes. As the harmonic ap-

proximation for the bending energy is proportional to ~ (k~), comparison with equation (II)yields the effective, thermally softened bending constant [iii

'~e? "G((k~))'~. (15)

Therefore our variational method is equivalent to a Hartree approximation which replacesthe non-linear Hamiltonian by a Gaussian with effective parameters that are determined self-

consistently.For a free membrane the structure factor is known to be v(k)

=(~k~)~~ Applying equa-

tions (12), (14) to this case we retrieve the well-known first order renormalization correction

of the bending constant: G((k~))m 1- 3/(47r~) In(k~/km;n) in agreement with the results of

[17, 18] (km;n is a lower frequency cut-off). For closed systems the exict form of the renor-

malization is as we will see below different. It is in general not possible to use the

renormalization derived for a free membrane in a system characterized by other physical con-

straints.

A second useful average which will be used below gives an explicit expression for the crum-

pling factor

C=

($)o=

iii + (vu)~)~/~)o (16)

which can be expanded for (k~) « 1

C(lk~))=

i + )lk~) £(k~)~ + Ollk~)~). (17)

Results.

With the averages equation (II) and equation (16) the free energy density can be written as a

functional of the structure factor v(k) (using relation Eq. (3) for d)

jjv(k)j=

(=

£ tt(k~) G((k~))

~~kin v(k)

kl.(18)

~~°


This expression has to be functionally minimized with respect to v(k) under one constraint,i~ cc

S/V=

Cte. which can be coupled to equation (18) by a Lagrange multiplier. The result

will be a minimal v(k) from which the equilibrium layer spacing and crumpling and the steric

force law can be calculated as functions of i~ and~. We note that the free energy per area

usually contains self-energy terms proportional to k) I-e- the number of degrees of freedom

associated with the base surface. These terms represent an insignificant additive constant in

the discussion of the free energy per area, but have to be omitted when going to the free

energy per volume; we will continue this discussion later when this point becomes relevant for

the calculation.

The minimization can be conveniently carried out by variationally minimizing

jjv(k)j=

ttjk~)G((k~)) ~~kin v(k)dk + ~i C((k~)) + ~2

((1))~)(19)

~~~(i~))°

under two constraints

~ ~~~~~~l~~~" ~(~)~~

~~~' ~~~~

The additional constraint on d will be removed later by df/dD=

0. We prefer the notation

D((1)) to stress that the layer density is in this context not preset as in zero order theories,

but a functional.

The result of df/dv(k)=

0 is

~~~~~4

/~2+ (4 ~~~~

with

a =

~~~ G((k~))~~=

~j/ (22)

~~~

~~(k~) G'((k2))

~j

c'((k2))j j~~

~~ D((i))D'((1))~~~~

° Gjjk2)) ~Gjjk2)) ' ~G(jk2))

where a > 0, i~> 0 while k(

can assume any sign. Whereas for given surfactant concentration

is and bare bending stiffness ~ the coefficient a is readily given by equation (22) [19], the

coefficients k(, i~ have to be evaluated from the non-linear equation system

~ arctan ~~+ arctan

~~~ ~~ l=27rpd~ (24)@~ @~ fi~

~ ~ ~(4

_~~~ ~° ~

4~~

kf k)k( +14~ ~~~~~ ~~~~

~~=

0 (26)

where equations (24, 25) correspond to equation (20) and C~~ denotes an inverse function.

The function f(ko, I, d; ~, i~) can be obtained by inserting the relations

(k~)=

j(k( 2i~)(1) (k( In hi + )k) (27)

/~~ kln v(k) dk=

)(k( 4i~)(1) (k( In hi + jk) In h2 + k) (28)o a


with the abbreviations hi=

i~/(k) k(k] + i~) and h2=

a/(k) k(k] + i~), and equation (22)into equation (19)

f(ko, I, d; ~, is=

d~~ (»(2a)~~l~ d~ (87r)~~ kl (i + in v (kc ))j (29)

Thus we have reduced the problem to the solution of three relatively simple equations. In partic-ular, solution of equation (24) and equation (25) is straightforward and reduces f(ko, I, d; ~, #~)

to f(d; ~, #~) so that we are left with the single equation df(d; ~, #~) lad=

0. At this point

we have to consider the self-energy. It is a harmless energy offset in problems where the total

projected area is constant. In the present problem, however, the number of lamellar layers is

allowed to vary and the offset would cause a spurious d~~ term in the free energy. In order to

subtract the self-energy we fix the quantities a, ko, I, and d at their physical values, consider

the limit k~ - cc and discard all diverging terms. After subtracting the divergences the free

energy density reads

f(ko, I, d; ~, is)=

d~~ (p(2a)~~ i~ d~ + (87r)~~k) In(1 k(kj~ + i~kj~)j (30)

The equation system equations (24-26) with f given by equation (30) defines the solution of

the problem to all orders in ~~~.

We start by solving analytically to first and second order in ~~~ Since ko/k~ and k/k~

are very small quantities we may expand the logarithmic term in equation (30) which then

becomes independent of k~ and equal to -(87r)~~ k(. Solving the simplified equation systemyields the well-known first order results for the structure factor [5],

k(m o, i~

m (8p)~~ ~~~ #) (31)

the crumpling factor C=

#sd m 1 + ci~~~ c2~~~ with [3],

the renormalization factor G m I gi~~~ + g~~~~ with [3],

gi =

-] ini(8»)-1/2 ~-i/21si (33)

and the free energy density [3]

f(~, Is) m (128»)~~ ~~~4i (34)

In second order, ak~ term with positive coefficient

k( m

~~~~#) (35)

128p

emerges in the structure factor which should be observable in systems with low bending stiffness

(~ m kT [10, 20] as a pronounced rounding or slight bump in the scattering structure factor

at low k.

The second order corrections to the swelling law and renormalization are

~~

~2 ~2~~~~~~~~ ~~~'~ ~~~~~~ ~~~~


~~~

~~~2

~/7r2

~~~~~~~~ ~~~'~ ~~~~~~' ~~~~

Both contain non-logarithmic terms whicl~ are directly related to film area conservation.

The free energy density up to second order in ~~~ reads

f(~Q, ~is) * b1 ~G ~~i~ ~2 ~G ~~i~ + ~3 ~G ~~i~ (~~)

with bi"

(128p)~~, b2"

3(10247rp)~~, b3 " (5127rp~)~~ The first term on the rhs does

not contain logarithmic renormalization terms which could originate from the non-linear partof the bending energy G and the crumpling factor C because these two contributions cancel

each other in second order. This indicates that Helfrich's results remain largely unaffected

even by second order terms and it might explain why Helfrich's high ~ model can in fact be

used for the interpretation of data taken in semi-rigid systems [20] this point has caused some

controversy in the literature [21]. The second term on the rhs of equation (38) is identical

to a term found by Golubovid and Lubensky [3] in their perturbation analysis and can be

rationalized as a non-local interaction term due to the surface area constraint. The last term

in equation (38) is proportional to if; Wennerstr6m and Olsson [22] have recently discussed

such terms although derived in a different theory from higher order elasticity terms in the

context of the lamellar to sponge transition. This term becomes significant at high surfactant

concentration.

However, the above approximations turn out to be unreliable at low bending rigidity. We

have therefore solved the equation system yquations (24-26) numerically. In a series of figures(1-3) we show numerical results for k( and k~ (Fig. I), the crumpling C, the renormalization G,and the free energy density f for realistic values of ~ and is. The swelling factor i~d in figure 2a

shows the typical logarithmic dependence on is which has been verified in experiment [10]for stiff film ~ =

5kT, but a systematic upward deviation for high dilution in the case of soft

membranes, ~ =1kT. This deviation should be measurable and characteristic for soft lamellar

phases. When comparing numerical and first order results we note significant differences in

the case ~ =1kT; this casts some doubt on the first order fitting procedure used in [10] to

estimate the value of the bending modulus in lamellar phases and we believe that the values

for the bending moduli (of the soft systems) reported there are underestimated by factors of

m 2-3. Indeed, this correction factor seems to reconcile the results of the measurements of ~

given in [10] with the results of alternative measurement techniques [23]. In figure 2b we show

the concentration dependence of the renormalization correction to the bending modulus. As

expected, higher anharmonic terms lead in the case of soft membranes to strong deviations from

the first order approximation. Finally, in figure 3 the free energy density as a function of the

bending modulus and the surfactant concentration is shown. At given is the steric repulsion

is always lower than predicted by first order approximation. For a realistic regime, is=

0.I,

lkT < ~ < 10kT, (Fig. 3a) we find that the approximation is valid down to some ~ m 5kT.

For softer systems the complex interplay of anharmonic corrections to the Hamiltonian and

the swelling corrections due to surface area conservation lead to deviations from the 1/~ force

law. However, as argued above, due to cancellation of renormalization and swelling terms up

to second order in ~~~ the scaling f cci( is practically unchanged even for small ~ =

1kT

(Fig. 3b).

Conclusions.

Finally, we want to discuss the shortcomings and merits of the presented approach. Monge

gauge cannot, as mentioned above, represent states with complex shape and topology


1-o 2.5

2- ~,,'

,","

i 5

o

o-z o.5

o-o o-o

0.0 o-Z 0.4 0.6 o-B I-o 0.00 0.05 o-lo 0.15 0.20 0.25

#~

a) b)

Fig. I. The coefficients k( and i~ in the scattering structure factor equation (21)as

functions of

the bending constant K and the surfactant concentration is. a) kl'10~us.

K~~ (solid line) and k~.10~

us.K~~ (dotted line) for is

=0.I. b) k( 2.5 x10~

us.#( (solid line) and i~ 10~

us.#( (dotted line)

for K =lkT (upper curves) and

K =5kT (lower curves).

1.6 1.0

o-B_,,,

---'"''"

.4

0.6'C ",

,',,"',

~"

, ,", 0 4 ,'', '

1-Z ",,,"

', ,', ,'

,0.2 ,'

,'"'-

---'

''--,,, ','

1-O 0.0 "

-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5

In(#~) ln(#~)a) b)

Fig. 2. a) The crumpling ratio G=

<ad of ensembles of undulating membranes forK =

lkT

(upper curves) andK =

5kT (lower curves)as a function of the surfactant concentration. Solid lines

denote accurate numerical solutions, and broken lines the respective first order approximations (Eq.(32)). The solid lines show a small deviation from the logarithmic law. b) The renormalization of

the bending constant G as afunction of surfactant concentration: numerical solutions (solid) and first

order appro~imations (broken, Eq. (33)), for theK =

lkT (lower curves) andK =

5kT (upper curves).


f ii o.7jf ii o.~j

o-B i-o

, ,

, ,

, ,

, ,

,' 0.8 ,'0.6 ,' ,'

,, ,

,,

,,

, ,

,' 0.6 ,', ,

0.4 ,",

, ,, ,

,,

,

,

,

,

o-Z

o-o o-o

o-o o-Z 0.4 0.6 o-B I-o 0.00 0.05 o-lo 0.15

K~~ ~~~

a) b)

Fig. 3. a) The free energy density f as afunction of the bending constant, at is

=0.I (solid

line); it deviates at lowK

visibly from Helfrich's law equation (34) (broken line). b) The free energydensity f

as afunction of the surfactant concentration for K =

lkT (upper curves) and K =5kT (lower

curves), where solid lines denote numerical solutions, and broken lines the corresponding Helfrich

approximation.

fluctuations. Therefore the Gaussian curvature f K dS which is coupled to the bending energyby the saddle-splay modulus k, does not enter the calculation. Inclusion of this term leads to

k dependent contributions to the structure factor and free energy density, f ccb(~, k) #(, and

is likely to be crucial for the still poorly understood lamellar to sponge transition [24]. This

requires a sophisticated state representation which includes topological defects. Work on this

important non-perturbative generalization will be presented elsewhere.

Nevertheless, the approach presented here is, within the validity of its assumptions, able to

provide a simple and consistent description of multilamellar phases in terms of structure factor,swelling law, renormalization of the bending constant and the steric force law as functions of

the surfactant concentration and the bending modulus. Its range of validity goes well beyondthat of low temperature theories [3, 5]. However, the emphasis of the presented treatment of

closed multilamellar systems is on principles rather than numbers. Changes in the constants

p or k~ affect although not strongly the numerical results without changing qualitativefeatures, thus somewhat restricting the predictive power of the model.

The results are in agreement with known observations, and reveal new features which are

related to the more accurate inclusion of layer crumpling, the constant area constraint and the

usually neglected coupling terms in the bending Hamiltonian. These should be observable in

the structure factor and swelling law of soft and dilute lamellar phases [9]. Our results also show

that Helfrich's first order steric force law is in fact also a good second order approximation,indicating that simple predictions of the Helfrich theory might be applicable even in semi-rigidregimes.


Acknowledgments.

It is a pleasure to acknowledge useful discussions with S. A. Safran, D. Roux, J. S. Huang,W. Helfrich, B. Ninham and R. Menes. P. P. acknowledges partial financial support from the

von Hoesslinschen Foundation of the City of Augsburg, Germany.

References

[Ii For reviews: Statistical Mechanics of Membranes and Surfaces; D. Nelson, S. Weinberg Eds.

(World Scientific, Singapore, 1989).

[2] David F., Leibler S., J. Phys. II IYancel (1991) 959; the authors discuss in detail the extensive

variables (real and projected film area) and their conjugate surface tension terms.

[3] Golubovid L., Lubensky T. C., Phys. Rev. B 39 (1989) 12110.

[4] The only exception known to the authors is [3] which gives a perturbative analysis up to second

order in K~~

[5] Helfrich W., Z. Naturforsch. 33a (1978) 305.

[6] Porte G., Delsanti M., Billard I., et al., J. Phys. II France1 (1991) l101.

[7] Yeomans J. M., Statistical Mechanics of Phase Transitions (Clarendon Press, Oxford, 1992).

[8] For a detailed review: Sornette D., Ostrowsky N., Micelles, Membranes, Microemulsions and

Monolayers; Gelbart W. M., Ben Shaul A., Roux D. Eds. (Springer, New York, 1993).

[9] Larche F. C., Appell J., Porte G., et al.,

Phys. Rev. Lett. 56 (1986) 1700.

[10] Roux D., Nallet F., Freyssingeas E., et al.,

Europhys. Lett. 17 (1992) 575.

[I Ii Helfrich W., J. Phys. IYance 46 (1985) 1263.

[12] Structure and Dynamics of Strongly Interacting Colloids and Supramolecular Aggregates in So-

lution; Chen S. H., Huang J. S., Tartaglia P. Eds. (Kluwer, Boston, 1992) pp. 265 and 237.

[13] Feynman R. P., Statistical Mechanics (Addison Wesley, Reading Mass., 1972).

[14] Safran S. A., Statistical Thermodynamics of Surfaces, Interfaces, and Membranes (Addison Wes-

ley, Reading Mass., 1994).

[15] Goldenfeld N., Lectures on Phase Transitions and the Renormalization Group (Addison Wesley,Reading Mass., 1992).

[16] Middleton D., An Introduction to Statistical Communication Theory (McGraw Hill, New York,1960).

ii?] Kleinert, H., Phys. Lett. 114A (1986) 263.

[18] Peliti L., Leibler S., Phys. Rev. Lett. 54 (1985) 1690.

[19] Note that is determines C((k~)) which in turn determines G((k~)).

[20] Safinya C. R., Roux D., Smith G. S., et al.,

Phys. Rev. Lett. 57 (1986) 2718.

[21] Gompper G., Schick M., Phase Transitions and Critical Phenomena, Domb C., Lebowitz J. Eds.,vol. 16 (Academic Press, London, 1994).

[22] Wennerstr6m H., Olsson U., Langmuir 9 (1993) 365.

[23] Roux D., private communication.

[24] Strey R., Jahn W., Porte G., Bassereau P., Langmuir 6 (1990) 1635, and references therein.

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