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
766 JOURNAL DE PHYSIQUE II N°5
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)
~~°
N°5 VARIATIONAL THEORY OF UNDULATING MULTILAYER SYSTEMS 767
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
768 JOURNAL DE PHYSIQUE II N°5
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~~~~~~~~ ~~~'~ ~~~~~~ ~~~~
N°5 VARIATIONAL THEORY OF UNDULATING MULTILAYER SYSTEMS 769
~~~
~~~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
770 JOURNAL DE PHYSIQUE II N°5
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).
N°5 VARIATIONAL THEORY OF UNDULATING MULTILAYER SYSTEMS 771
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.
772 JOURNAL DE PHYSIQUE II N°5
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.
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[5] Helfrich W., Z. Naturforsch. 33a (1978) 305.
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Phys. Rev. Lett. 57 (1986) 2718.
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