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HAL Id: jpa-00247642

https://hal.archives-ouvertes.fr/jpa-00247642

Submitted on 1 Jan 1992

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Elasticity and excitations of minimal crystals

R. Bruinsma

To cite this version:

R. Bruinsma. Elasticity and excitations of minimal crystals. Journal de Physique II, EDP Sciences, 1992, 2 (3), pp.425-451. �10.1051/jp2:1992142�. �jpa-00247642�

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Classification Physics Abstracts

61.30 68.10 82.70

Elasticity and excitations of minimal crystals

R. Bruinsma

Physics Department, University of Califomia, Los Angeles, U-S-A-, CA 90024 (Received 2 July 1991, revised 21 October I99I, accepted 22 November I99I)

Abstract. The elastic properties of the unusual crystals encountered in surfactant-rich solutions

are investigated. Triply-periodic minimal surfaces provide a convenient frame-work for the

understanding of such materials but, as is shown, degeneracy leads to vanishing elastic coefficients in the framework of the classical Helfrich energy. This degeneracy is lifted by higher-order

corrections and by finite temperature effects. We show that, as a result, thermodynamic stability

can be achieved at low levels of dilution but that with increasing dilution the P surface inevitably

melts. The degeneracy also leads to an unusual collective excitation spectrum which has a smectic- like undulation dispersion, except at very long wavelengths where it becomes sound-like. The

elastic moduli are found to have the same dependence on temperature and concentration as those of tethered stacked membranes and the shear moduJi are shown to have a temperature and

material independent ratio.

1. Introduction.

Crystals are constructed by the endless addition of the same building block : the unit-cell. In solid-state physics, the canonical unit cell is a near rigid structure made from a small number of atoms kept firmly in position by strong potentials. But, crystals are also encountered in

complex liquids such as oil/water/surfactant mixtures [II, block co-polymers [2] and in

biological systems [3], where the unit-cell is of a fundamentally different nature.

As an example, surfactant molecules dissolved in water can form a range of structures, such

as micelles, cylinders and stacked sheets, which are made out of surfactant bilayers [4]. In certain concentration ranges, X-ray diffraction demonstrates diffraction patterns consistent with cubic or diamond crystal structure. These crystals are believed to consist of a connected, triply-periodic biJayer extending across the sample [5]. The unit cells are very large, with lattice constants 100-200 h, containing Very large numbers of molecules which are allowed to diffuse inside the bilayer.

These unit cells do not reflect ordering at the atomic level. Instead, the origin of the crystal

structure is believed to be explained by the Helfrich bending-energy Hamiltonian [6] which

gives the energy cost of bending a membrane-like surface. If thermal fluctuation effects are not too strong, then the minimiZation of this Hamiltonian leads to the prediction that the location of the bilayer should be described as a « minimal » surface [7] : a surface of zero average curvature. Indeed, among the (large) catalog of minimal surfaces one finds triply periodic structures with cubic and diamond symmetry such as the simple cubic Schwartz P

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surface of figure I as well as many of the other crystal structures familiar from solid-state

physics [8]. Quantitative tests such as the relation between unit cell size and surfactant concentration [9] have provided further support for the « minimal-surface » model. It must be realized though that for unit cell sizes in the range 100-200 A and for film-thicknesses 6

= 20-30 h, significant corrections due to the finite membrane thickness must be present.

Fig. I. Unit cell of the P surface. The lattice constant is f(~ 200 h) and the membrane thickness is 6(~ 30 h). The triangles mark the flat-points.

If we nevertheless accept the « minimal-surface

» model, we could hope to calculate

thermodynamic response functions such as the elastic moduli using the Helfrich Hamiltonian.

This program immediately runs into difficulties. The P surface is only a, particularly symmetric, member of a continuous (two-parameter) family of minimal surfaces [10, iii- Numerical studies by Maggs and Leibler showed that, for instance, tetragonal deformation (see Fig. 2) does not violate the minimal surface constraint [12]. As a result, the crystal-

structure would drift over this family (baring external constraints). In particular, the elastic coefficients will be shown to be zero within the Helfrich Hamiltonian (Sect. 3) which is inconsistent with thermodynamic stability as well as with the requirement for long-range positional order in the crystal structure.

This ground-state degeneracy could be removed in three ways. The finite thickness of the membrane will produce, as mentioned, corrections to the Helfrich Hamiltonian [13, 14].

These higher-order terms need not respect the degeneracy of the Helfrich Hamiltonian and may stabilize the P surface, as already pointed out by Helfrich himself [15]. Next, as is well

c

Fig. 2. Tetragonally deformed P surface. For c/f less than a critical value somewhat greater than one, the surface remains minimal and is stable against collapse.

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known from magnetism, ground-state degeneracy can also be removed by thermal fluctua- tions. A P surface structure has, as we shall see, a complex spectrum of normal modes which

at non-zero temperature are thermally populated. The associated entropy may stabilize

certain crystal structures much like temperature tends to stabilize the BCC phase in solid-state

physics. A closer analog is perhaps found by noting that if we stretch or compress a polymer

we reduce its configuration space and thereby increase its free energy. This leads to elastic behaviour characterized by elastic coefficients which increase with temperature. If we think of

a membrane as a d

=

2 polymer then we could expect similar « rubber-elasticity » for the P surface. Finally, long-range interactions such as electrostatic and van der Waals interactions could also lift the degeneracy. However, in the presence of strong screening, we can neglect

electrostatic interaction while it is easy to see that attractive interactions like the van der Waals interaction would break the degeneracy in the wrong way : it would de-stabilize the P surface. We will neglect this last possibility in the following. A basic result of this paper is that the thermodynamic instability of the P surface is removed if we include the lowest order

correction term to the Helfrich Hamiltonian resulting from finite thickness effects. This correction term breaks the degeneracy of the groundstate of the Helfrich Hamiltonian in

exactly the right way so as to stabilize the P surface among the two-parameter family of minimal surfaces of which it is a member.

Another aim of this paper to investigate the elastic properties and the low-energy

excitations of the P surfaces. The low-energy spectrum controls the response to extemal

perturbations such as elastic deformations and changes in temperature (specific heat) as well

as the X-ray structure factor and the phase-diagram. In addition they are, as we shall see,

important to understand the thermodynamic stability of the P surface and the problem of the

melting of the P surface. The high-energy spectrum of membranes has been theoretically well

explored [16] but, except for the case of stacked membranes [17], the low-energy spectrum of

ordered structures is less well understood, although a global phase-diagram has been

proposed [18] (which included the P surface). It was also suggested [19] that the P surface upon melting could produce the isotropic L~ phase. Experiments on the P surface structures, such as small-angle X-ray structural studies [II and NMR [20], have centered on static

properties so there is little known about the low-energy spectrum.

The nature of the low-lying excitations of the P surface can be discussed in terms of the

solution of a quantum-mechanical problem involving a Schrbdinger-like equation defined on

a periodic surface imbedded in three-dimensional space. We find that over a large range of wave-vectors the low-lying modes are not sound-like but, instead, have an q~ undulation spectrum similar to that of smectic liquid crystals. At the lowest wave-vectors, this undulation spectrum transforms into a conventional 3-fold q~ acoustic-mode spectrum. The cross-over

wavevector is

qco =

(?/K f~)~'~ (l.1)

Here, K is the Helfrich bending energy, R K6~ with 6 (~ 30 h) the membrane thickness, and f (~ 100 hi is the unit cell size. The full spectrum is rich and complex (Fig. 3). Besides

the 3-fold undulation/sound spectrum, we can identify a 2-fold libration band of optical modes

and a I-fold « tight-binding » local-mode band. For wavevectors large compared to

I/f the usual single-membrane undulations [16] are of course still present.

The thermal properties of the P surface are discussed in section 4. Thermally excited

« collective

» modes as well as single-membrane short-wavelength fluctuations both lead to strong temperature dependence in the elastic moduli. The bulk-modulus B and shear-moduli l~i, 2 have a dependence on temperature T and lattice constant f given by

B

= a j R/f~ + a~ kB T/f~ (1.2)

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J~ B-Z-boundary

LI

Sound 'undulolion (ql (q~l

o q iIi

co q

Fig. 3. Collective mode dispersion (qualitative). The lowest 3 fold acoustic branch IL, T

,, T~) has a

sound-like dispersion for q S 6/f~ and an undulation dispersion for q a 6/f~. Next, there is a 2 fold libration (LI) band and finally a I fold local-mode (LM) band.

and

111,2 ~ ~ l,2(fl

I K~f~ fl2~B ~~f~) (~'~)

with ai, ~ and pi

~ constants. The constants Ci,

~ are geometrical invariants, Equation (1,2) thus predicts that (he

~ i/~~ ratio should be a temperature and material independent invariant.

As the lattice constant increases, the shear-moduli decrease. The maximum allowed lattice constant is

f~

~ (R/kB T)~'~ (l.4)

Since the surfactant concentration C~ is proportional to 6/f, equation (1.4) defines a

boundary in the phase-diagram such that for surfactant volume fractions C~s 6/f~, the P surface is unstable. In section 5, we conclude with a brief discussion of the experimental

consequences of the theory.

2. Stability and collective excitations of the P surface.

In this section, we will develop the classification of the collective excitations of the P surface

as a preliminary for a discussion of the elastic and thermal properties. We start with a bilayer of surfactants with a thickness 6 and a total area A, dissolved in water. If V is the sample

volume and if the membrane structure fills the sample homogeneously then the characteristic

spatial scale of the structure, f, must be of order VIA. The sample volume will be assumed to be homogeneously filled by the structure with a fixed amount of surfactant so f is fixed on average. If f » 6, direct intermembrane contact interactions should be weak. We also will

assume strong electrostatic screening so there are no strong long-range forces either. We can

then construct a phenomenological Hamiltonian by expanding the surface-energy in powers

of the membrane curvature. The result is [21]

~g

= ~g (1) ~ ~(2) ~ j2.1)

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with 3C~~~ the well-known Helfrich Hamiltonian (see Eq. (2.2)) and 3C~~~ the lowest-order correction term. As we shall see, this expansion is essentially in powers of (6/f). The Helfrich

Hamiltonian [6] is given by

3C~~~

=

d~« / (2 KH~(«

+ kK(«)) (2.2)

Here, «

= («1, «~) is a coordinate net spinning the membrane surface defined by Rim). The metric tensor of the surface g;jm ~~

~~ and g is its determinant. Next,

am; 3«j

Him is the mean curvature + at « while Ri and R~ are the principal radii of

2 R

i R~

curvature. Finally, K = I/Ri R~ is the Gaussian curvature. The bending energy K is material

dependent but known to be of order 10-100kBT. The Gaussian bending energy k is less well

known. We will only assume it to be positive and large compared to k~T. Because the

characteristic energy scale K is large compared to k~ T we will, in sections 2 and 3, discuss the

P surface in the T~0 limit. Later (Sect. 4) we will see that because of ground-state

degeneracy this view-point must be corrected. It should be recalled here that the membrane has a finite width, and the surface referred to above is, as usual, only a mathematical surface running through the middle of the membrane. Also, bending energies are treated as

phenomenological parameters which must provided by experiment or by a detailed,

microscopic calculation.

The second term in 3C~~~is a topological invariant if we assume the membrane surface to be closed. Then, according to the Gauss-Bonnet theorem,

ld~«/K(«) =4ar(1-G) (2.3)

with G the genus. For the P surface, G

=

3 per unit cell. We will always keep the number of unit cells N in the sample volume fixed so the second term of 3C~' will not enter in a discussion of thermodynamic stability. In particular, we will not discuss the difference in free energy

between the P surface and topologically different surfaces.

The next order correction JC~~~ can be found [13-15] either by expanding in higher-order

corrections in the curvature or by constructing explicit models of the bilayer :

3C~~~

=

4 d~« / K~(«)

+ ?~ j d~

« / K(«))~/A (2.4)

where R is expected to be of order K6~. Since K is of order I/f~, the small parameter of the expansion in equation (2.I) is (6/f)~. We thus must

assume (6/f) to be small compared to

one, which restricts us to dilute membrane structures. Note that the second term of

3C~~~ is, once again, a topological invariant. It could be questioned why no terms of the form H~ or (VH)~

were included in equation (2.4). The reason is that for KM k~T and

K » R/f( H

= 0 (see below) and such terms would be of higher order. Note that a term of the forrn of H~~~ is also required because otherwise the genus of the surface for which H is

minimized would diverge.

As long as 3C~~~ « 3C~~ and T

~ 0, we are allowed to minimize 3C by setting H

= 0, I-e-, by having Rim trace out a minimal surface. As mentioned, there are many triply-periodic

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minimal surfaces. We will assume the minimal surface to be a Schwartz P surface of lattice

constant f (Fig. I). The P surface unit cell has six tubes, of Gaussian curvature

l/f~, connecting a unit cell to its neighbors and eight « flat-points » where the Gaussian

curvature vanishes. Note that the length-scale f of the unit cell is not determined by the

minimization of 3C~~~ Instead, f is « set » by the extemally controlled surfactant concentration

as mentioned. This situation is in striking contrast with solid-state physics where the unit cell size is controlled by a minimum in the interatomic pair-potential. This indeterminacy in unit cell size is already suggestive of instability and it will play an important role in section 3. For very stiff membranes with K » kB T the characteristic length f is actually controlled by the inter-membrane potentials (such as hydration and van der Waals forces). We will always be

assuming that thermal fluctuations are strong enough to prevent this type of binding so

f is indeed concentration controlled.

For the experimentally realized cubic phases, the unit cell size f is actually not all that large compared to the membrane thickness 6, and the corrections to the Helfrich Hamiltonian

could be quite substantial. Nevertheless, if 3C~~~ would be comparable in magnitude to

3C~~~, then it would be hard to see why, experimentally, the minimal surface assumption works

so well in predicting the area per unit cell [9]. We thus assume that 3C~~~ is a good « zero-

order » Hamiltonian and include the largest finite thickness term 3C~~~ as a perturbation. We shall see however that without this « correction terra », the P surface is not stable and its

inclusion is essential.

2.I BLOCH STATES AND STABILITY.- Having reviewed the conventional argument why

P surfaces are encountered in surfactant systems, we now include 3C~~~ perturbatively and

investigate the excitation spectrum as well as the stability of the P surface in the presence of 3C~~~. Assume we allow small but arbitrary deformations of R(«), with R(« ) the unperturbed

P surface. It suffices to allow only displacements along the local normal h(«) :

Rim ~ Rim ) + v dim 12.5)

with v « f. This deformation increases 3C~~~ by an amount A3C

=

d~« If (Av 2 Kim ) v)~ (2.6)

2

with A the covariant Laplacian. In deriving equation (2.6), the condition Him

=

0 for

v = 0 must be used. The perturbation itself, 3C~~~, is given by Jc(2)

=

d2« vi 4 (K2(«) K(«) Kj(«)

v~ v>v + o(v2)) (2,7)

where K(( « ) is the extrinsic curvature tensor [25] of the P surface and where V, is a covariant derivative. Equation (2.7) is again only valid for minimal surfaces.

The quadratic term in equation (2.7) is in general small compared to A.1C, provided

K » 4/f( except

near degeneracies (see Sect. 3), but the term linear in v in JC~~~ is always important because it means that the Hamiltonian is minimized by a non-zero value Rim for v(«). For any non-zero R, the surface is thus no longer a true minimal surface.

We start by expanding vim) in a complete orthonormal set of eigenfunctions of the

eigenvalue equation :

l~i,

~ ? Kjw) v

= ii- (2.8)

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Since exact solutions of equation (2.8) are difficult to obtain, we will use variational methods to estimate A. Define the « Schrbdinger Hamiltonian » JC~:

JC~=-A+2K(«) (2.9)

whose expectation value for a (normalized) v («) is :

(3Cs)

"

d~" /

v *(") 3Cs v (" (2.10)

In terms of 3C~, equation (2.8) now reads 3C~v = iv- For any function v(«)

(3C~) m io (2. ii

where i~ is the lowest eigenvalue of 3C~. The proof is identical to that for quantum systems.

As an example of the usage of equation (2. ii ), if we equate v (« ) to a (normalized) constant,

v (« ) = A ~~( then we find

A~ < 2 (K) (2. 12)

with (K) the average Gaussian curvature of the P surface. From the Gauss-Bonet theorem, equation (2.3), applied to a periodic crystal it follows that

(K)

= 8 ar(N/A (2.13)

neglecting finite thickness effects. For the P surface A/N=«f~ with « =2.3 (so K is of order I/f~. Using this in equations (2.12) and (2.13) gives

A~ < 21.8/f~. (2.14)

The Schrbdinger equation thus contains at least one eigenstate with a negative eigenvalue.

Since I/f~ is the

« natural » scale for eigenvalues of the Schrbdinger equation, we conclude that the ground-state of 3C~ must actually be a deep-lying bound-state.

To find the relation between the eigenvalues of equation (2.8) and the excitation energies

of the P surface, let v~ («) be an eigenfunction with normalization :

ld~« / v~ vi, = 6~ ~,. (2.15)

Expanding

v (« ) = f jj a~ v~ («) (2.16)

1

(so aA has units of length) and inserting equation (2.16) in equation (2.6) gives

A3C=(jj (aA(~WA (2.17)

where w~

=

Kf~A~. A3C is diagonalized by the eigenstates of the Schr6dinger equation with

WA as the mode energy. We can consider w~ as the excitation spectrum of the P surface

because to lowest order in R/Kf~, JC~~~ only produces the earlier-mentioned «static»

deformation P of the P surface and does not alter the spectrum.

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