Measures of integration in calculating the effective rigidity of fluid surfaces

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Measures of integration in calculating the effective rigidity of fluid surfaces

W. Helfrich

To cite this version:

W. Helfrich. Measures of integration in calculating the effective rigidity of fluid surfaces. Journal de

Physique, 1987, 48 (2), pp.285-289. �10.1051/jphys:01987004802028500�. �jpa-00210441�

Measures of integration ⁱⁿ calculating the effective rigidity ^{of fluid}

surfaces

W. Helfrich

Institut für Theoretische Physik, ^Freie Universität Berlin, Arnimallee 14, ^{D-1000 Berlin} 33, F.R.G.

(Reçu ^{le 3} juin 1986, accept,6 le 6 aotit 1986)

Résumé.

La rigidité à la flexion de surfaces fluides est abaissée par des fluctuations hors du plan ; ^{les auteurs} qui

ont calculé cet effet trouvent des expressions qui diffèrent par ^un facteur numérique (1 ou 3). Nous comparons les

mesures de l’intégration (ou ^les représentations ^en modes) ^{dans le cas d’une} tige flexible, ^{nous montrons} pourquoi ^les

mesures de déplacement à ^{une dimension ne sont} pas adéquates, ^{et nous} examinons la mesure de la courbure pour les surfaces. Nous redérivons notre résultat antérieur (facteur numérique 1) par ^un couplage géométrique de modes de fluctuation de la courbure qui ^sont énergétiquement découplés.

Abstract.

Formulae differing by ^a numerical factor (1 ^vs. 3) ^have recently ^{been obtained} by several authors for the decrease of the bending rigidity ^of fluid surfaces caused by out-of-plane fluctuations. We compare the ^measures of

integration (or ^the representations ^{of the} modes) ^{for the} example of the flexible rod, ^show why one-dimensional

displacement measures are inadequate, and examine the curvature measure for surfaces. Our previous ^result (factor 1) ^is rederived in terms of coupling by geometry ^of energetically decoupled ^curvature fluctuation modes.

Classification Physics ^Abstracts

68.10E

87.20C

1. Introduction.

In the last two years ^a number of authors [1, 5] ^have

calculated the softening ^of fluid membranes and inter- faces caused by out-of-plane fluctuations. The effective

bending rigidity ^{K’ is} predicted ^to obey

where K is the bare (or local) bending rigidity, kB Boltzmann’s constant and T temperature. ^{The cutoff}

wave vectors qm;n ^and q max ^are governed by ^the

dimensions of the surface and of the constituent

molecules, respectively. The numerical factor a has been found to be unity by ^the present ^author [1] ^and

three by the others [2-5].

All the theories start from the bending elastic energy per unit ^area

where cl and c2 ^are the principal curvatures, co ^the spontaneous ^{curvature which we assume here to} vanish, and K the elastic modulus of Gaussian curvature cl c2, ^The integral of g ^over the surface area represents the total energy ^{of the} system. ^{We do not consider here} the complicated ^{case of} nonvanishing ^{surface ten-}

sion [2, 3]. ^{Central to} all calculations is a coupling ^of

fluctuation modes with each other or with a stationary

deformation of the surface. It is treated to first order either by elementary ^means [1, 3] ^{or in the} language ^of

field theory and renormalization [2-5]. (In ^one example

both methods are applied ^and give ^{the same result} [3].)

The basic disagreement among authors lies, briefly, ⁱⁿ

the « measure ». We mean the measure of integration

used to set up partition ^{functions and the measure or} representation ^{used to} define fluctuation modes and calculate the coupling between them.

The measure of most authors is displacement, ^either

normal to a flat base [2, 3] ^{or normal to a} curved base

representing the surface without short-wavelength ^fluc-

tuations [4, 5]. ^{Like any measure} referring ^{to a fixed} base, ^{the two} one-dimensional displacements ^omit

lateral motion. Such motion is inevitable as the real surface area over a given piece ^{of base varies} during out-of-plane fluctuations. In fact, ^the bending ^{energy is} generally calculated for the real area and on the

assumption that the surface is unstretchable, ^{i.e. of} fixed density. ^The neglect of lateral motion is particu- larly dangerous ^if displacement ^{itself is the measure.}

Our elementary calculation, ⁱⁿ its first version [1a], employs director fluctuation modes. Actually, they ^are

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01987004802028500

286

curvature modes as they satisfy ^{the stated} requirement

of being additive in the elastic strain which is curvature.

(At ^one point ^we wrongly ^invoke displacement ^{to deal}

with entropy. This is corrected in the Note added in

proof). ^{The curvature modes are} defined with respect

to a flat base area and are thought ^to couple through

the increase in real area over base area which they produce. ^In the second version [lb], concerning ^a spherical surface, ^{we start with} displacement ^modes.

The curvature measure is invoked to derive the changes

of mode entropy ^{due to} spherical ^curvature.

The purpose of the present ^note is twofold. Firstly,

the curvature measure is to be derived from the full

positional ^{measure for the} simple ^model system ^{of an} unstretchable rod, partly represented by ^{a stiff} polymer

chain. The rod, if restricted to a plane, ^{is in some}

respects equivalent ^{to a} membrane with curvatures in

one direction only. The model is of course not novel, but prepares for the ^use of the curvature measure in connection with surfaces. Also, it is used to indicate the failure of the one-dimensional displacement ^measures.

Secondly, ^{we wish to} deal with the curvature measure

in more detail and rigor than before. The curvature fluctuation modes of an originally flat surface are

described such that they ^are decoupled like those of the rod. The previous value of the effective rigidity (a =1 ) is obtained solely from the deformed geometry ^{of the} fluctuating ^{surface. It is} hoped ^{that the} exposition settles the controversy ^{about the measure.}

2. The rod in a plane.

An instructive, discrete model for the rod is the stiff

polymer. ^It permits ^{us to write down} precise partition

functions in terms of mass points. ^{For the sake of} simplicity, ^the monomers are taken to be single ^mass points, ^thus having ^{no moments} of inertia. We imagine

a sequence of ^N + 1 equal ^{masses. Their} position

vectors are

r, ( j

^0,..., N ) , ^with r. being ^{fixed. The}

bond vectors are a, = r, - rj - 1 ( j =1, ..., N ) . ^We adopt ^{a vector} ao ending ⁱⁿ ro ^{to fix the direction of the}

nearly straight ^chain. Bending ^vectors may be defined

as kj = (aj , Iaj _, 1 - ajlaj) a ( j = o, ..., N -1 )

where a is the time-averaged spacing ^of subsequent

mass points. ^{The vector} kj is the discrete equivalent ^of

the curvature vector of differential geometry ^{if a is} constant, ^{as we} will stipulate immediately. ^Cohesion

and stiffness of the chain _may be enforced by ^a potential energy of the type

The complete partition function of classical statistical mechanics then takes the form

where f3 =1/kB T. ^{The thermal de} Broglie wavelength

A of the mass points ^{is used to} express the volume of the

unit cell in configurational space, À 3N. The factor will be dropped ^{in the} following. By adding ^{the mass} points

one after the other it is easy ^{to see} that the position

vectors can be replaced by ^{the bond vectors as inte-} gration ^{variables so that} Zr ^becomes

The fluctuations of the bond lengths ^{are assumed to be}

very small. We integrate ^{them out and} put aj ^{= a}

constant. This leads to modifications of Zr ^and Za : ^The potentials u ( aj) ^{drop out and 8j is} restricted to the surface of a sphere ^{of radius a. An} interesting ^third

form of the partition function,

can now be obtained by substituting a2 dkj - 1 ^{for da,}

and omitting ^{constant factors.} Naturally, k, ^{is con-}

strained through ^{the fixed} lengths ^of _aj and a; + ^{1 so that}

its end point ^{describes a} sphere.

The various constraints imposed by ^{constant bond} length (or ^line density) ^{are difficult to handle in three-}

dimensional space. ^The situation is simplified ^{for the} polymer ^{in a} plane. ^Its partition ^{function may be}

written as

where k, ^{can be counted positive or} negative depending

on whether the polymer ^{turns left or} right. Equation (7) is, ^of course, ^an approximation ^{valid for stiff} enough chains, ^i.e. (k?) ..,:a -2 ^{(An exact form for more}

flexible chains could be obtained by using ^bond angles.)

A continuum version of (7) ^is

This is just ^the partition function of the unstretchable rod in a plane. ^{A finite set of} integration ^variables

kj ^{has now been} replaced by ^a continuum of variables k

k ( s ), s ^{being the arc length and L the total length}

of the rod. The symbol 0 marks the measure of

integration, ^{k. The two} partition functions, discrete and

continuous, ^are practically equivalent ^if

E being ^the bending rigidity of the rod.

The advantages ^{of the curvature measure in} writing

the partition function for the rod in a plane ^{are obvious.}

The constraint of no stretching ^is incorporated ^auto- matically, leaving k(s) ^{to be an} arbitrary ^function,

while the original ^function r (s) describing ^{the rod’s}

shape is under the constraint I drlds I ^{= 1. This is a}

formal argument against ^{the use of} displacement

measures. Moreover, ^curvature is identical to the elastic strain. If Hooke’s law is valid, the fluctuation modes of k(s) ^{as obtained} by the usual Fourier

expansion are.completely decoupled energetically. ^The

modes can therefore be treated separately. ^The only coupling that may remain between them is through ^the

deformed geometry ^{of the} fluctuating ^rod.

The simple example of the unstretchable rod in a

plane may ^{serve to} recognize ^{the most} conspicuous shortcomings of the one-dimensional displacement

mesures. We consider first displacement ^{normal to a}

base line [2, 3]. ^{The rod is} restricted to the xz plane ^and

taken to coincide with the x axis in the absence of fluctuations. The small distances II z ^{of the} fluctuating

rod from this base are expressed by ^u

^u (x ) . ^{Let us} imagine ^a short-wavelength mode ul being superim- posed ^{on a} region ^of practically ^uniform slope du2/dx

associated with a long-wavelength mode U2’ ^{If the}

amplitude ^{of the former} is very small, ^material _transport along ^the sloping ^line may indeed be ignored ^{as it varies}

with the square of the amplitude. However, ^{one has to} take into account that the material displacement ^as-

sociated with the new mode will be orthogonal ^{to the} sloping ^line. Therefore, ^the physical displacement ^is

smaller by the factor [1 ⁺ ( du2/dx ) 2] -1/2 ^{than that}

measured in u. Being proportional ^{to the} square ^of the

slope, ^the correction couples the modes in the lowest order possible, ^{so it cannot be} disregarded. ^{One could} hope ^{to avoid the failure} by choosing as measure the

displacement f ^{normal to a curved base line} [4, 5].

Vnfortunately, ^{even a single} fluctuation mode then ihV-61ves motions along ^the base which are proportional

to its amplitude. ^This is because the half-waves of, say,

f = f a sin ( qs ) ^{on the convex side} require ^more

material than those on the concave side of the curved base. The corrections of the displacement ^measure,

which also depend ^{on sin} (qs), produce ^a nonnegli- gible coupling of lowest order.

3. Curvature measure for surfaces.

It seems clear now that one should not use one-

dimensional displacement ^{measures to} deal with the

(possible) coupling ^of bending fluctuations either of rods or of fluid surfaces. How can one extend the curvature measure to unstretchable fluid surfaces ? The

only linear invariant to be formed from the two- dimensional tensor of surface curvature is its trace,

c = cl ⁺ C2 [6]. ^We adopt ^it as measure of integration.

One can in general ^{neither choose} c, ^and c2 separately

nor control the orientation of the principal ^{axes. A}

visual way of realizing ^{this is to build the surface} by placing ^mass points ^{one after the other} along ^an edge ^of

the surface. Evidently, ^the height ^{of a new mass} point

relative to some neighbours ^{that are} already ^{there is the} only ^available degree ^{of freedom} apart ^from fluidity

and determines the local curvature c. It is not possible

to propose for fluid surfaces ^a molecular model of similar precision ^{as that for the} polymer chain, though

lattice models _{may be} helpful ^{to some extent.}

We want to transform ^a flat and square piece ^of

surface coinciding ^{with the} xy plane ^{into a} slightly

undulated surface in roughly ^{the same location. Denot-} ing ^the positions ^on the flat surface by ^x

6, y

^{q, we} specify everywhere ^{on it the curvature c} ( ^{§ ,} q ^which

a surface element is to assume and look for a mapping r ( 6, q ) which also conserves local density ^{and is}

invertible. The three demands _can, in principle, ^be simultaneously ^{satisfied as the} fundamental theorem of surfaces allows for exactly ^three degrees of freedom in surface mapping [6]. ^The conservation of local density

may require the material to shear in its plane, ^{which is}

without consequences in ^a fluid. A one-to-one mapping

can be achieved, e.g., by stipulating that the material transport asociated with the transformation is irro- tational in the _xy plane. ^Other prescriptions ^{for the}

local rotations are equally acceptable. However, ^one-

to-one mapping itself is a prerequisite ^on physical grounds, since different parts of the fluid are indisting-

uishable which means that a particular shape ^{of the}

surface must not be counted more than once.

The three conditions imposed ^{on r} ( §, 11) may be summarized by ^the equations

where the subscripts designate derivatives. A Fourier

expansion ^of c ( 6, q ) defines surface fluctuation modes that are energetically decoupled ^like those of the rod in a plane. Accordingly, ^{the modes can be treated} separately and there is no need for the partition

function formalism. The decoupling may ^not be perfect

if the usual periodic boundary conditions are to be satisfied for r as well as for _c, here with an adjustable

square frame ^{to conserve total} membrane area. We

expect ^the effects of the boundaries to be negligible ^for

stiff and large enough ^surfaces. (It ^is certainly possible

to map any undulated surface back onto the flat square, but this _may involve the curvature mode of zero wave

vector.) ^The only important ^mode-mode coupling ^that

remains is by geometry, ^i.e. by ^the deformation of the surface. It will lead us to rescale lengths ^{and to}

renormalize directors.

To begin ^{with, the} ripples representing the fluctua- tions contract the adjustable square frame. ^The ratio of fixed real area A to variable base area A’ is, ^{to first} order, given by

1 -1

---

where the director n obeys

288

and is defined such that nz > 0. Equation (11) ^{has been}

derived many times and in different _ways [7]. ^{For a}

derivation in terms of curvature we may ^use the Fourier

expansion

where _p

( § , 7y) . ^{Also needed are the} equipartition

theorem

the director amplitude

the first-order ansatz

and periodic boundary conditions. In the last two

equations ^we ignore ^the difference between a wave

vector on the flat base and its counterpart ^{on the}

deformed surface. Evaluating ^{the sum in} (16) by ^means

of the usual integral ^and considering the cutoff wave vectors result in (11).

The consequences of rippling ^{for curvature} elasticity

may be conveniently ^{studied for a mode of minimum}

wave vector A with A = 2 ’IT / A - 112. ^{If no} other modes

are excited equations (14) ^and (15) prescribe :

where the minute base contraction due to the A mode is

disregarded. ^{This has to be} compared ^with the situation where all the other modes are released. Marking quantities that refer to the rippled ^surface produced by

the other modes with a prime ^and using [8] ^the generally ^valid relationship ^c _{= nx, x + ny, y,} ^{we have}

c =A.n =A’.n’ (18)

The wave vector and, ⁱⁿ compensation, the director

amplitude ^are rescaled in the last form which is a first- order approximation. Accordingly, (17) ^{may be re-}

written as

a L 1

The director has to be averaged ^{over the} ripples (except

the A mode) ^and subsequently brought ^{back to} unity.

The renormalized director amplitude of the A mode is

Inserting tA’ and A’ from (11) ⁱⁿ (19), ^{one arrives at}

, -

where

Comparison ^with (19) ^{shows K’ to be the effective} rigidity ^felt by ^{the A} mode. The result confirms

equation (1) ^{and, in} particular, ^{a = 1. The mean-}

square displacement uA’ ^{in z} direction associated with

tA’ ^is readily ^{seen to} obey

with the same effective rigidity.

If base area rather than real area is held constant and if wave vectors always ^{refer to the} base, ^{as in a similar}

calculation of the effective rigidity [la], ^{there is a} coupling ^{between modes} through ^{the increase in real}

area which appears ^to be energetic. Shrinking ^the

fluctuating ^{membrane, i.e.} rescaling ^{it in all} three space dimensions, ^{until its real area reaches the} size without fluctuations, leads back to coupling by geometry. (Note

that the bending ^elastic energy is scale invariant.) ^The previous derivation avoids the problems ^of periodic boundary conditions, ^{but is based on} poorly ^defined

modes. However, ^the differences between the two

methods, including ^small changes ^{in the cutoff} qmax’ do

not alter the effective rigidity in these first-order calculations.

The picture ^of coupling by geometry ^{cannot be} maintained in the _presence of a stationary ^curvature.

We may think, e.g. of ^a closed sphere ^{or a surface} kept

between concentric cylinders ^of sufficient spacing ^to permit out-of-plane fluctuations. As the fluctuations reduce the effective rigidity they ^will be enhanced by

the externally imposed curvature. The decrement of the rigidity ^{can now be} deduced from the increase of mode entropies ^{due to} the enhancement. We have shown for the sphere [1b] (and checked for the

cylinder) ^{that this method} gives ^{the same effective} rigidity ^as coupling by geometry.

The notion of curvature modes interacting ^{with an} externally imposed ^curvature may be used to disprove

Kleinert’s [5] ^{claim that} out-of-plane fluctuations also affect the modulus of Gaussian curvature, ^K. According

to the energy density (1) ^{there is no} interaction between pure saddle curvature, c2

cl, and ^{a curva-} ture fluctuation mode. If the presence of the saddle does not change ^the mean-square amplitudes ^{of the}

modes there can be no effect on K. For the total free energy of the fluctuating spherical surface Kleinert [5]

obtains the same result from calculating ^{rc’ and K’ in}

the displacement measure as we do from deriving

K’ alone in the curvature measure. The coincidence could be related to the fact that our calculation of the effective rigidity ^{of the} sphere [1b] ^is, formally, ^insensi-

tive to whether curvature or displacement ^is regarded

as measure.

4. Conclusion.

The present analysis confirms the idea that the natural

measure in dealing ^with fluctuations is the elastic strain.

The representation by ^{strain is} expected ^to give ^de- coupled fluctuation modes whenever Hooke’s law is valid. Incidentally, ^{it also} implies that the mean-square

amplitudes ^{of the modes are all} equal, ^i.e. independent

of wave vector. It is surprising ^{that even a} system ^as complex ^{as a} fluid surface fluctuating in three-dimen- sional space appears ^to obey these rules.

The curvature measure for surfaces _may eventually

need a better molecular foundation than that sketched above. Also, the residual mode-mode coupling ^caused by periodic boundary conditions _may be worth closer examination. However, ^{there can be} little doubt that

displacement ^{measures are unsuited for} studying ^the possible coupling ^between out-of-plane fluctuation modes. Actually, they may simulate ^a coupling ^where

none exists.

The situation is different if a fluctuating ^{fluid mem-}

brane is under lateral stress or subjected ^{to a} restoring

force that tends to keep ^{it in a} plane. Curvature may still be a good ^{measure, but now the} fluctuation modes

are coupled ^{even in this} representation.

Acknowledgment.

I am grateful ^to Dr. D. Forster for _numerous, mostly

controversial discussions.

Note added in proof: Papers ^{known to me} only ^as preprints ^{at the time of} writing reconsider the softening

of membranes [9] and introduce a concept ^{similar to} effective rigidity ^{into the} theory ^of strings [10]. ^Newer

papers deal with strings [11] ^{and membranes} [12, 13].

All authors find a

3 except ^{one who obtains}

a = 1 [13]. ^{See also a recent review} [14].

References

[1a] HELFRICH, W., ^J. Physique ⁴⁶ (1985) ^1263.

[1b] HELFRICH, W., ^J. Physique ⁴⁷ (1986) ^321.

[2] PELITI, ^{L. and} LEIBLER, S., Phys. Rev. Lett. 54

(1985) ^1960.

[3] SORNETTE, D., ^doctoral thesis, Nice, ^France (1985).

[4] FÖRSTER, D., Phys. Lett. 114A (1986) ^115.

[5] KLEINERT, H., Phys. Lett. 114A (1986) ^263.

[6] See textbooks on differential geometry.

[7] HELFRICH, W., ^Z. Naturforsch. ^30c (1975) 841 ; BROCHARD, F., ^DE GENNES, ^{P. G. and} PFEUTY, P.,

J. Physique ³⁷ (1976) 1099 ;

HELFRICH, ^{W. and} SERVUSS, ^R. M., Nuovo Cimento D 3 (1984) ^137.

[8] See, e.g., reference [1a].

[9] KLEINERT, H., Phys. ^{Lett. A 116} (1986) ^57.

[10] POLYAKOV, ^A. M., ^Nuclear Phys. ^{B 268} (1986) ^406.

[11] KLEINERT, H., Phys. Lett. B 174 (1986) ^335.

[12] DAVID, F., Europhys. ^{Lett. 2} (1986) ^577.

[13] FORSTER, D., Europhys. ^{Lett., submitted.}

[14] PELITI, L., Physica ^140A (1986) in press.