Membrane flexibility in a dilute lamellar phase : a multinuclear magnetic resonance study

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Membrane flexibility in a dilute lamellar phase : a multinuclear magnetic resonance study

Bertil Halle, Per-Ola Quist

To cite this version:

Bertil Halle, Per-Ola Quist. Membrane flexibility in a dilute lamellar phase : a multinuclear magnetic resonance study. Journal de Physique II, EDP Sciences, 1994, 4 (10), pp.1823-1842.

�10.1051/jp2:1994235�. �jpa-00248081�

J. P/i,I,,I ^{II Fian(e 4} II 994j 1823-1842 _OCTOBER 1994, PAGE 1823

Cla,,ification Pfi_{v-I-i (..;} Al) _iii

a( t,I

33.25 61.3() 68.10

Membrane flexibility in

_a

dilute lamellar phase

_{: a}

multinuclear

magnetic

_resonance

study

Bertil Halle and Per-Ola

Quist

Condensed Matter Magnetic Resonance Group, ^Chemical Center. Lund

University,

P-O- Box 124, S-22100 Lund, Sweden

(Ret en.(,(/ lo.lanum1> /994, ieiiie(/ 2/.fiui(> /994, a<.<.eptec/ I-I,rim(, /994)

Abstract. We demon~trate that the

bending

rigidity. _«, of amphiphilic bilayers _can be

accurately ^determined from the quadrupole

~plitting

in the NMR ~pectrum. ^We ~tudy the lamellar phaw in the ,y~tem water/SDS/pentanol/dodecane at

layer

volume fraction, in the range 0. I-I.

mea,uring

the quadrupole

,plittings

of _water

(~H),

counterions (~~Na), and «-deuteriated SDS l~H_). The ,plitting~ _are governed by the

layer

director fluctuation

(n [ ),

which _we calculate ,elf- con~i~tently

(allowing

for the finite thickne,; of the

bilayer)

from the _continuum elmticity theory for _a ~terically ~tabilized lamellar pha,e. Analy;i~ of the data yield; the _same re;ult.

« = 2.2 ₌ 0.2

lB

T, for all three nuclei. Since

(n) )

_i, determined by

;hort-wavefength layer

tluctuation mode;, thi; value ;houfd be clo,e to the bare (intrinsic) rigidity ^of the bilayer. In

contra;t, the

crumpling

renormafized rigidity previou;fy deduced from

light ,cattering

studio; of'

the

layer

undulation dynamic; in the ~ame,y~tem ^is con,iderably ~maller. our _« value is al,o much larger than the value recently ^{deduced from} the dilution behaviour of the X-ray Bragg ,pacing. We argue that thi~ di,crepancy I; due to _an incon;i~tency in the data rather than to a

deficiency in the theory.

1. Introduction.

The

physics

of

fluctuating

^{surfaces is}

currently

an area of intense theoretical and

experimental activity [1, 2].

Much of this work has focu~ed _on dilute

lyotropic

smectics> I,e., lamellar

phases

that _can be swollen _to very

large layer spacing; by

addition of solvent

(oil

or water)

[3

_].

Scattering

studies

[4-8]

of the Landau-Peierls

power-law decay

of the structure factor

[9]

around the

Bragg singularity

have established that dilute lamellar

phases

with electroneutral

layers

(or with

charged layers

in concentrated

electrolyte)

are stabilized

by

^steric

repulsion

between the

tluctuating layers,

with _an interaction energy

(per

unit

layer

area)

[10]

v(fi=j~~~~~~

^'

~, (i.ii

with 3 the

layer

thickness, f _the

(fluctuating) layer period

(the

separation

between the

layer midplanes),

and _K the

bending rigidity

of the

layers.

In _a

sterically

stabilized lamellar

phase, layer

fluctuations _are

entirely

controlled

by

the

bending rigidity. Experimental

mea~urements of this parameter are therefore crucial for the

understanding

of these systems. ^{While the}

power-law

_exponent in the _structure factor does _not

depend

on K, several other

experimental approaches

have been used _to determine

K.

Perhaps

the

simplest approach

^is to determine (from the

position

of the

Bragg peak)

the

average

layer period

d in _a dilution

experiment

_K can then be deduced from the nonideal

swelling

behaviour associated with

layer crumpling I1-13],

Another

approach

is _to

study

the

dynamics

of

layer displacement

and concentration tluctuations

by dynamic light scattering,

from which _{K can} be deduced with the aid of _a

hydrodynamic

model [14,

15].

The

bending

rigidity

has also been e~timated from electron

spin

resonance

lineshapes

of

spin-labeled

surfactants

[16(.

While the

existing experimental

data _on the

bending rigidity

of

sterically

stabilized lamellar

pha~es

are in essential accord, with _K values of order k~ T, ^there are

important quantitative di~crepancies [13].

These may reflect

experimental

limitations, inaccurate theoretical mode-

ling,

or fluctuation-induced

layer softening II 7-20]

_at

long wavelengths

(as

probed by light scattering).

In view of the central role

played by

the

bending rigidity

in

determining

the

pha~e

behaviour and tluctuations of dilute lamellar

phases,

new

experimental approaches

to this

problem

~hould be of value.

~

In the present

study

we have thus used the

spectral

line

~plitting

of

quadrupolar

nuclei in surfactants, counterions, and water molecules to

accurately

determine _K, The

quadrupolar splitting

method has ~everal

important advantages

the

experiments

are

;imple,

fa~t, and

accurate, and do not

require alignment

of the lamellar

pha;e,

Furthermore, most ;y,tem~

contain several nuclear

specie~

that _{can serve as} intrin~ic

probes

of

layer rigidity.

In section 3 _we derive the theoretical results needed _to obtain _K from the evolution of the

quadrupole splitting

with dilution, Since the literature contains _a

variety

oi results for the mean-square

layer

director fluctuation

(n( ),

_we rederive this

quantity

without the usual restriction to small volui~e iractions. We also demonstrate, in

Appendix

A, that all

layer

fluctuation modes that contribute to

(n (

_are

fully motionally averaged

_on the time scale of the

quadrupolar

interaction, In section 4 _we present our

experimental

results from the lamellar

phase

in the ,y~tem water/~odium

dodecylsulphate/pentanol/dodecane

and ,how that the

quadrupole ~plittings

irom surfactants, counterions, and water molecules

yield

the _same

bending rigidity,

_K

=

2.2 _± 0.2

t~

T. This value _i~

signiiicantly larger

than that (0.8 t.~

T)

deduced from

layer

undulation

dynamic~

a;

probed by dynamic light scattering

[14] or that

j0.4 t~

T) ^{deduced irom} the dilution behaviour of the

X-ray Bragg peak [12],

in both _cases at the _same

layer composition

_as in the present

study.

In section 5 and

Appendix

B _we argue that

our value i~ close to the intrin~ic

(bare) rigidity,

whereas

dynamic light ~cattering yields

a

rigidity

renormalized

by short-wavelength

fluctuations

[20].

We also demonstrate that the

large discrepancy

between the

K values deduced irom _our

quadrupole splittings

and from

X-ray

scattering [12]

is due _to inconsistent data rather than to a deficient

theory.

2.

Experimental.

2. SAMPLE PREPARATION.

Experiments

_were

performed

_on two series of

samples

^{in the}

lamellar

phase

of the quaternary system water/sodium

dodecylsulphate (SDS)/pentanol/dode-

cane, the

phase

^{behaviour of which has been}

thoroughly

studied at 21 °C [21,

2?]. Starting

N° lo LAYER FLUCTUATIONS IN A DILUTE LAMELLAR PHASE 1825

from _a concentrated lamellar

phase containing

water, SDS, and

pentanol,

_a series of

samples

were made

by

dilution with _a solvent mixture of dodecane and

pentanol.

The

composition

of the

starting

mixture _was the _{same, on a} mol basis, in both series 24. 8 mol water/mol SDS and 2.35 mol

pentanol/mol

SDS. In series I, where water ~H and counterion

~~Na splittings

_were

measured, the

starting

mixture contained

D~O (99.9 §~)

from

Sigma,

SDS («

specially

_pure ») from BDH Chemicals, and

pentanol

(~ 99

§b)

^from Aldrich. In series

II,

where

«-~H

_{SDS and}

counterion

~~Na splittings

_were measured, ^the

starting

mixture contained

H~O (doubly

distilled),

a-deuteriated SDS

(purified by repeated recrystallization

^from _aqueous

solution)

from

Synthelec,

Lund and

pentanol.

The solvent mixture _was the _same in both series 91.0

weight

percent ^dodecane (~ 99 fG) ^{from Aldrich} and 9.0

weight

percent

pentanol,

The

samples

used for NMR

experiments

were

prepared by weighing appropriate

amounts of the

concentrated lamellar

phase

and the solvent into

Pyrex

tubes, ^which were flame sealed

immediately

and then

gently

mixed for about a week. With

increasing

dilution the

viscosity

and

birefringence

of the

samples

_were

markedly

reduced.

The

bilayer

volume fraction ~ _was calculated _as the volume fraction of the

starting

mixture in the

sample, using specific

volumes

(ml/g)

of 1.00?

(H~O

), 0.905 (D~O), 0.85

(SDS),

1.228

(pentanol)

and 1.336 (dodecane ). This _assumes that

thibilayer compisition

_{is invariant}

under dilution. Our data suggest ^{that this is} _very

nearly

the _case (cf. Sect. 4). Each ~erie~

included II

composition;,

with j

ranging

from o-I _to I. The average

layer period

d (calculated as described in Sect. 3) i~

given

in table I in units of the

bilayer

thickness

6. Given the

assumption

of invariant

bilayer

thicknes~, 3 is

simply

the

layer spacing

d in the

starting

mixture (j

= ). With an estimated thickness of 3

= 35

1,

_the

_{layer spacing}

thus ranges from

351to 3501.

_(The

_precise

_{value of 6 is}

not

required

to determine the

rigidity

K).

2.2 NMR EXPERIMENTS. All NMR measurements were

performed

at 21 °C _{on a} Nicolet

Nic-360 spectrometer

equipped

with _a vertical saddle-coil

probe

^and

operating

at 55.54 MHz and 95.70 MHz for ~H and

~~Na.

_The ~H spectra were recorded with the

quadrupolar

echo

pulse

sequence («/2 ), r («/2

)~,

r acq, with _a 25 _~s ml?

pul~e

and _a

typical delay

^time

T of loo _~s. The

~~Na

_{spectra were} obtained from the

single-pul;e

free induction

decay following

a 20 _~s HI?

pulse,

with _an

acqui~ition delay

of

I/v~

to obtain

in-phase

satellite

peak~.

The

magnetic

field

inhomogeneity

_was shimmed to ca. 3 Hz. The

sample

temperature

was controlled

by

_an air-flow

regulator

^{(Stelar VTC 91)}

yielding

a

~tability

of _± 0.05 °C _or better. Before each

experiment

the

sample

was

equilibrated

at 21 °C in the

probe

for _at least 10 min.

The

quadrupole frequency

_{v~ was} mea~ured

directly

as the

splitting

between the satellite

peaks (arising

from the 90°

singularitie~)

in the ~H _doublet ;pectrum or a; the

separation

between the central and ;atellite

peak~

in the

~~Na triplet

;pectrum. ^On

going

from

concentrated to dilute

samples

the

lineshape gradually changed

from _an

isotropic powder

pattern to a ~pectrum characteristic of _a

partially aligned ~ample

^{with the}

optic

axis

perpendicular

to the

magnetic

field (as

expected

for _a

ph»e

with

negative diamagnetic susceptibility aniwtropy).

Due to

homogeneou~

(relaxation)

broadening

the

splitting

in _a

powder

_{-spectrum may} be

slightly

~maller than the _true

quadrupole frequency

_v~. A

complete line~hape

fit is then

required

to determine v~

accurately.

^Since, in the ,pectra ^recorded here, the

homogeneous

^linewidth was

alway,

much ,maller than the

;plitting,

a

line,hape

fit _{was not}

required.

Furthermore, the

layer rigidity

_wa~ determined from data obtained in the dilution range where the

~amples

are

~ufiiciently

well

aligned

that the

spectral peak

a~ymmetry I; very

~mall, whence the efiect oi

homogeneous broadening

_on the

splitting

I;

negligible

3.

Theory.

3.I LANDAU-PEIERLS-DE GENNES THEORY. The

phenomenological

static

description

of

thermally

excited elastic distortions in the smectic A

phase,

as

developed by

Landau

[?3],

Peierls

[24]

and de Gennes

[25],

is based _on the Hamiltonian

H

=

dr

fi

~" ~~ ^~

+ K

IV(

_u

(r

)]~

,

(3.1)

2

, 3z

with _u

(r)

the vertical

displacement

field

measuring

the

layer displacement along

the symmetry axis (z of the

phase.

The

integration

is _over the volume V of the

homeotropically aligned sample

_or, in the _case of _a

powder sample,

over the domain volume V. The

macroscopic

elastic _constants B and

Kj,

associated with

longitudinal compression

(at constant chemical

potential)

and director

splay, respectively,

are related _to the

microscopic

parameters in a

discrete-layer

model of the lamellar

phase

as

[10, 20]

B

=

d

~'~

,

(3.2)

f-

f _<1

Kj

=

), _(3,3)

with

V(f)

the

layer

interaction energy per unit _{area, K} the

layer bending rigidity,

and d the average

layer period.

In the presence of _an external

magnetic

field Bjj, as in _an NMR

experiment,

the fluctuation Hamiltonian also includes the interaction of the

diamagnetic susceptibility

AX of the lamellar

phase

with the Bjj ^field

[26, 27],

H~

=

~~ ~~

dr

[cos (2

p

n)(r

₊

cos~

_p

n~(r)]

,

(3.4)

2 _Ho

,

"

with p the

angle

between Bjj and the symmetry ^{axis. The} transver~e components ^{of the unit}

layer

normal (or

director)

_{n(r) are} related to the

displacement

field u(r) as

~i =

(fi,,

n, =

~ Viii ^~~

^"'

^(3.5)

where, in the last step, we invoked the harmonic (or weak

crumpling) approximation,

which is also

implicit

in

(3. Ii.

Fourier

expanding

the

displacement

field _as

u

(r

₌

z k(q

_exp

_(iq

r ),

(3.6)

q

and

applying

the

equipartition

theorem _we obtain with

(3.1), (3.4)

and

(3.5)

~

k~

^T

~~~ ~

vK

i(q~iA

)~ +

ql

₊

_(q

i If )~ F (p, _v~)1 ^'

~~'~~

where

F lP, _~

= sgn (AX llcos (2 p

cos~

~ +

cos~

_p

sin~

_~

,

(3.8)

N° lo LAYER FLUCTUATIONS IN A DILUTE LAMELLAR PHASE 1827

with _~ the

angle

between q~ and the _i axis. In

(3.7)

_we have also introduced two characteristic

lengths (25]

the smectic

penetration length

K ⁱ⁾ⁱ

A

=

fl

_(3.9)

B

and the

magnetic

^coherence

length

~jjKj

Ill

i~xi ~

^~3.'°1

3? CURVATURE FLUCTUATIONS. If the

layers

are continuous, fluctuations in the

layer

displacement

_u (r) _are

necessarily coupled

to fluctuations in the

layer

curvature and, hence, to fluctuations in the orientation of the

layer

^director n(r).

By modulating

the

anisotropic spin couplings

these orientational fluctuations

produce

ob~ervable eifect~ in the NMR

lineshape

and relaxation. The static manifestations of these effects _are contained in the second-rank orientational order parameter,

(F,)

=

(3(co~~ o)

-1)/2, or, in terms of director compo- nent~,

~~

_{~~ ~~ ~~~~~}

'

~~~

^{(3. Ii}

According

to (3.5) and

(3.6),

(N~)

~

iqi (("(q)(~) (3.12)

q

Inserting (3.7)

and

converting

the _sum to an

integral,

we obtain

(. T rh/ _no 2

r q~

(/i

₌ ^~

~

dq- dq~

d~

, ~ ~ ,

(3,

13)

4 _« K~ _{~jL- riL~} ii

lq~/A

)- + q~ +

(qi If l'F (p,

_~

where L~ and

L~

_are the

sample

or domain dimensions and _a is _a

short-wavelength

cutofi below which the continuum

description

fails

(a~

should be of the same order _a~ the _area

occupied by

a

~urfactant molecule _at the

layer midplane).

It i~ convenient _at this stage to

replace

the

penetration length

A

by

another characteristic

length,

the

patch length

L~ =

(dA )'?

_(3,14)

which i~ the transverse orientational correlation

length

and thus defines

[20]

the cross-over

from

single-layer

behaviour

(q~

_L~ » to collective smectic behaviour

(q~

_L~ « ). (The

patch length

is the surface

analogue

of the deflection

length

for _a confined

polymer

chain

[28].

It can be shown that if the

magnetic

coherence

length

is much

larger

than the

patch length,

f » L~

(3.15)

then the

magnetic

interaction has _no effect _on

(n( ),

Since f i~ of the order 10~ ^~ _m for

typical

lamellar

phases

in conventional

magnetic

fields and since L~ ^{is of the same order as}

d

[cf. (3.22)],

the

inequality (3.15)

is satisfied with _a wide

margin

even for

extremely

dilute

lamellar

phases.

(f does not

depend

on the

layer ~pacing

since both K~ and

AX

scale _as

lid-j We may ^{thus omit} the

magnetic

term in (3.13), which then involves

only

standard

integrals.

Since the domain dimensions in _a lamellar

phase

_are

typically

of order 10~^~ _m, the

following

conditions _are sati~fied in

virtually

all _cases of

practical

interest

d « L-,

(3,16)

L~

«L~

^{~ ~~'}

(3.17)

L,

~

in which case the result of the

integration~

in (3,13) become~

(n(

=

~~ ~

ln (I ₊

p~)

_{+ p arctan}

j, _(3,18)

4 _grK 2 _p

~~~~ L ~

p = n 2

j~ ^(3,19)

a

Unle~s the

layer

volume fraction is very close to I, _p is

~ufficiently large

that

13.18)

i~ well

approximated by

~ ~

~"~

4 _K ^~

~~ _~ ~~ ~~~

which differ,

by

le~s than %> from

(3,18

j _{if p ~} 3. The result

(3.

20) ha~

previously

been

given

(for the _case Bjj ₌

0) by

^Larchd et al.

[4). Usually,

however, _a less accurate version of

(3.20),

with ₊ In _p

replaced by

In _p, is u;ed.

The

paich length

_{L~ was} fir;t introduced

by

de Genne; and

Taupin (29(

within the _context of

a model of _a

~ingle layer

confined between _two

rigid plate~.

For this model

they

obtained _a re~ult for

(ii )

_{that differ}

_slightly

_irom

_(3.20)

_{in~tead of + In}

p

they

have In (p/m ). For the

limiting

case of

noninteracting layer; 16

= l~ the

patch length diverge;

and

(3,13) yields

in the ab;ence of _a

magnetic

field

(n )

=

~

In

~~

(3.21

"K a

This

~ingle-layer

result _was first derived

by

Helfrich

[30).

For _a

~terically

stabilized lamellar

pha;e

the

patch length

i~ obtained

by combining

(I. Ii, (3?), (3.3),

13.9)

and

(3.14)

~

~ K '/2

~ 3 _«

t~

T

~~ ~

(3.22)

3.3 LAYER _SPACING. To _account for the

dependence

of the

quadrupole splitting

_on the

layer

volume fraction _~b in the lamellar

pha~e

we need to know how the average

layer period

cl varies with ~b. As usual, we a~sume that the total

layer (midplane)

area, A, in the lamellar

phase

is fixed, I,e., the

layer midplane

is _a surface of inexten~ion. This is the _ca~e if the

layers

are

laterally incompre;~ible

(due _to strong

headgroup repulsion)

and of fixed chemical

compo;ition.

The base _area, I-e-, the

projection

of the mid~urface _on the base

plane

(perpendicular

to the symmetry axis of the

pha~el, A~,

is obtained from

elementary

differential geometry as

[20]

A~

I

⁼

(»=)

=

(Pi) (3?3>

N° 10 LAYER FLUCTUATIONS IN A DILUTE LAMELLAR PHASE 1829

This

ratio,

which is

nothing

but the first-rank orientational order parameter

(Pi)

=

(cos

o

)

of the

fluctuating layers,

is sometimes referred _{to as} the

crumpling

_parameter

[20].

The

layer

volume fraction _{~b can} be

expre~sed

_as

~

A~ d'

^~~'~~~

with d the average

layer period

and 3 the

(average) layer

thicknes~.

Invoking

the harmonic

approximation by writing (n-)

=

((I n) _)"~)

i

(n) ),

_we obtain from (3.23) and

(3.24)

d

=

,

(3.25)

~b

1

~

(n(

Combination

of _(3,19), (3.22) and (3?5) now

yields

p =

~ ~

(

~

)

~

3 ~B

a ~ j '

~,2

j

2

1

~

For

given

values of the dimensionle~s

quantities K/l~

T and 31a,

(3.18)

_or (3.20)

together

with

(3.26)

_can be solved

self-consistently

for

(n) ).

3.4 VALIDITY oF APPROXIMATIONS. The literature contains _a

variety

of results for

(fi ( ),

derived under different

approximations.

Before

applying

the

theory

to our

experimental data,

it is

appropriate

to examine the

validity

of these

approximations.

We

begin by observing

that if the

layer;

_are

incompres~ible

(a~

assumed)

there _can be _no curvature fluctuations _{at ~b}

= I, in which _ca~e

(3.24) implies

that 3

=

d(~b

= ). Ol'course, in real

layers

the

density

land

compo~ition)

_{may vary} somewhat in ~pace in response to curvature

fluctuations, just

as in _a one-component ^{smectic A}

phase [?5).

Indeed, _our data indicate weak

curvature fluctuations _even at ~b =

(cf. Sect. 4?). For the determination of the

layer rigidity

K,

however,

_{we use}

only

data in the range ~b ~ 0.65, where the

layers

do not interact

strongly

(as

compared

to ~b = ). The _{constant area}

approximation

should then be valid. Furthermore, since the variation of the chemical

composition

of the

layers (repartitioning

of

pentanol)

appears to be small and

mainly

confined to

higher

_~b

(cf.

Sect. 4,

),

it is reasonable to

regard

the

layer

thickness 3 _{as constant}

(independent

of _~b) in this range

(although

it _may differ from 3 _{at ~b}

= ).

Figure

show~ the effect of various

approximations

on the _curvature fluctuation

(n )

for

K =

2

l~

T and 31a

= 4. (These _are

e~sentially

the parameter ^{values that describe} our

data; cf. Sect. 4.2.) While the difference between (3.18) and (3.20) is

negligible

for

~b ~ 0.9,

only (3.18)

has the correct (within the model behaviour at ~b = : for

incompressible layers

the

longitudinal compression

modulus

h

becomes infinite at ~b = I, whence p =

0

by (3.9), (3.14)

and

(3.19)

and

(n[

=

0

by (3.18).

For the _~b

~ 0.65 data used _to

determine _K, however,

(3,18)

and

(3.20)

_are

virtually

identical (since _p w ).

In the derivation

[10]

of the ~teric

repulsion

(I. II ^from the Hamiltonian

(3.

II _a

large

p

(say,

_p

~

3)

_was assumed in _a step

analogous

to that

leading

from

(3.18)

to

(3.20).

The _use

~ 0.5

m~

£ 0.4

)

a 0.3

f

-.f~~

fl

.I

, ' ~~

~ _',

01 ill

~

0 0.2 0.4 0.6 0.8

layer volume fraction, #

Fig.

I. Tran,ver,e director fluctuation,

(n)),

calculated with _K 2 la T and 3/£1= 4 ii four

different level, of approximation : (I) u,ing (3.18) and (1,26) (11) u,ing (3,?~l) and (3.26) (ill _u,ing (3.201and (3?61 with the

(ii [ )

term neglected _; (lv i u~ing (3,201and (3.26) with IIII

(n( )

/2 ~ replaced by 1.

of (3?2), which follows from I. Ii,

together

with

(3.18)

i~ therefore _not

strictly

consi~tent.

However, since

(3.18)

_agrees with

(~,?0)

for _~b ~0.9 and is _exact

(within

the model) at

~b = I, this is of little consequence. Besides, _K is determined from data in the range

~b ~ 0.65.

The harmonic, _or weak

crumpling, approximation

has been invoked _at several ~tages ^{in the} derivation. In fact, it is

implicit already

in the Hamiltonian (3. ). While the theoretical fits to our data do not ~uggest _any

departure

from the harmonic

regime

(cf. Sect. 4.2),

figure

shows that the accuracy of the harmonic

approximation

deteriorates

rapidly

at lower volume fractions (~b _~ 0. I.

In (3.26), we have retained the term

(n))/2 although

it represents a

higher-order

contribution in _a

low-temperature expansion

in powers of

l~ T/(4

_wK ). As seen from

figure

I, this _term has a small but

significant

effect _over the entire _~b range. If this _term is discarded the

quadrupole splitting

_v~ decreases

linearly

with In

[(I

_~b

)/~b]

(cf. Sect. 4.2).

The

expression (3.22)

for the

patch length

_L~ associated with the steric

repulsion

I. is not limited to the dilute (cl w 3

regime,

as sometimes claimed. While the

original

derivation

[10]

apparently

assumed that cl » 3, it _wa; noted II

0]

that this

requirement

can be

dropped

^if the fluctuations _are

sufficiently

weak that the

layer

thickne~~ 3 _can be subtracted from the

layer period

d _as in

(3.22).

This ~hould be _a

legitimate procedure

here since the finite thickness

correction is _most

important

at

relatively high

volume fractions, where the fluctuation~ _are indeed weak. A compari~on of the curves Ill and IV in

figure

show; that the effect of the finite

layer

thicknes~ is substantial.

In conclusion, _we believe that the

approximations underlying (3,18)

_or

(3.20)

and

(3.26)

_are

justified

in the range 0,10

~ ~b ~ 0.65 for the present system. ^{At lower} ~b, the harmonic

approximation

breaks down. At

higher

~b, we expect ^the constant area

approximation

to fail and the

bilayer composition

to vary, with _a consequent

change

in the

layer

thickness.

Furthermore, at

higher

_~b the

patch length

_may reflect other

layer interactions,

such _{as van} der

Waals attraction, be~ides the steric

repulsion.

(At the

highest

volume fraction used to

determine _K, the average

~eparation

d 3 between

adjacent layers

i~ ca.

251.)

N° 10 LAYER FLUCTUATIONS IN A DILUTE LAMELLAR PHASE I83I

4. Results,

4. MOTIONAL _AVERAGING. The

quadrupole splitting

_v~ is

proportional

to the second-rank

orientational order parameter

(P~)

as~ociated with

layer

curvature fluctuations

v~ =

VI <P~i

=

VII

^'

^Ii>]

_, ^{(4. ii}

where (3, II was used in the second step. ^This

;imple

relation is valid

provided

that

layer

fluctuation modes

k(q~

of all _wave vectors q~ that contribute

significantly

to

(n(

_are

motionally averaged by

lateral diffusion of the

spin-bearing species

and/or

hydrodynamic layer

undulations at _a rate which exceeds the

quadrupole frequency

modulation a~;ociated with that mode. In

Appendix

A _we demonstrate that this is indeed the _case under the conditions of the present

,tudy.

4,2 BILAYER comPosiTioN, The

quantity v(

in (4.I) I; the

hypothetical quadrupole

splitting

in the absence of curvature fluctuation;

((P,)

= ). In the pre;ent

~tudy

the

quadrupole splitting

is obtained from

powder sample;

or from

~amples (partially)

oriented with the ;ymmetry ^axi;

perpendicular

to the

magnetic

field (~ince ~x _~

0).

In either _case

~<~

3

~

41(21

^~

(4.2)

The

spin

_quantum number I is for ~H and 3/2 for

~~Na.

_{The residual}

quadrupole coupling

constants k for the three nuclei,

partially averaged by

local motion; with respect to the local

layer

normal, _are not

accurately

known for the present system, ^For

D~O

and the Na+ counterion, k is

expected

to

depend ~trongly

on the

bilayer composition.

For the _a-

deuterons of SDS _we may write k ₌

Sx,

with _x

=

170 kHz and _a local order parameter S

(typically

in the range 0.18-0.24), which is

e~sentially independent

of

~ample composition

and interfacial curvature

[3

Il.

The measured

quadrupole splittings

are shown in

figure

2. The

~~Na splittings

are

virtually

identical for the two ~eries, _a~

expected

since the molar

composition

of the

~tarting

mixture is

25 j

@

i

15 _ o

~ .

o

° b

~

.

o

° bfl

-~

'0 o°

0A

~

u~ ^m ~

m

f~

Na ~ _>

"

~/ _~ ^m

5

~ a

W

> a .

0 0

0 _0.2

ayer

ig. 2, -

variation of he

(li) in ~eries I

2.2

a

* O

_it _~

i1.8

_*

~

~ ~

~ *

~

, ^~

(

~ ^°

(

IA

g

⁴

© 4

> w

I

₄

fi

~

0 0.2 0.4 0.6 0,8

layer volume fraction, #

Fig. 3, As in

figure

I, but all

quadrupole splittings

normalized _to unity for the _mo,t dilute sample.

the _same. The normalized

splittings,

shown in

figure

3, reveal

only

^~mall differences between

the three nuclei,

presumably reflecting

minor variations with _~b of the

bilayer composition,

^If

v((SDS

is taken to be

independent

of ~b, then

v((Na)

is reduced

by

10 fl _on

going

from

~b = to ~b = 0. I. The variation of g (Na ^{with the} alcohol/SDS ratio has been measured in the lamellar

phase

of the ternary system water/SDS/decanol

[P.-O. Quist, unpublished

results ]. In the range 1-?.5 mol decanol/mol SDS, where the

bilayers

are free from structural defects

[3

II,

k(Na)

decreases

roughly linearly

with the decanol/SDS ratio. If the

(relative)

variation is the same in the present system, a 10 fl reduction of k

(Na

) would

correspond

to a modest increa~e of the

pentanol/SDS

ratio in the

bilayer

from 2.35 to 2.67. This variation in the

bilayer

composition

should increase k

(D~O

with

decreasing

~b. However, the fast deuteron

exchange

between water and

pentanol

should have _an

opposite

effect

(since pentanol

outside the

bilayer

is in _a

locally isotropic

environment). As ~een from

figure

3, these two effects appear ^{to cancel}

out.

4.3 BENDING RIGIDITY. With

(n(

obtained from (3,18) and (3.26) we can describe the

dependence

of the

quadrupole splitting

_{v~ on} the

layer

volume fraction _~b with three parameters

v(, K/l~

T and 31a. We

begin by analyzing

the SDS data.

Although

the three parameters could, in

principle,

be determined from _a nonlinear

least-square~

^fit to the

(v~,

_~b data, the strong ^{covariance of}

v(

and 31a makes it _more

meaningful

to

adjust

v(

and

K/l~

T for different fixed values of 31a. Furthermore, since the theoretical

description presented

ⁱⁿ section 3 is _not

expected

to hold at the smallest

layer separations (cf.

Sect. 3.4),

we include in the fit

only

data from _~b

~ 0.65.

As shown in

figure

4, excellent fits (reduced

x~

~ l?) _are obtained for _a wide range of

31a values~

corresponding

to _K

=

1.8-2.8

l~

T. However, _{we can}

impose

certain

physical

constraints _on 31a. First,

(P,)

at ~b =

obviously

cannot exceed I, which

implies

that

31a ~ l.5. Thi~ also ensures that S

~ 0.18, _as

expected.

Second, the

bilayer

thicknes~ _cannot

exceed _ca.

401 (including

the

hydrocarbon

tails of SDS) and the cutoff

length

_a cannot

rea;onably

be ~maller than _ca. 5

1,

_{whence 31a}

~ 8.

IA larger

value of 31a would also

imply

unreasonable value~ for S and

(P~)

at ~b I).] With 1.5~31a~8 _we arrive _at

K =

2? _± 0?

k~

T. The fit with 31a

=

4

(yielding

_{K =} 2.19

k~

T and

v(

=

25.7 kHz) is shown in

figure

5.

N° 10 LAYER FLUCTUATIONS IN A DILUTE LAMELLAR PHASE 1833

3.0

K/kBT

2.5

10S 2.0

1.5 ,

~j___~

~'~

"~~'~~"""~'~~'~~~~~~~at

_#=1

~'~

0 5 10 15 20

61a

Fig,

4, Re;ults of fitting the _two parameter, K/l~ T and

v(

to the

(v~,

~ data from a-deuteriated

SDS. The local (C-D bond) order parameter S and the curvature fluctuation order parameter

(Pj)

at ~ in the _mo,t concentrated ,ample _are obtained from

v(,

The reduced chi-square

X'of

the lea~t-square~ fits i~ al;o given. The vertical line~ delimit the physically acceptable _range ^of 61a.

25

fi

m _o

m 20

~

ii

~ 15

~~0

_{0.2 0A 0.6 0.8}

layer volume fraction, #

Fig.

5. Re;ult of two-parameter ^nonlinear fit, according to (3. IX i and 13,26), to the SDS

quadrupole

,plitting,. Data points ^denoted by _open

symbol,

_were noi included

in the fit.

As shown in

figure

I, the

higher-order

(in

l~

T/K) term

(n( )/?

in

(3?6)

has _a small but

significant

effect.

Neglecting

this _{term, we} obtain

v~ = a p In ~

), _(4.3)

~b

with

fi ^~w ~i

_{~ ~} ~~

3~.)T ^'~

^~~'~~

<1

4 _wK

"~

3

l~

T ~ (4,5)

J()UR~AL DL PHi ~IQUE II -T 4 ~'l(> ()LT()HER l'iLM

The linear fit of (4.3) to the SDS data I, ;hown in

figure

6. The

high degree

of

linearity

supports ^the

validity

of the theoretical

de~cription

for data _in the range 0.10

~ ~b ~ 0.65. Since

(n [

is

roughly proportional

to ~b (cf.

Fig.

or table II, the

neglected (n ) )/?

_term would

es;entially

^contribute

only

to _a in (4.3j without

affecting

the

linearity

of the

plot.

With

&la

= 4 _we obtain from (4.4) and (4.5 _K

=

?,16

k~

T and

v(

?5. kHz, in clo;e agreement

with the re,ult, from the nonlinear fit in

figure

5. We _note al;o that _a fit of _{1>~ a +} p In _~b to the _;ame data exhibit; _a

;y,tematic

deviation from

linearity

and

yield; ,ignificantly

different

parameter ^values.

Table

give,,

for each

,ample

in the serie;, the order parameter

(P?),

the director

fluctuation

(») ),

the

layer period

d, and the

patch length

_L~ (the latter two in units of the

bilayer

thickne;; Al. The results for d/&,

L/3

and

(»~

^{in column, ~-5 of table} were

Table I.

San>p/c t.omj>o,titian.v

(.retie.; II) £i»</

cjiia»titie.i

c-ah iilate</ _{f> on>} ii><> pa>.an>ete>..;

e.itiacted

Ji.om

the

fit

in

Jiguie

5.

wt% ;olvent _~b cl13 (")

L~/3

l'~j

(n[) (I (1>[

^(~)

(Pi)

(~)

0 (1) (0) (0) 0.085 0.872

9.9 0.873 (1,19) (0.26) (0.082 0. 16 0.826

19.9 0.754 (1.42

(0.

57)

(0.

138 0, 147 0.780

30.0 0.639 1.70 0.96 0.176 0.177 0.735

39.9 0.534 2.07 1.45 0.206 0.204 0.694

49.8 0.433 2.58 2.16 0.235 0.235 0.648

60.1 0.334 3,38 3.25 0.264 0.265 0,603

70.2 0.244 4.71 5.06 0.296 0.298 0.553

75,1 0.201 5,75 6,48 0.314 0.313 0.531

80,1 0.158 7.38 8,70 0.336 ~l.3 ~9 0,492

85.1 0. 17 10, 12.4 0.362 0.360 0.460

l'~) Calculated from

K/(B

T and Ala.

(hi

Calculated from

i>[,

25

o

20

I

#

~

15

~~2.5 _{1.5 0.5 0.5 1.5 2.5}

In i(I j)/~i

Fig.

6. Re,ult of linear lit, according to (4.3), _to the SDS

quadrupole

,plitting,. Data point, ^denoted by _open ,ymbol, _were not included _in the tit.

N° 10 LAYER FLUCTUATIONS IN A DILUTE= LAMELLAR PHASE 1835

obtained from

(3.18), (3.22),

(3.25) and

(3?6),

with 31a and

K/l~

T taken from the fit in

figure

5. Since the;e

equations

_are not

expected

to be

quantitatively

accurate at the

highest

volume fractions

(cf.

Sect.

3.4),

the entries

corresponding

to data

points

not included in the fit

are enclo~ed in

parenthese~.

The re;ult~ for

(n (

_and

(Pj)

in the last _two columns of table I were obtained from (4. ), with

v(

from the fit in

figure

5. The;e results should thus be accurate also at the

higher

volume fractions,

provided

that

v(

does _{not vary} with _~b (a reasonable

approximation

for SDS). As

expected

from

figure

5, the _two sets of

(n))

values differ

significantly only

for _~b

~ 0.65. The value

(n)

=

0.085 at ~b =

corresponds

to _a root-

mean-;quare

angular

director fluctuation of 17° away from the symmetry ^{axis of the}

phase.

The

analysi~

of the counterion

~~Na

and _water ~H

~plittings yields

_very similar result~ _as for SDS with fits of the _;ame

quality. Taking

1.5

~ 31a _~ 8 31a should be the _same for all nuclei, of course), _we obtain _K

=

2. _± 0.2

1~

T from the counterion

splittings

and _K

=

2? _± 0. ?

1~

T

from the water

splittings.

The nonlinear fits _to the counterion and _water

splittings,

with

31a

= 4, are shown in

figures

7 and 8.

11

9

~

m

$

Q

~

>

5

~

fraction, #

Fig. 7. - ;ult

,plitting;, ^Data denoted by open ,ymbol, were not ncluded in the fit.

o.9

°'? °

,

~- o

o

I

n

>

~~0

0.2 0A 0.6 0,8

layer volume fraction, #

Fig. ~. Re;ult of two-parameter nonlinear fit, according to (3.18> and (3.26), _to the D~O quadrupole ,plitting,. Data point, ^denoted by open ,ymbol, were not included _in the fit.

5. Discussion.

It is of interest _to compare our value for the

bending rigidity,

_K

=

2.2 _± 0.2

l~

T, with the results obtained

by

Nallet et al

[14]

from

dynamic light scattering

studies of the _same ~ystem (at the _same

composition

and

temperature).

These studies

probe

the

dynamics

of

layer

displacement

fluctuations

(undulation

mode) _or of

coupled displacement

and concentration

fluctuations

(baroclinic mode).

In the

experimental

_wave vector

regime,

the relaxation

frequencies

of these modes _are

proportional

to

Kj

and

fi, respectively.

For _a

sterically

stabilized lamellar

phase,

either mode _can thus be used _to determine the

bending rigidity

K. Nallet _et al. obtained

K = 0.8

l~

T from

Kj

and _K

= 2.4

l~

T from

fi.

_The

di~crepancy

between the _K values deduced from the two modes, also encountered in other system;

II 3],

may be due _to deficiences in the model used to calculate the

hydrodynamic

coefficient

entering

the baroclinic relaxation

frequency

and/or to the

sensitivity

of the undulation mode

experiment

to

sample misalignment IF.

Nallet,

personal communication].

Recent

light scattering

studies

(13]

of the undulation mode in ;everal other dilute lamellar

pha;es yield

_even lower

rigiditie;

_:

K/l~

T

=

0.2-0.3.

When the small

rigidities

deduced from the undulation

dynamics

_are

compared

with the

much

larger rigidity

obtained here from

quadrupole splittings

it ,hould be borne in mind that the _two

techniques probe layer

fluctuations _on

widely

different

length

scale~. The

quadrupole

~plitting, being

determined

by

the director fluctuation

(n) ), essentially

reflect~

single-bilayer

behaviour, I-e-, fluctuations with

wavelength~

~horter than the

patch length

(cf.

Appendix

B).

Dynamic light ~cattering,

_on the other hand,

probes

tluctuation~ _on

wavelengths

of order 10~ ^~ _m, much

larger

than the

patch length (cf.

^Tab. I). ^{It has} been

suggested (20]

that the latter type ^of

experiment

reflects _an effective

rigidity

_K

~, renormal17ed

by

curvature fluctuations _on shorter

wavelengths (essentially

within the

patch

_area

L() ^according

to

3 1.~ T

~R ~~'

8 ~

~ ~ ~~ ~~~P~~~~ ~~'~

with L~ ^defined

by (~,22)

and

K~j the intrinsic (bare)

rigidity

that determine~

(i>( ).

In view of

(3.19), (3.20)

and (3, II, _{we can express} thi~ (after

replacing

⁴

by

_w ⁱⁿ the

logarithm)

_more

compactly

as

~R

(~?)

_~0, (5.2)

with

(P~)

the second-rank orientational order parameter ^{of the}

fluctuating layers.

With

Ku =

2.2

lB

T and

(P~)

from table I, we find that _K~ varies from

1.91~

T at ~b = to

1.0 kB Tat ~b =

0. I. A

dependence

of _{K~ on ~b} is in fact

suggested by

the data in

figure

6 of

reference

[14],

where the individual values of the undulation «diffusion coefficient»

D~

= K/

(d~ yield (with

_~

=

l.4 cPl _K~ values

ranging

from 0.6

k~

T _to

1.81.~

T

(for

the _most concentrated

sample).

The

length

scale

dependence

of the effective

rigidity predicted by

(5.I)

implies

that the traditional continuum

description

of

layer

fluctuations, _a~ outlined in section 3, is not

strictly

valid _on

length

scales shorter than

L~. Generalizing

the continuum

theory

to include _a

wavenumber-dependent rigidity,

_{K(cj~ ), we} show in

Appendix

B that the

rigidity

deduced from

qu3drupole splittings

i~ close _to, but

slightly

;maller than, the bare

rigidity

K<I

Recently,

^Roux et al,

[12]

deduced _K from the nonideal dilution behaviour in the present system (at ^the same

layer composition),

with the average

layer period

d obtained from the X- ray

Bragg peak position, Unfortunately,

the results

given

in table I of reference

[I?]

_are