Viscoelastic relaxation of insoluble monomolecular films

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Viscoelastic relaxation of insoluble monomolecular films

J.C. Earnshaw, R.C. Mcgivern, P.J. Winch

To cite this version:

J.C. Earnshaw, R.C. Mcgivern, P.J. Winch. Viscoelastic relaxation of insoluble monomolecular films.

Journal de Physique, 1988, 49 (7), pp.1271-1293. �10.1051/jphys:019880049070127100�. �jpa-00210808�

Viscoelastic relaxation of insoluble monomolecular films

J. C. Earnshaw, ^R. C. McGivern and P. J. Winch

The Department of Pure and Applied Physics, ^{The Queen’s} University ^of Belfast, Belfast BT7 1NN, ^Northern

Ireland

(Requ le 8 d6cembre 1987, révisé le 7 mars 1988, accepté ^{le 11 mars} 1988)

Résumé.

On étudie les monocouches du mono-oléate de glycérol à l’interface eau-air en utilisant les ondes

capillaires ^excitées thermiquement qu’on observe dans une grande plage de nombres d’onde par diffusion

quasi élastique de la lumière. Par une nouvelle procédure d’analyse des résultats on obtient ab initio 4 propriétés viscoélastiques de la surface : les modules élastiques ^{de surface et} les viscosités qui gouvement ^les cisaillements normaux à la monocouche (~ tension) ainsi que la dilatation dans le plan de la couche. Ces

mesures permettent ^la première comparaison rigoureuse des modules de tension et de dilatation avec leurs valeurs d’équilibre classiques. Plusieurs effets suggèrent que ^ces deux modules subissent des processus de relaxation différents : pour les modules élastiques ^{on trouve} des différences entre les valeurs dynamiques ^et statiques dans différents états de la monocouche, ^et pour les viscosités, ^{on trouve des} comportements différents suivant l’état de compression de la monocouche. Ces effets dépendent également ^{de la} fréquence.

Dans la monocouche complètement comprimée, le module de cisaillement transverse est caractérisé par ^une relaxation exponentielle, ^{avec un} temps ~ ⁹ 03BCs. Ce temps de relaxation décroît exponentiellement lorsque ^la

monocouche est dilatée, ^et atteint 100 ns pour des aires par molécule de 60 Å2. On peut ^exclure rigoureusement ^la présence de processus plus lents. Le module de dilatation est généralement ^{moins bien}

déterminé que celui de cisaillement transverse ; toutefois, dans l’ état dilaté de la monocouche, les données démontrent l’existence d’une relaxation beaucoup plus lente, ^{vers 03C4 ~ 290} 03BCs. On discute brièvement les mécanismes moléculaires associés à ces relaxations.

Abstract.

Glycerol mono-oleate monolayers ^at the air-water interface have been investigated by quasi-

elastic light scattering ^from thermally ^excited capillary ^{waves over a} wide range of ^wave numbers. Using ^a relatively novel data analysis procedure four surface viscoelastic properties ^were deduced ab initio from the

light scattering data : surface elastic moduli and viscosities governing shear normal to the monolayer (~ tension) and dilation in the film plane. The tension and dilational modulus were compared ^with classical, equilibrium values in the first rigorous comparison of its kind. Various effects suggested ^{that the two moduli}

were affected by ^rather different relaxation processes : discrepancies between the light scattering ^and equilibrium values of the two elastic moduli occurred in different states of the monolayer, ^{and the two surface}

viscosities (both ^zero for the clean subphase) behaved very differently ^on monolayer compression. ^These

effects were observed to be frequency dependent. ^{In the} fully compressed monolayer ^{state the} transverse shear modulus was characterised by ^an exponential relaxation, of time scale ~ 9 _03BCs. This relaxation time fell

exponentially ^on monolayer expansion, reaching ^{100 ns} for molecular areas ~ 60 Å2. Slower processes than these were rigorously excluded. The dilational modulus was generally less well determined than that affecting

transverse shear. However in the expanded monolayer ^state, the data sufficed to demonstrate much slower relaxation, ^{03C4 ~ 290} 03BCs. Possible molecular mechanisms are briefly discussed.

Classification Physics ^Abstracts

68.10

62.30

47.35

1. Introduction.

The viscoelastic properties ^{of a} molecular film at a

liquid ^{surface are} of considerable interest, ^both intrinsically ^{and for the} light ^which they may shed upon the phase ^{behaviour of these} systems. They ^are macroscopic manifestations of the intermolecular interactions within the film, ^and perhaps ^between

the film and the adjoining fluid. Even for a surface which is isotropic within its equilibrium plane ^there exist, ⁱⁿ principle, ^five separate independent ^surface

moduli [1], ^{each of which may} comprise ^{elastic and}

viscous components [2]. ^In practice ^this variety ^has

not been exploited ⁱⁿ experimental ^{studies of am-} phiphilic monolayers. ^{The commonest surface} properties observed have been the surface tension

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019880049070127100

1272

and that surface viscosity relating ^to shear strain within the monolayer plane [3]. This paper ^concerns the determination from a single experimental ^obser-

vation of four different surface properties, compris- ing ^two viscoelastic moduli. These correspond ^{to the}

surface tension and to the dilational modulus of the

interface, together ^{with the} corresponding ^surface

viscosities.

The surface of _any liquid ^is continually roughened by ^molecular agitation. ^The random surface can be Fourier decomposed ^{into a} complete ^{set of} capillary

modes. Such thermally ^{excited waves, whilst of} microscopic amplitude, ^scatter light appreciably.

The spectrum ^{of the} light ^scattered by ^{a surface}

mode of defined wavenumber reflects the temporal

evolution of that _wave, and thus carries information upon the relevant liquid ^{or surface} properties [4].

Systems studied include those in which the tension of the surface or interface and the viscosity ^{of the bulk}

fluid or fluids were inferred from the light scattering

data (see [4, 5] ^{for recent} reviews). ^{Here we will be}

concerned with specifically surface effects such as

arise in molecular films.

Capillary waves are governed by ^the restoring

force of surface tension and are damped by ^viscous dissipation within the bulk liquid. ^{However other,} specifically ^surface viscoelastic properties ^influence capillary ^{waves. In} particular ^Goodrich [6] postu- lated that the surface tension and the dilational elastic modulus of a molecular film must be ex-

panded ^as viscoelastic moduli. This has been verified

on the basis of microscopic arguments [7, 2]. ^The temporal evolution of capillary ^{waves is} thus govern- ed by ^four specific ^surface properties, ^two being ^the

tension (yo) and dilational modulus (so) ^{and two} being viscosities (y’ ^and E’) associated with these

quantities.

Monolayer ^covered liquid ^surfaces (including fatty

acid [8-11], lipid [10-13] ^and polymer [9,14] ^mono- layers) ^{have been studied} by light scattering ^from capillary ^{waves. In} nearly ^{all cases these} studies have involved spreading ^a very expanded monolayer ^and subsequently monitoring ^the changes ^{in the} light scattering spectrum ^{as the} monolayer ^was compres- sed. Both correlation techniques [12, 8, 10] ^and spectrum analysis [9, 11] have been used to estimate the capillary ^wave frequencies (wo) ^and damping

constants (r). From these two observables _up to four surface properties have been extracted. Such

analyses ^have incorporated essentially arbitrary ^as- sumptions ^or data from other experiments:

(1) information from other experiments ^concern- ing ^{some of the} properties (e.g. yo, eo) ^{has been}

used to constrain the interpretation process [8-10].

Unfortunately it appears that the values of yo and Eo affecting ^the capillary ^waves probed by light scattering may ^not be identical with those found in classical experiments. ^For example molecular relax-

ation processes may involve ^a yo(w) which differs from the equilibrium tension value [15]. Similarly,

diffusive interchange ^of surfactant molecules be- tween surface and subphase ^{will cause e to be} frequency dependent [16] ;

(2) similarly, ^{whilst a} particular ^surface property

(e.g. y’) may be ^zero for the case of a specific amphiphile, ^{this cannot be assumed to be true in} general ;

(3) the various surface properties affect the dis-

persion behaviour of w with q. Measurement of wo and T as functions of q ^should thus, ⁱⁿ principle,

determine these properties better than an obser- vation at a single q [17]. Unfortunately, frequency dependence ^of these surface properties may modify

the dispersion ^behaviour [15]. ^The characteristic effects of the various properties ^{will thus not be} easily discernible.

Unfortunately ^{there can in} general ^{be no} unique interpretation ^{of two} experimental observables in terms of four physical properties, except perhaps ⁱⁿ

very special circumstances.

The monolayers used in this work are formed of the neutral lipid glycerol monooleate (GMO). ^Both

mono- and bimolecular films of this lipid ^have previously ^{been studied} by light scattering. ^Black lipid ^{films of GMO} provided ^{the first} unambiguous

evidence for the existence of an interfacial viscosity

associated with the modulus governing shear normal to the interface plane (tension) [18]. Subsequent

studies of such films have shown viscoelastic relax- ation of this modulus [15]. ^The phase transitions of GMO (-- 16° C) in both bi- and monomolecular films have been studied [19, 13]. ^The present exper- iments are restricted to room temperature, ^well above this transition, where the classical surface pressure-area isotherm of GMO is well established

[20-23].

This paper ^{concerns two} major points :

(1) ^a relatively ^novel approach ^{to data} analysis ^for monolayer ^{covered surfaces} [24, 25], ^an extension of

a method successfully applied ^to interfaces [26, 27]

for which only ^the interfacial tension and the viscosi- ties of the bulk fluids affect the capillary ^{waves. For}

the monolayer ^{case the} physically interesting par- ameters of the fit are the two surface moduli mentioned above (each comprising elastic and visc-

ous portions). ^{Here we} demonstrate the practicabili- ty ^of extracting all four surface properties ^{ab initio}

from a single observation of the spectrum ^{of the} scattered light. ^Where possible, ^{the surface} proper- ties thus deduced are systematically compared ^with

data obtained by ^classical methods ;

(2) ^the frequency dependence ^of the surface mod-

uli. Viscoelastic relaxation is clearly demonstrable

for the transverse shear _case, and can be inferred for

the dilational case. For transverse shear the variation of the relaxation rate with molecular packing ^has

also been measured. These observations, the first of their kind, open ^{a new} field of interfacial viscoelas-

ticity.

2. Theoretical background.

Various fluctuations, ^both transverse (or capillary)

and longitudinal (or dilational), ^{in a} monolayer may scatter light. However, ^the intensity ^scattered by longitudinal ^waves is, ⁱⁿ general, very much lower than that due to the transverse waves [9]. ^{We thus} only ^consider scattering by ^the capillary ^{waves. The} theory ^{is well} established [4] ^and only ^{a brief}

summary of the relevant points is necessary.

2.1 CAPILLARY WAVES. - Capillary ^{modes are}

characterised by ^{a surface wave number} q ( = 2 1T /A). ^{Such a surface} fluctuation can be described by ^the departure ^of the surface from its

equilibrium plane :

Experimentally ^{waves of} real q propagating ^{in the}

x direction are selected for observation, ^{their tem-} poral ^evolution being characterised by ^the frequency

w (= w 0 + i F ). This is related to q ^{via the} dispersion equation, ^{which is} [4, 28] :

where _y is the surface tension, q the viscosity ^and

p the density ^{of the} liquid, ^and

In equation (2), ^E is the dilational modulus of the

monolayer,

A being ^{the molecular area} in the film.

The spectrum ^{of the} light ^scattered by ^{the ther-} mally ^excited capillary ^waves reflects their temporal

evolution. For a monolayer covered surface the spectrum ^{is an} explicit function of the surface

properties y ^{and e} [4] :

While the spectrum ^is approximately Lorentzian, ^the

deviations of P (w ) from that form are well estab- lished [29, 26]. ^{The several surface} properties ^affect

P ( w ) ^{in different and} characteristic fashions. As will be seen below, ^{this fact} permits ^a single experimental

observation of the spectrum ^{to be} analysed ^{in terms}

of the four properties affecting ^P ( w ). ^{In our} exper- iments we measure the field correlation function of the scattered light, ^{which is} the Fourier transform of

P(w).

2.2 SURFACE PROPERTIES.

By analogy ^{with the}

three-dimensional _case, e corresponds, ^for isotropic strain, ^to the modulus of hydrostatic compression ⁱⁿ

the plane ^{of the film} (K), whereas for uniaxial strain it also involves [30] ^the corresponding shear modulus

(S ) :

Capillary ^waves restricted to the x - z plane ^involve

uniaxial compression ^and dilation, ^{so that the} appro-

priate ^{value of E will be} Buniaxial. The value of

e appropriate ^{to the} measurement of a monolayer

isotherm will depend upon the ^{exact nature} of the strain applied [31]. ^In practice, apart perhaps ^from highly compressed, ^{« solid »} mono layers, Swill usually ^be negligible. Despite ^these complications,

8 will here be referred to as the ^« dilational modu- lus » for brevity.

As pointed ^out by ^Goodrich [6], ^{both Band}

y may be extended ^to incorporate viscous .effects.

Following ^{the usual} rheological convention we may write (there ^{is no} established notation in this field) :

Here yo and Eo ^are elastic moduli describing ^the

response of the system ^{to shear normal to the} interface (tension) ^and dilation within the plane ^of

the interface respectively, y’ ^{and E’} being ^the corresponding ^surface viscosities. These viscosities

are not the conventionally ^{measured «} surface vis-

cosity ^» which refers to shear within the plane ^{of the}

interface [3], ^{but are distinct} quantities appropriate

to well-defined strains applied ^{to the} monolayer.

The nature of the quantities here called y’ ^and

£’ is not as yet fully understood. They may be

regarded ^as macroscopic hydrodynamic ^variables [6]

or as surface excess properties arising ^{from the} microscopic inhomogeneity ^{of the fluid} interfacial

region [7, 2]. Theoretical predictions ^{of their} magni-

tudes are not available.

The dilational modulus affects the capillary ^waves

because these waves couple ^to longitudinal ^fluctu-

ations of the film [28]. ^This coupling ^{has been}

treated elsewhere [4, 32] ^{and we} simply present

some results useful for the discussion of experimental

data. As Eo ^increases (relative ^{to a fixed} yo) ^{both the}

wave frequency wo and the damping ^r vary (Fig. 1), passing through maxima when Eo/,yo ^{= 0.16. For}

£o ? yo both wo and r vary slowly with Eo. The

surface viscosity ^{s’ reduces the} magnitudes ^{of these}

1274

Fig. ^1.

The variation of capillary ^wave frequency ^and damping ^as functions of Ea/ yo, computed ^{for three}

different wave numbers and assuming ^{yo =} 72.75 mN/m.

The variations are shown for these different values of e’ : 0 (- ), ^{10- 4} (- -) ^{and 5- mN s/m} (- - -).

variations. In contrast the surface viscosity y’ sys-

tematically increases r and reduces _Wo.

2.3 VISCOELASTIC RELAXATION. - In conventional

rheological ^notation [33], ^an oscillatory ^stress (T(t) ⁼ T * e‘ w‘) and strain (u (t ) ⁼ u* eiwt ) ^{are re-}

lated via a complex dynamic ^modulus G*(w):

In the present ^{context the} storage ^modulus G’(co)

can be identified with _yo (or 80) ^{while the loss}

modulus G"(w) corresponds ^to w y’ (or toE’).

There are no microscopic ^{theories of} interfacial

rheological relaxation. However for linear viscoelas-

ticity, combinations of simple rheological ^models

can be found which exhibit arbitrary frequency dependences ^of G*. We mention only ^{two such}

models : the Voigt viscoelastic solid and the Maxwell fluid. For the Voigt model both G’(w ) ^and G " (to )/ w ^{are constant.} The Maxwell fluid, ^corre- sponding ^{to a} single exponential relaxation (of ^time

constant T), ^{is described} by

where Ge ^{is the} equilibrium (w 0 ) ^elastic modulus,

and G the strength ^of the relaxation process. The Maxwell model interpolates reasonably ^{between the}

low frequency ^{viscous and} high frequency ^elastic

behaviour of a viscoelastic fluid [34].

3. Experimental ^methods.

3.1 LIGHT SCATTERING.

Our heterodyne spec- trometer has been described in detail elsewhere [35].

Briefly, light ^{from an Ar+ laser} (A ^{= 488} nm) ^was spatially ^{filtered to ensure a Gaussian} intensity profile (TEMO. mode) ^and illuminated the liquid

surface. Light ^scattered by thermally ^excited capil- lary ^{waves was detected} using heterodyne ^methods

to measure the small frequency shifts of the scattered

light (- kHz ). The train of photodetections ^{from the}

detector was processed by ^{a multi-bit} photon ^cor-

relator (Malvern K7025), interfaced to a PDP11/34A

minicomputer.

Autocorrelation of a heterodyne mixture of two

optical fields, corresponding ^to intensities Is (scat-

tered field) ^and Ir (reference field), yields [36]

where g (’) (,r ) ^and g(2)(T ) ^are the first and second order correlation functions of the scattered field

respectively. Provided that Iris sufficiently greater than I, ^{the third term of this} equation ^{dominates the} time-dependence. ^In practice, ^our experiments ^in-

volved ratios of IS/Ir 10- 3, ^{so that the self-beat}

term was much smaller than the random noise on the observed correlation functions. Such 7s//r ^{ratios are}

much lower than those usually ^advocated [37] ^{for the}

rather different case of monotonically decaying

correlation functions. The low Is/Ir ^ratios proved entirely satisfactory here, probably ^{because our} experiments employed ^methods recently ^described

for rapid ^data acquisition [38], involving amplifi-

cation of the small modulation of G(,r).

The wave number q ^{of the} capillary ^{mode ob-}

served is used as an input ^to the direct fitting

method : P ( w ) ^{is evaluated as a function} of the four surface properties ^{at a} specific q ^{value. A} good

estimate of q is thus central to the success of this

approach ^{to data} analysis. ^{In the} present ^work q ^was determined to 0.25 % [35].

The light scattering ^was significantly ^affected by

instrumental effects which arose from the finite extent of the laser beam on the surface [39]. ^The

observed spectrum ^{was a} convolution of P (w ) ^with

an instrumental function. For a laser beam of Gaussian profile ^{the observed} correlation functions

can be written as

where j3 is the standard deviation of the instrumental function in the frequency ^{domain and} f ( T ) ^{is the}

time dependence ^{of the} correlation function ex-

pected ^{from waves of the} selected, central q ^value.

3.1.1 Data analysis.

^{Two different} analyses ^were applied ^{to our} correlation data. Both involved the

use of non-linear least-squares fitting ^with appropri-

ate mathematical forms. The wave frequency wo and

damping ^{r were} determined using equation (13)

with an exponentially damped ^{cosine time} depen-

dence :

where the phase term 0 (small ^and negative ^for

these data) ^{accounts for most of the} deviations of

P(w) ^{from an exact} Lorentzian form [12]. ^Such

estimates of _wo and T are known to be unbiassed

[40].

The exact spectral ^{form of} equation (5) ^{was also}

used in fitting experimental ^{data. The} applicability

of such analysis ^to experimental ^{data is as} yet ^{in its} infancy : systems involving only yo and q ^{have been} investigated [26, 27], ^but the extension to film- covered surfaces or interfaces is relatively ^novel [13].

We refer to this approach ^{as « direct} fitting ^{». The}

observed correlation functions were analysed ⁱⁿ

terms of the four surface properties (yo, ^y’,

EO, e’) by fitting ^with equation (13) using ^{a time} dependence ^defined by the Fourier transform of P (w ) formulated as a function of these four proper- ties. The viscosity ^and density ^{of the} subphase ^were

assumed to have their accepted values. In detail we set

and use [26]

in equation (13). ^{Here the} parameter ^{A allows for} the possibility ^{of a self-beat or} second-order contri- bution to the observed correlation functions. Fitted values of A were always very small, ^if not zero, ^as

expected ^{from our low} 7s//r ^{ratios. In} analysing ^the

data presented below, ^{we therefore fixed A = 0.}

Detailed discussion of the sensitivity of the direct

fitting ^to the various surface parameters ^{is deferred}

to section 5. Tests with simulated data [24, 25]

suggested ^{that this} analysis ^{was stable} and robust.

For data of reasonable quality (random ^{errors on} G (,r ) - 1 % ^of amplitude A) ^{the surface} properties

used to generate ^{the data were} adequately ^recov-

ered. For only ^one combination of surface properties

were erroneous solutions found in these tests [25],

and even then the majority of fits gave ^accurate estimates of the properties. ^{This situation involved}

small values of y’ ^{and zero e : in the} poor fits

yo and E’ ^were excessive (the ^latter greatly so), ^while

the instrumental linewidth {3 ^was underestimated.

Constraining {3 ^{to be of} physically reasonable magni-

tude (> ^{half of} value found for good fits) ^{avoided the}

apparent secondary minimum in the sum-of-squares hyperspace quite efficiently. ^{In our} experiments,

several correlation functions were recorded under identical conditions at each molecular area studied.

Application ^of appropriate constraints to those few functions which gave surface properties differing

from the rest always yielded ^internal consistency ⁱⁿ

the experimental ^{results. Such} ambiguities only

occurred in cases where both the surface _pressure and y’ ^{were small.}

3.1.2 Instrumental effects.

^{The form of the instru-}

mental function is basic to the data analysis ; ^we

have therefore measured it for our apparatus. ^The time dependence ^{of the observed} correlation func- tion is of the form

where f (T ) ^{is the exact} theoretical form (Eq. (16)

with A ⁼ 0) ^{for waves of the} selected q ^value.

Experimental correlation functions for light ^scattered by capillary ^{waves on the free surface of water were} analysed (Fig. 2). G ( T ) ^{was divided} by f(T),

evaluated for the known properties ^{of water,} yielding h ( T ). ^{These estimates of} h ( T ) ^{were somewhat} noisy, particularly around the zero-crossing points ^of G(,r). ^However h ( T ) ^was clearly ^{linear in} T 2, ^the slope giving j8 ^{= 4 480} s-1 (within ^{2 %} of that from the direct fitting procedure). ^{This Gaussian} h ( 7- ) ^is

shown in figure 2, together ^with [/(T).A(T)],

which _agrees excellently with the observed data.

This comparison ^{involves no} fitting ^of f(,r). ^We

conclude that a Gaussian function provides ^an acceptable description of the instrumental effects upon G ( T ).

It has been suggested [39] ^that f3 ^can be calculated

a priori, given ^certain experimental parameters. ^The measured value just quoted agrees reasonably ^with

that computed ^{for our} apparatus. However because various other effects (e.g. vibrations, ^electronic distortion) might ^affect f3 ^we prefer ^to keep ^{it as a}

free parameter of the data fitting. We have exper-

imentally ^{confirmed the} theoretically predicted [39]

dependences ^of /3 upon q, yo and the laser beam diameter.

3.2 MONOLAYERS. - Our Langmuir trough ^was

’

machined from a single ^{block of} PTFE, ^{and was}

17 cm x 22 _cm, the subphase being ^{rather shallow} (depth - ³ mm) ^to damp ^out any long wavelength perturbations ^{of the} liquid ^surface. The radius of

curvature of the liquid ^{surface at the} laser beam was

> 1 km, having negligible ^{effect on the value of}

q selected by ^the optical apparatus [41]. ^The trough

1276

Fig. ^2.

^The measurement of the instrumental line-broadening function. A correlation function measured for the clean surface of water (q ^{= 395.0} cm-1) ^{is shown} (x), together ^{with the exact} theoretical correlation function f ( T) expected

for water (- - -). ^The chain line (- - -) shows the Gaussian form h ( T ) ^obtained by dividing the measured data by

the theoretical function. The full line ( ) indicates the product of the theoretical function and the Gaussian instrumental form [ f ( T ) . h ( T ) ] .

was mounted on a massive brass base and sur-

rounded by ^a perspex housing provided ^with optical

windows. Both base and housing ^had provision ^for

thermostatic control. The entire apparatus ^was mounted on a massive (2.3 tonne) optical ^{table with}

vertical and horizontal vibration isolation.

The glycerol-monooleate (purity > ^{99 %, 1-isom-}

er, Nu Chek Prep Inc) ^{was used as} supplied, ^{with no}

further purification. ^{It was} spread ^{as a solution in}

hexane of known concentration (- ¹ mg/ml). ^The subphase ^was pure ^water (resistivity > 18 MH cm),

from a Milli-Q (Millipore) ultrafiltration unit. Before the monolayer ^was spread ^the liquid ^{surface was} repeatedly swept ^and aspirated using ^{a Pasteur} pipette ^{with a low vacuum line to remove} any surfactant contamination. The monolayer ^isotherms

were generated by successive additions of 3 f.LI aliquots ^{of solution.} After each addition three mi- nutes were allowed to elapse before classical and

light scattering measurements were initiated, ^to

allow the solvent to evaporate ^{and the} system ^to

equilibrate.

The constant-area (or successive addition) ^method

of tracing ^out monolayer ^{isotherms has} recently

been discussed by McDonald and Simon [42], ^who

argue certain advantages ^{over the more usual com-} pression technique, amongst which is the fact that

collapse ^{occurs at the} equilibrium pressure of the

lipid, metastable states not being observed. However in our studies two other problems ^{led to} dissatisfac- tion with the compression ^{method :}

(1) ^when using ^{a barrier to} compress ^a monolayer

to high surface pressures (a ⁴⁰ mN/m) ^some prob-

lems were encountered with film leakage ;

(2) ^{as the} barrier approached ^the illuminated area

of the surface the meniscus might have altered the surface curvature and thus the q ^studied.

The method of successive additions completely ^eli-

minated both of these problems. As will be discussed below the successive addition method gave ^essen-

tially identical results to compression (after ^relax-

ation of the monolayer ^to equilibrium following compression).

For comparison with the results inferred from the

light scattering observations the surface tension was

measured with a 2.5 cm square roughened platinum

foil Wilhelmy plate ^{and a} Joyce ^Loebl microbalance.

The system ^{was tested} using ^clean liquids. ^The

surface pressure ^was determined to better than

± 0.1 mN/m.

4. Results.

Light scattering observations were carried out as a

function of molecular area at a fixed q (different ^for

each film) ^{for five} separate monolayers ^{at T =}

20 ± 0.1 °C. For four films light scattering ^measure-

ments were continued in the collapse ^{state over a}

range of q (220 ^{to 1 500 cm’} 1). During these exper-

iments, ^at each surface concentration or q value

studied, ^five separate correlation functions were

recorded consecutively ^{with no} selection. The data shown are the averages of the fitted parameters ^for these five functions, ^{the errors} being ^{standard errors}

on the mean.

Equilibration ^{of the} monolayer ^{after each addition}

of lipid ^{solution was checked} by observing ^the

classical surface pressure. The q ^{scan in the} fully compressed ^{state lasted - 1} hr. after first reaching

the collapse ^{state. Over this time the classical}

7T tended to decrease slightly (- ^{0.6 mN/m over 1-} 2 hr.), presumably ^{due to} slow relaxation effects in the monolayer. ^{It was to} avoid such problems ^that light scattering experiments ^{were carried out for a} single q ^{for each} of several separate monolayers. ^As

will be seen below, ^these experiments gave entirely

consistent results.

The measured classical (Wilhelmy plate) ^{w - A} behaviour, ^{shown in} figure ^{3 as a cubic} spline interpolation ^{of the measured} points, agreed ^well

Fig. ^3.

The measured ^{7r -} A isotherm inferred classi-

cally ( ) ^and by light scattering. ^{The 03C0 values} (x)

determined by the direct analysis of the correlation functions (observed at q ^{= 320.6} cm-1) agree excellently

with the classical data. Values (9) ^found using ^the approxi-

mate relation (Eq. (21)) ^{with the} capillary ^{wave fre-} quencies ( wo ) ^are discussed in section 5.

with literature data [20-23], all of which involved

compression ^{methods. The} only significant ^differ-

ence involved a small extension at low 7T’ to rather

higher ^{A than for some} isotherms. In particular, ^the collapse ^area (28.5 A2) ^{accorded well with} published values, although ^{the observed} collapse pressure

(verified for several monolayers) ^{was some 2 mN/m}

above certain literature values (e.g. [23]). ^{It seems} unlikely ^that this difference arose from relaxation processes within the monolayer, ^{as the} collapse

pressure fell _very slowly with time. The discrepancy might ^{be due to a} temperature difference, ^most published ^data being ^{for 25 °C} (not ²⁰ °C).

We first present experimental ^{data for a} single

GMO monolayer ^{at one} surface wavenumber:

q ⁼ 320.6 cm- 1. Correlation functions observed at various surface concentrations are shown in figure 4, illustrating the basic data underlying ^the present analysis. The noise on the individual correlation functions was typically ^{2 % of the} amplitude ^{of the} time-dependent part ^{of the} function, falling ^some-

what (- 1 ^{% in} fully compressed state) ^{with the}

increase in scattered intensity ^{as the} tension fell on

monolayer compression. ^The changes ⁱⁿ frequency

and damping of the functions with surface concen-

tration are clearly ^evident.

Whilst we do not place great emphasis ^{here on the} capillary ^wave frequencies, the variations of coo and r are shown in figure ^{5. The} reproducibility ^{of both}

w o and r varied with the monolayer ^{surface concen-} tration, ^due partly ^{to the} decreasing ^{noise on the}

data upon monolayer compression, ^{but also to}

fluctuations within the monolayer itself. The damp- ing ^{often increased} dramatically (compared ^{to that}

for the clean subphase) upon the addition of even a

single ^{3 RI} aliquot ^of film-forming ^solution (i.e.

A - 700 A2). We believe that these variations, ^and

the rather greater ^scatter in wo ^at low surface concentrations arose from the existence in the ex-

panded monolayer ^of separate regions having ^rather

different lipid concentrations, causing fluctuations in the light scattering signal. Differences in the values of the various surface properties ^{in this} region support ^this suggestion (see ^Sect. 5).

We turn now to the principal results of this paper : the surface properties yo, y’, Eo and £’ derived by

the direct fitting ^method (Eq. (13) ^with Eq. (16)).

The parameters of the fit were initially ^{not con-} strained, apart ^from requiring ^{the four surface} properties ^{to be} positive. ^{In most cases these initial}

fits yielded consistent values of the surface properties

for the five individual correlation functions observed.

However for very expanded monolayers (A >

75 A2) ^{this was not} always ^the case, probably ^{due to}

the separated regions just mentioned. Averaging ^the

correlation functions at these large ^{molecular areas}

led to poorer fits, ^shown by correlated, oscillatory

residuals, ^not evident for the individual correlation

1278

Fig. ^4.

Various correlation functions observed at q ⁼ 320.6 cm-1 for a monolayer ^{of GMO :} (a) ^{the clean} subphase surface ; (b) ^{a rather} expanded monolayer, ^{at a}

molecular area of 140 A2 ; (c) ^{in the} fully compressed state, ^{at A =} 28.1 A2. The lines show the best fit functions derived from the direct analysis. The residuals of these fits

are also shown.

functions (Fig. 4). ^In practice, ^at all surface concen-

trations, ^{the data were} analysed ^as separate ^corre- lation functions.

Surface pressure values (Fig. 3) ^{were derived} using light scattering values for the tensions of both the monolayer covered surface and the clean sub-

Fig. ^5.

Capillary ^wave frequencies ( wo ) ^and damping

constants ( r), ^as functions of surface concentration for q ⁼ 320.6 cm-1. The errors upon wo ^are smaller than the

plotted points. ^{The lines are} discussed in section 5.

phase. For the clean subphase yo agreed ^{well with}

the accepted ^{value. At} large ^{surface areas} per molecule (A > ⁷⁵ A2) ^the light scattering ^tension

values tended to be somewhat scattered, ranging

between 70 and 74 mN/m, compared ^{to a value for}

the clean subphase ^{of 73 mN/m at 20 °C. The} problem ^{is that} identified in tests using ^simulated

data (see ^Sect. 3) ; ^{as in that case,} constraining f3 ^{to exceed a} reasonable value led to consistency ⁱⁿ

the fitted values of the surface properties. ^{At these}

molecular areas the mean values of the surface pressure ^were consistent with the classically

measured data. For more compressed monolayers

yo ^was quite precisely determined : _e.g. at a surface pressure of 35 mN/m the fractional error in tension

was - 0.3 %.

To show the surface properties ^{derived from} light scattering ^at large ^{molecular areas, the} remaining

data are plotted against ^surface concentration (here

we write T S =1 /A ^to avoid confusion with the

capillary ^wave damping). ^For TS ^s greater ^{than the} collapse value, ^the monolayer properties ^{were, with-}

in _errors, identical to those at that point. However,

for clarity these data are plotted separately.

The light scattering estimates of the dilational modulus eo (shown ⁱⁿ Fig. 6) ^are those returned by

the direct fitting method, ^{and are not related to the} logarithmic derivative (Eq. (4)) ^{of the} light scattering

03C0- A isotherm. This derivative would not differ

significantly from the classical variation shown. Also

plotted ⁱⁿ figure ^{6 are} eo values measured by light scattering for q ^{= 533.8 cm-1} (for ^{a different mono-} layer). ^These data will be compared later with the classical variation.

Fig. ^6.

The dilational modulus _Eo as a function of

r,. The classical variation (-) ^is compared ^with light scattering values of Eo (shown ^{for two} monolayers ^studied

at the same temperature). ^The classical so goes ^{to zero at} the collapse ^{area. One} monolayer (x) ^{was studied at}

320.6 cm-1, ^{the other} (0) at q ^{= 533.8} cm-1: for clarity

error bars upon the latter data ^are omitted.

Figure 7 shows the variation of y’, together ^with

the differences between light scattering and classical surface pressures :

emphasizing ^the agreement ^of yo with the classical values. The e’ data are shown in figure ^8. Despite

the relatively poor precision ^{of the} data, ^{the two} surface viscosities clearly behaved very differently

upon monolayer compression, presumably dupe ^to

differences in the underlying ^molecular mechanisms.

Fig. ^7.

The variations with TS ^{s of the difference}

AlT (see Eq. (18)) ^{and the} transverse shear viscosity ^{of the} monolayer, y’, ^observed for q ^{= 320.6} cm-1. Zero values

are indicated by ^{vertical downward arrows. The line} represents the behaviour expected, ^{based on} analysis ^of

the data of figure ¹² (see ^Sect. 5).

There _are, of _course, no classical values with which these data can be compared.

The y data for the other compression experiments

are shown in figures ^{9 to 12. The same form of}

variation as for the q ^{= 320.6} cm-1 results was

evident in all the data :

9 at large molecular areas å 7T was compatible

with _zero, whereas close to, ^{or in the} fully compres- sed state it tended to be non-zero (effect increasing

with q) ;

9 in all cases the transverse surface shear viscosity (y’ ) ^rose rapidly ^{from zero} for the clean subphase ^to

a maximum around 100 A2/molecule, thereafter fall-

ing towards the collapse point (28.5 A2lmolecule).

The ^E data for the higher q ^{values are not} shown, ^as

the precision of determination of both elastic and

viscous parts decreases with q. The features shown

for the q ^{= 320.6} cm-1 data were broadly repro-

duced at higher q : Eo substantially ^{exceeded the}

1280

Fig. ^8.

The variation of the dilational monolayer viscosity, e’, ^{observed at 320.6} cm-1. Vertical arrows

indicate zero values of the viscosity. ^e’ changes by ^an

order of magnitude ^{as the} monolayer ^is compressed ; ^the

variation is discussed in the text.

classical value at small F,, ^{where E’ was} small, ^while

at rs 2: ^0.02 molecule/A2 both co and E’ ^were poorly determined, ^but larger than in the expanded ^mono- layer.

In the fully ^{condensed state the} physical properties

found by ^direct fitting ^were quite consistent for the several monolayers ^{studied. The data were thus} averaged ^to improve ^the precision, and hence the

reliability ^{of the} subsequent analysis. The slow drifts in the Wilhelmy plate pressures ^over these rather

long experiments ^{had to be allowed for. As} light scattering observations were initiated at each q value, the classical _pressure was noted : over the

experimental ^{durations 03C0 decreased} roughly linearly. ^The change in classical tension was sub- tracted from the light scattering yo values. To ensure

the absence of any other systematic drifts, light scattering observations were repeated ^at the q ^value

studied first (the lowest) ^after completion ^{of each}

scan over q : the surface properties ^{inferred from the}

two separate ^{sets of data at} this q only ^{differed in the}

value of yo obtained.

We concentrate here upon the trends in tension and the associated surface viscosity. The observed

Fig. ^9.

^The variations with F, of A 7r and y’ ^for

q ⁼ 533.8 cm-’. Zero values are indicated by ^vertical

arrows.

yo values increased by - ^{1 mN/m} as q increased,

whereas y’ ^decreased by ^{about an order of} magni-

tude. Figure ^{13 shows the} frequency dependence ^of

y, rather than that upon q. The frequencies ^{used in}

the analyses ^{were the} capillary ^wave frequencies ( w o ). ^{The several} fully collapsed monolayers yielded essentially ^identical frequencies ^for capillary ^waves

of given q. The behaviour of the dilational modulus

was not at all well determined : all that can be said is that both elastic and viscous parts ^{were of} large magnitude. ^No variations of e with q ^{could be}

discerned.

5. Discussion.

5.1 ANALYSIS OF THE CLASSICAL ISOTHERM.

MacDonald and Simon [42] ^have recently argued

that in a 7T - 7B plot ^{of the isotherm} subtle effects indicative of monolayer phase changes ^{are more} evident, ^and thermodynamic analysis ^{is more direct.}

Here it also eases comparisons with the other surface

properties (Figs. ^{6 to} 12). ^{We thus} replot the classical

data in figure ^14.

Fig. ^10.

The variations of å 7T and y’ with TS ^for

q = 745.0 cm-1.

The ir - TS variation is basically piece-wise ^linear

for GMO (as ^{found for DMPC} [42]). ^{For 0.01} 7"g ^0.015 molecule A2 ^it approximately ^conforms

to a perfect gas law. There are then two successive linear regions, ^{the break in} slope occurring ^{at about}

40.5 A2. The slopes ^of the v - F, plot ^{in these two} regions ^{are - 4.5 kT and 7 kT} respectively. Analysis

of published ^{isotherms for GMO} [20-23] ^confirms

both these gradients ^{and the location} of the break-

point. ^{These features seem to be} well-founded,

whereas the extent and magnitude ^{of the initial}

increase at low rs ^{seem somewhat more variable.}

The all-trans GMO molecule is about 22 A long [43], while the collapse ^{area of 28.5} A2 implies ^a

lateral dimension about 5 A. Thus the ^{7r -} rs plot departs ^{from zero at an area} (- ⁸⁸ A2 ) ^{rather less}

than that of a GMO molecule lying upon the surface, probably ^{due to some} rotational isomerisation in the

acyl ^chain. The molecular area at the break in slope,

41 A2, is much less than this, ^and the molecules

presumably ^{can no} longer lie upon the surface, being

forced to adopt ^{a more «} upright » orientation, ^the

Fig. ^11.

The variations with F, of ð.7T’ and y’ ^for

968.5 cm- 1.

hydrocarbon ^chains beComing detached from the

liquid ^surface.

The change ⁱⁿ slope ^{of the 7T -} F, plot ^at

- 41 A2 indicates a change ⁱⁿ monolayer behaviour, perhaps sufficiently ^{distinct to merit the} description

« phase change ^». MacDonald and Simon [42] ^have

advanced an apparently generally applicable ^in- terpretation of the observed linear 03C0- TS ^be-

haviour. Upon compression ^{of an} amphiphile ^mono- layer the number of possible configurations ^{of each} hydrocarbon ^chain (W) ^{is reduced, the} polar ^head

group being ^{tied to the water} surface. Thus the entropy ^{of the} film, ^{which is a function} of molecular area, decreases and w increases. Assuming ^{that the}

number of configurations ^{available to an n-carbon} acyl ^{chain is}

where A is the molecular area in the monolayer ^and

g is ^a proportionality constant, MacDonald and Simon [42] ^{derive an} expression

which offers an immediate interpretation ^{of the}

linear ^{7r -} T plot.

1282

Fig. ^12.

The q ^{= 1 160} cm-1 data for A 7T and y’.

The gradient of 7 kT in the condensed monolayer phase ^is attributed to some 7 C-C bonds freely undergoing rotational isomerisation. The 18-C atom oleic acid moiety ^{of the GMO} molecule has a double bond at the 9-10 position, ^{so that the} configurational

freedom _{may be} restricted to that half of the GMO molecule remote from the water surface. Where

d7r/dF, - ^{4.5 kT the.} acyl ^{chain is more} restricted.

This cannot be due to increased _pressure from

adjacent molecules, ^and presumably arises from a

tendency of the molecules in the more expanded monolayer ^to partially recline upon the surface, reflecting ^the appreciable ^adhesion between the

hydrocarbon chain and water. The configurational

freedom of that part ^{of the} acyl ^chain lying upon the water would be considerably reduced, explaining ^the

lower d7rldF,.

5.2 CONSISTENCY WITH WAVE FREQUENCY AND DAMPING. - As discussed above (Sect. 2), ^these

four surface properties ( yo, y’, Eo ^and e’) all affect the propagation ^of capillary ^{waves to some extent.}

We have checked the consistency ^{of the surface} properties ^deduced by ^{the direct} fitting ^{with the}

Fig. ^13.

^The frequency dependence of yo and

y’ ^for fully compressed monolayers. ^{The lines} represent ^a least-squares ^{fit to the data} using the functional forms of the Maxwell model (Eqs. (10) ^and (11)).

observed ( coo, T ) ^{of the} capillary ^waves by ^substitut- ing ^the experimental values of E and y into the

dispersion equation (Eq. (2)) ^and solving ^for wo and r. The variations of _mo and r thus found, ^{shown as} cubic spline interpolations ⁱⁿ figure 5, agree well with the observed frequencies ^and damping ^con-

stants. The only significant differences occur be- tween 0.01 Ts ^0.02 molecule/A2, ^{where the} pre- dicted variation falls slightly ^{below the observed}

r values. Decreasing Eo by ^{1 mN/m} (within ^the experimental error) ^{would remove the} discrepancy.

We conclude that the direct analysis ^method yields

values of the surface properties ^{which are} entirely

consistent with the observed propagation ^of capillary

waves upon the monolayer covered surface.

The present experiment yields ^{non-zero values for}

all four surface properties, ^{which do not} always

concur with the classically determined values (e.g.

Eo). ^In general ^{it would be} impossible ^to uniquely analyse the mo and r data ^a priori ^{in terms of the}

surface properties. ^{There must} inevitably ^{be am-} biguities in any such analysis. ^{It is not} sufficient to

show that _wo and r values can be interpreted ⁱⁿ

terms of two surface properties assuming ^{others are}

equal ^to their classical values (e.g. yo, so) ^{or are zero}

(e.g. y’). Nearly always, ^other combinations of

surface properties will exist which could account for