Thermal instability in lamellar phases of lecithin : a

planar undulation model

A.G. Petrov, G. Durand

To cite this version:

A.G. Petrov, G. Durand. Thermal instability in lamellar phases of lecithin : aplanar undulation model. Journal de Physique Lettres, 1983, 44 (18), pp.793-798.<10.1051/jphyslet:019830044018079300>. <jpa-00232264>

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L-793

Thermal instability in lamellar phases of lecithin :a planar undulation model

A. G. Petrov (1) and G. Durand (2)

(1) Institute of Solid State Physics, Academy of.Sciences, Sofia, Bulgaria(2) Laboratoire de Physique des Solides, Université de Paris-Sud 91405 Orsay, France

(Re~u le 9 juin 1983, accepte le 20 juillet 1983)

Résumé. 2014 En chauffant un échantillon multilamellaire de lécithine d’0153uf hydratée, en géométrieplanaire (lamelles normales aux plaques), on observe d’abord une instabilité à striation périodique,puis les domaines « en épi ». Nous montrons qu’une contraction normale des lamelles chaufféespeut produire une instabilité d’ondulation des lamelles en géométrie planaire comme en géométriehoméotrope. En supposant un ancrage fort des lamelles sur les plaques, l’absence de contrainte surla surface extérieure des lamelles localise les ondulations près des plaques. La période spatiale etle seuil de ces ondulations sont estimés. L’observation d’échantillons de DLL hydratée, de diversesépaisseurs, nous permet d’identifier les striations périodiques avec cette instabilité d’ondulationslocalisées.

Abstract. 2014 Under heating, a hydrated egg lecithin multilamellar sample in a planar geometry (lamel-lae normal to the plates) shows a periodic striation instability, followed at higher heating by the so-called « ear-like » domains. We first demonstrate that a normal lamellae contraction under heatingcan result into a lamellar undulation instability in a planar sample, as well as in a homeotropic geo-metry. Assuming a strong anchoring of the lamellae on the plates, with the force free condition on thelamellae outer surface, the undulations are localized close to the plates. The spatial period and thethreshold are estimated. Observations of hydrated DLL samples of various thicknesses allow usto identify the periodic striations with this localized undulation instability.

J. Physique - LETTRES 44 (1983) L-793 - L..798 15 SEPTEMBRE 1983,

Classification

Physics Abstracts61.30E - 62.20

The so-called « ear of wheat-like » domains have been observed [1] in hydrated egg lecithinmultilamellar samples subjected to an A.C. electric field. A further study of this phenomenon [2]has demonstrated that : a) the ear-like domains are generated by Joule heating and b) below thethreshold for ear-like domains, periodic striations occur above a lower heating threshold oftypical temperature jump AT ’" 1 °C. To explain this observation we propose in this letter alayer undulation thermal instability analogous to the one currently observed in smectic liquidcrystals under dilation [3], or else in heated lamellar lecithin phases [4], but in a different geometry.

Let us first resume the fmding of reference 2.

a) At room temperature (T = 27 °C) a lamellar phase of hydrated egg lecithin (10 % wt.H20) placed in between two transparent Sn02 coated parallel electrodes (sample thicknessd ~ 5-50 ~m) takes spontaneously an « homeotropic » alignement, with the layers parallel to

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:019830044018079300

L-794 JOURNAL DE PHYSIQUE - LETTRES

the electrodes. The first effect of a low frequency (1 kHz) A.C. electric field seems to reorient thelamellae perpendicular to the glass plates, i.e. to induce a homeotropic to planar transition.After that, for voltage of about 60 V, most of the effects observed in planar monodomains arejust due to Joule heating, from the relatively large A.C. current (10-100 mAjcm2 in the aqueousphases) flowing through the sample. This was demonstrated by the observation of a non aqueouslamellar phase prepared with synthetic dilauroyl lecithin (DLL, Fluka) and ethylene glycol :in absence of water, the current through the sample was much lower, and no instability could beinduced up to applied voltages - 175 V. On the other hand, by purely heating the sample, insta-bilities in the form of rows of focal conics, merging into « ear-like » domains under further heating,were observed.

b) Below the threshold heating for structure disruption by focal conics, at a much lowerAT - 0.5 to 1 ~C, quasi periodic striations occur. An example of these striation is shownon figure 1, which represents the picture of a thin (d = 20 ~) planar monodomain of hydratedDLL (10 % wt. H20) at T = 27 ~C, seen between crossed polarizers under a microscope. Picture1 a is more or less uniformly dark, because polarizer and analyser, parallel to the edges, are orient-ed along the optical eigen-axes of the planar uniaxial texture. More precisely, the lamellae arenormal to the plates and parallel to the small side of the picture. This is deduced from the obser-vation of the direction of easy motion of bubbles in the sample. Heating is usually achieved byremoving the thermal filter of the microscope. For a temperature increase of AT = 1 ~C (measur-ed by thermocouples inside the sample), one observes (Fig. lb) the periodic striations as a sys-tem of dark lines parallel to the optical axis, separated by brighter lines. The average period(distance between three adjacent lines) varies from 4-7 ~m, several times lower than the widthof the ear-like domains. This instability is reversible, i.e., it disappears slowly by lowering thetemperature. At higher thermal excitation, the periodic striations have an organizing action onthe ear-like domains. The focal conics are arranged in rows, with the ellipses in the plane of thesample and the hyperbolae along some of the periodic striations.

Resuming these observations on DLL, we look now at the periodic striations. By rotatingslightly the polarizer, we note a lateral shift of the dark lines. As these lines are oriented perpendi-cular to the lamellae, it is reasonable to think that the periodic striations could be explained by athermal undulation instability of the lamellae seen from the side. Undulation instability underheating was indeed observed [4] in lamellar lecithin samples with very low water content (2 % wt.)in the homeotropic geometry. The thermal thickness expansion coefEcient j8 of lipid bilayers in egglecithin is known [5] to be negative :/!==2013 2 x 10’~/~C. This contraction is one order ofmagnitude larger than, say, the thermal volume expansion of water (0.207 x 10- 3/~C) and isprobably related to the increased disorder of lipid chains in the bilayers. When heated betweentwo fixed parallel plates, the lamellae are submitted to a dilative strain. Above the instabilitythreshold, the lamellae undulate to fill better the space. Elastic free energy is now stored also aslamellae curvature energy rather than uniform lamellae dilation. The new point in the presentplanar geometry is that the lamellae are not parallel but perpendicular to the fixed boundaryplates. Let us show that, provided the lamellae anchoring is strong enough on the electrodes,one can also observe an analogous undulation instability in planar orientation, but localizedclose to the plates.Our geometry is described in figure 2. z is the normal to the lamellae. The plates are parallel

to y (in the plane of the lamellae) and z. x is the direction of observation. The striations are observ-ed parallel to z, i.e. can be described as an undulation u ~ exp(iqy) [q/~y, u//z], where u is thenormal distortion of the lamellae. We call B ~ 108 cgs [6] the constant pressure elastic modulusof the 1-dimensionnal « crystal » of lamellae. Let us call K (~ 10-6 cgs) the Franck curvatureconstant of the layers. The quantity £ = (KjB)1/2 is the penetration length, of the order of thedistance between lamellae. The measured [4] ~, is in the range of 100 A for almost dry samples,and could be a little larger in our case, to account for the higher water content of our samples.

L-795THERMAL UNDULATIONS IN LAMELLAR LECITHIN

Fig. 1. - Planar texture of hydrated DLL : d = 20 gm, T = 27 "C. The lamellae are vertical parallelto the left side elongated bubble. a) Before heating ; b) after heating (AT - 1 ~C), showing periodic stria-tions. The small side of the picture corresponds to 250 um.

L-796 JOURNAL DE PHYSIQUE - LETTRES

The problem of layer contraction under heating, with fixed boundary plates, is exactly the sameas the one of layers with fixed spacing, attached to linearly expanding plates along z, whichsimplifies the notations. We write the elastic free energy of the lamellar system as [7] :

The first term represents the layer compression, in presence of a layer tilt. Expanded in u, itcontains a third order term which will lead to the instability. The last term is the layer curvaturecontribution. In the linear regime, u obeys the Euler equation :

We look for a solution u ~ exp(iqy) W(x) V(z).This implies the relationship :

- - .........--

The free edges of the layers, in contact with water or some desordered part of the sample, do nottransmit uniaxial forces i.e. au/az = 0 at these boundaries. Calling L the z-extension of the planarmonodomain, V must be of the shape

writing now W = exp(kn x), we find the dispersion relationship : kn = q2 ± ~Tc/~L. When theplates expand, we assume that the lamellae stick to the plates and remain parallel to x, y (stronganchoring). To calculate the exact profile W(x), we must Fourier analyse the disturbance u =ATz on the ~,. To simplify, we keep only the first term V 1 = Vo i sin 7rz/L. In the simplest 2-

1 + ~ / ~ B1/2dimensionnal case (q = 0), we get ~ == ± 20132013 ~ , which shows that the layer disturbance(R )~ g

~/2 ~,L y

propagates only over a distance C ~ (L~,)lj~ from the plates. C is in the range of a few ~m, for typicalL ~ a few hundred lim. In thin samples (d 2 C ~ 5 ~m), all the bulk is under dilative stress.In thick samples (d &#x3E; 2 C), just the two boundary layers close to the plates are under stress (seeFig. 2).Assume now a small layer undulation u’ ’" exp(iqy) exp(kx) uniform in z. It obeys the free edge

condition ~u/~~ = 0. In absence of dilative stress, one must have kl = q2, i.e. a small perturba-tion of lamellae orientation on the plates is attenuated in the bulk on its wave length. We can nowmake a simple prediction for the threshold undulation instability.

a) Thin samples (d ~ 2 C). There exist a uniform dilation r~u/az ~ ~ AT. The undulation

appears when the non ... and compression - 1 au ~_u’ 2

+appears when the non linear coupling term between tilt and compression - 2013~ 2 au [~ax) +appears w en tenon lnear coup lng term etween tl t an compressIon - 2 À,2 oz ax +

au’ 2 1 aau’ a2u’ 2+ ~ay) _ compensates for the additional curvature energy - - 2 + f2013r) ? , i.e. for :

Wj 2~~; ~;J

L-797THERMAL UNDULATIONS IN LAMELLAR LECITHIN

Fig. 2. - Planar geometry, with linearly expanding plates. The lamellae are under dilative stress on boun-dary layers of thickness (AL) 112 close to the plates. Two undulations along y can be amplified above thre-shold to relax these stresses.

The undulation penetrates all the bulk, to relax all the stresses, as soon as q 7r/~. At lowestthreshold q can be taken equal to zero, which gives : # LlT ’" À.2 ~2/d z. The appearance of theinstability should depend very much on the boundary orientation fluctuations.

b) Thick samples (d &#x3E; 2-C). q must be small enough for the undulation to penetrate the stressedboundaries, but large enough to leave the unstressed central part unperturbed - this means

À. 2 ir2 7r2 Aqt - 1. As qd &#x3E; 1, the threshold is defined by : ~8 AT ~ . - = 1t2 2013, independent from thethickness d. The undulation instability should appear with a period - C = (ÀL)1/2. With thepreviously quoted figures, one expects P LlT ’" 10-3 for L ~ 200 Ilm and A - 100 A, i.e. AT ~1 ~C for ~ ~ ~ I = 2 x 10’~C.and~ ~ 1.5 gm. These figures compare well with the observed data.

Finally, we must expect in absence of undulation a direct non linear effect of the layer tilt in thex direction, to relax locally the dilative stresses, in the case of thick samples, at the junctionbetween the stressed boundary and the central unstressed region. This could lead to the nucleationof layer dislocations or focal conic lines and could explain the nucleation of the ear-like domains.To check our model, we have measured the periodic striation threshold in samples of 7 %

hydrated DLL of various thicknesses corresponding to the case of thick sample. A T seems toincrease with d, from AT = 0.75 ~C for d = 5 ~m up to AT = 1.8 °C for d = 110 ~m. However,this increase compares with the large dispersion of threshold determination, which can be ashigh as 100 %. We can consider that the threshold is more or less thickness independent. Moreaccurately, we find that the striation period is equal to 6 ± 1 ~m independent of thickness for dvarying from 5 to 110 ~m. The lateral extension L was comparable for these samples, in the rangeof 200-300 flm. In practice, the period is found equal to 4 C. These two results on the thresholdand the period are in agreement with the prediction of the planar undulation model for thicksamples. To be complete, let us mention that the ear-like domain period, observed at higherheating, is practically equal to d between 110 and 20 pm, and slightly larger (16 pm) for the thinner5 ~m sample. The best check of our model, however, is the observation, for d = 110 ~,m, of twosets of striations. The first one is focused just below the upper plate, the other one just above thelower plate. Varying the focus, one sees that two dark lines from the lower striations collapse intoone dark line of the upper one and vice versa. This can easely be understood : the two layer

L-798 JOURNAL DE PHYSIQUE - LETTRES

undulations close to the plates must be out of phase, to cancel their residual distortion in theunstressed central part of the sample.

In conclusion we have shown that, in presence of a strong anchoring, a planar lamellar systemundergoing a heating contraction, can present an undulation instability which relaxes the dilativestresses localized close to the plates. The expected behaviour (threshold, period, localization)of this lamellar undulation instability compares well with our observations on the lamellarhydrated phase of DLL. Further observation of this instability is necessary to check other pointsof our model, like for instance the strong anchoring hypothesis. We believe however that theobservation of this thermal instability can provide useful information on the mechanical pro-perties of lipid lyotropic phases and, possibly, on biological lamellar textures like myelin orchloroplasts.

References

[1] VAL’KOV, S. V., CHUMAKOVA, S. P., Kristallographia 27 (1982) 1198.[2] PETROV, A. G., NAYDENOVA, S., to be presented at the fifth Liquid Cryst. Conf. Soc. Countries, Odessa

(1983).[3] DELAYE, M., RIBOTTA, R., DURAND, G., Phys. Lett. A 44 (1973) 139.;

CLARK, N. A., MEYER, R. B., Appl. Phys. Lett. 22 (1973) 493.[4] POWERS, L., CLARK, N. A., Proc. Nat. Acad. Sci. 72 (1975) 840.[5] RAND, R. P., PANGBORN, W. A., Biochem. Biophys. Acta. 318 (1973) 299.[6] EVANS, E. A., HOCHMUTH, R. M., in Current topics in membranes and transport, Vol. X, A. Kleinzeller

and F. Bronner, eds. (Acad. Press., N. Y.) 1978, p. 1.[7] RIBOTTA, R., DURAND, G., J. Physique 38 (1977) 179.