# Edge Dislocation Arrays in Swollen Lamellar Phases

Frédéric Nallet, J Prost

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Frédéric Nallet, J Prost. Edge Dislocation Arrays in Swollen Lamellar Phases. EPL - Europhysics Letters, 1987, 4 (3), pp.307-313. 10.1209/0295-5075/4/3/009. hal-04186347

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*Europhys. Lett.*, **4** (3), pp. 307-313 (1987)

# Edge Dislocation Arrays in Swollen Lamellar Phases.

F. NALLET and J. PROST

Centre de Recherche Paul Pascal, Domaine Universitaire - F-33405 Talence, France

(received 11 February 1987; accepted in final form 10 April 1987)

PACS. 61.30. – Liquid crystals.

PACS. 61.70J – Etch pits, decoration, transmission electron-microscopy and other direct observations of dislocations.

PACS. 62.20D – Elastic constants.

**Abstract.** – We study the array of edge dislocations arising in wedge-shaped smectic samples. This allows the measurement of the smectic penetration length $\lambda$, together with more microscopic quantities. Our results on the swollen lamellar phase of a quaternary mixture close to a microemulsion support larger $\lambda$ values than those recently obtained by high-resolution X-ray scattering on this system.

Quaternary mixtures of water, oil, surfactant and cosurfactant (*microemulsions*) exhibit a rich variety of phases. Among these, several isotropic ones, where the interface between water and oil is thought to be highly curved towards one or the other and lamellar (smectic) phases, where water and oil lamellae are separated by an essentially flat interface [1]. The latter may often be swollen by water or oil with the reticular distance increasing from a few nanometers up to nearly a micrometer in some cases [2, 3]. The swelling stops at the occurrence of a first-order transition to an isotropic phase. It is of interest to know the behaviour of the elastic constants characteristic of smectics as swelling proceeds. The elastic constants measured in a given experiment can be profoundly modified by structural defects such as dislocations [4]. Unless the experimental regime is such that dislocations cannot move [5], it is hard to assess the exact influence of defects on apparent elastic constants values. In this letter, this drawback is bypassed by the direct observation of the defects themselves. The first part is devoted to the theoretical presentation of a model: we show that the smectic penetration length $\lambda$ [6] controls the detailed spatial distribution of edge dislocations in a suitably prepared sample. In a second part, results obtained with a lamellar quaternary mixture are given and discussed.

In contact with a clean, polished glass wall a smectic liquid crystal often tends to orient with layers parallel to the wall (homeotropic alignment). In a cell of variable thickness there is a conflict between the homeotropic alignment requirement and the need for a constant layer spacing: as a result, defects arise. The simplest is the edge dislocation, i.e. the abrupt termination of one (Burgers vector $b=1$) or more (Burgers vector $b>1$) smectic layers. In a wedged cell an array of such edge dislocations is expected and has already been observed [7, 8]. We argue in the following that the ideal array is not always made of dislocations with Burgers vectors equal to unity.
We consider a typical Newton rings geometry: the convex part of a spherical lens (radius of curvature R) is set on a plane plate. We choose cylindrical coordinates with the origin at the contact point, the (r, θ)-plane corresponding to the surface of the plate. The sample is confined between the two surfaces z = 0 and z = h(r) = r²/2R (r ≪ R). Let l be the (slowly varying) distance between two adjacent dislocations, b their Burgers vector, and a the layer spacing. Clearly

$$ \frac{(r+l)^2}{2R} = \frac{r^2}{2R} + ba, \qquad (1) $$

thence, as long as $l \ll r$, $l = Rba/r$: the measure of the distance between dislocations is also a measure of their Burgers vectors.

Let us now show that b will in general be a function of position. To that end, we shall calculate and minimize the elastic energy per unit length of the array of dislocations (¹).

The orders of magnitude of the relevant lengths (R, h(r), λ, ba) in this experiment allow us to estimate the total elastic energy in two steps, studying first the strain field of a dislocation, then that due to the slowly varying thickness of the cell. More precisely:

The strain field due to the very presence of a dislocation is taken equal to the solution obtained by de Gennes [9] in the case of an isolated dislocation in an unbounded medium. This procedure makes sense because the strain field, decaying as $\exp[-x_{\parallel}^2/4\lambda|x_{\perp}|]$ ($x_{\parallel}$ is a coordinate parallel to the layers and perpendicular to the dislocation core, $x_{\perp}$ a coordinate perpendicular to the layers), is negligible outside the parabola $x_{\parallel}^2 = 4\lambda|x_{\perp}|$. Thus, whenever the distance l is large compared to the range of the parabola (i.e. $l \gg \sqrt{8\lambda h}$ for a dislocation confined on a thickness h), strain fields of neighbouring dislocations are not overlapping and each dislocation may be viewed as isolated. Furthermore, Pershan [10] considered the case of a dislocation confined on a thickness h between two parallel planes. As far as its contribution to the elastic energy is concerned, the strain field is controlled by a large wave vector cut-off and, therefore, depends very little on the actual boundary conditions, provided that $h \gg b^2 a^2 l^2 \pi^2 \lambda$: the unbounded-medium approximation is legitimate.

Between two adjacent dislocations, outside the parabolas, the strain field is not zero: there is a fixed number of smectic layers for a varying cell thickness. Let $\bar{h}/a$ be that fixed number; the strain field u is constrained by the relation

$$ \int_{0}^{h(r)} \partial_{z} u \, dz = h(r) - \bar{h} \qquad (2a) $$

and is a linear function of z (curvature controlled strains are of the order of $\lambda^2 r^2/R^2 l^2$ ($\le 10^{-7}$) relative to compression ones and, therefore, negligible). Hence

$$ u(r, z) = \frac{h(r) - \bar{h}}{h(r)} z. \qquad (2b) $$

The constant $\bar{h}$ is obtained by expressing that, at equilibrium, the strain field must be antisymmetrical with respect to each dislocation, i.e. the Peach-Koehler force is zero. With $r_1, r_2 = r_1 + l$ being the locations of two adjacent dislocations of Burgers vectors $b_1, b_2$, we

<footnote>(¹) Due to the geometry of our experiment, the array is made of circular dislocation loops. However, the radii of these loops are very large compared to the radius of curvature of the layers close to the dislocation core (typically $10^4$ times larger), so that for all practical purposes one can deal with a straight dislocation line. The energy of a dislocation will be calculated per unit length of this line.</footnote>
have $u(r_i^-) = b_i a/2$, $u(r_i^+) = -b_i a/2$ and eventually, for slowly varying Burgers vectors, $\bar{h} = [h(r_1) + h(r_2)]/2$.

The energy of the array of dislocations may now be obtained as a sum of the contributions from each dislocation, since we already noticed that dislocations do not interact. These contributions are the following:

i) Far-field energy $\mathcal{E}_F$. It originates from the strain field inside the parabola and is given by

$$ \mathcal{E}_F = \frac{B\lambda ba}{2} \qquad (3) $$

$(B$ is the compression elastic modulus and is related to $K$, curvature elastic modulus and to $\lambda$, smectic penetration length, by $B = K/\lambda^2$).

ii) Core energy $\mathcal{E}_c$. It originates from distortions near the singularity of the smectic order, at scales shorter than $ba$, where the far-field elastic energy integral has to be cut off to prevent UV divergences. According to Kleman and Williams [11, 12], the core energy of a large Burgers vector dislocation can still be described by a continuum theory, since an edge dislocation of Burgers vector $b$ can be viewed as $a + \pi$, $-\pi$ disclination pair of length $ba$, the energy of which is calculated in a *nematiclike* description

$$ \mathcal{E}_c = \frac{\pi}{2} K \ln \frac{b}{2b_c}. \qquad (4) $$

Note that the (short wavelength) cut-off scale here is $b_c a$, with $b_c a$ *priori* of the order of unity, whereas it was $ba$ in the case of $\mathcal{E}_F$; $b_c$ is a microscopic quantity which cannot be calculated in this continuum description: it should be considered as characteristic of the system just as $\lambda$ in this type of approach. Note also that, for large $b$, $b_c$ is expected to be independent of $b$ apart from a possible dependence on its parity; on the other hand, for smaller $b$, we cannot exclude a $b_c(b)$ dependence.

Equation (4) can be rewritten:

$$ \mathcal{E}_c = \frac{\pi}{2} K \ln \frac{b}{2} + \mathcal{E}_\mu, \qquad (5) $$

$\mathcal{E}_\mu$ is a direct measure of the short-scale core energy. (In our case this scale is given by the thickness $\delta$ of the water-surfactant lamella).

iii) Wedge confinement energy $\mathcal{E}_w$. It originates from the strain field outside the parabolas and is given by the integral

$$ \mathcal{E}_w = \frac{B}{2} \int_{r_1}^{r_2} dr \int_{0}^{r_2} dz (\partial_z u)^2, \qquad (6) $$

where $u$ is the strain field of eq. (2b).

To lowest order in $l/(r_1 + r_2)$ we get

$$ \mathcal{E}_w = \frac{B l^3}{12R}. \qquad (7) $$

The energy per unit length of the array of dislocations is then

$$ \frac{\mathcal{E}}{l} = \frac{\mathcal{E}_F + \mathcal{E}_c + \mathcal{E}_w}{l} = \frac{B l^2}{12R} + \frac{\pi B \lambda^2}{2l} \ln \frac{b}{2} + \frac{\mathcal{E}_\mu}{l} + \frac{B \lambda b a}{2l}. \qquad (8) $$
Minimizing $\mathcal{E}/l$ with respect to $b$ (recall that $l = Rba/r$), we obtain the following results:

As long as $r$ is smaller than some value $r^*$, the optimal Burgers vector is a constant, equal to the minimum value allowed $b_{min}$ (this minimum value is not necessarily equal to the smallest possible value $b=1$ because of the unknown small $b$ behaviour of the dislocation core energies, eq. (5)). The local reticular distance of the array $l$ decreases as $r$ increases as

$$l = \frac{R(ab_{\min})}{r}. \qquad (9)$$

For $r$ larger than $r^*$, $b$ becomes a function of $r$:

$$b = \frac{r}{a} \cdot \left( 3\pi \frac{\lambda^2}{R^2} \right)^{1/3}, \quad \mathcal{L} \left( \frac{r}{r_0} \right), \qquad (10)$$

where

$$r_0 = 2\alpha \exp \left[ 1 - 2 \frac{\mathcal{E}_\mu}{\pi K} \right] \left( \frac{R^2}{3\pi\lambda^2} \right)^{1/3} \qquad (11)$$

and $\mathcal{L}(r/r_0)$ is the one-third power of the largest root of the equation $x = \ln r/r_0 + 1/3 \ln x$. In this case (2), $l$ increases as

$$l = (3\pi\lambda^2 R)^{1/3} \mathcal{L}\left(\frac{r}{r_0}\right). \qquad (12)$$

Thus an experimental determination of the law $l(r)$ is a new way to measure the smectic length $\lambda$; it gives also by means of $r_0$ (eq. (11)) the ratio of the molecular quantity $\mathcal{E}_\mu$ to the thermodynamic one $K$; the value of $b_{min}$ provides some hints concerning the small $b$ behaviour of the core energy.

The sample cell consists of a 4 mm thick, rectangular (25 × 50 mm²) glass plate, polished to better than half an optical wavelength, epoxied onto the lower part of a brass oven and of a 80 mm diameter glass plane-convex lens held by the upper part of the oven. O-rings insure the tightness of the cell. Two lenses were used, with radii of curvature $R = 528$ mm and $R = 1063$ mm. The four samples studied are quaternary mixtures of dodecane, sodium dodecylsulfate (surfactant), water and pentanol (co-surfactant), known, after Roux and Bellocq [13], to be located in the smectic A (or $L_\alpha$) region of the phase diagram. Their compositions (weight fractions) are given in the table. The first two samples have compositions close to those of the samples labelled $x = 0.54$ and $x = 0.62$ in Safinya et al. [14]: the oil weight fractions are the same and surfactant or cosurfactant compositions are equal within a few percent. The layer spacings given in the table are thus directly taken, or extrapolated (3), from the data of Safinya et al.

<footnote>(2) In the derivation we assumed $l \gg \sqrt{8\lambda h}$. With $l$ given by eqs. (9) or (12), and $h = r^2/2R$, this inequality is satisfied in the experiments reported below.</footnote>

<footnote>(3) The extrapolation, which concerns the last two samples, is based upon the empirical relation $(a(x) - a(0))^{-1} = \alpha x^{-2} + \beta x^{-1} + \gamma$ with $a(0) = 3.82$ nm, $\alpha = 7.3763 \cdot 10^{-4}$ nm⁻¹, $\beta = 1.8970 \cdot 10^{-1}$ nm⁻¹, $\gamma = -2.2793 \cdot 10^{-1}$ nm⁻¹ deduced from the data of ref. [13]. Such a relation proceeds from a geometrical model of the swelling with oil, assuming a mean area per polar head constant.</footnote>
<table><caption>TABLE. — $x_0$ dodecane weight fraction, $x_A$ pentanol weight fraction (water/surfactant weight ratio: 1.55), $R$ radius of curvature of the lens, $\lambda$ smectic penetration length $\lambda = \sqrt{K/B}$, $r_0$ parameter in eq. (12), $ab_{min}$ parameter in eq. (9), $a$ estimated [14] (5) layer spacing.</caption><thead><tr><th>x<sub>0</sub></th><th>x<sub>A</sub></th><th>R (mm)</th><th>λ (nm)</th><th>r<sub>0</sub> (mm)</th><th>ab<sub>min</sub> (nm)</th><th>a (nm)</th></tr></thead><tbody><tr><td>0.540</td><td>0.138</td><td>1063</td><td>28</td><td>3.7</td><td>102</td><td>11.5</td></tr><tr><td>0.620</td><td>0.126</td><td>528</td><td>43</td><td>2.7</td><td>186</td><td>16.3</td></tr><tr><td>0.672</td><td>0.118</td><td>528</td><td>44</td><td>2.5</td><td>175</td><td>22</td></tr><tr><td>0.750</td><td>0.106</td><td>528</td><td>113</td><td>2.9</td><td>311</td><td>42</td></tr></tbody></table>

Samples are prepared by squeezing a drop of the mixture between plate and lens. The cell is then mounted onto an optical translation stage and observed between crossed polarizers by means of an optical polarizing microscope. Good homeotropic alignment is obtained by heating the sample into the biphasic lamellar/isotropic regime and slowly cooling down to room temperature. After annihilation of unwanted defects, the array we are interested in remains. The defects appear as bright, curved lines over a dark background. Near the centre, where the thickness of the sample is less than about 20 µm, the contrast against background is too low to allow the observation of the dislocation loops. On the other hand, far enough from the centre (thickness larger than about 100 µm) focal conics disorganize the dislocation array. Nevertheless this leaves a number of about 200 to 300 adjacent dislocation lines, the locations of which can be recorded as a function of their rank (see figure). We then fit the theoretical distribution based upon eqs. (9), (12) to the observed one, using the MINUIT procedure [15], with $\lambda$, $r_0$ and ($ab_{min}$) as adjustable parameters. Fitted curves are shown on the figure and resulting parameters given in the table.

Our results unambiguously show evidence of the predicted behaviour for the Burgers vector. Near the centre of the cell, the Burgers vector is constant: dislocations are getting closer as $r$ increases. At some distance to the centre, the Burgers vector starts increasing with $r$: dislocations are getting away. Our experiments correspond to an increase of the Burgers vector from about ten to thirty. As seen on the figure, eqs. (9), (12) provide a good fit of our data. The smectic length $\lambda$ and the parameter $r_0$ are rather safely extracted from the fit, since they are related to the mean slope and to the (upward) curvature of the data. Using different fitting procedures and with an estimated uncertainty on the measurements of the dislocation locations of about 10 µm, we obtain a scatter smaller than 10% for $\lambda$, 20% for $r_0$ (4). On the other hand, the product of the layer spacing $a$ by the minimum Burgers vector $b_{min}$, related to the (downward) curvature of the very first part of the data, is ill-defined: a safe measurement of this parameter would require the observation of dislocations closer to the centre of the cell.

The ratios $\xi_u/K$, deduced from $r_0$ and $\lambda$ (eq. (11)), are found significantly smaller than unity which shows that practically no correction to the Kléman-Williams core energy is needed. On the other hand, our $\lambda$ values are considerably larger than those obtained by Safinya et al. [14]. The good fit of our theoretical expression to experimental data, together with the stability of the $\lambda$ values with respect to variations in the data analysis give us reasonable confidence in our values. We find no obvious reason for the discrepancy. A subtle

<footnote>(4) These include: i) continuous variation of the product $ba$; ii) integer variation of $b$, $a$ being set to its estimate value; iii) integer variation of $b$, $a$ being allowed to adjust to an optimal value; iv) variation of $b$ restricted to even values, $a$ allowed to adjust. All fits are equally acceptable. On the other hand, we can exclude variations of $b$ by integer steps larger than three.</footnote>
<figure><img src="image_2.png" /><figcaption>Fig. 1. – Position *vs.* rank of dislocations in two samples (dodecane weight fraction *x* = 0.54: triangles; *x* = 0.75: circles; symbols usually every five dislocations).</figcaption></figure>

one could be a *strong* wave vector dependence of *K* in the range (10⁻² ÷ 10⁻¹) nm⁻¹ [16]. Alternatively, one could question the X-ray results obtained on polycrystalline samples.

To conclude, let us point out that the direct observation of dislocation arrays provides a simple measure of the smectic penetration length λ. Aside from its simplicity, the advantage of the method lies in the fact that it is truly static: all the (numerous) macroscopic variables of the quaternary mixture are in equilibrium. This feature, shared with X-ray scattering, is absent in undulation instability experiments. On the other hand, the characteristic wavelength of this measurement is ill-defined: curvature strains arise at the scale of the layer spacing *a*, whereas compression ones are at the scale of the thickness *h* of the sample [17].

We did not discuss in this letter the behaviour of λ as one proceeds through the phase diagram. It is not possible at present to draw definite conclusions, with regard to Helfrich theory [18] for instance, from our too scarce data. New experiments are being performed to complete our work in this respect, which will be reported on in a forthcoming publication.

Quite interesting on a chemical physics point of view are also the more microscopic information as concerns the *short-scale* core energy $\mathcal{E}_u$ and the minimum Burgers vector $b_{min}$. For instance, the smallness of $\mathcal{E}_u$ suggests that there is no discontinuity in the water-surfactant membrane: this is only possible for *even* Burgers vector dislocations. A better knowledge of $b_{min}$ and $\mathcal{E}_u$, for small *b*, requires the extension of the experiment to thinner parts of the sample: this program is currently under way.

***

We are grateful to A.-M. BELLOCQ and P. BOTHOREL for stimulating our interest in the physics of *microemulsions*. It is a pleasure to acknowledge fruitful discussions with D. ROUX and C. R. SAFINYA.