Observation of edge dislocations in ordered polystyrene latexes

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Observation of edge dislocations in ordered polystyrene latexes

P. Pieranski, L. Strzelecki, J. Friedel

To cite this version:

P. Pieranski, L. Strzelecki, J. Friedel. Observation of edge dislocations in ordered polystyrene latexes.

Journal de Physique, 1979, 40 (9), pp.853-859. �10.1051/jphys:01979004009085300�. �jpa-00209170�

Observation of edge dislocations in ordered polystyrene ^latexes

P. Pieranski, ^L. Strzelecki and J. Friedel

Laboratoire de Physique ^des Solides, Université Paris-Sud, ⁹¹⁴⁰⁵ Orsay, ^France.

(Reçu ^le 24 janvier 1979, accepté ^{le 29 mai} 1979)

Résumé. 2014 La géométrie ^{coin de} Grandjean-Friedel-Cano ^est utilisée pour obtenir ^un système bien contrôlé de dislocations coin dans les latex ordonnés. Des observations sont faites au microscope métallurgique ^{par réflexion} grâce à la modification de la couleur de lumière réfléchie spéculairement. L’énergie de la déformation élastique

est calculée. On vérifie _que la configuration observée de dislocations correspond ^au minimum de cette énergie élastique.

Abstract.

The Grandjean-Friedel-Cano wedge geometry ^{is used to} produce well controlled edge dislocation patterns in ordered latexes. Due to the strain field, the color of the Bragg ^reflected light ^{is found to be} modified, making the dislocations easily ^{visible in a} metallurgical microscope. An estimate of the elastic energy stored in this strain field is made. The minimum energy configuration of the dislocation is found to agree with that observed

experimentally.

Classification

Physics ^{A hstracts}

61.70 - 82.70

6t.70

82.70

1. Introduction.

Latexes of charged polystyrene

balls in _aqueous solution have been actively investigat-

ed for _{many years.} Under appropriate conditions of size, charge, ^{volume fraction,} ionic concentration and temperature, ^{the balls are found to} crystallize

due to their electrostatic repulsive interaction into a

classical Wigner crystal (see ^{for example one of the}

most recent articles of Tokano and Hachisu [1]).

Because the interparticle spacing ^{lies in} the range of a few tenths of a micron, ^the Bragg reflexion of the visible light provides ^a way ^to monitor the crystalline

structure by ^a direct observation in metallurgical microscope.

For a perfect monocrystal, ^the colour and the

intensity of the reflected light give ^us the information about the interplanar spacing and about the orienta- tion of the crystallographic ^axes [2].

In polycrystalline samples, ^the grain boundaries, subboundaries, ^or isolated dislocations [7] ^{should be} particularly well visible because the colour and the

intensity of the reflected light should be strongly perturbed by these defects.

In this _paper we describe a method of producing

well controlled and characterized edge dislocations in these crystalline latexes, using ^the classical Grand-

jean-Friedel [3] ^{method of a} wedge shaped ^boundaries

geometry, which has been extensively ^{used to} study

the defects in liquid crystals. (Recently, this method has been used by Meyer et ^al. [4] for observation of the

edge dislocations in Smectic liquid crystals.) ^We

report the first observations and discussion of the static properties of these defects in a system ^{of ordered} polystyrene ^balls.

2. Experimental.

^2.1 PREPARATION OF THE SAMPLE.

The latex chosen for this study ^was prepared using the emulsion polymerization ^method

under the specific reaction conditions [5] ^{so that}

the particles obtained had a diameter of - 0.1 _gm.

After the purification ^{with an ionic} exchange ^resin [6],

the crystal formed with the interplanar spacing ^of

0.21 _gm (see ^section 3) ^{so that a brilliant} orange

Bragg ^{reflexion was} observed for incident light per-

pendicular ^to the container wall.

For the microscopic observation the latex sample

was transferred into ^a cuvette made of a pyrex tube T

( - ^{1 cm in} diameter) (Fig. 1 ) ^{and a} glass plate ^P epoxied

to the bottom. The cuvette was carefully ^cleaned

and rinsed with distilled water before introducing ^the

latex sample.

2.2 GENERAL ASPECT OF THE SAMPLE.

The obser- vations were made using ^{a reversed} metallurgical microscope (Reichert MeF).

The sample ^{texture as} observed under microscope

was generally polycrystalline. The size of the crystal-

lites depended strongly ^on the ionic concentration of the sample. The addition of ionic impurities ^favour-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01979004009085300

854

Fig. ^1. - Schematic of the experimental set-up.

ed the formation of large crystallites ; ^{we observed}

indeed that in samples ^{with a low} melting temperature

(T,, -- ⁶⁰ OC) ^the average size of crystallites ranged

between 100-1 000 _gm. On the other hand, ^these weak crystals ^were easy ^to destroy by ^{a mechanical} perturbation. ^{In the} recrystallization process the size of the initial _germ crystallites ^increased progressi- vely (during ^{a few} hours) from the initial size of the order of a few _gm.

A particular polycrystalline ^texture is visible in

figure ^{2. The} crystallites ^are separated by grain

boundaries

the black walls in the photograph.

The crystallites have been identified to have the f.c.c. structure and their close packed (111) planes ^are parallel ^{to the} glass boundary. Very ^few crystallites

have different orientations. For example, in the central

crystallite ⁱⁿ figure ^{2 the} (111) planes ^are oblique

with respect ^{to the} glass boundary ^{and a} periodic

system ^of steps ^{forms at} this interface. By measuring

Fig. ^2.

Polycrystalline ^{texture of} the sample.

the distance 1 ⁼ 4 pm between the steps, ^{we cal-} culate the tilt angle

v ⁼ d111/1 ^{= 0.21} gm/4 gm ⁼ 0.05 rad ⁼ 3 deg.

(the interplanar ^distance d111 ^is determined in sec-

tion 3).

2 . 3 CONTROL OF THE SAMPLE THICKNESS.

Using

the simple system represented ⁱⁿ figure ^{1 a} glass sphere (S) ( - ^{4 mm in} diameter) is introduced in the cuvette (T) and maintained close to the bottom glass plate (P). The distance between the sphere ^{and the} plate ho ^can be controlled continuously ^{with accu-}

racy of 0.01 _{gm in} the _range between _{0-3 gm.} This fine displacement control is obtained by applying ^a

current I to a small electromagnet ^{E which attracts} magnetically and bends the elastic support ^{of the}

sphere.

Independently, ^the optical displacement stage pro- vides a rough control of the sphere position.

3. Results. - 3 .1 DISLOCATION PATTERNS. - When the sphere ^is introduced in the sample ^{close to the}

bottom glass plate, ^the polycrystalline ^{texture forms}

with crystallites of the order of 10-100 _{gm, in} size, except ^{in the} region ^{below the} glass sphere, ^{where a}

very regular system of the concentric rings is observed

(Fig. 3).

In spite ^{of the} general aspect ^which strongly ^resem-

bles that of Newton rings (Fig. 3f) ^we identifiy easily

the edge dislocations as lines across which the colour of the reflected light ^varies very sharply. ^{On the} photo- graphs (Fig. 3a-e) ^{this is} quite visible due to the presence of a green filter ; the internal edge ^{of each} dislocation, ^{which in fact is} red, appears in the photo- graph ^as black. The external edge, ^{on the} contrary, ^is

green and consequently ^is bright ^{in the} photograph.

The colour varies monotonically ^{between two} adja-

cent dislocations, ^{which means that the} interplanar spacing ^varies continuously ^{in this} space. This obser- vation is compatible ^{with a} simple model of the dislo- cations which ^are _necessary to fill the volume of the

wedge shaped ^container represented ⁱⁿ figure ^4.

For the production ^{of such a} configuration ^{it is}

essential to assume that the crystallographic planes

are parallel ^{to the} glass plates, ^which agrees with the generally observed fact that, ^{when the} crystal

structure is f.c.c., ^the (111) planes ^are parallel ^{to the} glass ^walls (section 2.2).

When crossing ^the dislocation line the _average

discontinuity ^{in the} interplanar spacing ^can be estimat- ed as Ap - po/n, ^where po is the unperturbed ^inter- planar spacing and n is the number of crystallogra- phic planes corresponding ^to the local thickness of the sample. The colour contrast should then decrease when the thickness of the sample is increased. We observe this in fact in figure 3a-e where we present ^the dislocation patterns for different spacings ho.

For a thin sample, ^where only ^{a few} crystallogra-

Fig. ^3.

a-e) Edge dislocation patterns in latex observed in

metallurgical microscope with green filter. The ^{textures are} obtained for different thicknesses of the gap in the ^{centre :} ho ^{= 0.2 J.1m} (a), 0 - 6 gm (b), 2.0 g.m (c), 3.2 gm (d), 10 gm (e) -

f ) Newton rings obtained for a space between the sphere ^and plane

filled with water (Âlight ^{= 0.55 gm).} The gap thickness in the centre 0

ho ;--- 0.2 pm.

phic planes ^are present, ^the intensity ^of light ^reflected

from the latex/glass boundary ^is predominant ^{and the}

effect of dislocation is superposed ^on the Newton

rings pattern. ^{This is} observed indeed in the centre of figure ^{3a where} only ^one crystallographic layer ^is

present.

Using ^the pattern ^{of Newton} rings in green light (Â ^{= 0.55} pm), for the space between the plane ^and sphere ^{filled with water} (refractive ^index nW ⁼ 1.33),

the thickness h of the gap between the sphere ^and

plane ^was ploted ⁱⁿ figure ^{5a as a} function of the

distance r from the centre 0 (Fig. 3f ).

856

Fig. ^4.

Simplified schematic of the dislocation distribution in the regular wedge. ^The wedge angle ^is exagerated. ^The approximate

real geometry ^can be recovered by the five fold dilation of the x dimension. a and b are two düfferent possible dislocation configu-

rations.

Fig. ^5.

a) Solid lines and points ; distance between the plane

and the sphere ^h plotted ^{as a} function of the distance r from the centre. Open circles ; dislocation radius measured from figure ^3a.

b) Distance between the plane ^{and the} sphere ^{at the} position ^{of a} dislocation (determined ^from figure 5a) plotted ^{as a} function of dislocation number.

Knowing the radius _rm of the dislocation loops (measured ⁱⁿ figure 3a) ^{we have} determined graphically

the local thickness hn ^{of the} gap corresponding ^to

each dislocation. The variation of the thickness h,,

in a function of the dislocation number n (Fig. 5b)

fits well with a simple ^law

where _po ⁼ _{0.21 gm} is the unperturbed interplanar spacing. This distance _po is larger ^{than the} particle

diameter (0 ;:t 0.1 gm).

When increasing ^the gap thickness ho (by applying

a current I to the electromagnet) ^the dislocations move

toward the centre 0 and collapse. (The description

and discussion of their motion will be presented elsewhere.) ^In figure ^{6 we have} plotted ^the displace-

ment Ah needed to collapse An dislocation (or ^to

create An crystallographic planes ^{in the} centre).

Fig. ^6.

Thickness variation Ah _necessary to collapse ^{An dislo-}

cation loops (or ^{to create An} crystallographic planes).

The interplanar ^distance Ah/An ^{= 0.22} gm cal- culated from figure ^{6 is} slightly ^{different as} compared

with the value found above. The discrepancy ^between

these two results is larger ^{than the errors} of the linear law fits in figures ⁵ and 6. The possible ^{source of this}

difference can be the variation of the particle ^concen-

tration due to the slow solvent evaporation. (In ^these

first experiments ^{no care was taken to} prevent ^the

evaporation ^of water.) ^{We have also} observed that

some particle concentration gradients ^can persist

for a long ^time (a ^few hours) ^{in the} sample. ^This

introduces another source of the error if care is not taken to make the measurements in the same region

of the sample.

Although the existence of dislocations is evident

from the observations described above, ^{we have not}

established exactly their vertical position. ^{Due to the}

finite field depth ^{of the} microscope objective ^this

determination is _very difficult. As a consequence, ^we cannot discriminate between configurations ^{4a and b.}

4. Theoretical.

The main feature of the dislo- cation patterns visualized in figure ^{3 is the} global regularity ^{of their radial} distribution which for thin samples (Fig. 3a-c) resembles that of Newton

rings. ^The position ^{of the} dislocations is correlated to the local thickness h of the _{gap via} ^an elastic interaction with the rigid boundaries ; ^the dislocation distribution in equilibrium should minimize the elastic defor- mation _{energy W} stored in the crystal.

4.1.1 Calculation of the elastic _energy (simplified)

- The calculation of this _energy can be simplified considerably ^{in the case of a} very thin sample ^{with a}

small wedge angle a ; ^{as we can see} in figure ⁴ (a ^or b)

there are two distinct regions ^{in the} wedge :

a) ^the vicinity of each dislocation of width ô,, -- npo where the deformation is singular,

b) the bands of width z- po/a between dislocations where the strain field is regular ^{and can} be described in terms of a dilation or compression.

It should be possible ^to neglect ^the contribution from the singular region (a) if its width is much smaller than the distance between dislocations npo po/a

or na « 1. If this condition is satisfied there is no

difference between the two configurations represented

in figure ^4.

The deformation _energy stored in the band of the thickness h between _npo and (n - 1) p., ^{can be}

written as follows :

where E is the Young ^modulus.

One calculates then

The _energy minimum

is obtained for Xdin ⁼ po/(2 a), where the thickness of the edge ^{is the} following :

We can consider that, ^{for n} large enough, ^our

estimate (5) agrees with the experimental ^obser-

vation (1). ^{For the case of small n, a more} sophisticated

model should be elaborated by taking ^{into account}

the interfacial energies.

4.1.2 Method of image dislocations.

^{In the above}

estimates we have neglected the contribution to energy from the vicinity of the dislocations. For a more detailed calculation, ^{let us consider a collection} of image dislocations represented ⁱⁿ figure ^{7 which}

have to be introduced because of the rigid boundary

conditions [7]. (The elastic modulus (- 10’ cgs) ^{of the}

latex crystal ^{is in} fact smaller by ^a few orders of

magnitude than that of the limit glass ^surfaces (- ^{1 U11} cgs).)

Fig. ^7.

Collection of the image dislocations which produces

the stress field equivalent ^{to that in a} wedge ^{of an} angle ^{a with}

the rigid boundary conditions. a and b correspond ^to configurations

a and b in figure ^4.

This collection of dislocations can be considered

as a typical polygonized ^{structure or as an ensemble}

of the cylindrical dislocation walls [8].

In this last case we observe that the stresses compen- sate at distances p,12 ^{a so} that it is sufficient to include

only ^{the self} energy of the central dislocation and the _energy of its interaction with the image dislocations in the region ^{of radius} po/2 ^a.

For simplicity ^of computation, ^{we will} suppose that all dislocations have their Burgers ^vectors po

parallel (this results in an increase of the elastic

energy).

858

With this last simplification ^{we have}

where K ⁼ 1

v, ^v is the Poisson ratio, y is the elastic modulus and _ro is the dislocation core radius.

Replacing the summation by ^an integration ⁱⁿ (6),

one obtains :

the contribution of the first two terms in this expression corresponds ^to the elastic _energy estimated simply

above (form (4)).

The third term represents ^the energy in the vicinity

of the dislocation which has been neglected previously.

For small an the second term dominates.

Another possible collection of image dislocations

(creating ^the strain field equivalent ^{to that in} figure 4b)

is represented ⁱⁿ figure ^{7b. This} configuration ^{is in}

fact similar to that considered previously ^{so that the} corresponding ^elastic energy ^can be calculated directly

from _eq. (6) :

The Burgers ^{vector is} now p’ 0 ⁼ 2 po ^{and the} angle

a’ ⁼ 2 a. The radius of the dislocation core r’ 0 ^is

however difficult to compare with _ro because the dislocation is now in contact with the solid boundary

and one should therefore take into account the inter- facial energies.

In the limit of small an (an ^« 1) when the second term in _eq. (7) dominates, ^one finds that the energies

of the two configurations

are the same for reasons which were presented ⁱⁿ

section 4.1.1.

4.2 RESTORING FORCES.

The formula (3) ^{for the}

elastic _energy stored in the wedge -gives ^{us the indi-}

cation about the restoring ^{force F} acting ^{on the dislo-}

cation when it is displaced ^{from the} equilibrium position Xdin ;

This elastic force is essentially determined by ^the

factor Ea/n which, ^{in our} experiment, ^{is a function}

of the distance r from the centre. We can estimate this factor as follows :

The thickness of _{the gap} between the sphere ^of

diameter Do ^{and the} plane ^is approximately :

where ho ^is the thickness of the _gap in the centre.

The wedge angle ^{a is then} given by ^the slope :

while the number n of the crystallographic planes ^is given simply by :

The elasticity ^{factor can} be written then as :

This function presents ^{a maximum for} r Max ⁼ ,Iho ^Duo.

We can reasonably ^{assume that the} irregularities

on the dislocation contour are mainly determined by

the elastic energy. In figure ^3a the dislocation loops

are the most regular ^{in the centre} because in this

sample ho - ^{0 and} r Max -+- 0.

In figure ^{3c we show a} photograph of the dislocation pattern ^{for the} sample ^with ho = 10 po. ^{The dislo-}

cations loops ^are irregular ^{in the centre as} expected.

Their shape improves ^when going ^{out from the}

centre.

A more detailed discussion of the shape ^{of the}

dislocation loops ^and of their creation and anni- hilation _processes will be presented ^{in a} separate article on dynamical properties ^{of the} edge dislocations in ordered latexes.

5. Conclusions. - In section 4 we have proposed

two different dislocation configurations ^{which to a}

first approximation ^{for a thin} wedge with small

angle ^{have the same elastic} energies ^{but which are}

very different when the singular regions ^{in the} vicinity

of dislocations are considered. We believe that ^a

more detailed study ^{of these} singular regions ^would produce very interesting information about the dislo- cation structure (which ^should correspond ^to partial

dislocations in an f.c.c. crystal) ^{and the core} energies.

The aim of the present paper ^was both to give ^first experimental evidence for the existence of the edge

dislocations in ordered latexes ^as well as to demonstrate that by using ^a very simple experimental

set-up ^{it is} possible ^{to obtain} crystalline layers ^with

a well-known number of the crystallographic planes

which can be made as small as needed.

In particular ^we hope ^{that the} thermodynamic properties ^of crystals ^can be studied when reducing

the dimensionality from three to two.

It would be also interesting ^to understand how far the analogy between the edge dislocations in smectics [4] and ordered latexes can be pursued.

Acknowledgements.

The authors ^are indebted to Dr. S. Mitaku for a kind introduction to the physics

and chemistry ^of polystyrene ^latexes.

We thank Dr. C. Colliex for the electron-microscope

determination of the latex particle diameters, ^and

Dr. R. Reich for the use of this metallurgical ^micro- scopic equipment.

Finally, ^we greatly acknowledge ^{Prof. P. Chaikin}

for numerous discussions about different problems

related to this article.

References

[1] TAKANO, K., HACHISU, S., ^{J. Colloid} Interface ^{Sci. 66} (1978) ^130.

[2] HILTNER, ^P. A., KRIEGER, I. M., J. Phys. ^{Chem. 73} (1969) ^2386.

[3] FRIEDEL, G., ^Ann. Phys. (1922) ;

GRANDJEAN, F., ^C.R. Hebd. Séan. Acad. Sci. 172 (1921) ^71.

[4] MEYER, R. B., STEBLER, B., LAGERWALL, S. T., Phys. ^{Rev. Lett.}

41 (1978) ^1393.

[5] MITAKU, S., Private communication.

[6] AMETANI, K., FUJITA, H., Japan ^J. Appl. Phys. ¹⁷ (1978) ^17.

[7] Dislocations by FRIEDEL, ^J. (Pergamon Press) ^1964.

[8] ^{The stress} fields of different dislocation walls are considered in detail in Theory of Strengthening by dislocation groups,

by LI, ^{J. C.} M., published in Electron Microscopy ^and

Strength of Crystals, ^edited by Thomas and Washburn

(Interscience Publishers) ^1963.