Formation of periodic crack structures in polydiacetylene single crystal thin films

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Formation of periodic crack structures in polydiacetylene single crystal thin films

J. Berréhar, C. Lapersonne-Meyer, M. Schott, J. Villain

To cite this version:

J. Berréhar, C. Lapersonne-Meyer, M. Schott, J. Villain. Formation of periodic crack struc- tures in polydiacetylene single crystal thin films. Journal de Physique, 1989, 50 (8), pp.923-935.

�10.1051/jphys:01989005008092300�. �jpa-00210968�

Formation of periodic ^crack structures in polydiacetylene single crystal ^{thin films}

J. Berréhar (1), ^C. Lapersonne-Meyer (1), ^{M. Schott} (1) and J. Villain (2,*)

(1) Groupe ^de Physique des Solides de l’ENS, Université Paris VII, Tour 23, ² place Jussieu,

75251 Paris Cedex 05, ^France

(2) I.F.F., K.F.A., ^D-5170 Jülich, ^F.R.G.

(Reçu ^{le 8} septembre ^1988, accepté ^sous forme définitive le 19 décembre 1988)

Résumé. 2014 Nous décrivons ici les conditions d’apparition d’un réseau de fractures périodiques

dans des films monocristallins de polydiacétylène. Les films dont l’épaisseur ^est supérieure ^{à une}

certaine épaisseur critique présentent ^ces fractures, tandis que les plus ^{minces ne sont} pas fracturés. Un modèle basé sur la théorie élastique linéaire rend compte ^{de cette} épaisseur critique.

Abstract.

The formation of a regular pattern ^of periodic ^cracked ridges ^on single crystal polydiacetylene films is described. Films thicker than a certain critical thickness are cracked while thinner ones are not. A model in the framework of linear elasticity theory ^is developed, ^which

accounts for this transition as a function of the film thickness.

Classification

Physics ^Abstracts

46.30N

68.60 - 82.35

Introduction.

Many ^molecular crystals ^of substituted diacetylenes undergo ^{a solid state} polymerization, according ^{to reaction} (1), under various possible agents ^{such as} heat, ^UV light, X, y ^or e- irradiation.

In several _cases, the material remains single crystal throughout ^the polymerization ^which proceeds through ^a monomer-polymer ^mixed crystal ^{state of} continuously increasing polymer

content.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01989005008092300

Fully ^oriented conjugated ^{chains are obtained} by formation of covalent bonds between the

diacetylene ^monomer molecules. The polymerization process thus leads ^to macroscopic polymer single crystals presenting highly anisotropic properties ^{and which can} be modelised

as quasi one-dimensional solids. A typical example is the well-known bis(p-toluenesulfonate)

of 2,4-hexadiyne-l,6-diol (abbreviated ^name TS-6) for which the side-groups ^{R and}

R’ are identical, ^{with R :} -CHZ-0-SOZ- 0 ^{-CH3. A large amount of} experimental

data are available for this diacetylene and for the corresponding polymer. ^In particular, ^the crystal ^{structure of the monomer} [1] and of the polymer [2] ^is known. Both crystals ^are monoclinic, of space group P2 1 / ^{and the polymeric} chains grow along ^{the b} crystal ^{axis. It can}

be stressed that the main relative difference ( ’" ⁵ %) between unit cell parameters ^{of the}

monomer and of the polymer ^{is observed} along the chain direction : at room temperature,

bmonomer

^{5.178 À and} bpolymer

^4.910 Â, the relative differences along ^{a and c axis} being ^less

than one percent. ^Elastic properties ^{of TS-6} during polymerization ^{are also} known. The stiffness components ^were determined from sound velocity measurements [3] : ^the longitudi-

nal stiffness (along the chain direction) ^increases by ^{a factor 6.5} during polymerization

whereas the shear components ^are only weakly ^affected by ^the polymerization.

The present ^work mainly ^{concerns the} polymerization ^{of TS-6 in a} geometry specific ^{to our}

own experimental method : the polymerization, ^induced by low energy electron irradiation, ^is limited to a certain thickness of the monomer crystal. ^A polymer film is thus obtained on its

monomer substrate and must accomodate both the unit cell parameters discrepancies ^{and the}

elastic properties modifications.

In this paper, ^we wish to describe first the experimental conditions of formation of quasi- periodic arrays of cracks ^over the whole film surface. A critical film thickness Rc ^{exists and}

films thicker than Rc ^are cracked while thinner ones are not. A model in the framework of linear elasticity theory ^is presented in the second part ^{which accounts for most of the} experimental results, ⁱⁿ particular ^for the transition « uncracked-cracked » as a function of the film thickness.

1. Expérimental part.

1.1 EXPERIMENTAL METHOD. - Single crystal thin films of poly-TS-6 ^{are obtained} using monoenergetic electron irradiation as the polymerizing agent [4]. ^{When the} polymeric ^chains

grow parallel ^to the irradiated surface, which is the case of TS-6 cleaved along ^the

b . c plane, ^the polymer film thickness is given by the range of penetration of the électrons in the sample and thus related to the incident electron energy [5]. Electron energy is chosen between 0.5 and 5 keV, typical ^{current densities are} kept in the range 1-100 nA/cm2. The

polymerization ^{carried at room} temperature is monitored by ^the optical transmission of the

crystal (analyzing wavelength corresponding ^to the maximum absorption ^of long chains).

Doses needed for complete polymerization of the film are in the order of 1 C/cm3. It must be noted that these doses are a hundred times less than the typical ^doses for which radiation

damage is observed on poly-TS-6 crystals [6]. ^{The same} experimental ^{method can be} applied

to film preparation ^{of other} polydiacetylenes. Some of them are even more reactive to electron polymerization ^than TS-6, ^{so that the} polymerization dose is three orders of

magnitude ^less than the radiation damage threshold. Thus the formation of periodic ^crack

structures observed in polydiacetylene thin films under certain experimental conditions as

described and studied below must not be interpreted ^{as a radiation} damage ^effect.

By ^the experimental ^method itself, starting ^{from a} single crystal ^monomer plate ^M 100 ^....m thick), single crystal polymer films P of thicknesses R ranging ^{from 500 to 5} 000 Â

are easily ^obtained by adjusting the incident electron energy. In all ^cases studied _up to now,

the films remain on their monomer substrate, ^no spontaneous splitting ^occurs (if wanted, they

can be separated ^{from the substrate} by ^selective dissolving ^{of the} monomer). ^{These films are}

thus prepared ^{in a sort of} epitaxial ^{situation and one could} expect ^{them to be in a stretched}

state since the preferred ^{unit cell} parameters ^are those of the monomer substrate at least at the monomer-polymer ^interface.

1.2 OBSERVATION OF PERIODIC CRACK STRUCTURES. - How strain is relaxed in the film is a

question of interest and the film thickness appears ^to be an important parameter in the strain

relaxation _process.

The films were observed under electron and optical microscopy (for ^{a few films,}

observation of the sample surface with an optical microscope ^{was made at} every stage ^{of the} film préparation : freshly ^{cleaved monomer,} partially ^and totally polymerized ^{film on its} substrate, ^and polymer film alone after dissolving ^the monomer).

When films are prepared ^{with an} incident electron energy below 2 keV (film thicknesses R = 1 500 Â) ^the sample surface is very smooth, showing ^no observable alteration : the only

defects seen on the optical ^{or electron} micrographs (see photo 1) preexisted ^on the cleaved

monomer surface. But when thicker films ^are prepared (R 2:: ^{2 000} Â, electron energy

2:: 2.5 keV), ^{their surfaces} present ^a very different aspect (photos ^{2 and} 3) : ^a regular pattern

Scanning ^electron micrographs ^{of a} 1500 Â (photo 1) ^{and of a 2 500 À} (photo 2) thick film. Optical

microscope pictures ^of 4 400 Â thick films in a one step (photo 3) ^{and in a two} steps (photo 4)

polymerization procedure. ^b axis is horizontal for all photos.

over the whole sample surface of rows of cracked ridges, quasi-periodically spaced (in ^the order ^{of 1 to 20 m} apart) ^and perpendicular ^{to the chain} direction. In the case of TS-6, ^we

have stressed that the chain direction coïncides with the direction of greatest discrepancy

between the monomer and the polymer lattices. It will be argued in the discussion below (cf.

Sect. 3.2) that this last direction is indeed the one of importance ^to predict the direction of cracks.

These ridges ^or cracks appear after ^an irradiation dose in the order of 6-8 x 10 - 2 C/CM3 i.e.

12 to 16 times less than the dose needed for complete polymerization. This dose could be evaluated for each sample ^{since a} reproducible ^accident (a ^small slope rupture) appears ^on the

polymerization kinetics, ^{as seen} in the insert of figure ^1. This accident coïncides with the cracks formation (it ^has been checked that the sample’s surface is smooth before this accident and ridged immediately after).

Fig. ^1.

Polymerization ^kinetics (followed ⁱⁿ transmission) ^for 4400 A thick films. (A) ^One step polymerization with 4.5 keV e - => cracked sample. ^{Insert : initial} part ^{of curve} A, magnified. ^Crack

formation occurs between the two arrows. (B) ^Two steps polymerization with 2 then 4.5 keV e =>

uncracked sample.

Cracks thus appear long ^{before the} polymerization ^{is achieved} (polymer ^{content less than} 10 %) ^{when the} sample ^is microscopically heterogeneous ^{since it consists} mainly ^{in the}

monomer matrix containing ^short polymer ^chains [7] ^{in low} concentration. Then the cracks formation does not require (from ^an energetic point ^of view) ^the breaking ^of many covalent bonds.

We have now prepared ^{thick films} (R > ³ 000 À ) ^{with no surface} alteration. Polymerization

must then necessarily ^{be achieved in at least two} steps. ^{The first one is the} polymerization ^{of a} partial ^thickness R, using electrons of incident energy ^low enough ^{to be sure that no crack}

formation occurs (R! s ^{1 500} Á, Ei s ² keV). ^{The final} polymerization conditions can then be chosen in function of the final wanted film thickness, determined by ^the highest ^electron

energy ^used. Figure ^{1 shows the} polymerization kinetics of 4 400 A thick films. A ^one step

polymerization (A) ^{with 4.5 keV} electrons leads to a cracked sample ^{whereas a} two-steps

polymerization (B) gives ^{a film with a} very good ^surface quality (photo 4). After the first

polymerization with 2 keV electrons, ^{a -} 1 500 À coating of uncracked polymer prevents ^the final polymer ^film (4 400 À thick) ^from cracking. Though the total dose needed for complete polymerization ^is notably higher in this last _case, the absence of surface alteration proves

again that when cracks occur (at ^lower doses) ^{they should not be} assigned ^{to radiation} damage.

To summarize, ^{the main} experimental facts that should be accounted for are :

-

the possible formation of periodic ^crack structures over the whole film surface ;

- the existence of a critical thickness Re under which films are uncracked ;

-

and the fact that to prevent ^a thick film from cracking, ^an uncracked film coating ^is

sufficient.

2. Model.

2.1 CRACKS. - We consider (Fig. 2a) ^{a film of} polymer ^{P on its monomer} substrate M. Both materials are single crystals ^{and are assumed to be in} epitaxy for small thickness R

(R ^- 1500 Â). ^{However, this} epitactic ^state becomes unstable for large R because the

preferred ^unit cell size parallel ^to the surface is bo in the « substrate » M and b, «-- bo ^{in the film. We assume for} simplicity ^{that this} discrepancy ^occurs only ^{in one direction x,}

the direction ^x being the b axis of the crystal ^{for which, as mentioned above, the main} discrepancy between unit cell parameters ^is observed. Then it can occur that the state of minimum energy is constituted by ^a periodic array of macroscopic, straight ^cracks perpendicular ^{to x} (Fig. 2b). This is the effect we wish to study ^below.

Fig. ^2. - (a) ^The (polymer) film P and the (monomer) substrate M. (b) The cracked structure.

It is worth recalling ^{the case of} epitactic layers of, say, ^a metal chemisorbed by ^another

metal. In that case the natural misfit is compensated by ^{« misfit} dislocations » (Fig. 3a) ^which

allow the density ^{to be different at} the external surface and in the substrate. This model is

hardly applicable here since diffusion of such large molecules is presumably negligible, ^{so that}

the total mass of an atomic layer parallel ^{to the surface is} independent ^{of its} depth. ^The

presence of misfit dislocation would therefore imply ^a global shrinkage ^{of the} layer (Fig. 3b).

A detailed comparison ^{of the} free energy of figure ^{2b and} figure ^{3b will not be} presented, ^but

several mechanisms can make the structure of figure ^3b unfavourable.

2.2 ASIMPLIFIED CALCULATION. In this section, ^{the free} energy of figures ^{2a and 2b are} compared. It is assumed that figures ^{3a and 3b have a} higher free energy ^{or cannot} be realized.

Neglecting ^{for the moment} interactions between cracks, ^{the structure is} expected ^{to crack if}

the free energy of a crack is negative. ^{The exact} calculation of this energy would involve the calculation of the strains around the crack. In order to avoid this tedious exercise, ^a

variational method will be used. We shall assume a certain form of the strain which depends

on a limited number of parameters. ^The energy should then be minimized with respect ^to

Fig. ^3. - (a) Misfit dislocations. This configuration ^is possible ^{in an} epitaxial layer, ^{but not in a} polymerized ^film. (b) ^A possible effect of polymerization. The contribution of vertical bonds to the surface energy is larger ^{than in} figure 2a because the number of unsatisfied bonds at the top of the P film is the same, but there are additional unsatisfied bonds at the top ^{of the M} substrate. Thus, if vertical bonds have a high energy, this ^structure is not favoured.

these parameters. ^{The two} parameters (Fig. 4a) ^are the range fbo of the strain and the maximum angular distortion 0. At a distance larger ^than Qbo ^{from the} crack, the strain is assumed to vanish. In the strained region the strain parallel ^to the surface is assumed to

depend only ^{on the} height, ^{and to be a linear} function of the height (Fig. 4a).

Fig. ^4. - (a) ^{In an} (unstable) configuration where atomic layers ^{are not curved} by cracks, the energy is

given by (6) if strains are small. (b) ^A pair of cracks. In section 3.1, ^this pair ^is supposed ^{to be in its}

ground state.

Let L be the length ^{of the} sample in the crack direction, ^{and let h be the} height ^{of the crack.}

Then crack formation implies ^an energy increase

which essentially corresponds ^{to broken} horizontal van der Waals interactions. There is also a

shear energy increase

where A, ^{as well as} C, ^{is a constant.}

Finally ^{there is a} free energy decrease due to shortened horizontal bonds. It will be assumed

to be a quadratic ^{function of the lattice} mismatch Ab. The free energy change ^{due to cracks is}

thus written as

where K is a constant. At a height hx above the origin ^{of the} crack, bo - b

(hxlt) tg ^{0 -w}

hx9/P. Inserting ^{the mean value x}

^{1/2 and} X2= 1/3 into (3), ^{one obtains :}

where

The problem ^{is to minimize}

with respect ^{to 0 and} f. Cracks occur at equilibrium if the absolute minimum of (6) (as ^a

function of f. Cracks occur at equilibrium if the absolute minimum of (6) (as ^a function of f and 0) ^is negative.

In principle (6) should also be minimized with respect ^{to the} depth ^{h of the cracks, but it}

will be seen that this depth coincides with the film thickness. A, C, K, ^{b are assumed to be}

uniform within the film, and this is of course only approximately ^{true in the} problem

considered here since the degree ^of polymerization ^{should vanish} continuously ^{at some} depth

in the order of R under the surface.

2.3 MINIMIZATION OF THE APPROXIMATE FREE ENERGY (6).

^We minimize with respect ^to 8, and then the resulting function of f with respect ^to f.

Minimization with respect ^{to 0} yields

Insertion into (6) yields

Since this is a decreasing function of h, ^{h should be as} large ^as possible, ^i.e. equal ^{to the film}

thickness R of the polymerized ^film. Expression (8) should be minimized with respect ^to f.

The minimum of (8) ^satisfies dw/d(l/f)

^{0, or}

Insertion into (8) yields

Cracks will form if this energy is negative, i.e., using (5)

Thus, ^{for a} given degree ^of polymerization, ^a transition is expected ^{when the} depth ^{R of the} polymerized film is varied : for R - hc ^{no cracks are} present. ^For R :> hc ^{there are.}

3. Discussion.

3.1 ELASTICALLY MEDIATED INTERACTION BETWEEN CRACKS.

The free energy of ^a system ^{of n} parallel ^cracks (Fig. 2b) ^is equal ^{to n times} the energy of ^a single ^crack (approximately given by (10)) plus ^a correction. This correction is the interaction between

cracks, ^{which is} calculated in the appendix ^{and shown to be} repulsive. ^{This can be seen also} by

the following simple argument. ^{Consider a} pair ^{of cracks} (Fig. 4b) ^{in its state} of lowest free energy 2 EÓ ^{0. The uncracked state} is assumed to have zero energy. We need ^two

assumptions :

a) ^{that the} system ^{has a} symmetry plane (whose cross-section is the dash-dotted line) ; b) ^{that the strain} perpendicular ^{to the} surface is negligible. ^{This is correct in the} polymerization experiments ^{described in} part ^1.

In that case one can replace ^one half of the system (for instance that on the right ^{hand side}

of the symmetry plane) by ^a system without crack. The result is a system ^{with one crack with a} free energy EÓ (Fig. 4a). Now, ^this system ^is certainly ^not in its lowest energy ^state. Therefore the lowest energy of a single ^{crack is} Eo EÓ ^and the interaction energy is

2 Eo - 2 Eo > 0.

Now, it is difficult to explain the formation of a periodic array of cracks if the interactions

are repulsive ^{as will be} argued ^{now. The first} crack will form at a place ^where polymerization

turns out to be stronger. ^{The next cracks will} form, under the effect of increasing polymerization, very far from the first one (since ^the interaction certainly vanishes with

distance) ^and again their location is determined by fluctuations of the degree ^of polymeri- zation, ^and presumably random. One thus reaches a state where the sequence of cracks has

essentially random distances Ll, L2, L3, ^..., Ln,

^{Then the} repulsive interaction will dominate fluctuations and the next crack will appear halfway between both cracks whose distance is the longest. One may expect this process ^to continue until complete polymeri-

zation. Then the distance between cracks will be close to the value which minimizes the free energy, but important fluctuations are unavoidable since crack formation is initially ^{a random}

process. In fact, ^{each interval} Ln will divide into 2p’ _interval, where pn ^{is the} integer ^which

minimizes the free energy density ¡(À), ^{defined as a} function of the distance À between cracks. Thus

These relations generally ^determine Pn unambiguously ^when f ^{and the function} f(f) ^{are known. The structure will be rather} irregular. The observed structures are not

perfectly regular ^{and therefore not} inconsistent with this scheme.

However, ^{the whole} array of cracks appears very abruptly ^{at a} polymer ^{content of}

10 % and this seems inconsistent with the above description. ^{On the other hand,} this feature

might easily be understood if there were an interaction U between 2 consecutive cracks of the type displayed ⁱⁿ figure ⁵ (full line). Indeed the first crack appears ^as before when the

polymerization ^is sufficiently developed for the crack free energy Eo ^{to be zero. But then, 2}

cracks will at once appear ^at distance fo ^{since the} corresponding energy is

is negative. These 2 cracks will generate ^{2 new} cracks and the chain reaction produces very

Fig. ^5.

Interaction energy U between cracks ^{as a} function of the distance. Dashed line : positive, repulsive interaction, ^which probably arises if A, C, K and bl ^{are fixed} (see Sect. 3.1). Full line : interaction needed to explain the formation of a periodic structure. Such a shape probably arises from increased polymerization ^{in the} neighbourhood ^{of a crack} (Sect. 3.2).

soon a periodic ^{structure with} period Qo, ^where Qo ^minimizes U(f). ^In reality, ^{the cracks}

appear before Eo ^{vanishes, so that the} period ^{is not} exactly fo.

3.2 INTERACTION BETWEEN POLYMERIZATION AND CRACK FORMATION. - In the previous

section the degree ^of polymerization has been assumed to be fixed. This is correct if crack formation is much faster than polymerization. ^{Since our} description ^fails to account for rapid cracking it will be argued here that crack formation favours polymerization, ^while development ^of polymerization hinders the formation of further cracks.

Formation of a polymer involves several steps [8-10]. ^{The first step} is the creation of a

dimer. It is an endothermic process, which ^occurs essentially only under the effect of irradiation. Now, the dimer has a free radical at each end. These free radicals are chemically

active and act at once on the ^« next » monomers, producing formation of a trimer, ^a tetramer,

etc. This chain reaction is fast, but, ^if polymer ^{content is still} small, stops ^after polymerization

of a few _monomers, say about 20 [7]. ^The oligomer ^{formed is a} stable molecule [9]. ^Plausible

chain termination reactions can be written but none has been experimentally demonstrated _up

to now. When the first crack appears, strain is relaxed and the lattice parameter ^{in the} direction of the crack is reduced in the vicinity of the crack. The chain initiation (dimer formation) and termination processes ^{are not} much dependent upon crystal strain but the chain propagation ^{rate is} greatly enhanced when strain is relaxed. This was shown by y-rayes

polymerization ^studies [11] ^and since y-rays ^act through Compton secondary electrons, ^the polymerization process induced by low energy electrons is ^not expected ^to be very different.

Polymerization ^thus speeds up producing ^a further increase of (bo - hl) in ^the vicinity ^of

the crack, ^{so that condition} (11) becomes satisfied and new cracks form on both sides of the first one. A chain reaction similar to that described at the end of the previous ^section produces

the appearance of the whole crack array in ^a short time. Neglecting any spatial fluctuation of the polymer concentration, the location of each crack depends only ^{on the} previous ^{one, so}

the structure is periodic. However, ^{some random} fluctuations should be present.

While crack formation facilitates polymerization, increasing polymerization hinders crack formation. And indeed we have shown in part ¹ that, ^if polymerization ^{is first induced in a} layer thinner than Rc and later in a thicker layer, ^{cracks do not occur. We} attribute this effect to the fact that a completely polymerized crystal ^{can no} longer be cracked since this would

require breaking covalent bonds. The relation with the previous algebra is that C becomes very large ^{with a} high polymerization. ^{Indeed we have mentioned} above that the longitudinal

stiffness increases by ^a factor 6.5 _upon polymerization. ^{For a weak} degree ^of polymerization,

there are few polymer chains and they ^are fairly ^short. Presumably, they ^{are not broken} by

cracks, ^which obliges ^{the crack} edge ^{to be} irregular ^to accommodate the local length ^{of the} polymer. Thus, C is rather small at the beginning ^{of the} polymerization process, and this makes cracking possible. ^To summarize, polymerization ^{has two} opposite effects : it increases

(bo - bl ), ^and this favours cracking, but it also increases C, and this makes cracking impossible. Thus, ^detailed predictions ^are certainly ^difficult.

The mismatch (bo - bl) between the monomer and the polymer lattices is indeed a relevant parameter. ^{We have a few} preliminary ^{results on another} diacetylene ^IPUDO (for ^{which the}

lateral group R is -(CHz)4-OCONHCH (CH3)z) ^{of a different} crystal ^structure [15]. ^The

direction of greatest mismatch upon polymerization in unit cell parameters ^{is the c} axis of the

crystal ^{which is} perpendicular ^to the chain direction. In IPUDO films prepared ^{with our} method, cracks appear perpendicular ^{to the c} direction and parallel ^{to the} polymer ^chains.

Though ^the longitudinal ^stiffness along the chain direction surely ^increases ùpon polymeri- zation, ^{it seems} impossible up ^{to now to} prevent the film from cracking, except parallel ^{to the}

chain direction.

(bo - bl ) depends ^{on the} degree ^of polymerization ^{and also on} temperature. The variation of the unit cell parameters ^with temperature is known for TS-6 [12-14] : bo - bl decreases with temperature. ^{It can} then be checked whether Rc ^as predicted by equation (11) ^also depends

on temperature. ^Film preparation ^{at low} temperatures (77 ^{K,-5 T room} temperature) using

the ^same experimental method is started now and the first results indeed show that ^a higher

value of Rc ^is observed when the polymerization is carried at low temperature (T

200 K).

Because of the interaction between polymerization ^{and crack} formation, ^it is also difficult to predict ^the period ^À (Fig. 2b). ^A plausible guess is that ^À is of the order of magnitude ^{of the}

interaction range 2 f of cracks, given by (9)

It is interesting ^{to note} that this length ^is proportional ^to h and does not depend ^on (bo - bl) ^nor C. A and K are presumably ^more weakly dependent ^{on the} polymer

concentration.

From our first experimental observations the prediction ^seems quite ^{correct but an} analysis

of the inter-cracks distances distribution (mean ^{value À,} dispersion) ^is required ^to bring ^some

more quantitative information.

Conclusion.

The formation of approximately periodic arrays of cracks is ^a novel phenomenon since, ⁱⁿ

contrast with other defect ordering phenomena in irradiated materials [16, 17], it involves irreversible effects and diffusion does not occur.

With assumptions ^all supported by what is known about the studied material, ^{the model} developed here has shown that for a film thickness lower than the critical thickness

Re a stable uncracked structure existed and that /P > Rc ^{was a} sufficient condition of stability

of the cracked structure. And this of course accounts for the experimental ^results reported

here.

Most of the model predictions ^{could be tested} experimentally ^{at least} qualitatively. They ^all

show that the relevant microscopic parameter is the main mismatch bo - bi between the

monomer and the polymer lattices. Since it depends ^on the chosen diacetylene ^{itself, on the} degree ^of polymerization ^and usually ^on temperature, control of crack formation seems thus

possible ^{in this} system.

Appendix.

Interaction between cracks.

The long range behaviour of the interaction between cracks ^can be obtained if it is assumed that the strain arises from a periodic ^{array of} dipoles ^of edge dislocations perpendicular ^{to the}

surface and centred ^on the surface (Fig. 6). ^{In this scheme, a fictive} crystal symmetric ^{of the}

real one with respect ^to the surface is added and the dislocation dipoles ^introduce additional,

fictive atoms inside the cracks. The dipoles ^{are formed} by dislocations below the surface and

image dislocations with opposite Burgers ^vector above the surface. The image dislocations are

¡1ecessary ^to avoid large strains below the surface and a logarithmic divergence of the energy.

fhe Burgers ^{vector is} equal ^to the atomic distance and parallel ^to the surface. If the height ^of

he crack is h and if 2 0 is the angle between the edges ^{of the crack} (Fig. 2b) the number of additional fictive atoms at the surface is 2 h 8 _per crack, ^{and this} is also the number of dislocations below the crack. Thus, 2 0 is the density of dislocations along ^the symmetry plane

of the crack.

Fig. ^6.

^{Cracks seen as} produced by edge dislocations. The upper part ^{of the} crystal is fictive as well as

the atoms in the cracks. The fictive atoms allow the dislocation dipoles ^to be introduced and thus a

divergence of the energy ^to be avoided which would otherwise occur for uncompensated dislocations.

The energy contains :

1) ^{the same term Ch} per unit length ^{of crack as before} (Eq. (1)) ;

2) the elastic energy gain ^{due to} crack formation in the irradiated region. The energy gain

per unit length of crack is proportional ^to the number 2 h 0 of dislocations and to the thickness h’ of the irradiated zone, assumed to be smaller than h. Thus it is equal ^{to - K’} hh’ 0, ^where

K’ is a constant. For h’ :::. h this energy saturates to - Kh2 0. These formulae are easily

checked if one remarks that the volume below a dislocation dipole contributes nothing ^{to this} part of the energy because the integral ^on the coordinate x parallel ^to the surface vanishes, namely

Here, 2 d is the distance between the two dislocations of the dipoles and y is the distance ^to

the surface. The calculation of the energy involves integration ^{over x, y and d ;}

3) ^an elastic energy loss which ^can be regarded ^as the energy of the system ^of dislocations inside each crack. For the sake of simplification ^this system ^{can be} approximately replaced by

2 dislocations at distance h with Burgers ^vectors h B and - h 0 (the ^exact calculation would

give ^{the same} result). ^The corresponding interaction [18] may be approximated by

A’h 202(j + p In h ), ^{where the} coefficients A’ and _p depend ^on the elastic constants. The constant 1 between the brackets is generally ^{omitted when} only ^the long distance behaviour is of interest. Thus, ^the interaction has not the form (2), ^{where not} only ^the logarithm ^is missing,

but h2 is replaced by ^hi. The Ansatz which has been used to obtain (2) ^is certainly not correct if hli ^{becomes too small, but is} hopefully reasonable when (9) ^is satisfied ;

4) the interaction between dislocations from different cracks. This interaction is of elastic

origin ^as all other contributions except ^{the lst one.} The dislocations of a given ^{crack will} again

be replaced by ² dislocations at distance h, and this is now a good approximation ^if

flh ^is large. ^The interaction energy between ^{two nearest} neighbours ^is [18]

which has also not the same form as the f-dependent ^{term in} (4). ^{The four} contributions can

be collected to write the energy per unit surface ^area of a periodic array of cracks with period

2 f and depth ^{h as}

For h h’, h’ should be replaced by h in this formula.

Minimisation with respect ^{to 0} yields

and

(A.1) ^{is an} increasing ^{function of h, and} therefore the minimum of the energy does ^not

correspond ^{to h} > h’. When K’ is small, w ^is always positive ^as found in the text by ^another approximation, ^{and for a certain value} K,’ the minimum of w vanishes. As seen from (A.2), K,’ ^can be obtained as follows. Let the curves Fe ^{of the} (h, Y) plane be defined by

Then Kc ^{is the smallest} value of K’ for which the line Ai ^of equation

has a common point ^{with a curve} Fe ^{for some value of f. Since all curves} Fi ^{are above} FOO , ^{this value is f}

^{oo in} agreement with the discussion o f section ^{3.1. The} intersection point corresponds ^{to the} largest allowed value of h, ^{which is the} thickness h’ of the irradiated film.

Thus the present approximation yields ^{the same} qualitative ^{results as the} simpler approximation

of section 2, ⁱⁿ spite ^{of serious} differences in the details.

References

[1] ^{AIMÉ J.} P., LEFEBVRE J., ^BERTAULT M., SCHOTT M., WILLIAMS J. O., J. Phys. ^{France 43} (1982)

307.

[2] ^{KOBELT D. , PAULUS E. F., Acta} Crystallogr. ^{B 30} (1974) ^232.

[3] ^REHWALD W., VONLANTHEN ^A. and MEYER W., Phys. Status Solidi (a) ⁷⁵ (1983) ^219.

[4] ^{BERRÉHAR J.,} LAPERSONNE-MEYER C. and SCHOTT M., Appl. Phys. ^{Lett. 48} (1986) ^631.

[5] ^BERRÉHAR J., HASSENFORDER P., LAPERSONNE-MEYER C., ^SCHOTT M., ⁱⁿ preparation.

[6] READ R. T. and YOUNG R. J., J. Mater. Sci. 19 (1984) ^327.

[7] ^{ALBOUY P. A., KELLER} P., ^{POUGET J.} P., J. Am. Chem. Soc. 104 (1982) ^6556.

[8] BÄSSLER, H., ^Adv. Polym. ^{Sci. 63} (1984) ¹ and references therein.

[9] ^{SIXL H., Adv.} Polym. ^{Sci. 63} (1984) 49 and references therein.

[10] NIEDERWALD H. and SCHWOERER M., ^Z. Naturforsch. ^{A 38a} (1983) ^749.

[11] BAUGHMAN R. H., J.C.P. 68 (1978) ^3110.

[12] ^{AIMÉ J.} P., Thesis, Université Paris VII (1983).

[13] ^{AIMÉ J.} P., ^BERTAULT M., ^LEFEBVRE J., ^SCHOTT M., ⁱⁿ preparation.

[14] ^{BLOOR D., DAY R. J., ANDO D. J., MOTEVALLI} M., ^Brit. Polym. ^{J. 17} (1985) ^287.

[15] ^BERTAULT M., ^TOUPET L., private communication.

[16] ^KRISHAN K., ^Radiat. Eff. ⁶⁶ (1982) ^121.

[17] ^JAGER W., ^EHRHART P. , SCHILLING W., Proc. Int. Conf. on Non-linear Phenomena in Materials Science (Aussois, France) Sept. 1987, Eds. G. Martin and L. P. Kubin (Trans. ^Tech. Pub.)

1988.

[18] NABARRO F. R. N., Theory ^of crystal dislocations (Clarendon Press, Oxford) 1967, p. 92.