Christophe Tromas

Yves Gaillard

Y Gaillard † ‡

J Woirgard

Nucleation of dislocations during nanoindentation in MgO

Keywords: plastic deformation, dislocations, nucleation, plasticity, nanoindentation nanoindentation, plasticity, homogeneous nucleation, dislocation, pop-in

come

In many materials, incipient plasticity during nanoindentation is characterized by an initial displacement burst called pop-in. Pop-ins have been observed in ceramics [1][2][3], semiconductors [4][5] or metals [6][7][8]. Gerberich and co-workers have ascribed pop-in phenomenon in passivated metal surfaces to oxide layer breakage [4,9]. In plastic crystalline materials, this yielding, which makes the elastic-plastic transition, is commonly associated to the homogeneous nucleation of dislocations [5,[6][7]. This assumption is supported by the very high stress that can be reached, due to the nanometre scale contact area between the indenter and the material, in regions of materials small enough to be defect free. The hypothesis of homogeneous nucleation of dislocations during nanoindentation has been corroborated during the last few years by several atomistic calculations in metals [10][11][12]. However, there is still a lack in experimental data's on the real nucleation process. In particular, the experimental knowledge of the exact nucleation site of the dislocations and of the specific slip systems activated during the pop-in is of great interest for a better understanding of incipient plasticity during nanoindentation.

In this work, nanoindentations have been performed on a (001) surface of magnesium oxide (MgO) single crystals. MgO is a fcc material with rather simple plasticity, as there are only six kinds of slip systems instead of twelve for a classical fcc crystal. These systems can be divided into two groups: four <110>{110} 45 slip systems, with slip planes inclined at 45° form the surface (001) and two <110>{110} 90 slip systems with slip planes perpendicular to the (001) surface. These specific slip systems give rise to surface deformations around an indent organised as a rosette arms pattern [13][14][START_REF] Tromas | Encyclopedia of Materials: Science and Technology[END_REF]. In this study, a spherical indenter with a large radius of curvature (10 µm) was used rather than a Berkovich one with the purpose of producing a less concentrated dislocation structure, for an easier identification. Furthermore, the expression of the stress field under an elastic sphere-plane contact (Hertz contact) is well known [START_REF] Huber | ) Formatted: Spanish (Spain-Modern Sort) Deleted[END_REF][START_REF] Johnson | Contact mechanics[END_REF]. It allowed the calculation of the resolved shear stress just before the initial stage of plasticity. In the case of the spherical indenter, multiple pop-ins are observed on the loading curves. The dislocation structures generated during the first pop-in have been determined through the observation of the slip line pattern on the surface by atomic force microscopy, and by the nanoetching technique. These structures have been found to be very simple as they are limited, just after the first pop-in, to one or few dislocation pile ups. The activated slip systems have been fully characterized and the stress field at the beginning of the first pop-in has been calculated. The results show unambiguously that the dislocations generated during the first pop-in are all nucleated at the point of maximum resolved shear stress in their respective slip system.

Experiments

Nanoindentations have been performed on a freshly cleaved (001) surface of MgO, with a NHT apparatus from CSM Instruments. This apparatus was equipped with a diamond spherical indenter with a 10 µm radius of curvature. This radius has been determined by indenting a fused silica sample of known Young's modulus (72 GPa), and controlled by direct observation by AFM. The resulting surface deformations were observed by AFM. The nanoetching technique [START_REF] Gaillard | [END_REF] was also used to investigate the dislocation structure in the volume. This technique consists in revealing the emergence points of dislocations by chemical etching. The etching sequence is repeated after removing thin layers of material by chemo- mechanical polishing (CMP). The nanometre size etch pits are observed by atomic force microscopy. This sort of tomography of dislocations allows identifying the 3D organization of individual dislocations around a nanoindentation imprint [START_REF] Tromas | Encyclopedia of Materials: Science and Technology[END_REF][START_REF] Gaillard | [END_REF]. AFM observations were performed in tapping mode on a D3100 apparatus from Digital Instruments.

Results and discussion

Nanoindentation curves

When indenting MgO single crystals with a Berkovich indenter, a pop-in is always observed [3]. The deformation is purely elastic before the pop-in and elasto-plastic after the pop-in [19]. The critical load for the pop-in is different from one indentation to another, and ranges from 0.4 mN to 2.8 mN with the Berkovich indenter. However, as observed on figure 1a where different nanoindentation curves obtained with a Berkovich indenter have been superimposed, the pop-in always makes the transition from an elastic deformation curve to a master curve representative of the elasto-plastic deformation regime. The tip of the Berkovich indenter used in this study is fairly blunt, and can be approximated by a 360 nm radius sphere, as confirmed by the Hertzian fit performed on the elastic part of the curves in figure 1a.

With the 10 µm spherical indenter, multiple pop-ins are observed as shown on figure 1b. As for the Berkovich indenter, when the nanoindentation is stopped before the first pop-in, the force-penetration curve does not show any hysteresis, demonstrating thus that the deformation is purely elastic before the first pop-in. The deformation after the last pop-in is also described by the master curve of the elasto-plastic regime. With this spherical indenter, the first pop-in occurs at variable loads, ranging form 8 mN to 30 mN. However, during the first pop-in, the penetration depth is several orders of magnitude lower than the radius of the indenter. This point ensures that the elastic contact before the pop-in can be described as a Hertzian contact.

In figure 1b, the elastic deformation curve, as calculated from the Hertz formulation [START_REF] Johnson | Contact mechanics[END_REF], has been plotted in dashed line. The master curve of the elasto plastic regime, also plotted in dashed line, has been deduced from nanoindentations where the pop-ins occurred at very low loads. The multiple pop-ins, whatever being their apparition loads, make the transition from the elastic curve to the continuous elasto-plastic one. Nanoindentation curves stopped after one or several pop-ins have shown that the deformation between two successive pop-ins remains purely elastic. So, with the 10 µm spherical indenter, the transition from the elastic deformation to a continuous elasto-plastic regime is ensured by successive plastic displacement bursts interspaced by elastic loadings.

Dislocation structures

Nanoindentations stopped before the continuous elasto-plastic regime, that is after the first or second pop-in for example, exhibit surface deformations limited to few slip lines, without real indentation imprint reproducing the indenter shape. In particular, when the indentation is stopped after the first pop-in, a single slip line is generally observed, as shown on the indentation A on figure 2a, with a very small cavity on one side of the slip line. This slip line is oriented along a [010] direction and corresponds, according to the slip systems of MgO, to dislocations lying in a {110} 45 plane. The step height of 6.4 nm measured by AFM indicates that the slip lines corresponds to the emergence of 30 dislocation half loops. This structure, though very simple compared to those observed around classical Berkovich nanoindentations in MgO, recalls the rosette arms pattern because of the slip line orientation. Cleavage steps are also observed on the surface. A first chemical etching on the indented surface presented on figure 2b shows that two slip systems are actually activated around the observed indent: the <110>{110} 45 slip systems corresponding to the previously observed slip line, and a <110>{110} 90 slip system, revealed by the etch pits alignment along the 1 10

    direction.
This second system does not produces visible surface deformations as it corresponds to to the surface, so the etch pits are all at the same place as on the surface etching pattern.

However, the etch pits alignment along the [100] direction has moved 200 nm toward the indentation axis. This clearly proves that the corresponding dislocations are all in a convergent ( ) 45 10 1 plane, that is a plane lying below the indented area [START_REF] Gaillard | [END_REF]. This dislocation structure is schematized on figure 3.

Resolved shear stress

A common assumption made for the interpretation of experiments about incipient plasticity during nanoindentation, is that nucleation of dislocations is supposed to occur where the principal shear stress reaches a value close to the theoretical strength. This consideration does not take into account the slip planes that are really activated. A more realistic approach, in the case of a single crystal, is to make the hypothesis that the pop-in corresponds to the nucleation of dislocations at the point of maximum resolved shear stress (RSS). In the previous part, it has been shown that the dislocation structure around the nanoindentation A (stopped after the planes. In the following, the RSS will be calculated for both of these systems

The deformation before the pop-in is fully elastic. So, as a spherical indenter was used, all the stress tensor components ij σ can be calculated in the material before the pop-in. In particular, the contact radius a c just before the pop-in is given by the formula [START_REF] Johnson | Contact mechanics[END_REF]:

1 3 c * 3FR a 4E   =    
with F the applied load, R the spherical indenter radius (10 µm) and E* the reduced Young's modulus defined by: system observed around the nanoindentation A and is plotted on figure 4a. A maximum is observed along the indentation axis at a depth of 0.44a c , with a c the radius of the contact area.

MgO ind 2 2 ind MgO 1 1 1 E E E * -ν -ν = + With ν ind =0.
For the indentation A, this maximum just before the pop-in is 4.1 GPa and is located at a depth of 328 nm. Let's make the hypothesis that under such a high stress, glissile dislocation loops are homogenously nucleated in the ( ) 45 10 1 plane on the indentation axis. These loops then extend until they reach the surface, where they produce a slip line. As they glide along a slip plane inclined at forty five degrees from the surface, the knowledge of the nucleation site allows to predict the formation of this slip line at a distance of 0.44a c from the indentation axis. In the case of the nanoindentation A, the slip line is predicted to appear 328 nm from the indentation axis. This slip line and the contact circle are schematized on figure 5a.

For the <110>{110} 90 slip systems, the RSS is ( )

xx yy 1 2 σ -σ or ( ) yy xx 1 2
σ -σ according to the system considered. This RSS has been calculated for the <110>{110} 90 slip systems observed on the nanoindentation A, and is plotted on figure 4b. This RSS presents four identical maxima at a depth of 0.65a c , a radial distance of 1.1a c from the indentation axis and angular positions relative to the [100] direction: θ=0, θ=π/2, θ=π, and θ=3π/2. The projections of these four sites on the surface are indicated by numbers 1 to 4 on figure 5a. In the case of the indentation A, just before the pop-in, this RSS maximum is 431 MPa, and is located at a depth of 477 nm, and at a distance of 818 nm from the indentation axis. Let's consider, in the case of the nanoindentation A, the nucleation of dislocation loops in the (110) 90 slip plane at two of these points of maximum RSS. These loops extend in their slip planes until reaching the surface. The emergence points of these dislocations are predicted to be aligned along the dislocation structure nucleated during the first pop-in, based on the positions of the RSS maxima, has been superimposed to the AFM image the surface etching pattern around nanoindentation A, at the same scale (same image as in figure 2b). The scheme fits perfectly with the observed dislocation structures.

A closer study of the etch pits alignments around the nucleation site 1, indicated by an arrow on figure 5b, have been carried out. A higher magnification of this alignment on figure 5b shows that the etch pits are distributed by pairs on either side of the projection of the nucleation site. This is consistent with the hypothesis of dislocation loops that extend on either side of this nucleation site. These dislocation loops have been schematized on the magnification of figure 5b. Furthermore, an elastic calculation of the dislocation positions in this pile up after removing the indenter gave the same distribution as that observed experimentally [21]. These results validate the initial hypothesis of dislocation nucleation at the point of maximum RSS. Furthermore, these results have been found to be reproducible with other nanoindentations stopped after the first pop-in.

For the second set of dislocations of the activated <110>{110} 90 slip systems, even if the position of the etch pits line is in good agreement with the predicted nucleation site 2, one may argue that dislocations are not distributed in pairs because the etch pits are all on the same side of the nucleation site 2. However, in this case, these dislocations may have moved toward the dislocations lying in the ( ) 45

About a nucleation criterion

The results presented in this article do not provide a specific criterion for dislocation nucleation. They only prove that the dislocation nucleation, associated to the pop-in phenomenon, occurs at the point of maximum resolved shear stress for the considered slip system. They also cannot explain why the pop-in load is different from one indentation to another. However, this last point must be put into perspective, with reference to the previous results. The resolved shear stress seems to be a more accurate parameter than the pop-in load to study the dislocation nucleation during a nanoindentation test. The maximum RSS is found to be 0.5P m for the <110>{110} 45 slip systems and 0.05P m for the <110>{110} 90 slip systems, with P m the mean contact pressure:

3 1 3 3 2 m R * FE 6 3 2 P         π =
with F the applied load, E* the reduced Young's modulus, and R the radius of the spherical indenter. So, the maximum of the RSS varies rather slowly, as the cube root of the applied load. This can explain, if the nucleation of dislocation is a thermally activated phenomenon, why the pop-in load can occurs in such a large range. Finally, it is also surprising that the <110>{110} 45 slip system is activated at the same time as the <110>{110} 90 slip system, because for a given load, the maximum of the RSS is ten times lower for the last one. However, as observed for the nanoindentation A, there are only about ten dislocations in this last system, compared to thirty in the <110>{110} 45 slip system. For some other indentations also stopped after the pop-in, no dislocations were found in the <110>{110} 90 slip system.

Deleted: homogeneous

Deleted: In the previous part, the sites of nucleation of dislocations have been unambiguously determined. However, these results do not provide a specific criterion for dislocation nucleation.

Conclusion

For nanoindentations performed on (001) MgO surface with a spherical indenter of 10 µm of radius of curvature, the incipient plasticity is characterised on the load-displacement curves by a staircase yielding. These successive pop-ins, interspaced by elastic loading sequences, make the transition from an initial elastic deformation regime to a continuous elasto-plastic one.

Nanoindentations have been stopped after the first pop-in, and the dislocation structures have been characterized by AFM observation of the surface deformation and by using the

  The nanoindentation test, though originally developed for the determination of local mechanical properties, provides a new interesting tool to study the early stages of plasticity.

Formatted

  with a Burgers vector parallel to the surface. According to previous study on the dislocation structures around nanoindentation in MgO[START_REF] Tromas | Encyclopedia of Materials: Science and Technology[END_REF][START_REF] Gaillard | [END_REF], this last system, which accommodates the lateral deformation, is composed of two sets of dislocation half loops of opposite Burgers vectors depending on their position with respect to the indentation axis. This etching pattern also differentiates unambiguously the cleavage steps from the slip lines as no etch pits are produced along the cleavage steps. The figure2cpresents a second chemical etching pattern obtained around the same indent A after removing a 200 nm thick layer of material by CMP. As expected, the etch pits alignment along the 110     direction is nearly the same as that observed on the initial surface. It corresponds indeed to dislocation perpendicular

  in) is limited to dislocation pile ups in a (110) 90 and a convergent ( )

  07 and E ind =1141GPa the Poisson ratio and Young modulus for the indenter (diamond) and ν MgO =0.18 and E MgO =310 GPa for MgO. Just before the pop-in, for the nanoindentation A, the applied load is about 13.8 mN, and give rise to a contact radius a c =744 nm. A new formulation for the stress field under a Hertzian contact has been recently proposed byHanson et al. [20]. This formulation, which leads to an easier numerical calculation, has been used in this study. In the following, the stress induced by the indenter is calculated in Cartesian coordinates, with the origin at the intersection between the surface and the indentation axis, and the following axis orientation: x'x along [100], y'y along [010] and z'z along [001]. The RSS acting on a <110>{110} 45 slip system is () 100] surface direction. This RSS have been calculated for the [ ]( )

  the surface projection of the nucleation site. Such dislocation emergence points alignments are schematized by dotted lines on figure 5a, in the case of dislocations nucleated at sites 1 and 2. In the figure 5b, the predictive scheme of the Deleted: reaching the surface and producing a slip line Deleted: The position of the Deleted: and the (101)45 slip plane Deleted: , that is at Deleted: in the case Deleted: of the nanoindentation A Deleted: homogeneous Deleted: Two of these possible points alignments, associated to the nucleation

10 1

 1 slip plane to interact according to the dislocation reaction in MgO: This reaction, which has been widely observed around nanoindentations in MgO, results in a sessile dislocation segment oriented along the [ ] 1 11 direction. This assumption is reinforced by the four etch pits of the . Finally, the position of this second set of dislocations in the ( ) 90 110corroborates the assumption of a nucleation at the site 2, and the hypothesis of nucleation at the point of maximum resolved shear stress.

Figure 1 :

 1 Figure captions:

Figure 2 :Figure 3 :

 23 Figure 2: Nanoindentation A, stopped after the first pop-in: (a) AFM image of the surface deformation; (b) Chemical etching pattern on the indented surface; (c) chemical etching pattern after removing a 200 nm thick layer of material

Figure 4 :

 4 Figure 4: (a) Resolved shear stress for the <110>{110} 45 slip system and (b) for the <110>{110} 90 slip system, represented in the (100) plane containing the indentation axis. The nucleation of a glissile dislocation loop is schematized on figure (a).

Figure 5 :

 5 Figure 5: (a) Schematic of the site of maximum resolved shear stresses projected on the surface. The predicted position of the slip line and of the etch pits alignments relative to the indentation axis are also indicated. (b) Superposition of the previous schematic to the surface nanoetching pattern (see figure 2b), at the same scale. A magnification of the [110] etch pits alignments emphasizes the etch pits position with respect to the supposed nucleation site.

  Figure1

http://mc.manuscriptcentral.com/pm-pml Philosophical Magazine & Philosophical Magazine Letters


G Kermouche

J L Loubet

J M Bergheau

Keywords: polymers, indentation, mechanics of materials, mechanical properties viscoplasticity index, strain rate sensitivity Cone indentation, Polymeric materials, Viscoplasticity index, Strain rate sensitivity /mc.manuscriptcentral.com/pm-pml

The increasing use of polymeric materials as thin coatings requires appropriate mechanical characterization methods taking account of time-dependent effects. A well-known result is that a constant indentation strain rate allows to determine the mechanical properties of such solids from indentation tests. To do that, most people assume a power law creep behavior and use the viscoplasticity index to determine the strain rate sensitivity. In the first part of this paper, we investigate the influence of the indentation strain rate on the hardness of power law creep solids and elastic-viscoplastic solids. It is shown that the assumption of power law creep solid is not sufficient to describe the behavior of polymeric materials in indentation. In the second part, we propose a new index, the EVP index, which gives a better approximation of the strain rate sensitivity than the viscoplasticity index. In the last part, we apply this method to nanoindentation tests on PMMA and polycarbonate samples.

be used to identify their mechanical properties in such conditions and, in recent years, the indentation technique has been developed for that purpose.

The indentation of isotropic elastoplastic material has been widely investigated over the past few years {Tabor [1], Loubet et al [START_REF] Loubet | Microindentation techniques in Material Science and Engineering[END_REF], Hill et al [START_REF] Hill | Proc. R. Soc. Lond. A[END_REF], Dao et al [START_REF] Dao | [END_REF], Cheng and Cheng [5]} and many laws from experimental studies and simplified models have been proposed. Recently, the development of nanoindentation techniques has made it possible to finely characterize the mechanical properties of surfaces {Oliver and Pharr [6], Loubet et al [START_REF] Loubet | Mechanical properties and deformation of materials having a ultra-fine microstructures[END_REF], Bec et al [START_REF] Bec | [END_REF]}. Tabor [1] and Hill [START_REF] Hill | The mathematical theory of plasticity[END_REF] have pointed out that the real definition of hardness is the mean pressure under load and that the value of the hardness of ductile solids is essentially a measure of their plastic properties. They showed that the mean pressure can be related to the yield stress of the material by a simple factor which depends on the properties of the indented material and on the shape of the indenter. It was also shown experimentally that, for self-similar indenter {conicalpyramidal tips}, the indentation of elastoplastic materials follows the principle of geometric similarity. This principle gives a solid theoretical framework which has led to the development of many approximate solutions of elastoplastic indentation problems [START_REF] Bucaille | [END_REF] [11]. It is now possible to extract the yield stress and the strain hardening exponent of such solids.

However, currently available theories developed for time-independent materials cannot suitably be used to measure the mechanical properties of time-dependent materials such as polymeric materials. Indeed, the principle of geometric similiraty is not always satisfied. For power law creep solids, Lucas [12] has observed that the mean pressure is constant during the indentation test when indentation strain rate ḣ/h is constant, where h is the penetration depth and, ḣ the penetration rate. The same result has been obtained by Cheng and Cheng [5] with the help of a dimensional analysis. In a previous paper [START_REF] Kermouche | Cone indentation of time-dependent materials: the effects of the indentation strain rate[END_REF], we have shown that the principle of geometric similiraty is satisfied during cone indentation experiments of time dependent materials {viscoelastic solids, elastic-viscoplastic solids}, only if the indentation strain rate is constant. This result is of primary importance for glassy polymers for which the assumption of power law creep solids is not acceptable. Indeed, it is much appropriate to consider an elastic-viscoplastic behavior [14].

In the first part of this paper, we compare the effects of the indentation strain rate on the mean indentation pressure of power law creep solids and elastic-viscoplastic solids that follow the Bingham-Norton law [START_REF] Lemaitre | Mechanics of Solid Materials[END_REF]. Then, we explain that the viscoplasticity index [START_REF] Bertrand-Lambotte | [END_REF], defined only for power law creep solids, does not equal to the strain rate sensitivity in the case of elastic-viscoplastic solids {such as glassy polymer}. We thus propose a new index, which allows to obtain a better estimation of the strain rate sensitivity than the viscoplasticity index. In the last part, we apply this method to nanoindentation experiments performed on PMMA and PC {polycarbonate}.

2 The effects of the indentation strain rate on hardness values

Power law creep solids

The assumption of power law creep behavior is very often well adapted to the modeling of anelastic time-dependent materials. The strain rate sensitivity of such materials can be described by a simple power law relation :

σ = K εm ( 1
)
where σ is the uniaxial stress and ε is the uniaxial strain rate. Consistency K and strain rate sensitivity m are materials constants. Cheng and Cheng [5] have shown using dimensional analysis that the mean indentation pressure is given by :

p m = Π δ K ḣ h m (2)
where Π δ is a dimensionless function of strain rate sensitivity m and of cone angle β. The same kind of results has been suggested by Storakers [17]. Equation [START_REF] Loubet | Microindentation techniques in Material Science and Engineering[END_REF] shows that the mean pressure is constant if and only if ratio ḣ/h is constant which agrees with the results of Lucas [12]. An approximation of Π δ is given in reference [START_REF] Kermouche | Cone indentation of time-dependent materials: the effects of the indentation strain rate[END_REF].

Elastic-viscoplastic solids

Glassy polymers can be modelled as elastic-viscoplastic solids following a Bingham-Norton law [START_REF] Lemaitre | Mechanics of Solid Materials[END_REF]. This assumption seems to be well suited to model the loading part of instrumented indentation experiments [14] on such materials. The parameters of the Bingham-Norton law without strain hardening are • Yield stress at infinitely low strain rate : Y ,

• Consistency : K,
• Strain rate sensitivity : m.

For glassy polymers, Y is assumed to be equal to zero. The uniaxial constitutive equations of this model are:

εp = σ K 1 m (3)
σ = Eε e (4)
ε = ε e + ε p (5)
The cone indentation of elastic-viscoplastic solids at constant indentation strain rate has been investigated in a previous paper [START_REF] Kermouche | Cone indentation of time-dependent materials: the effects of the indentation strain rate[END_REF]. They showed that the mean pressure, or indentation hardness, can be expressed starting from a purely elastic state for high indentation strain rates and a purely elastoplastic state for low values. The transition between the two states depends on mechanical properties such as consistency, strain rate sensitivity and Young's modulus and also on cone angle β. For a Berkovich tip {β = 19.7 • } and glassy polymers, we proposed the following approximate expression:

p m = Π δ K ḣ h m 1 + 5.586 E * Π δ K ḣ h m (6)
where E * = E/(1 -ν 2 ). Equation above shows that in the limit as ḣ/h → 0, p m reduces to the mean contact pressure of power law creep solids and in the limit ḣ/h → +∞, p m reduces to the mean contact pressure of elastic solids , p m = E * /5.586 [18].

Comparison of these two models

We consider the mechanical properties of PMMA at ambient temperature which has been previously identified by Bucaille [START_REF] Bucaille | Simulation numérique de l'indentation et de la rayure des verres organiques[END_REF]. Bucaille showed that for PMMA and Berkovich tips, there is no influence of the strain hardening on hardness values. Hence, we do not consider the strain hardening of this material. The corresponding mechanical properties are given in table 1.

[INSERT TABLE 1 ABOUT HERE]

The curves p m -ḣ/h have been computed using equation 2 and 6 and are plotted in figure 1.

It is clearly shown that the viscoplastic assumption can not be used to model the behavior of PMMA in indentation for the common experimental indentation strain rate range. Moreover, we can remark that the error of hardness determination using the viscoplastic assumption increases with the indentation strain rate. When ḣ/h = 0.0001s -1 the error is about 23 % and when ḣ/h = 1s -1 , it is up to 50 %. Hence, elastic properties of glassy polymers have to be taken into account to determine time-dependent mechanical properties such as the consistency and the strain rate sensitivity.

[INSERT FIGURE 1 ABOUT HERE]

3 Identification of the strain rate senstivity

The viscoplasticity index

The viscoplasticity index [START_REF] Bertrand-Lambotte | [END_REF] is defined as the slope of graph ln(p m ) to ln ḣ/h . It is written

I v = ∂(ln(p m )) ∂ ln ḣ h (7)
As suggested by Lucas et al [12] and Cheng and Cheng [5], this index is a very good mean to determine the strain rate sensitivity for power law creep solids. Unfortunately, when the indented solid exhibits an elastic-viscoplastic behavior, much attention has to be paid to the interpretation of the viscoplasticity index, because I v varies with ḣ/h. According to equations ( 6) and ( 7), we can write:

I v = m 1 - 5.586p m E * (8)
Equation [START_REF] Bec | [END_REF] shows that when ḣ/h → 0, I v reduces to the strain rate sensitivity m. In return, when ḣ/h → +∞, I v is quasi-null. Consequently, the viscoplasticity index under-estimates the strain rate sensitivity of the material. To illustrate this, the [INSERT FIGURE 2 ABOUT HERE]

3.2 A new index to estimate the strain rate sensitivity

The principle of geometric similarity being satisfied during a cone indentation at constant indentation strain rate, a representative stress σ r and a representative strain ε r , characterizing the stress and strain distributions, and corresponding to each other on the uniaxial stress-strain curve can be defined [1][13]. A representative strain rate εr corresponding to the strain rate of a uniaxial test can thus also be defined. For glassy polymers, the following relation can be applied: σ ∝ εm [START_REF] Jonas | [END_REF]. Assuming that the representative strain rate of the indentation test is

given by εr ∝ ḣ/h [START_REF] Bucaille | Simulation numérique de l'indentation et de la rayure des verres organiques[END_REF], it comes:

σ r ∝ ḣ h m (9)
where σ r is the representative stress of the indentation test. Let us now introduce a new index I ev {called EVP index} :

I ev = ∂(ln(σ r )) ∂ ln ḣ h (10)
It is now necessary to estimate representative stress σ r of indentation test. Tabor [1] suggested to relate σ r to the hardness of the materials:

σ r = p m γ p (11)
where γ p is a function of the cone angle. For a berkovich indenter, γ p is about 2.8. The Tabor's representative stress is well adapted to the indentation of rigid plastic solids. Unfortunately, when elastic deformations are not negligible, this definition does not work. Moreover, we can note that equation (11) leads to I ev = I v .

A more general approximate solution to berkovich indentation of elastoplastic solids has been given in a previous paper [11]. It is written: For rigid viscoplastic solids {p m /E → 0}, the equation ( 12) reduces to equation 11 with γ p = 2.8.

σ r = 0,
Hence, I ev equals to I v in the case of power law creep solids. For elastic-viscoplastic solids, the definition of representative stress σ r given in equation ( 12) yields to:

I ev = I v 1 + 0, 783 σ r 0, 087E (13)
Equation [START_REF] Kermouche | Cone indentation of time-dependent materials: the effects of the indentation strain rate[END_REF] shows that the value of I ev cannot be lower than I v . Moreover, combining equations ( 13), ( 12), ( 7) and ( 6), it is easy to show that I ev cannot be greater than strain rate sensitivity m. Hence, we have:

m > I ev > I v (14)
Consequently, the EVP Index is always a better approximation of strain rate sensitivity m than the viscoplasticity index in the case of Bingham-Northon elasticviscoplastic solids. In figure 3, EVP index I ev and viscoplasticity index I v are plotted according to indentation strain rate ḣ/h. I v and I ev are computed using equations 6 and 12 and using mechanical properties of PMMA {table 1}.

[INSERT FIGURE 3 ABOUT HERE]

Application to nanoindentation experiments

Nano-indentation tests on PMMA and PC samples have been performed with the help of a Nano-indenter XP {MTS r }. The aim of this experimental work is to show that the EVP index is easy to use on nanoindentation results and that it gives relevant results for these two polymers. No polishing procedure is used on these materials. The roughness of samples, as taken, is below 0.1 µm peak to peak. The tests conditions are:

• Continuous Stiffness {∆h=1,5 nm ; ω=32Hz}

• Max. Load: 15 mN

• ḣ/h : 0.0015 s -1 , 0.005 s -1 , 0.05 s -1

• Berkovich tip Oliver and Pharr model [6] is used to estimate the contact area between the indenter and the solid:

h c = h -0.75 F K c -h 0 ( 15
)
where h c is the contact depth, h is the penetration depth, h 0 is the indenter tip defect, F is the load applied on the indenter and K c is the contact stiffness. The contact area is given by A c = 24.5h 2 c and the hardness is computed as F Ac . The calibration method of Loubet [START_REF] Loubet | Mechanical properties and deformation of materials having a ultra-fine microstructures[END_REF] is employed to avoid thermal drift errors and to estimate h 0 . For such solids, it has been shown that the Oliver and Pharr model gives acceptable values of hardness [21]. The nanoindentation results are summarized in table 2. σ r is computed using equation (12). For glassy polymers such as PMMA, the Young's modulus measured by nanoindentation experiments depends on the continuous stiffness frequency [22]. Hence, it does not always correspond to the one which has to be used in equation (12). For this reason, the value proposed by Bucaille [START_REF] Bucaille | Simulation numérique de l'indentation et de la rayure des verres organiques[END_REF], E=4200 MPa for PMMA and E=2400 MPa for PC, are used to compute σ r .

[INSERT TABLE 2 ABOUT HERE]

the curves p m -ḣ/h and σ r -ḣ/h are plotted in graph 4. The viscoplasticity index and the EVP index are determined by a power law fit of these curves. For PMMA, I v is about 0.077 and I ev is about 0.1. The reference value of PMMA strain rate sensitivity is about 0.1 {see table 1} which agrees well with I ev . For PC, it gives 0.039 for I v and 0.053 for I ev . According to reference [14], the strain rate sensitivity of PC is about 0.05.

[INSERT FIGURE 4 ABOUT HERE]

Discussion

The experimental results are in good agreement with equation ( 13) which shows that the value of I ev is greater than the value of I v for elastic-viscoplastic solids. Moreover, we can observe that for PC and PMMA, the value of I ev determined using nanoindentation experiments is very close to the reference values of the strain rate sensitivity. For PMMA, we can remark that the viscoplasticity index leads to an error up to 25 %. For these two materials, the EVP index gives thus a better approximation of the strain rate sensitivity than the viscoplasticity index.

The use of I ev is based on the assumption that the indented material follows a Bingham-Norton elastic-viscoplastic behavior with Y = 0. If Y does not equal to 0, the development of section 3 are no more valid and thus the use of I ev may lead to wrong estimations of strain rate sensitivity. However, it can be shown analytically that equation ( 14) is still valid for Y > 0. Hence, we recommend to use I ev rather than I v to estimate strain rate sensitivity of elastic-viscoplastic solids.

Nevertheless, it is important to keep in mind that the EVP index is not always a good approximation of strain rate sensitivity of materials. It highly depends on the experimental indentation strain rate range and on the mechanical behavior of the indented solid. If the assumption of Y = 0 is correct for glassy polymers like PC or PMMA, it is not the case for metallic materials. Consequently, care has to be taken concerning the use of this index.

Conclusions

In this paper, it has been shown that the assumption of power law creep solids is not always correct to describe the indentation behavior of time-dependent materials such as glassy polymers. In fact, it is necessary to take into account the influence of elastic properties on instrumented indentation hardness measures. Thus an elastic-viscoplastic formalism seems to be much adapted. Consequently, it points out that viscoplasticity index I v , defined only for power law creep solids, may lead to wrong estimations of the strain rate sensitivity.

For this reason, we propose a new index, EVP index I ev , which can be used instead of the viscoplasticity index to estimate the strain rate sensitivity. This index is based on a power law fit of the representative stress -indentation strain rate curve. For that, we use a previous work to estimate the representative stress from instrumeted indentation results {hardness and Young's modulus}. Obviously, the EVP index will only give an approximation of the strain rate sensitivity. Nevertheless, it is shown theoretically that the estimation of the strain rate sensitivity will be better with I ev than with I v . PMMA and PC nanoindentation results are in good agreement with these conclusions.

The use of EVP index is as easy as the use of the viscoplasticity index. It can be employed directly after nanoindentation experiments at different indentation strain rates and it does not require any inverse methods or complex solving operation to obtain a first approximation of the strain rate sensitivity.

F

  to estimate the strain rate sensitivity of glassy polymers using conical/pyramidal indentation 1 IntroductionThe increasing use of polymeric materials as thin films deposited on substrates requires appropriate techniques of mechanical characterization. Obviously the conventional tensile test cannot 1

•

  Young's modulus : E, • Poisson's ratio : ν,

  I v -ḣ/h are plotted in figure2in the case of a power law creep solid and in the case of an elastic-viscoplastic solid.

  087p m 0, 243 -0, 783 pm E

• 5 tests for each ḣ/h 7 Page

 7

Figure 1 :Figure 2 :

 12 Figure 1: Comparison of PMMA hardness using a viscoplastic model {power law creep} and an elastic-viscoplastic model

Figure 3 : 14 Page

 314 Figure 3: Viscoplasticity index I v and EVP index I ev vs ḣ/h for an elastic-viscoplastic solid representative of the behavior of PMMA {K=205 M P a.s -1 , m=0.1, E=4200 MPa}

Figure 4 : 15 Page

 415 Figure 4: Comparison between I v , I ev and strain rate sensitivity m for PMMA and PC, A : determination of I v using a power law fit of the p m -ḣ/h curve, B : determination of I ev using a power law fit of the σ r -ḣ/h curve

Table 1 :

 1 Comparison of PMMA hardness using a viscoplastic model {power law creep} and an elastic-viscoplastic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2 Viscoplasticity index I v versus the indentation strain rate ḣ/h for a power law creep solid {K=205 M P a.s -1 , m=0.1} and for an elastic-viscoplastic solid {K=205 M P a.s -1 , m=0.1, E=4200 MPa} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3 Viscoplasticity index I v and EVP index I ev vs ḣ/h for an elastic-viscoplastic solid representative of the behavior of PMMA {K=205 M P a.s -1 , m=0.1, E=4200 MPa} . . . . . . 14 Mechanical properties of PMMA [19]

	o r
	P
	e
	e r
	R
	e
	v i e
	w
	O n l
	y

4

Comparison between I v , I ev and strain rate sensitivity m for PMMA and PC, A : determination of I v using a power law fit of the p m -ḣ/h curve, B : determination of I ev using a power law fit of the σ r -ḣ/h curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Table 2 :

 2 Results of nanoindentation on PMMA and PC samples. σ r is computed using equation

	(12)

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Joseph Gastaldi
email: gastaldi@crmcn.univ-mrs.fr

Thomas Schenk

L Mancini

Holger Klein

Juergen Haertwig

S Agliozzo

Jose Baruchel

Marc De Boissieu

L Mancini ¶

H Klein §

J Härtwigg#

S Agliozzo ||

J Baruchel#

M De Boissieu£

SYNCHROTRON X-RAY DIFFRACTOMETRY AND IMAGING OF STRAINS AND DEFECTS IN ICOSAHEDRAL QUASICRYSTAL GRAINS

Keywords: defects, quasicrystals, strain

published or not. The documents may come

Whether the quasicrystal growth has to incorporate defects to ensure the stability of the quasicrystal lattice, or not, remain a challenge which set the supporters of the energy influence [START_REF] Steinhart | The Physics of Quasicrystals[END_REF] against those of the entropy effect [START_REF] Shechtman | [END_REF].

It has been reported that some quasicrystal grains are probably highly perfect since they give rise to anomalous X-ray transmission [3], but this assumption has not been confirmed by direct observations of the amount of strains and defects contained in these grains by using X-ray topography [4] [5].

Otherwise, investigations of defects by Transmission Electron Microscopy (TEM) in many as-grown and annealed grains have concluded to a minimum dislocation density of about 10 10 to 10 11 m/m 3 [4], which is substantial. Comparing to what is known for crystals, such a dislocation density is a limit below which it is difficult to detect these defects by TEM. Besides it has been determined that it does not disturb too much the anomalous X-ray transmission [START_REF] Batterman | [END_REF] [8]. Therefore, there is a need of further investigations using X-ray topography which is a technique appropriate to the characterization of defects at a low density level. Indeed, this technique allows to image defects in relatively thick grains (0.2 -1 mm) precisely when they are not too numerous (dislocation density < 10 8 m/m 3 ) [START_REF] Authier | ray and Neutron Dynamical Diffraction: theory and applications[END_REF]. Likewise it is possible to get contrasts from strain fields (up to a long range of about a few tens of micrometers) of individual defects the appearance of which is tightly related to the dynamical interaction of X-rays inside the grain lattice and are thereby highly indicative of the degree of lattice perfection. Using the X-ray beam delivered by the third generation synchrotron improves on one hand the spatial resolution of the X-ray topographs owing to the smallness of the source of these synchrotrons and on the other hand introduces additional useful phase contrasts generated by the high transverse coherence of this beam [START_REF] Cloetens | [END_REF].

Previously we have demonstrated that this technique is suitable for quasicrystals [START_REF] Authier | ray and Neutron Dynamical Diffraction: theory and applications[END_REF] and studied the degree of perfection of about ten icosahedral quasicrystal grains of various alloys (Al-Pd-Mn, Al-Cu-Fe, Zn-Mg-Y). Therefore we disclosed that some of them are much more perfect than the others [4] [5]. In this work we have concentrated on another slice of such a grain of the Al-Pd-Mn alloy. Similarly, we have performed an extensive Synchrotron X-ray Topographic investigation of defects in this grain, combined with Phase Contrast Radiography and High Resolution X-ray Diffraction examinations. Very few two-lobes contrasts associated to pores and no loop-shaped contrasts [12] were observed on X-ray topographs, this time, but straight line segments and band contrasts have been identified and attributed to events occurring after the growth process.

Experiments:

The most perfect grain of a polygrained Al-Pd-Mn quasicrystalline sample, having a size of about 2x2x0.3mm 3 , was selected according to the narrow (less than 10 arc second) Full Width at Half Maximum (FWHM) of the rocking Curves (RC) recorded from some of its areas, (50x50) µm 2 large, by synchrotron X-ray High Resolution Diffraction (HRD). This grain was investigated further, by this technique, varying the X-ray energy in order to detect anomalous X-ray transmission. In addition, Phase Contrast Radiographs and X-ray Topographs were exposed simultaneously, so as to image defects and their surrounding strain field which are accountable of both the enlargement of RCs and the disturbance of the anomalous X-ray transmission. The thin sample (0.3 mm thick)was prepared by the AMES Laboratory. It was cut out of a quasicrystal , grown by the Bridgman technique using a mother alloy of the composition Al 70 Pd 20 Mn 10 . Most of the X-ray experiments were carried out at the ID19 station of the European Synchrotron Radiation Facility (ESRF) in Grenoble (France) † . This beamline, dedicated to X-ray imaging and HRD, is characterized by a very long source-to-sample distance (150 m) and a small angular source size seen from a point of the sample (0.1 -1 µrad) which ensure a high geometrical resolution on X-ray topographs (0.1 µm). The small source size (≅ 100µm) combined to the long source-to-sample distance † Complementary experiments were performed at the Italian synchrotron, ELETTRA, in Trieste, in similar conditions. provides on the other hand the large transverse coherence length (≅ 40µm for λ=0.516Å) necessary for the observation of phase contrasts on the radiographs. The beam was monochromatized by two perfect Si crystals diffracting in the (+,-) non-dispersive setting. The spectral width was ∆λ/λ ≅ 10 -4 . The X-ray topographs were recorded either on films (Kodak SR) or on plates (Ilford L4 25µm) whereas phase-contrast radiographs were captured either on High Resolution films (Kodak SR) or by a CCD based camera developed at the ESRF and named FReLoN (Fast Read Out Low Noise) [13].

'[Insert figure 1 about here]'

Results:

Figure 1 is an X-ray topograph, which shows the spatial distribution of strain and defects inside the whole volume of the selected grain. Black, grey and white regions can be seen in the image. They correspond to different parts of the grain which are continuously misoriented as proved by the necessity to rock the grain along the Bragg peak by an amount of 0.05 ° for their complete imaging. Thereby, lattice curvatures with a radius of about 2 m for the entire grain and of about 0.5 m for a region of (50x50) µm 2 were evidenced. Besides, two-lobes, linear and band contrasts, respectively marked TL, L i and B i in Figure 1, can be distinguished in these regions. The few two-lobes contrasts present are associated to the pores visible in phase contrast radiographs, as previously shown [12]. As for linear and band contrasts, they are certainly related to defects located in the volume of the grain as indicated by the section topographs in Figure 2b which are true optical cuts of the sample at the level of the arrows in figure 2a.

'[Insert figure 2 about here]'

This is suggested, in this figure, by the line contrast Li and band contrasts Bi which are spreading in the middle of the section topograph image instead of limiting themselves to the sides. By the way one has to notice that there are no more Pendellösung fringes in this section topograph than in all other ones we recorded, either from other places in the grain or using other diffraction vectors. The only fringe sets we disclosed in section topographs were in restricted parts of the grain volume and were probably related to fault sequences like local condensation of phasons (Figure 3).

'[Insert figure 3 about here]' '[Insert figure 4 about here]'

Otherwise, anomalous X-ray transmission was detected, measured and imaged (Figure 4), in this grain, as it was observed in a previous one of the same family [4]. Moreover very narrow rocking curves were recorded from some of its (50x50) µm 2 areas and FWHM down to 0.0009° were measured (Figure 5). It was also noticed that displacing the incident X-ray beam 50 µm side-away change the FWHM value up to 100%.

'[Insert figure 5 about here]' '[Insert figure 6 about here]'

We tried to determine the directions and the displacement vectors of lines Li as well as those of band Bi, respectively labelled lL i , dL i for L i and lB i , dB i for B i . To this intent we recorded X-ray topographs using twelve different diffraction vectors g j (Figure 6). In both cases we faced with the same difficulty, i.e. it was impossible to recognize lines L i and bands B i in all of the various X-ray topographs so obtained, impeding us to characterize fully each of these two defects. For the determination of the line directions lL i and lB i , we need to analyze three X-ray topographs, recorded with three different diffraction vectors g j in which defects are visible. This is required in order to dispose of at least three different projection images allowing to reconstruct the inclination of the defects in the grain volume. This was possible only for lines L 1 because they appeared grouped in a family unlike the others (Figure 1) and thereby were easier to find again. It was determined that the direction of these lines was along the [-1/-1 0/1 0/0] two-fold axis inclined at about 75 degrees relatively to the normal of the grain surface which is along one of the other two-fold axes ([0/0 0/0 0/1]) (Figure 6). As for the determination of the displacement vectors dL 1 associated to these defects (Burgers' vectors bL 1 in case of dislocations) the application of the invisibility criterion g j .dL i = 0 was somewhat unsuccessful as discussed later. Again we get only some information for lines L 1 . As they were visible in reflections g j oriented along [-1/0 0/-1 0/0], this means that the displacement vector dL 1 is not in the direction of the line lL 1 [-1/-1 0/1 0/0] which is perpendicular, i.e. L 1 are not of the pure screw type. As for the determination of the exact direction of dL 1 , some ambiguity is remaining because lines L 1 are not visible in too many reflections (at least 6:…).

Discussion:

Unlike the previous one we investigated [4] [5], this new "highly perfect" Al-Pd-Mn quasicrystal grain allowed us to distinguish, clearly, individual defects in X-ray topographs. However, a complete characterization of all of them was not possible (determination of lL i , dL i …) despite their low density. This density was estimated to be of about 10 8 -10 9 m/m 3 from the various X-ray topographs we recorded from the same region as that of figure 2a. Although this value is indeed lower than the density of defects in the previously studied "highly perfect" Al-Pd-Mn grain [4] [5] and much lower than that reported by TEM observations [4], it remains close to the upper limit of the possibilities of defect characterization by X-ray topography [START_REF] Authier | ray and Neutron Dynamical Diffraction: theory and applications[END_REF]. Therefore, as this technique is very sensitive to low strains [START_REF] Authier | ray and Neutron Dynamical Diffraction: theory and applications[END_REF] and allows to image defects by their strain fields extending up to a few tens of micrometres, one can guess that when the number of defects is as large as 10 8 -10 9 m/m 3 , some overlapping strain fields can disturb the visibility of the corresponding contrasts and leads to the failure of the invisibility criterion g j .dL i = 0. Moreover, it can also be supposed that the observed lattice curvatures, although they have a large radius (0.5m -2m), will increase this effect. Nevertheless, as long as straight defects L i and bands B i (up to more than 100 µm) are clearly distinguished in many X-ray topographs and even if it has been impossible to fully characterize them, they can be tentatively compared to dislocations and phason faults observed by TEM [14][15][16] [17].

L i defects are looking like the straight dislocation segments reported by Caillard et al. as well in as-grown grains [15] as in deformed ones at high temperature [17]. They are much longer (≅ 100 µm instead of ≅ 0.1 µm) but this is certainly due to the difference of thickness between our sample (300µm) and those investigated by these authors who used TEM. Despite that they did not give any indication, it can be thought that the thickness of their samples was at most a few 0.1 µm which results in an equivalent ratio: defect length/ thickness of the sample. Besides, L i defects can be considered as likewise oriented as those observed by Caillard et al., provided we suppose that the identification of L 1 defects to one of the same axis, we made in section III, can be extended to all of them. Carrying on this hypothesis is the fact that two L i defects are sometimes linked, making an angle close to that of two two-fold directions (L 1 L 2 , L 1 L 3 , in figure 1) ‡ . Pursuing the comparison, for instance, ‡ As it can be estimated directly, by comparing the topograph in figure 1 to the stereographic projection along the incident X-ray beam (figure 6), neglecting the Bragg angle and bearing in mind that L 1 is oriented along the [-1/-1 0/1 0/0] two fold axis with the discussion of the ability of these defects to climb, a property of quasicrystal dislocations revealed by Caillard et al. [14], is difficult because we have been unable to determine their displacement vectors.

Nevertheless, one can guess that they are the result of climbing defects (dislocations?) generated by stress centres existing in as-grown quasicrystal grains as it has been suggested also by Caillard, et al. [15]. Similarly we could suppose that L i defects arise from the climbing of polygonal loops which intersect the sample external surfaces and leave dislocation segments inside the quasicrystal grain. Stress centres could be pores and precipitates as we previously evidenced and discussed [12] [18]. Indeed, during the annealing of quasicrystal single grains, we observed defects with loop-shaped contrasts enlarging while precipitates were growing and pores shrinking.. Despite these defects had not a segmented aspect on X-ray topographs [11] [12], it can yet be supposed that their loop-shaped contrast, which were difficult to analyse [18], results from the sum of the unresolved strain fields of concentric polygonal dislocation loops like those observed by TEM by Caillard et al. [15]. This consideration would corroborate one of the two hypothesis we brought forward in 1997 after having also identified by TEM analysis straight dislocation segments inside the same sample in which we precisely observed loop-shaped contrasts by synchrotron X-ray topography [START_REF] Reinier | [END_REF] [START_REF] Gastaldi | [END_REF]. The climbing of such giant defects, over a large distance, has already been observed in crystals during "in situ" studies, carried out by synchrotron X-ray topography, of the growth of recrystallized pure aluminium grains. It was then clearly seen that long screw dislocations, emitted by moving grain boundaries, climbed immediately into segmented helical dislocations [21]. Afterward, the pitches and diameters of these helical dislocations increased rapidly until they disappeared when reaching the external surfaces of the samples, leaving thereby mixed dislocations segments inside the grains [22]. It was thus calculated that only a very small vacancy supersaturation is necessary to allow climbing of such long dislocations (1.000005 for a 1mm long screw dislocation at 500°C [23]), a condition which is easily fulfilled, either during very slow cooling [23] or annealing at constant temperature, owing to the thermal fluctuations arising from the furnace temperature controller. In our case of interest, an estimation of both the supersaturation of vacancies needed and that standing in quasicrystal grains during solidification, annealing or cooling, can be made likewise using values of ∆H f , b, µ, … given in the literature for AlPdMn quasicrystals (see Appendices 1 and 2). Similar results are thus obtained which sustain our hypothesis. The vacancy supersaturation S (T) could be high enough during solidification (the temperature gradient is of about 10 to 30K/cm at the level of the interface) as well as the after-growth cooling, to activate Bardeen-Herring sources and generate large dislocation loops like those we observe together with Caillard et al.. Moreover, in electron microscopy experiments of Caillard et al. [13] both the gradient and fluctuations of temperature inside the electron microscope furnace might be greater than in ours, which is consistent with their observations of loops of smaller size [a ∆T of 30-35K is sufficient to activate a Bardeen-Herring source of a few 0.1µm length (Appendices 1and 2)]. Our observations are also corroborated, insofar as the better temperature control of our furnace is compatible with the growth of larger loops [the activity of Bardeen-Herring sources of about a few 10 µm long can be triggered by a ∆T in the 0.1-2K range (Appendices 1and 2)]. Therefore, it can be supposed that the L i defects are the remnants of climbing concentric polygonal dislocation loops, like those reported from TEM studies [15]. These loops appearing as loop shaped defects in X-ray topographs owing to the poor spatial resolution inherent to this technique, especially when defects are too numerous [START_REF] Authier | ray and Neutron Dynamical Diffraction: theory and applications[END_REF]. Nonetheless, if L i defects were partially resolved it is certainly due to the improvement of the perfection of the investigated quasicrystal grain, during the after-growth annealing [5]. This is corroborated on one hand by the unusual very narrow rocking curves we recorded from this grain (section III) and on the other hand by the very few number of pores observed in the corresponding X-ray topographs (Figures 1 and2) because pores are known to be numerous in the volume of as-grown grains [12] and to shrink during annealing [24].

As for contrasts B i , they are certainly not associated to the double contrasts of L i defects because they appeared so, in all X-ray topographs where they were visible, independently of the diffraction vector g j we used. Moreover their contrast line are much larger (Figures 1 and2) than those of L i defects. Therefore B i contrasts are probably linked to a different type of defect. They could be thought to limit some planar defects (phason walls, stacking fault [14]) for example, because some fringe sets were disclosed in some section Xray topographs (Figure 4) despite the insufficient degree of perfection of the studied quasicrystal grain (section III).

Accordingly, it appears that this AlPdMn icosahedral quasicrystal grain, though being the most perfect one we have ever studied up to now (section III), contains still a lot of strains and defects. Furthermore, the synchrotron X-ray topographic investigation of this grain seems to indicate that they are the results of stresses, either external (thermal stresses…) or internal (pores…) acting after the growth of grains. This phenomenon is well known in the case of the growth of both semiconductor [25] and metallic crystals for which it was demonstrated that the dislocation density (length) was increased by climbing, thanks to the activity of Bardeen-Herring or Amelinckx-Bontinck sources ¶ during cooling [22] [26]. We already put forward this proposal [5] [11] and Caillard et al. have recently emphasized the feasibility of such a dislocation mechanism in quasicrystals [17]. Therefore one can wonder whether strains and defects are mainly generated in quasicrystals after growth or during the growth process itself. For metallic crystals the situation is clear: no or very few strains and defects are induced in these grains by the moving growth interfaces either during recrystallization or solidification as it was proven by "in situ" synchrotron X-ray topography studies [START_REF] Reinier | [END_REF] [27] [28] [29]. No such studies have been carried out, up to now, during the growth of quasicrystals but we have started one which is in progress. The challenge is not only to determine when strains and defects are generated in quasicrystal grains but also to assess if they are constituent elements of the quasicrystal lattice and hence intrinsic to the icosahedral structure. This could help to better understand as well the long range propagation of the quasicrystalline order as the quasicrystal stability.

Conclusion:

The main results of this extensive Synchrotron X-ray Topographic investigation, combined with Phase Contrast Radiography and High Resolution X-ray Diffraction examinations, of defects in a newly selected perfect grain are: -very few two-lobes contrasts associated to pores and no loop-shaped contrasts [12] were observed in this case on X-ray topographs, but straight line segments and band contrasts together with some fringe patterns, which are not Pendellösung fringes, have been identified. These observations confirm that real quasicrystals are far from being "highly perfect" (no lattice curvature, very few dislocations: much less than 10 7 m/m 3 ) as it was believed when their ability to transmit anomalously X-ray was evidenced.

-line segments which have been difficult to fully characterize and were supposed oriented along two fold axes, could be considered as the result of the climbing of polygonal dislocation loops as those observed, by TEM, by Caillard et al. [15]. Thus they would ¶ Bardeen-Herring [START_REF] Bardeen | Imperfections in nearly perfect crystals[END_REF] sources are like Frank-Read sources (anchored edge-dislocation segments) but are acting in the plane perpendicular to the gliding plane (climbing plane) instead inside the gliding plane itself. Amelinckx-Bontinck [START_REF] Amelinckx | [END_REF] sources differ from the Bardeen-Herring sources by the starting dislocation segment which has a mixed character with a strong screw component. indicate that most of the strains and defects observed in quasicrystal grains at room temperature are the result of stresses (external and internal) acting after growth. This hypothesis has to be checked by an "in situ" observation of the quasicrystal growth.

Acknowledgments:

The authors are very grateful to S. W. Kycia for having kindly provided them with this new "highly perfect" grain, to M. Kleman, V. Dmitrienko for helpful discussions and to the technical staffs of the ESRF and ELETTRA for their assistance. In order that an edge dislocation segment in a vacancy environment, pined at its two extremities, acts as a Bardeen-Herring source of loop dislocations, the osmotic force induced by the vacancy supersaturation S (T) must exceed the dislocation line-tension force. Thus: S (T) ≥ exp { (µbv a kT / 2πL(1-ν)) (ln α/ 1.8b) } [28] By taking the atomic volume v a = b 3 (b = Burgers' vector module), the coefficient α = 4, an average value of the shear modulus µ = 60 GPa according to the Tanaka et al. measurements [START_REF] Tanaka | 18 Deleted: 19 Deleted: 20 Deleted: 21 Deleted: 22 Deleted: 23 Deleted: 24 Deleted: 25 Deleted: 26 Deleted: 27 Deleted: 28 Deleted: 29 Deleted: 30 Deleted: 31 Deleted[END_REF], (1-ν) = 0.75 [15]

  -line detector that consists of a scintillator converting X-ray into visible light, coupled to a CCD Deleted:

  supersaturation S (T) which could run in an AlPdMn quasicrystal during solidification or annealing:S (T) = C T /C (e)TC T = actual concentration of vacancies in the grain at a temperature T, C (e)T = concentration of vacancies given by the thermal equilibrium at the same temperature T,S (T) = exp [-∆H f (1/ T 1 -1/ T 2 )/ k] T 1 > T 2 [32] ∆H f =vacancy formation enthalpy, k = Boltzman constant, T 1 ,T 2 = temperatures. By taking the value ∆H f = 1.22 x 10 -19 Joule given for AlPdMn by Sato et al. [33], T 1 = 1133K and k = 1.38 x 10 -23 , we have calculated the vacancy supersaturation S (T) prevailing in an AlPdMn quasicrystal for 5 values of the temperature gradient ∆T = T 1 -T 2 which could exist, either close to the interface during solidification or in the grain during annealing: supersaturation S (T) necessary for the activation of Bardeen-Herring sources in an AlPdMn quasicrystal:

Figure 1 :

 1 Figure 2

Figure 2 :

 2 Figure 2: Synchrotron monochromatic X-ray topographs recorded from the central region of the studied grain. λ=0.516Å, g = τ 3 (0/2 0/0 0/0), rocking angle 0.025°. a) extended beam topograph. b) section topographs, recorded at the level of the arrows using a beam width limited to 20µm in the vertical direction, which are thereby true optical cuts of the grain.

Figure 3

 3 Figure 3Synchrotron monochromatic X-ray section topograph displaying a set of fringes F. Rocking angle 0.09°, λ=0.516Å, g = τ 3 (0/0 0/2 0/0).

Figure 4

 4 Figure 4Synchrotron monochromatic X-ray topographs recorded from a region of 600x600 µm 2 perpendicularly to, a) the forward diffracted beam I 0 , b) the diffracted beam I h . Rocking angle 0.02°, λ=1.033Å, g = τ 2 (0/2 0/0 0/0).

Figure 5 a

 5 Figure 5 a) Rocking curve (continuous line) recorded from the region of 50x50 µm 2 marked by a black square in the topograph of figure b and its Gaussian fit (dotted line). b) Synchrotron monochromatic X-ray topograph recorded from a region of 1x1 mm 2 . Rocking angle 0.025°, λ=0.516Å, g = τ 2 (0/2 0/0 0/0).

Figure 6

 6 Figure[START_REF] Wollgarten | Lectures on Quasicrystals[END_REF] Stereographic projection of the studied grain showing the directions of respectively, the incident beam (along [0/0 0/0 0/1]), the 12 diffraction vectors g j used (red circled) and the lines of contrasts L 1.

  and a mean value of b // = 035 nm[17], we have estimated the vacancy supersaturation necessary to activate Bardeen-Herring sources, in an AlPdMn quasicrystal, at a temperature close to the melting temperature, 1133 K, during its growth by solidification, for 3 lengths L of the edge dislocation segment.

	S (T)	2.2012	1.1130	1.0050	Deleted: ,
					Deleted: ,
					Deleted: but that
					Deleted: .
		F o r F o r F o r			Deleted: 30
		P P P
		e e e			Deleted: 31
		e r e r e r
		R R R
		e e e
			v i e v i e v i e
			w w w		Deleted: 26
			O n l O n l O n l y y y	Deleted: 32 Deleted: 13 Deleted: 15
	L	0.1µm	1µm	30µm

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S Etienne
email: serge.etienne@eeigm.inpl-nancy.fr

L David

Long Term Physical Aging of Amorphous Polymers

Keywords: glass transition, polymers Keywords (user supplied): Secondary relaxation, Physical ageing, Amorphous polymers secondary relaxation, amorphous polymers, glass transition, mechanical spectroscopy, physical aging, structural relaxation, cooperative motions

The dynamic properties of amorphous polymers is investigated during long term aging by means of high resolution low frequency mechanical spectroscopy. Two polymers are considered in this work, namely atactic polystyrene and quenched amorphous poly(ethylene naphtalate). It is shown that during aging at temperature higher than that of secondary relaxation but lower than that of primary one (if tested at 1Hz), the strength of the secondary relaxation first increases and then decreases for long aging times. This phenomenon is explained on the basis of concepts of (i) the cohesion heterogeneity of the amorphous network at the nanometric scale and (ii) cooperativity of molecular motions responsible for the secondary relaxations.

Introduction

Physical aging in the glassy state (or structural relaxation) is known to affect the physical properties of amorphous systems and amorphous polymers in particular. Although numerous studies have been devoted to the relationships between structural state and molecular mobility, a precise and general picture of the effects of aging on the different degrees of freedom in polymers is not known yet. Experimental and theoretical approaches of the aging effects on the α (main) relaxation are main keys for the glass transition understanding. Moreover, the influence of structural state on the β, γ, δ, ... (i.e. secondary) relaxation processes is of interest since the β relaxation is a major aspect of the glassy state physics. Actually, the β relaxation process was first proposed to originate from motions within 'islands of mobility' [1] and to be a specificity of the glassy state [2]. Such assumptions were studied and checked by dielectric spectroscopy on a set of molecular glasses and liquids [3]. More recently, the β relaxation was considered as the elementary motion for the more complex α relaxation [START_REF] Pérez | Physique et Mécanique des Polymères Amorphes[END_REF][START_REF] Etienne | [END_REF][6], thus inferring a relation with structural state and thus structural relaxation. The aim of this work is to show that physical aging does influence the sub-T g relaxation process in a complex way when it shows a degree of cooperativity (i.e. when the activation parameters have no longer physical meaning in term of energy barrier). It is shown that the aging condition (temperature and duration) may result in specific aging effect on the secondary relaxation process.

Materials and techniques

Materials

Two amorphous polymeric materials were investigated in this work, namely atactic polystyrene (PS) and poly(ethylene naphtalene 2,6 dicarboxylate) (PEN for short). In the case of PEN, fast quenching from the liquid state was necessary to avoid crystallization. The glass transition temperature Tg, assessed by DSC (heating rate 10 °C/min), is 95°C and 120°C (midpoint) for PS and PEN, respectively. The molecular weights M n and M w of PS are 93000 and 240000g/mol; and that of PEN are 22000 and 38000 g/mol.

Techniques

The dynamic shear modulus G* = G'+jG''= G.exp(jϕ) was measured using a high resolution mechanical spectrometer [7], based on an inverted pendulum operated in the forced oscillation mode, and working in the low frequency range (10 -4 Hz up to 1 Hz), which allows to separate the secondary and main relaxation processes more clearly. The strain amplitude does not exceed 10 -5 . Such a low level ensures that the mechanical excitation does not modifies the structural state of the specimens. This mechanical spectrometer makes it possible to study the dynamic shear modulus by frequency and/or temperature scanning. For constant frequency measurements, the heating rate was 1°C/min. The long term aging procedure in isothermal conditions was carried out in the spectrometer, thus ensuring reliable results.

In addition, calorimetric measurements (DSC model TA 2920) were performed to characterize the energy released on rejuvenation of aged specimens and estimate the so-called fictive temperature according to a procedure long ago proposed by Moynihan et al [8].

Experimental results

The aging effect was studied by monitoring the complex shear modulus by frequency scanning at constant temperature and also by temperature scanning at constant frequency.

PS

Figure 1 presents the complex evolution of the phase lag ϕ measured at a constant temperature (80°C, i.e. not far below Tg) and constant frequency (0.316Hz) as a function of aging time at 80°C. During the shortest aging times, the decrease of the value of ϕ is attributed to the classical aging effect on the α process inducing a reduction of the long range molecular motions. Nevertheless, it appears clearly that the β relaxation strength first increases for longer times and then decreases for ultra long term aging.

Figure 2 displays the associated evolution of the dynamical behavior of PS observed during aging at 80°C. The increase of the secondary relaxation strength is again clearly observed on aging. This observation also is confirmed by the comparison of isochronal measurements of the specimen i) after aging and ii) after erasing of thermal history as shown in Figure 3, where the enhancement of the β process on aging is underlined.

The structural relaxation was characterized by calorimetric measurement (heating and cooling rate =10°C/min) as shown in Figure 4. The overall effect of long term aging (86°C during 7 months) appears clearly by calculating the fictive temperature T f . The fictive temperature of PS is decreased down to T f =89°C after long term aging, that is to be compared to T f =100°C after rejuvenation.

PEN

In order to go a step further, another amorphous polymer was considered where the secondary and main relaxation processes are clearly separated, namely PEN as shown in Figure 5. Actually, there are two sub-Tg relaxation processes in PEN, namely β and β* (according to the usual nomenclature agreed in the case of PEN). Actually, the low temperature secondary process β can be separated into two individual components, namely β 1 and β 2 . These low temperature secondary processes correspond to localized motions of chain segments, since the activation entropy is rather low for both components. In addition, the β 2 component was shown to be affected by the presence of adsorbed water. As a consequence, the β 1 and β 2 processes will not be considered further in this work. The β* process in PEN is believed to be a true secondary relaxation process but with cooperative character, since the activation entropy is found to be about 200J/mol.K. Thus, the aging was carried out at intermediate temperature (105°C) i.e. between that of T α and T β * . It is observed that during the long term aging the strength of the secondary relaxation is first decreasing, then increases (Figure 6a), and then decreases at very long aging duration (Figure 6b). The upturn attributed to the α relaxation is only visible on the reference state, and does not appear on the curves obtained after aging.

The evolution of the β * relaxation strength during long term and ultra long term aging is summarized on Figure 6. It is obvious that the β * relaxation strength goes trough a maximum for an aging duration about 3.10 4 minutes.

Discussion

It is shown that the β* relaxation of PEN exhibits a behavior similar in many aspects to that of β relaxation in PS and the high temperature component of the β relaxation in BPA-PC [9], PEEK and PES [10]. Isothermal physical aging at an intermediate temperature, i.e. between T β (1Hz) and T α(1Hz) appears to first increase the strength of the β relaxation, even if it slows down the dynamics of the α process and eventually the β process. In agreement with these previous findings, this study leads to the conclusion that with aging, the domains of cooperatively rearranging regions responsible for the main relaxation process evolve and transform into domains of high mobility for the less cooperative sub-T g relaxation motions ('that could correspond to 'islands of mobility' [1][2][3][START_REF] Johari | The glass transition and the nature of the glassy state[END_REF]). For the longest aging times, the amorphous structure continues to evolve and the amplitude of the β-relaxation decreases again. In other words, the rearranging regions where the α and β relaxation processes develop are different but should be related in the sense that structural relaxation induces a conversion of part of the molecules involved in the α motions into active species responsible for the β (or β*) events. This suggests that an enhancement of β relaxation is occurring after a structural change when the characteristic size of the density fluctuations or 'cooperatively rearranging regions' (structural size) is corresponding to the characteristic size of the implied molecular motions (dynamic size). As a result, such a complex phenomenon cannot be described with a 'quasi punctual' defect picture of amorphous matter [START_REF] Pérez | Physique et Mécanique des Polymères Amorphes[END_REF]. Aging below T g induces a decrease of the structural characteristic sizes, and a transient correspondence of structural and dynamic sizes for the β-relaxation can be observed when this process is close to the main relaxation and/or when it exhibits a cooperative character. This contrasts with the well known effects described by Johari [START_REF] Johari | The glass transition and the nature of the glassy state[END_REF]. Furthermore, it is interesting to address the spatial location of main and secondary processes at the nanometric scale. In fact the heterogeneous nature of the glassy structure was proposed a few years ago to explain the excess of vibrational density of states as shown by low frequency Raman scattering and inelastic neutron scattering [START_REF] Saviot | [END_REF]. According to this concept, cohesive domains (with size about 4 nm in polymers) are separated by less cohesive zones that could be identified to the cooperatively rearranging regions. It has been shown that the contrast within the network structure resulting from cohesive domains and soft zones is decreased by physical aging in a monotonous way, the soft zones becoming more and more ordered. Taking into account the complex effects of aging on β (or β*) processes, this means the decrease of the cohesive contrast occurring within the soft zones is responsible for the conversion of part of the α rearrangements of limited extension into secondary relaxation events exhibiting cooperative character. This proposal could explain why the secondary relaxation strength goes through a maximum upon aging. Within the frame of the energy landscape, during the first step of the structural state evolution, the system can be understood as falling into a minimum presenting a large number of small energy barriers (β* motions). Ultimately, after ultra long term aging, the system falls into a deeper minimum with higher energy barriers (the characteristic time of the α processes is largely increased, and simultaneously, the strength of the β* process is decreased). Evolution of the phase lag ϕ at 80°C and 0.316Hz during aging of polystyrene. Ultra-long term aging measurements ( ) indicates that the β relaxation strength goes through a maximum. The doted line is drawn as a guide for the eye.

Figure 6a

PEN isochronals (0.316Hz) after aging at different times at 105°C showing the increase of the β* relaxation amplitude after long term aging. The rising due to the α relaxation appears only on the reference state curve. It does not appear on subsequent curves because (i) the α relaxation dynamics is slower and (ii) the upper temperature of the scans was limited to the aging temperature (105°C) in order to avoid any change in the aging rate due to the measurements. Figure 6b PEN isochronals (0.316Hz) after aging at different times at 105°C showing the decrease of the β* relaxation amplitude after ultra long term aging.

Figure 1 .

 1 Figure 1.

Figure 2 .

 2 Figure 2. Frequency scanning at 80°C of polystyrene showing the β-relaxation enhancement on aging.

Figure 3

 3 Figure 3 Temperature scanning at constant frequency (1Hz) of polystyrene after aging 21 days at 80°C (solid circles) and subsequent rejuvenation due to fast cooling from T=125°C after the first heating scan (open circles).

Figure 4 .

 4 Figure 4. Calorimetric measurements of PS on heating (at 10K/min). First scan: after aging 7 months at T g -15°C, second scan: quenched state.

Figure 5 .

 5 Figure 5. PEN mechanical relaxations measured on heating (1K/min) at 10Hz.

Figure 7

 7 Figure 7 Evolution of the phase lag at 105°C and 0.316Hz during aging of PEN. The β* relaxation strength goes through a maximum close to 3.10 4 min.

  105 °C (min)Etienne et al, Figure7.

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