PROGRESS IN X-RAY DIFFRACTION OF RESIDUAL MACRO … · 2018. 1. 10. · PROGRESS IN X-RAY DIFFRACTION OF RESIDUAL MACRO-STRESS DETERMINATION RELATED TO SURFACE LAYER GRADIENTS AND

PROGRESS IN X-RAY DIFFRACTION OF RESIDUAL MACRO-STRESS DETERMINATION RELATED TO SURFACE LAYER GRADIENTS AND

ANISOTROPY

S.J. Skrzypek I, A. Baczmaiiski 2 ‘Faculty of A4etallurgy and Materials Science “Faculty of Physics and Nuclear Techniques

University of Mining andMetallurgy, Al. Mickiewicza 30, 30-059 Krakbw - Poland

Abstract ‘The well known sin2p method for macro-stress measurement elaborated for Bragg-

Brentano geometry has some disadvantage e.g. penetration depth of X-ray beam varies during m.easurement. This is particularly important in the case of coatings or surface layers where residual stresses appear with large gradients, The new version of the sin’v method, named g- sin’w (based on the grazing angle scattering geometry) was applied for determination of the macro-residual stresses in TiN coatings deposited on sintered WC and on sintered high speed steel. These types of samples present a wide range of residual stresses i.e. from large tensile to large compressive. Using the g-sin2w method the stresses are determined for chosen near surface layers for which effective penetration depth remains almost constant during measurement. Anisotropic elastic constants were used for calculations. The measured results were compared with a model based on thermal shrinkage.

Introduction The most universal sir& diffraction method enables measurement of the macro-stress

tensors and the elastic properties of polycrystalline materials. If a known stress can be applied during measurement, diffraction elastic constants (DEC) and elastic constants or mechanical compliances can be measured [l-4 J.

The diffraction methods have several important features such as a non-destructive and non-reference character, possibility of stress analysis for multiphase and anisotropic materials. The conventional well-known sin’ p method has some disadvantage i.e. penetration depth of X- ray rad.iation varies during experiment when the w angle is changing. This is particularly important for thin coatings, films or surface layers where the residual stresses have large gradients which can reach several hundreds MPa/p.m. Till now the gradients of residual macrostresses in surface layers were measured using different wavelengths [5,6] or by gradual removing of surface layers and following measurements [7].

In this work the g-sin’ly method based on the grazing incidence angle X-ray diffraction (MD) geometry is applied to macro-residual stresses (RS) measurement in TiN coating. Van Acker et al. [8] and Quaeyhaegen and Knuyt [9] have introduced the first approaches to this geometry in term of residual macro-stress measurement. Using this method the non-destructive analysis of the residual stresses for different penetration depth bellow the sample surface can be performed. The problems concerning penetration depth, diffraction intensity distribution,

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critical angle of total reflection for GD geometry were also discussed by Huang [lo] and Goehner [ 111.

In the presented geometry the penetration depth can be easily chosen by incidence angle (o: in Figs 1 and 2) and it is almost constant during measurement for wide range of \I, and 8 angles (Eq.3 and 4). This method was used for determination of the RS in TiN coatings deposited on sintered WC and on high-speed steel (HSS). These kinds of samples presented a wide range of residual stresses i.e. from large tensile to large compressive.

Diffraction methods of lattice strain measurement

The conventional diffraction sin2t+o method for the determination of residual stresses is based on measurement of the interplanar spacing for various directions of the scattering vector characterised by the q and I,V angles (Fig. 1). In the diffraction experiment, the mean interplanar spacing fikl) averaged only for the reflecting grains having the scattering vector normal to the Qzkl) crystallographic planes is measured. Moreover, in the case of surface measurement (by X-ray radiation) the absorption should be taken into account in calculations.

The average value of the lattice strain fikl, in the Lj direction (Fig. 1) is defined a.s:

< d(+7v)>(hkl) - d(h”,) < w4vb(hkl) = __ (1)

d”

where: &@kI) is the interplanar spacing for (hkZ] planes in the stress-free material and the (h~~~ average is defined for grains volume as in Eq.2.

E’ig.1. Geometry of g-sin2 vmethod. The (hkI) spacings aremeasured along LS axis in I-, system and the stresses d, are defined with respect to the S-sample system. The incident angle a is fixed during measurement. The orientation of s8cattering vector is characterised by the np and w angles. For the u.nit incident (KO) and diffracted (Qvectors the scattering vector is equal to [12]:


In good approximation, the measured (J&l) is averaged according to:

(2)

0

where; the < . . >(hkIj average is calculated over all volume of the reflecting grains in the beam path from surface to the depth t for which G, = 95% of radiation is absorbed (see Eq.4 and CMlity ]:12]), ,u is the linear coefficient of absorption and Z(X) is the function of path length vs. depth (x) and the a and 0 angles are defined in Fig. 1,

The conventional method for stress determination, called sin’ry has been elaborated for symmetrical Bragg-Brentano geometry and can be applied to “w” and “Y;’ goniometers. In both cases tbe orientation of the scattering vector varies, however, the planes indexes (hkl) are kept constant during dm measurement. It can be shown, that different penetration depths of measurement versus the tilt angle w appear for these geometries [ 13,141 and causes problems in stress measurement in the case of stress gradients, non-uniform microstructure or texture in the near surface volume. The importance of stress gradients in surface layers was recognised by Perry [6] and Kra.uz and Ganev [5]. This problem can be solved using the new geometry i.e. a new version of the sin’vmethod based on grazing angle scattering geometry.

The new method, called g-sin2 vI/ [ 13,14 J is characterised by small and constant incident angle (cr- in Fig. 1 and 2) and by different lengths and orientations of the scattering vectors (Fig. 1). In contrast to the conventional sin” y method, the measurements are performed for different fhkl) planes usiig appropriate values of 8 fikl) angles. The interplanar spacing

Using some simplifications, the depth (t) for which fraction G, (fraction of total intensity of the X-ray beam) is absorbed is called effective depth of penetration (EDP) which depends on the difI?action geometry and can be calculated from formulas:

a) for the sin”ty and psin’vmethods : t = - Zn(~ - Gx) cos( y) sin( t9)

2P ,

b) for the cvsin’ty and p/c+sir?vmethods : t = - ln(l - Gx) I I ’

’ sin(Q+ y) + sin(O- y) I

(4)

c) for the g-sin2ry method : t= - ln(l - Gx)

i

I I P- sin a ’ sin(2 y f a) I

where, for every method the appropriate relations between angles 0, w and a were considered [ 14,151. The difference between penetration depths vs. sin’ty for different geometries are presented in Fig2 The main advantage of the g-sin’w method is almost constant penetration depth for fixed small angle a (a = 1 - 10 degrees) and large range of I+V angle (Eq.3). Moreover, the required penetration depth of X-ray radiation can be easily chosen by changing the a angle and the stresses can be determined for different thickness of layers under the sample surface (Eq.4c and Fig.2).

Fig.2. The penetration depth t vs. sin2ry 20 .

calculated from Eq.4 for G,=O. 95 and for 18- --.:;::- 16-

.-.* different geometries. Absorption of Cu

*. -* 5 . sin’ y 14-

*. --* Kcli rladiation in TiN material (J = 561

*‘. *.

cm-‘) ‘was considered. In the case of the -y2- o-sin UC’\. -*.. * \ . . \

g-sin’ v, method calculations were SIO- *. .

- 8- \ .

. . . performed for two different a angles i.e.

. a=6 \

1 and 6 degrees. 6- \

4- g-sin’y ‘, 2- a=1

_-----w--w---- 0' ' 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

sin’ w

The g-sin’p method is based on the non-standard geometry of measurement for which the s;:cl(q, v/)>~kl, interplanar spacing in directions defined by the q and I,V angles are measured for different hkl reflections, The crucial point of the work is correct interpretation of the eixperirnental data using properly calculated (or measured) elastic difYraction constants and layer thickness. The influence of crystal anisotropy on interpretation of the stress measurement will be considered here.


Determination of residual stresses from diffraction measurement For a general stress state and for an quasi-isotropic sample, the average lattice strain in the L3

direction is equal to [l-3];

-C &‘(qo, w / >(hkJ) = SI (hkl)(Lrfl + C& + & + 2 s.2 (hkl)(oi, cm2 p + O$~ sin’ v, f ~7:~ sin 44 sin’ w 1

I I (5) + 2 s.2 (hkUc& cos’ y + - s2 (hkl)( oi3 cos 9 + &sin cp) sin 2ry

2 where: sl(hkl) and s.$hkt) are the diffraction elastic constants for a quasi-isotropic

polycrystal and d, macrostresses are defined with respect to the S system (Fig. 1.). Another definition of the DECs may be introduced for a textured sample [Z, 18,191;

< E’(P~ YV) +a/+) = Fij OL, r P,I v’) CT; Pa)

The FV coefficients are not the tensor components because they relate the stresses dV expressed in S system with the strains cE$3>@kl, defined along & axes of the L-frame. Obviously, using the appropriate transformation, the FV elastic constants can be easily calculated from the R, ones, i.e. :

Fij (atkl, 4, vv, f(i9~ =R,, (hkl, 9, ~9 f@rO~ mi Y nj ( W where the y matrix transforms stresses from the S to the L system (Fig. l), i.e, :

o- 1 I= y mi y nj cz i. and R,, are DECs for textured polycrystal defined in the following section. IEn

For the biaxial stress state in a qasi-isotropic sample the well known formula for the measured interplanar spacing can be obtained:

< dtp, y) >(hkll = {sl @W(a;l f & + ; s2 (hW& cm2 P + dz2 sin2 v + (6)

t& sin 2~) sin2 I// P ihlli+d ,,,lli The s&kJl and s$hkQ elastic constants depend on the single crystal constants, grain-matrix

interaction and hkl reflection. In the case of textured material the FiJ depends also on orientation distribution. Moreover, for biaxial stress state and for isotropic material, Eq.6 proves the linear character of the

< a(qj v) +ikkg = {$I (hkl)(a:l+ o&J + i s2 (hkl)(cri, cm2 p + dJ2 sin2 q +

0i2 sin 293) sin’ I+v )a *+a *

where for cubic structure :

(7)

and a”=d ’ h2+k2+12 (hkl) d In the above equation the recalculated lattice constants a* and c~, are used

instead of interplanar spacings $ fikl) and okl in the fitting procedure. Of course the experimental flk!, depends on the used reflection and orientation of the scattering vector because it is directly recalculated from (hkg using appropriate crystallographic relation (e.g. for cubic structure Eq.7). The same type of average, i.e. through volume of difkacting crystallites, is used for the recalculated lattice constants and the measured interplanar spacings.

It should be stated that in the presented method the s&ikl) and s$ikC) constants used for one ~(9, +~w vs. sin’ly graph depend on the hkZ reflection. Moreover, in the case of textured material, the RG constants which depends on the orientation distribution function must be used instead of s#ikl) and s$%kq. Consequently, the diffraction elastic constants depend on the I,V angle and ekl, vs. sin’ly graphs are not linear.

For calculated diffraction elastic constants s@zkJ), s.$hkJl or FY , the experimental lattice constant,s

where all the above quantities are defined with respect to the L-&me. Finally, in the case of Voigt model the d&-action elastic constants do not depend on the hkl

reflection for an quasi-isotropic sample and they are equal to:

For cubic crystals the above elastic constants are calculated and they are usually expressed by components of single crystal compliance tensor sijk [3, 211:

‘5 = soh11 +2~,,22)+~osl,22s1212 and s = 2w212 (s1111 -s1122)

3s Ill1 - 3k22 -I- 4SIZI2 2 3 Sllll - 3JII22 + 4SIZl2

where: SO = siril- ~112~ - 2 ~1212 In the case of textured material more general DEC i.e. Rij’ are defined (see Eq.5a,b) [2].

The calculation of [cklijr] is based on the texture mnction of all crystallites from the irradiated volume which contribute here to the average:

Rg = [c’j’~~ where [Cfqkl/ = ~C$H Ci$)f(g)dg (12) E

In this equation the c+‘(g) single crystal stifI?nesses (expressed in the L system) are integrated over the whole orientation space E and f(91 is the orientation distribution function characterising texture.

Reuss model [22] In this approach we assume that the local gQ-stress is homogenous across the sample, i.e.,

u;l=d,. Using the sYkl ’ compliance tensor for single crystal, we can write the following equations in the L system:

-5+33 = s’ 338 y (-J.. = s)33. (p 31 cl (13)

< &‘33>(hkl) = < sl33ij>(hklj ok’ where; the average is calculated over all diffracting grains having scattering vector

normal to the fikg plane i.e.:

where: 5 is the angle of rotation around scattering vector perpendicular to the fhkl) planes.

The diffraction elastic constants depend on the hkl reflection used. For the cubic slymmetry and isotropic sample they can be expressed by:

SI = -‘WI >(hkl) and S2 = 2 (

where: I&= @?f+h2~+kzf)/(h2+~+?)2 and ~0 is given by Eq. 11.

Consequently, using the Reuss model the DECs in textured material may be expressed as:

(17)

where: f(g) is given by Eq. 12 and r is the angle of rotation around the scattering vector perpendicular to the @k() planes.

Experimental results The g-sin’w geometry and from five to eight diffraction lines (i.e.: ( Ill>, (2001, (2201,

(3111, (2221, (400) and (4203) were used for determination of the residual stresses in TiN coatings deposited on sintered WC carbide (sample no.V30) and on sintered high speed steel (HSS, sample no.T31). The coatings were produced by the CVD method in the case of V30 sample and by PVD for the other one [23,25]. Cu Kar radiation was used with a Philips di@ractometer (X-Pert MPD) and CoKa with a Bruker (D8 Advance) diffractometer.

The diffraction patterns were recorded for cp = 0 and cp = 90. For verification the additional measurements for cp = 180 degrees were performed using the Bruker (D8 Advance) diffractometer and the splitting of the curves were not observed. It proves that the nonlinearities in the sin2v are not caused by the shear stresses (i.e., o13 = 023 = 0).

The profiles of dieaction peaks were corrected for absorption and Lorentz-polarisation fa.ctors [3,12]. The asymmetric Q diEaction geometry for grazing incidence angles larger than the critical angle of total reflection (0.28 degree) was applied in experiments. A grain size of TiN was small enough, -1-2 pm [25]. Therefore, according to Hart et aZ.[24], correction due to refraction can be neglected.

The diffraction elastic constants (sI and s2 ) were calculated from single crystal elastic constants (S~III= 2.17 lo3 GPa-‘, s1z22= - 0.38 10” GPa-‘, s12j2= 1.49 10” GPa-’ [3]) using the Voigt and Reuss approaches.

The macro residual stresses determined for different penetration depths are presented in the Table I and 2. The experimental lattice parameter (a(p, v)>ek~) (recalculated from measured

However, both approaches give approximately similar values of the macrostresses (see Table 1 and 2). It should be stated that the linear regression can be used only for the Voigt method because the diffraction elastic constants (.sI and s2 or R,) do not depend on the hkl reflection.

Talb.1. Residual biaxial stresses (&I, = d& in TiN coatings deposited on WC carbide substrate (V30). The g-sin2v method was used and the penetration depth was calculated using Eq.4 for C&=0.95 and ,u = 561 cm-’ for Cu Kai . The thickness of the coating was 5 pm.

Grazing angle (a)

Wgl

1

3

6

Stress Stress Average Penetration (oL=dzz) (dll’J22) stress depth

Wal NW (&=d,z) (9 Reuss method Voigt method NW u-4

15s2+37 1550136 1566 0.9

1242539 1194&38 1218 2.6

1107&43 1045&41 1076 4.8-5.2

The tensile biaxial stress state in the coating of the V30 sample arises due to sample cooling from the high temperature (1173 K) of the CVD process to room conditions (AT=900 K). This stress is caused by the difference of thermal expansion coefficients for TiN (CX~ = 9.35 x 10d) and WC carbide (a~=6 x lo&). The thermal origin residual stress (&II = d22 = 1760 MPa) calculated by a simple model is larger then the maximum value obtained on the upper layer of the coating [ 131. By the same calculation for the PVD process for TiN on HSS steel (CX~S =17x10”) for AT=500 K the thermal origin residual stress was; dll = dam = -2251 MPa.

R.esults in Table 1 and 2 show a gradient of residual stresses in first case caused by a relaxation process in the intermediate layers between TiN and the substrate. For TiN deposited on HSS steel by the PVD method little gradient was found. The measured macro-residual stresses are quite close to the calculated thermal case. The results from Tab.2 obtained with the D%Advance diffractometer are additionally presented in Fig.3.

The biggest deviation of the measured points from calculated ones (from linearity in case of the Voigt model and from approximated points for the Reuss model) appeared for the lowest Bragg angle diffraction lines i.e. for ( 1 1 1 } and (200). It is their nature, but could be caused additionally by stacking faults which are common for this type of crystal lattice which result in shifts in the 0 angles.


4.255- I {111)

4.220-c I 4.220 ! 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

4.260

4.255

4.250

4.245 G- ; 4.240

4.235

4.230

4.225

sin’ v sin’ \v

Fig.3. The (,,~o lattice parameters (refers to Tab.2, sample T3 1) calculated using Eq.8 are fitted to the experimental points for different grazing incident angles : a) a = I" (t = 0.6 pm), b) a = 6” (t = 3.3 urn). For better visualisation the calculated p~,~ values are connected using continues line for Reuss method and dashed line for Voigt method. The measurements were curried out for cp = 0 (e) and 180 degrees (+)

Tab.2, Residual biaxial stresses (d~l = d22> in TiN coatings deposited on high-speed steel su.bstrate (sample T3 1). The g-sin2 y method was used and the penetration depth was calculated using Eq.4 for G,=O.95 and p = 561 cm-’ for Cu Kar and p = 837 cm-’ for CoKa. The thickness of the coatings was 5.1 urn.

Grazing

i

Philips (X-Pert MPD) Bruker (D8 Advance) angle ,,d’ Reuss method, oll = cr22 [MPa] Reuss method oll = oz2 [MPa]

kl (hCUKa1) @COKal) depth ,,t” textured q-isotropic textured q-isotropic depth ,,t”

b-4 +-120 MPa +-120 MPa +-130 MPa +-130 MPa [Pm1 1 0.9 -3532 -3562 -3177 -3074 0.6

2 1.8 -3482 -3556

3 2.6 -3575 -3607 -3207 -3207 1.75 (T)

4 3.3-3.5 -3572 -3652

6 4.8-5.2 -3662 -3737 -3588 -3496 3.2-3.4(T)


Conclusions A significant gradient of the residual stresses in a TiN coating deposited on WC carbide

(Table 2) was found. The decrease of the measured stress versus sample depth probably results from relaxation processes in the interface layers between TiN and the substrate. The effect of stress heterogeneity was observed using the g-sir? y method based on the grazing angle scattering geometry. The conventional sin’w and m-sin2p methods are not valid for the study of stress gradients,

For the g-sin” v geometry the penetration depth of the X-ray beam is much smaller than in the conventional sin2v methods (Tab. 1 and 2, Fig.2). Using the new geometry the stresses are: determined for a chosen volume below the surface which can be easily changed by choice of incident beam angle (Fig.2). The advantage of this geometry is a constant penetration depth during the experiment. The methodology of experimental data treatment presented, enables real non-reference and non-destructive stress measurements to be made, on the grazing angle scattering geometry.

The correct diffraction elastic constants should be used for proper interpretation of the experimental data. These constants have to be calculated for different hkl reflections and various sample orientations. It is readily seen in the Fig.3 that the crystal anisotropy creates the rmnlinearities on the strain vs. sin’w plot. From a theoretical point of view these nonlinearities can be easily modeled using the Reuss approach.

The results provided in Tab.2, which were obtained in two laboratories with two different diffractometers and wavelengths are satisfactorily close to each other.

Acknowledgements; This work has been partially sponsored by European Commission under the PECO Program

(grant CIPA 92-3032) and by the University of Mining & Metallurgy fkom Cracow - Grant No. Il. 11.110.230. Authors acknowledge MSc. K. Chrusciel for his contribution in calculations.

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PROGRESS IN X-RAY DIFFRACTION OF RESIDUAL MACRO … · 2018. 1. 10. · PROGRESS IN X-RAY DIFFRACTION OF RESIDUAL MACRO-STRESS DETERMINATION RELATED TO SURFACE LAYER GRADIENTS AND