Deformation mechanism in NiAl single crystals at low temperatures

Intermetallics 57 (2015) 93e97

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Deformation mechanism in NiAl single crystals at low temperatures M.Z. Butt a, *, Dilawar Ali b a b

Rafi M. Chaudhri Chair, Centre for Advanced Studies in Physics, GC University, Lahore, 54000, Pakistan Department of Physics, GC University, Lahore, 54000, Pakistan

a r t i c l e i n f o

a b s t r a c t

Article history: Received 20 July 2014 Received in revised form 2 September 2014 Accepted 14 October 2014 Available online

Available data on the temperature dependence of yield stress and of associated activation volume of high-purity NiAl single crystals between 76 and 205 K have been analyzed within the framework of a solid-solution hardening model, which is based on the concept of depinning of an edge-dislocation segment from several randomly dispersed, isolated, point defects simultaneously. The vacancies in NiAl single crystals act as point-defect obstacles to the stress-assisted thermally-activated glide of edge dislocations, and their concentration is estimated as about 10 at.%. The product of yield stress and associated activation volume (z0.146 eV) is found to be independent of temperature and yield stress, as envisaged in the model. © 2014 Elsevier Ltd. All rights reserved.

Keywords: A. Aluminides B. Yield stress B. Plastic deformation mechanisms D. Point defect

1. Introduction Theories of plastic flow in binary solid-solution crystals can be divided into two groups. In the models of first group, the rate process of yielding is assumed to be breakaway of a dislocation segment from individual, point-like, obstacles [e.g. Refs. [1‒3]]. The models in second group are based on the concept of depinning of an edge-dislocation segment from several, randomly dispersed, isolated, point defects simultaneously [e.g. Refs. [4‒18]]. Computer simulation predicts that the critical concentration of point defects below and above which depinning process will be of “individual”and “collective”- type, respectively, lies in the range 10
5 e 10
4 [19]. In 2001, Brunner and Gumbsch [20] studied the temperature dependence of yield stress of high-purity NiAl single crystals with stoichiometric composition over a wide range of temperature between 76 and 325 K. The residual resistivity ratio of the crystals was 12 and the dislocation density 4  1011 m
2. Tensile tests performed at a constant plastic shear-strain rate 1  10
4 s
1, were combined with stress-relaxation experiments. The tensile axis was close to <111> direction (so-called “soft” orientation) which facilitated plastic deformation primarily by the motion of <100> dislocations on {011} slip plane. They observed that the yield stress (t) of NiAl single crystals decreases rapidly with increasing temperature from 76 to 230 K (region I), and then it decreases slowly up to 320 K

* Corresponding author. Tel.: þ92 42 99214601. E-mail addresses: [email protected], [email protected] (M.Z. Butt). http://dx.doi.org/10.1016/j.intermet.2014.10.006 0966-9795/© 2014 Elsevier Ltd. All rights reserved.

_ T increases (region II). Similarly, the strain-rate sensitivity (vt=vg) rapidly with decreasing temperature below 230 K, reaches a maximum at about 100 K and decreases again below this temperature. Brunner and Gumbsch [20] assumed that the high concentration of above equilibrium vacancies in NiAl single crystals at temperature below 1000 K as well as the low mobility of vacancies makes these point defects to act as localized obstacles to dislocation glide. The stress necessary to overcome the short-range interaction of these immobile vacancies with dislocations in the glide plane determines the flow stress of NiAl single crystals. Brunner and Gumbsch [20] analyzed the experimental data of region I in terms of Fleischer's model [1] of solid-solution hardening, which is based on depinning of a dislocation segment from an individual, point-like, obstacle as the rate process of plastic deformation. However, the mean obstacle spacing of only 3b determined from Fleischer's model appears to be far too small to be considered realistic, and the idea of depinning of a dislocation segment from an individual, point-like, obstacle (vacancy in this case) as the rate process of plastic deformation was therefore ruled out by them in view of the too small mean obstacle spacing and the limited flexibility of dislocations. Based on the atomic simulation, Schroll and Gumbsch [21] have also concluded that the tremendous decrease of yield stress with increasing temperature cannot be explained on the basis of dislocation interactions with isolated structural point defects alone. In order to explore an acceptable picture of deformation mechanism for the observed temperature dependence of yield stress and of strain-rate sensitivity in NiAl single crystals referred to above, the data have been re-analyzed within the framework of

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ButteFeltham model [12,13], in which the rate process of plastic deformation is based on depinning of an edge-dislocation segment from a number of randomly dispersed point-like obstacles simultaneously. 2. ButteFeltham model The unit activation process of yielding in ButteFeltham model [12,13] of solid-solution hardening involves the stress-assisted, thermally-activated detachment of edge-dislocation segments from short rows of closely spaced solute-atom pinning-points. The mean spacing between neighboring solute atoms, denoted by circles in Fig. 1, is taken to be l ¼ b/c1/2, where b is the lattice parameter or the length of the Burgers vector, and c is the solute concentration expressed as an atomic fraction. To facilitate slip under an applied shear stress t at a given deformation temperature T, the length of an edge-dislocation segment L (¼AB as shown in Fig. 1), after unpinning, must extend to that of the arc ABC with maximum displacement nb (n being a numerical constant) adequate to remove most of it from the short-range stress field of the initial pinning-points. The activation energy for the formation of arc ACB (Fig. 1) is given by the relation [13]:


 W ¼ Wo x
1=2 ex1=2

(1)

(evW/vt)T or kT [v(lng_ )/vt]T, is given by Ref. [13]:

v ¼ vo expðmkT=Wo Þ

(5)

Here vo ¼ (1/4)b3n2 (Gb3/Uc1/2)1/2 is the activation volume associated with the critical resolved shear stress (CRSS) to at T / 0 K. Combining Eqs. (4) and (5), one finds that, for a given alloy, the product t v is constant, i.e.

tv ¼ to vo ¼ Wo

(6)

On using the expressions for macroscopic parameters

to ¼ 4Uc1/2/nb3 and Wo ¼ n (Uc1/2Gb3)2, the microscopic parameters n and U of slip envisaged in the ButteFeltham model are given by the formulae:

2
 n3 ¼ Wo Gb3 ð4G=to Þ U ¼ Wo2




n2 Gb3 c1=2



(7) (8)

The length of the edge-dislocation segment Lo involved in the unit activation process of yielding at T / 0 K is related to the yield stress to as under:

Lo ¼ ð4Gn=to Þ1=2 b

(9)

Also, the activation volume vo in terms of Lo can be expressed as: with the yield criterion:

W ¼ mkT;

_ ¼ 25±2:3 m ¼ ln ðg_ o =gÞ

(2)

Here x (¼t/to) is the ratio of the CRSS of the crystal deformed at temperature T to that expected as T / 0 K, Wo is a positive constant specific to the material, g_ is the shear rate of the crystal with typical values in the range 10
3 to 10
5 s
1, g_ o is a constant of the order of 107 s
1, and k is the Boltzmann constant. For rather low temperatures, where diffusional processes are dormant in the crystal, Eq. (1) is reduced to [13]:

W ¼ Wo ln ðto =tÞ

(3)

where Wo ¼ n (Uc1/2 Gb3)1/2 and to ¼ 4Uc1/2/nb3. This in conjunction with Eq. (2) leads to:

t ¼ to exp ðemkT=Wo Þ

vo ¼

1 nLo b2 4

(10)

3. Comparison with experiment The data points in Fig. 2 denote the values of CRSS, t, of NiAl single crystals deformed by Brunner and Gumbsch [20] in the temperature range 75e205 K and represented here as a function of temperature T in semi-logarithmic coordinates. The line drawn through the data points by least-squares fit method is encompassed by the relation:

ln t ¼ 6:38e1:19  10
2 T

(11)

(4)

The temperature and concentration dependence of activation volume v, customarily defined as.

Fig. 1. Detachment of an edge-dislocation segment from a short row of solute atoms, and its movement to a new pinning site in a stress-assisted thermally-activated process.

Fig. 2. Temperature dependence of the CRSS of NiAl single crystal in semi-logarithmic coordinates. Data points were taken from Brunner and Gumbsch [20].

M.Z. Butt, D. Ali / Intermetallics 57 (2015) 93e97

with a linear correlation coefficient r ¼
0.998. One can also rewrite Eq. (11) as:

  T t ¼ 593 exp
84

(12)

This provides to ¼ 593 MPa and Wo ¼ 0.181 eV. Similarly the activation volume v data obtained by Brunner and Gumbsch [20] from their stress-relaxation experiments performed on NiAl single crystal, is denoted by points in Fig. 3 as a function of temperature T in semi-logarithmic coordinates. The line fitted to the data points by least-squares fit method is represented by the expression:

ln v ¼ 48:6  10
2 þ 1:2  10
2 T

(13)

with a linear correlation coefficient r ¼ 0.999. Eq. (13) can be rewritten in its equivalent form as

  T v ¼ 1:63 exp 84

95

Referring to Fig. 4, the activation volume v has been depicted as a function of CRSS t in double-logarithmic coordinates. The straight line fitted to data points by least-squares fit method is given by.

ln v ¼ 6:86e0:995 ln t

(17)

with a linear correlation coefficient r ¼
0.993. This slope of the lnv
lnt line being close to
1 points to

tv ¼ constant

(18)

in accord with ButteFeltham model [12,13]. The numerical value of t v ¼ 0.146 eV is close to Wo ¼ tovo ¼ 0.181 eV. The reason for somewhat lower value for t v compared with Wo ¼ tovo has already been pointed out above. Finally, Fig. 5 depicts the dependence of the product t v on (a) temperature T and (b) CRSS t. One can readily note that the product t v is independent of both T and t, as envisaged in ButteFeltham model [12,13]. The average value of t v ¼ 0.146 eV is close to that of Wo represented by the dashed line above the data points.

(14) 4. Discussion 3

which helps to determine vo ¼ 1.63 b and Wo ¼ 0.181 eV. It is interesting to note that the dashed straight line somewhat above the data points (Fig. 3), represented by the expression:

with a linear correlation coefficient r ¼ 0.997, corresponds to the values of to and Wo derived from the CRSSeT data (Fig. 2) leading to vo ¼ 2.0 b3. One can readily note that the functional form of veT is the same in both the cases (Eqs. (14) and (16)). However, a slight difference in the value of vo can be attributed to the variation in the mobile dislocation density in the crystal, which is invariably assumed to be constant during the stress relaxation test.

On using the value of to and Wo (Table 1), one can evaluate the microscopic parameters of slip in NiAl single crystals. In the model expressions to ¼ (4Uc1/2/nb3) and Wo ¼ n (Uc1/2Gb3)2, the term Uc1/2 can be replaced by Uc1/2 ¼ U* which is now the binding energy per interatomic spacing along the dislocation line segment involved in the unit activation process [18]. Taking G ¼ 7  104 MPa, b ¼ 0.288 nm, and Gb3 ¼ 10.45 eV for NiAl single crystals [22,23], one can find the values of n, U*, Lo and vo (Table 1) from Eq. (7)e(10). To find the concentration of vacancies in the NiAl single crystal, we shall first determine the force Fo ¼ tobLo acting on a dislocation segment of length Lo to unpin it from the vacancies close to it, and then take it to the saddle-point configuration. With the values of to, Lo and b given in Table 1, one finds Fo ¼ 7.72  10
10 N. Atomistic simulation by Schroll and Gumbusch [21] show that the interaction force of <100> edge dislocation with a vacancy is 1.6  10
10 N. Therefore the edge-dislocation segment of length Lo will detach itself from about 5 vacancies simultaneously.

Fig. 3. Semi-logarithmic representation of the temperature dependence of activation volume in NiAl single crystal. Data points were taken from Brunner and Gumbsch [20].

Fig. 4. Dependence of the activation volume on the CRSS of NiAl single crystals in logelog coordinates.

ln v ¼ 69:3  10
2 þ 1:2  10
2 T

(15)

or

  T v ¼ 2:0 exp 84

(16)

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M.Z. Butt, D. Ali / Intermetallics 57 (2015) 93e97

Fig. 6. Dependence of dislocation length L and number of vacancies N interacting with it on deformation temperature T of NiAl single crystals.

Fig. 5. Dependence of the product of CRSS and associated activation volume of NiAl single crystal on (a) deformation temperature and (b) CRSS. Data points were obtained from the measured values of CRSS and strain-rate sensitivity given in Brunner and Gumbsch [20].

This c¼

helps

to

determine

planar !

Number of vacant lattice sites in area L2o Total number of lattice sites in area L2o

vacancy   ¼

55 1616

concentration ¼ 0:1, i.e. 10%.

Literature survey shows that Ni vacancy concentration in NiAl compound crystals is at variance due to the specimen's thermal or mechanical treatment history as well as its shape and size, i.e. whether bulk, coarse powder, or thin film [24‒29]. The vacancy concentration derived in the present work is not inconsistent with that reported by Collins and Sinha [28], who observed that Nivacancy concentration in milled NiAl compounds with compositions very close to stoichiometric 50:50 composition, is of order 1‒ 10 at.%. As far as triple defects (2VNi þ NiAl) formation in stoichiometric NiAl compound [27, 28] is concerned, Pike at al [24] have

Table 1 Microscopic parameters of slip in NiAl single crystals. G ¼ 7  104 MPa, b ¼ 0.288 nm, Gb3 ¼ 10.45 eV. T (K)

to (MPa)

Wo (eV)

n

U*(meV)

Lo(b)

vo (b3)

75e205

593

0.181

0.52

11.6

15.7

2.0

found from microhardness measurements that solid-solution hardening rate in NiAl compound for Ni vacancies (G/4) was significantly larger than that for Ni antisites (G/12). It is worthy of note although outside the scope of this work that vacancies in nominally pure metals also play a significant role in determining their yield strength. For instance, the yield strength of polycrystalline Al specimens annealed at 500  C and aged for six months at room temperature is found to be 31% higher than that of the annealed but un-aged specimens. It is because the vacancies migrated to the cores of edge dislocations during natural ageing pin the dislocations and make their movement rather difficult [30]. On the other hand, the yield strength of Mo polycrystals annealed at 700  C is reduced by 20e30% on natural aging for six months due to migration of vacancies to the cores of screw dislocations during the course of aging. Consequently, the Peierls field/dislocation core geometry is modified leading to lower Peierls energy per interatomic spacing along the length of screw dislocations trapped in the Peierls valleys, and hence reduction in the yield strength [31]. Finally, reference to Fig. 6 shows that as deformation temperature T increases CRSS t decreases, and the length of the dislocation segment L involved in the unit activation process increases in accord with the relation L ¼ b (4Gn/t)1/2. Since the inter-vacancy distance l ¼ b/c1/2, the number of vacancies interacting with the dislocation segment L is therefore given by N ¼ (L/l). Table 2 lists the values of L and N as a function of T, which shows that L (b) increases from 16 to 51 and N increases from 5 to 16 as T is raised from 0 to 200 K. The lines fitted to the data points in each case (Fig. 6) show an exponential rise in L and N values with the increase in temperature.

Table 2 Length of edge-dislocation segment L and number of vacancies N interacting with it for various temperatures. T (K)

t (MPa)

L (b)

N

0 50 100 150 200

593 327 181 100 55

16 21 28 38 51

5 7 9 12 16

M.Z. Butt, D. Ali / Intermetallics 57 (2015) 93e97

5. Conclusions From the forgoing data analysis in terms of ButteFeltham model of solid-solution hardening, we conclude that the rate process of plastic deformation in NiAl single crystals with soft orientation in the temperature range 75e205 K is the breakaway of edgedislocation segment from several randomly dispersed vacancies simultaneously. References [1] Fleischer RL. Rapid solution hardening, dislocation mobility, and the flow stress of crystals. J Appl Phys 1962;33:3504e8. [2] Fleischer RL. Substitutional solution hardening. Acta Metall 1963;11:203e9. [3] Friedel J. Disloctions. Oxford: Pergamon; 1964. p. 379. [4] Mott NF, Nabarro FRN. Report of a conference on the strength of solids. London: The Physical Society; 1948l. [5] Riddhagni BR, Asimow RM. Solid-solution hardening in concentrated solutions. J Appl Phys 1968;39:4144e51. [6] Riddhagni BR, Asimow RM. Solid-solution hardening due to the size effect. J Appl Phys 1968;39:5169e73. [7] Feltham P. Solid solution hardening of metal crystals. J Phys D Appl Phys 1968;1:303e8. [8] Labusch R. A statistical theory of solid solution hardening. Phys Status Solidi 1970;41:659e69. €rtung. Acta Metall [9] Labusch R. Statistische theorien der mischkristallha 1972;20:917e27. [10] Kratochvil P, Lukac P, Sprusil B. On solid solution hardening in crystals with randomly distributed solute atoms. Czech J Phys B 1973;23:621e6. [11] Nabarro FRN. The theory of solution hardening. Phil Mag 1977;35:613e22. [12] Butt MZ, Feltham P. Solidesolution hardening. Acta Metall 1978;26:167e73. [13] Butt MZ, Feltham P, Ghauri IM. On the temperature dependence of the flow stress of metals and solid solutions. J Mater Sci 1986;21:2664e6. [14] Butt MZ, Feltham P. Solidesolution hardening. Rev Deform Behav Mater 1978;3:99e149. [15] Nabarro FRN. Stress equivalence in the theory of solution hardening. Proc Roy Soc Lond A 1982;381:285e92.

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