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Intermetallics 57 (2015) 93e97

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Intermetallics

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Deformation mechanism in NiAl single crystals at low temperatures

M.Z. Butt a, *, Dilawar Ali b

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

a r t i c l e i n f o

Article history:Received 20 July 2014Received in revised form2 September 2014Accepted 14 October 2014Available online

Keywords:A. AluminidesB. Yield stressB. Plastic deformation mechanismsD. Point defect

* Corresponding author. Tel.: þ92 42 99214601.E-mail addresses: mzbutt49@yahoo.com, mzakriab

http://dx.doi.org/10.1016/j.intermet.2014.10.0060966-9795/© 2014 Elsevier Ltd. All rights reserved.

a b s t r a c t

Available data on the temperature dependence of yield stress and of associated activation volume ofhigh-purity NiAl single crystals between 76 and 205 K have been analyzed within the framework of asolid-solution hardening model, which is based on the concept of depinning of an edge-dislocationsegment from several randomly dispersed, isolated, point defects simultaneously. The vacancies inNiAl single crystals act as point-defect obstacles to the stress-assisted thermally-activated glide of edgedislocations, and their concentration is estimated as about 10 at.%. The product of yield stress andassociated activation volume (z0.146 eV) is found to be independent of temperature and yield stress, asenvisaged in the model.

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Theories of plastic flow in binary solid-solution crystals can bedivided into two groups. In the models of first group, the rateprocess of yielding is assumed to be breakaway of a dislocationsegment from individual, point-like, obstacles [e.g. Refs. [1‒3]]. Themodels in second group are based on the concept of depinning of anedge-dislocation segment from several, randomly dispersed, iso-lated, point defects simultaneously [e.g. Refs. [4‒18]]. Computersimulation predicts that the critical concentration of point defectsbelow 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 withstoichiometric composition over a wide range of temperature be-tween 76 and 325 K. The residual resistivity ratio of the crystals was12 and the dislocation density 4�1011 m�2. Tensile tests performedat a constant plastic shear-strain rate 1 � 10�4 s�1, were combinedwith stress-relaxation experiments. The tensile axis was close to<111> direction (so-called “soft” orientation) which facilitatedplastic deformation primarily by the motion of <100> dislocationson {011} slip plane. They observed that the yield stress (t) of NiAlsingle crystals decreases rapidly with increasing temperature from76 to 230 K (region I), and then it decreases slowly up to 320 K

utt@gmail.com (M.Z. Butt).

(region II). Similarly, the strain-rate sensitivity (vt=v _g)T increasesrapidly with decreasing temperature below 230 K, reaches amaximum at about 100 K and decreases again below this temper-ature. Brunner and Gumbsch [20] assumed that the high concen-tration of above equilibrium vacancies in NiAl single crystals attemperature below 1000 K as well as the low mobility of vacanciesmakes these point defects to act as localized obstacles to dislocationglide. The stress necessary to overcome the short-range interactionof these immobile vacancies with dislocations in the glide planedetermines the flow stress of NiAl single crystals.

Brunner and Gumbsch [20] analyzed the experimental data ofregion I in terms of Fleischer's model [1] of solid-solution hard-ening, which is based on depinning of a dislocation segment froman individual, point-like, obstacle as the rate process of plasticdeformation. However, the mean obstacle spacing of only 3bdetermined from Fleischer's model appears to be far too small to beconsidered realistic, and the idea of depinning of a dislocationsegment from an individual, point-like, obstacle (vacancy in thiscase) as the rate process of plastic deformation was therefore ruledout by them in view of the too small mean obstacle spacing and thelimited flexibility of dislocations. Based on the atomic simulation,Schroll and Gumbsch [21] have also concluded that the tremendousdecrease of yield stress with increasing temperature cannot beexplained on the basis of dislocation interactions with isolatedstructural point defects alone.

In order to explore an acceptable picture of deformationmechanism for the observed temperature dependence of yieldstress and of strain-rate sensitivity in NiAl single crystals referred toabove, the data have been re-analyzed within the framework of

M.Z. Butt, D. Ali / Intermetallics 57 (2015) 93e9794

ButteFeltham model [12,13], in which the rate process of plasticdeformation is based on depinning of an edge-dislocation segmentfrom a number of randomly dispersed point-like obstaclessimultaneously.

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 segmentsfrom short rows of closely spaced solute-atom pinning-points. Themean spacing between neighboring solute atoms, denoted by cir-cles in Fig. 1, is taken to be l ¼ b/c1/2, where b is the latticeparameter or the length of the Burgers vector, and c is the soluteconcentration expressed as an atomic fraction. To facilitate slipunder an applied shear stress t at a given deformation temperatureT, the length of an edge-dislocation segment L (¼AB as shown inFig. 1), after unpinning, must extend to that of the arc ABC withmaximum displacement nb (n being a numerical constant)adequate to remove most of it from the short-range stress field ofthe initial pinning-points.

The activation energy for the formation of arc ACB (Fig. 1) isgiven by the relation [13]:

W ¼ Wo

�x�1=2ex1=2

�(1)

with the yield criterion:

W ¼ mkT ; m ¼ ln ð _go= _gÞ ¼ 25±2:3 (2)

Here x (¼t/to) is the ratio of the CRSS of the crystal deformed attemperature T to that expected as T/ 0 K,Wo is a positive constantspecific to the material, _g is the shear rate of the crystal with typicalvalues in the range 10�3 to 10�5 s�1, _go is a constant of the order of107 s�1, and k is the Boltzmann constant.

For rather low temperatures, where diffusional processes aredormant in the crystal, Eq. (1) is reduced to [13]:

W ¼ Woln ðto=tÞ (3)

whereWo¼ n (Uc1/2 Gb3)1/2 and to¼ 4Uc1/2/nb3. This in conjunctionwith Eq. (2) leads to:

t ¼ toexp ðemkT=WoÞ (4)

The temperature and concentration dependence of activationvolume 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-activatedprocess.

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

v ¼ voexpðmkT=WoÞ (5)

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

tv ¼ tovo ¼ Wo (6)

On using the expressions for macroscopic parametersto ¼ 4Uc1/2/nb3 and Wo ¼ n (Uc1/2Gb3)2, the microscopicparameters n and U of slip envisaged in the ButteFeltham modelare given by the formulae:

n3 ¼�Wo�Gb3

�2ð4G=toÞ (7)

U ¼ W2o��

n2Gb3c1=2�

(8)

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

Lo ¼ ð4Gn=toÞ1=2b (9)

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

vo ¼ 14nLob2 (10)

3. Comparison with experiment

The data points in Fig. 2 denote the values of CRSS, t, of NiAlsingle crystals deformed by Brunner and Gumbsch [20] in thetemperature range 75e205 K and represented here as a function oftemperature T in semi-logarithmic coordinates. The line drawnthrough the data points by least-squares fit method is encompassedby the relation:

ln t ¼ 6:38e1:19� 10�2T (11)

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

M.Z. Butt, D. Ali / Intermetallics 57 (2015) 93e97 95

with a linear correlation coefficient r ¼ �0.998. One can also re-write Eq. (11) as:

t ¼ 593 exp�� T84

�(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 performedon NiAl single crystal, is denoted by points in Fig. 3 as a function oftemperature T in semi-logarithmic coordinates. The line fitted tothe data points by least-squares fit method is represented by theexpression:

ln v ¼ 48:6� 10�2 þ 1:2� 10�2T (13)

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

v ¼ 1:63 exp�

T84

�(14)

which helps to determine vo ¼ 1.63 b3 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:

ln v ¼ 69:3� 10�2 þ 1:2� 10�2T (15)

or

v ¼ 2:0 exp�

T84

�(16)

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

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

Referring to Fig. 4, the activation volume v has been depicted asa function of CRSS t in double-logarithmic coordinates. The straightline 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 thelnv � lnt line being close to �1 points to

tv ¼ constant (18)

in accord with ButteFelthammodel [12,13]. The numerical value oft v ¼ 0.146 eV is close to Wo ¼ tovo ¼ 0.181 eV. The reason forsomewhat lower value for t v compared withWo ¼ tovo has alreadybeen 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 productt v is independent of both T and t, as envisaged in ButteFelthammodel [12,13]. The average value of t v ¼ 0.146 eV is close to that ofWo represented by the dashed line above the data points.

4. Discussion

On using the value of to and Wo (Table 1), one can evaluate themicroscopic parameters of slip in NiAl single crystals. In the modelexpressions to¼ (4Uc1/2/nb3) andWo¼ n (Uc1/2Gb3)2, the term Uc1/2

can be replaced by Uc1/2 ¼ U* which is now the binding energy perinteratomic spacing along the dislocation line segment involved inthe 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 Fo¼ tobLo acting on a dislocationsegment of length Lo to unpin it from the vacancies close to it, andthen take it to the saddle-point configuration. With the values ofto, 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 theinteraction force of <100> edge dislocation with a vacancy is1.6 � 10�10 N. Therefore the edge-dislocation segment of length Lowill detach itself from about 5 vacancies simultaneously.

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

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

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

Table 2

M.Z. Butt, D. Ali / Intermetallics 57 (2015) 93e9796

This helps to determine planar vacancy concentration

c ¼

Number of vacant lattice sites in area L2oTotal number of lattice sites in area L2o

!¼�

5�516�16

�¼ 0:1, i.e. 10%.

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

Table 1Microscopic parameters of slip in NiAl single crystals.G¼ 7� 104MPa, b¼ 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-solutionhardening rate in NiAl compound for Ni vacancies (G/4) wassignificantly larger than that for Ni antisites (G/12).

It is worthy of note although outside the scope of this work thatvacancies in nominally pure metals also play a significant role indetermining their yield strength. For instance, the yield strength ofpolycrystalline Al specimens annealed at 500 �C and aged for sixmonths at room temperature is found to be 31% higher than that ofthe annealed but un-aged specimens. It is because the vacanciesmigrated to the cores of edge dislocations during natural ageing pinthe dislocations and make their movement rather difficult [30]. Onthe other hand, the yield strength of Mo polycrystals annealed at700 �C is reduced by 20e30% on natural aging for six months due tomigration of vacancies to the cores of screw dislocations during thecourse of aging. Consequently, the Peierls field/dislocation coregeometry is modified leading to lower Peierls energy per inter-atomic spacing along the length of screw dislocations trapped inthe Peierls valleys, and hence reduction in the yield strength [31].

Finally, reference to Fig. 6 shows that as deformation tempera-ture T increases CRSS t decreases, and the length of the dislocationsegment L involved in the unit activation process increases inaccord with the relation L ¼ b (4Gn/t)1/2. Since the inter-vacancydistance l ¼ b/c1/2, the number of vacancies interacting with thedislocation segment L is therefore given by N ¼ (L/l). Table 2 liststhe 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 raisedfrom 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 increasein temperature.

Length of edge-dislocation segment L and number of vacancies N interacting with itfor various temperatures.

T (K) t (MPa) L (b) N

0 593 16 550 327 21 7100 181 28 9150 100 38 12200 55 51 16

M.Z. Butt, D. Ali / Intermetallics 57 (2015) 93e97 97

5. Conclusions

From the forgoing data analysis in terms of ButteFelthammodelof solid-solution hardening, we conclude that the rate process ofplastic deformation in NiAl single crystals with soft orientation inthe temperature range 75e205 K is the breakaway of edge-dislocation segment from several randomly dispersed vacanciessimultaneously.

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