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Grain size dependence of the plastic deformation kinetics in Cu
Materials Science and Engineering A341 (2003) 216 /228 www.elsevier.com/locate/msea
Grain size dependence of the plastic deformation kinetics in Cu Hans Conrad * Department of Materials Science and Engineering, North Carolina State University, Raleigh, NC 27695-7907, USA Received 11 June 2001; received in revised form 4 April 2002
Abstract Data on the effect of grain size d in the range of nm to mm on the plastic deformation kinetics of Cu at 77 /373 K are analyzed to determine the influence of grain size on the strain rate-controlling mechanism. Three grain size regimes were identified: Regimes I (d :/10 6 /10 3 m), II (d :/10 8 /10 6 m) and III (d B/ /10 8 m). A dislocation cell structure characterizes Regime II, which no longer occurs in Regime II. The absence of all intragranular dislocation activity characterizes Regime III. The following mechanisms were concluded to be rate-controlling for o :10 5 10 3 s 1 : (a) Regime I , intersection of dislocations ; (b) Regime I , grain boundary shear promoted by dislocation pile-ups ; and (c) Regime III , grain boundary shear. The major effect of grain size on the intersection mechanism in Regime I is on the mobile and forest dislocation densities; the effect in Regime II is on the number of dislocations and on the number of grain boundary atom sites; the effect in Regime III is on the number of grain boundary atom sites. The transition grain size from one regime to another depends on the strain rate and temperature. Crystallographic texture is also important. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Copper; Dislocations; Grain boundaries; Thermally-activated processes; Hall-Petch constants
1. Introduction There has developed in recent years a renewed interest in the influence of grain size (especially in the submicron range) on the mechanical properties of materials. Two major stimulants have led to this interest: (a) superplasticity; and (b) the desire for ultrahigh strength with reasonable ductility. It is anticipated that these properties may be enhanced by a reduction in grain size. In addition to possible beneficial effects on mechanical properties, novel physical properties can result as the grain size is reduced to the nanometer range, where the number of atoms situated in the grain boundaries comprise a significant fraction of the total [1]. The influence of grain size d on the flow stress s (or hardness H :/3s) of crystalline materials at low homologous temperatures (T B/0.25 TM) is usually considered in terms of the Hall-Petch (H-P) relation, s si KH-P d 1=2 ;
* Tel.: /1-919-515-7443; fax: /1-919-515-7724 E-mail address: [email protected] (H. Conrad).
(1)
where si is the lattice friction stress and KH-P the socalled H-P constant. The H-P relation has been reported to be obeyed down to a grain size of /20 nm, below which there generally occurs a decrease in the constant KH-P, and in some cases ultimately a negative value has been reported, i.e. grain size softening; see Ref. [2]. It has been proposed [3,4] that the mechanism responsible for the H-P relation (i.e. grain size hardening) is the pile-up of dislocations at a grain boundary, giving a stress concentration which activates dislocation generation and/or motion in the adjacent grain. The grain size softening in the low nanocrystalline range (so-called inverse H-P effect) has been attributed by various investigators to: (a) grain boundary diffusion creep (Coble creep) [5,6]); (b) grain boundary sliding [7]; and (c) grain boundary shear [2]. Since in general the flow stress depends on strain rate o and temperature T , the H-P constants si and KH-P (and any departure from the normal H-P behavior) may vary with these test parameters. The objective of the present study was therefore to examine the effects of these parameters on the grain size dependence of the flow stress of Cu over the wide grain size range from nanometers to millimeters and to identify the mechan-
0921-5093/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 0 9 3 ( 0 2 ) 0 0 2 3 8 - 1
H. Conrad / Materials Science and Engineering A341 (2003) 216 /228
isms governing the corresponding plastic deformation kinetics. Cu was chosen because desired data for the analysis are available over the wide grain size range of interest. The analysis was restricted to low homologous temperatures (T B/0.25 TM) to reduce the possible complicating influence of lattice diffusion on recovery and grain growth. A Hall-Petch plot of the effect of grain size on the flow stress of Cu at 300 K and o 10 5 10 3 s 1 is presented in Fig. 1. The data points for the freestanding, vapor-deposited (VP) and electrodeposited (EP) films scatter about the dashed line representing an extrapolation of results by Hansen and Ralph [8] for bulk Cu with d /11/250 mm. The flow stress of the compacted-powder specimens (VP-C and EP-C) tend to fall significantly below the extrapolation for d 5/25 nm (d 1/2 ]/200 mm 1/2) and suggest a grain size softening (i.e. a so-called inverse H-P effect) for d 1/2 ]/200 mm 1/2. As discussed by Koch and Narayan [18] the compacted-powder specimens may contain processing defects (including porosity, non-uniform grain size and contamination), which can lead to a reduction (or increase) in the flow stress. They questioned whether the inverse H-P effect was a real phenomenon or merely reflected the increased difficulty of preparing nanocrystalline materials free of undesirable defects. Another perspective on the effect of grain size on the flow stress of Cu is provided in Fig. 2, which is a plot of log s vs. log d. This plot suggests three grain size regimes: (a) Regime I, d :/10 6 /10 3 m; (b) Regime
Fig. 1. H-P plot of the effect of grain size d on the flow stress s of Cu at 300 K and o 10 5 10 3 s 1 : Symbols " and X are for hardness with s /H /3; the remaining symbols are for the flow stress at o :/0.01 in tension. VP, vapor-deposits; EP, electrodeposits; VP/C, compacted VP powders; EP-C, compacted EP powders. Data from Refs. [5,6,8 / 16].
217
II, d :/10 6 /10 8 m; and (c) Regime III d B/10 8 m. In Regime I, the flow stress increases significantly with decrease in grain size; in Regime II the effect of grain size appears less pronounced; in Regime III a decrease in flow stress with decrease in grain size is suggested. The existence of three regimes is supported by dislocation behavior. In Regime I dislocations form cell structures whose size lc in Cu is given by [19]: (2)
s 21mb=lc ; 10
2
where m /4.21 /10 N m is the shear modulus and b /2.56 /10 10 m the Burgers vector [20]. A plot of Eq. (2) is included in Fig. 2. This indicates that Regime II occurs when the grain size becomes smaller than the cell size, i.e. when d B/ /10 6 m. Hence, no dislocation cells are expected to form during plastic deformation when d B/ /1 mm, thereby terminating Regime I and beginning Regime II. Intragranular dislocation activity however continues in Regime II until, due to elastic interactions, the separation of dislocations becomes greater than the grain size. According to dislocation theory the spacing ld between dislocations due to elastic interactions is given by [24]: t mb=2pld (screw)
and
(3) t mb=2p(1 n)ld (edge); pffiffiffi Plots of Eq. (3) (taking M 3 [20] and M /3 [21 /23]) are also included in Fig. 2. These indicate that Regime III occurs when the spacing between dislocations becomes larger than the grain size, i.e. when d B/ /10 nm. This grain size is in accord with computer simulations [25,26], which gave that intragranular dislocation activity no longer occurred in Cu when d 5/8 nm. The computer simulations [26] also indicated that for d /10 nm there occurs a grain size regime where the flow stress is weakly dependent on the grain size, which is in accord with the behavior in Regime II shown in Fig. 2. Finally to be noted regarding Fig. 2 is that the maximum stress obtained to-date with reduction in grain size in Cu is about 1/6 the theoretical shear strength given by tth /m/ pffiffiffi 16 [27], taking M 3:/ The present paper focuses on three sets of data pertaining to the plastic deformation kinetics in the three regimes indicated in Fig. 2, namely: (a) those by Cao and Conrad [28,29] for Regime I; (b) those by Embury and Lahaie [11] for Regime II; and (c) those by Cai et al. [6] for Regime III. The first set is representative of Regime I and provides kinetic data desired for the analysis. The second two sets are the only data known to the author which provide plastic deformation kinetics data for Regimes II and III, respectively, in Cu. In view of the limited data available for the latter two regimes, the strain rate o to be considered in the present analysis is of the order of 10 5 /10 3 s 1 (except for the data of Cai et al. [6], where o :10 7 10 6 s 1 ) and the strains
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Fig. 2. Log s vs. log d for Cu at 300 K and o 10 5 10 3 s 1 : Open symbols and * are for o :/0.01; filled symbols and X are for o :/0.2. VP, EP, VP/C and EP/C have the same meaning as in Fig. 1. Data from Refs. [5,6,8 /17].
o are for the most part of the order of /1%, with a maximum of /20%.
2. Plastic deformation kinetics
1=2
tb ambrb ;
where a :/0.5, m (4.21 /1010 N m 2) is the shear modulus and b (2.56 /10 10 m) is the Burgers vector. Inserting rb given by Eq. (4) into Eq. (5), one obtains, tb amb(bg=b)1=2 d 1=2 ;
2.1. Regime I (d /10 6 /10 3 m) This is the grain size regime of most bulk specimens and has received considerable attention in the past. It thus provides a reference framework for the other two regimes. For Cu (and other FCC metals) the flow stress in Regime I obeys the Hall-Petch relation [8,17], with the H-P constant si increasing significantly with strain and with decrease in temperature. The constant KH-P also tends to increase, but is less dependent on these parameters. The H-P constant KH-P has been attributed to the influence of grain size on the average slip distance of dislocations [30] or on the geometrical dislocations needed to accommodate the non-uniform deformation at grain boundaries [31]. In both models the corresponding dislocation density due to grain boundaries rb is given by,
(5)
(6)
giving KH-P /amb (bg /b )1/2. The value of KH-P derived employing Eq. (6) has been found [30] to be in accord with that measured on polycrystalline Cu at 300 K, namely /5 N mm 3/2 [8,17,30]. The flow stress of single and polycrystalline Cu at T B/0.25 TM has been attributed to the intersection of dislocations [28,29,32 /36]. The strain rate for this process is given by [37], g g0 exp [DG(t )=kT]; e
(7)
(4)
where g0 is the pre-exponential and DG the Gibbs free energy of activation, which is a decreasing function of the effective stress t t t e m : t is the applied resolved shear stress and tm the long-range internal stress opposing the motion of the glide dislocations. The preexponential for the intersection process is given by, r b nD rm b2 nD ; g0 m (l )2 b (8) l l
where b /0.25 /2.0, g the resolved shear strain, b the Burgers vector and d the grain size. rb is related to the resolved shear stress tb by the well-known (and experimentally established for Cu [19,23]) relation between shear stress and dislocation density
where rm is the mobile dislocation density, l* the spacing between the forest dislocations, b the Burgers vector and nD :/1013 s 1 the Debye frequency. It is expected that rm :/(0.1 /0.5)r [28,32/36], where r is the total dislocation density, which is given by,
rb bg=bd;
H. Conrad / Materials Science and Engineering A341 (2003) 216 /228
r (t=amb)2 ;
219
(9)
with a :/0.5 [19,23]. Assuming that the effect of t on g0 (through its influence on rm) is small compared to its effect on DG , and thereby that g0/ :/constant, we obtain for intersecting forest dislocations [37,38], DG (Q cvt )=(1c);
(10)
and v @DG=@t bl x (3=2)v;
(11)
where Q * / /Tv (@t */@T ), v kT(@ln g=@t); c / /(T / m) dm/dT , v* the activation volume and x * is the activation distance, i.e. the obstacle width, which is of the order of b . The activation enthalpy DH * pertaining specifically to the short-range stress field of the obstacle 1=2 is given by Q *. Further, since l rf :r 1=2 where rf is the dislocation forest density, one obtains from Eq. (9), l : 0:5mb=t:
(12)
The variation of x * with t , or with t*, will depend on the form of the force f vs. activation distance curve for the intersection process, i.e. on the form of the DG vs. t * curve. An analysis of the plastic deformation kinetics of polycrystalline Cu with d /18 mm (d 1/2 /7.4 mm 1/2) in terms of Eqs. (7) /(12) was performed by Conrad and Cao [28,29] for the temperature range of 77/300 K and a tensile strain rate o 1:7 10 4 s 1 : Plots of the tensile flow stress sf and the long-range internal stress sm (the latter being obtained by decremental unloading tests [39]) vs. the true tensile strain o are presented in Fig. 3. The flow stress values are in accord with those by Hansen and Ralph [8] and Ono and Karashima [17] for
Fig. 3. True tensile stress vs. true strain for Cu (d /18 mm) at 77 /300 K. sf is the flow stress (/o 1:7 10 4 s 1 ) at constant defect structure corresponding to the long-range internal stress sm. Data from Cao and Conrad [28,29].
pffiffiffi Fig. 4. Apparent activation volume v 3kT @ln o=@s vs. temperature for several strains in Cu. Data from Cao and Conrad [28,29].
the same grain size and thus are considered to have the same values for the H-P constants si and KH-P. The variation of the apparent shear stress activation volume v MkT @ln o=@s with temperature for several strains obtained by strain rate change (jump) tests is pffiffiffi shown in Fig. 4. M 3 [20] is the Taylor orientation factor relating the resolved shear stress t to the tensile stress s . To be noted in Fig. 4 is that v decreases with strain, and for a given strain increases with temperature (i.e. with decrease in flow stress), as expected for the dislocation intersection mechanism. The values of the physically-significant activation volume v * / /@DG / @t /(3/2)v range from 420 to 2040b3, which magnitudes are in the range of those expected for the intersection mechanism. Since for this mechanism v * /l*x *b / 1=2 A *b (A * is the activation area and l rf ( /t/
Fig. 5. /kT@ln o=@s vs. 1/s for Cu. Data from Cao and Conrad [28,29].
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amb )), one expects that v* 8/1/t, or the equivalent (kT @ln o=@s)81=s: A plot of kT @ln o=@s vs. 1/s is presented in Fig. 5. The data points lie on straight lines through the origin whose slopes ( /x *mb2/3) increase slightly with temperature. The values of x* obtained from the slopes of the straight lines range from 3.1b at 77 K to 4.2b at 300 K. Considering that some reduction in x * occurs with applied stress, these values are in accord with those (7.19/2.4b ) measured on Cu by TEM using the weak beam technique [40] for the dislocation dissociation width w /mb2/16pgSF at zero stress, where gSF /50 /10 3 J m 2 is the stacking fault energy for Cu [40 /43]. The increase in x * with temperature can then be attributed to the greater temperature dependence of gSF whereby (1/gSF)dgSF/dT / /8 /10 4 K 1 [40 /43] compared to that of the shear modulus m whereby (1/m) dm/dT / /4 /10 4 K 1 [44,45]. Plots of DH* and of DG vs. temperature obtained by employing Eq. (10) and taking DH * /Q * are given in Fig. 6. Both energies increase with increase in test temperature as required by Eq. (7) for a relatively constant g0 ; the difference between the two representing the entropy DS * /3 /10 4 eV K 1, which is a reasonable value. The slope of the straight line for DG gives g0 7:1 106 s 1 : The value of the Helmholtz free energy DF * /1.03 eV ( /0.23mb3) is obtained from an estimate of the temperature T0 at which s* /0, based on the temperature dependence of sf and sm in Fig. 3. The derived values of DF * and DS * are in accord with those for the intersection of dislocations in Cu [32 /34]. The value of the mobile dislocation density rm g0 =b2 nD 1:1 1013 m 2 derived from the pre-exponential g0 is also reasonable. The rate -controlling process in Cu with grain size in Regime I tested at 77 /300 K and o :10 4 s 1 is therefore concluded to be the intersection of forest dislocations by the gliding dislocations.
Of interest here is the relationship between the H-P equation and the rate-controlling mechanism. The above analysis indicates that the same rate-controlling mechanism governs both si and KH-P. As discussed above, the effect of grain size on the flow stress is through an increase in the dislocation density according to Eqs. (4) /(6). The manner in which the additional dislocations due to the presence of grain boundaries contribute to the flow stress is considered in two models. One model [8,30] assumes that the additional dislocations are mostly distributed uniformly throughout the grain interiors, so that the total dislocation density r within the grains is, (13)
r ri rb ;
where ri is essentially that which occurs in single crystals, and sb that due to the presence of grain boundaries. Inserting Eq. (13) for r into the relation t /ambr1/2, one obtains, t amb(ri rb )1=2 amb(ri bg=bd)1=2 ;
(14)
or t2 (amb)2 ri (amb)2 rb t2i t2b t2i (Kd 1=2 )2 :
(15)
The other model [46,47] for the effect of grain size on the flow stress assumes that the additional grain boundary dislocations are concentrated in the vicinity of the grain boundary and therefore the flow stress in this region differs from that in the grain interior. This is essentially a composite model and gives, 1=2
1=2
t ambri ambrb ;
(16)
or t ti tb ti Kd 1=2
(17)
Some support for the composite model is provided by the TEM observations of Hansen and Ralph [8], which reveal a smaller dislocation cell size and a higher dislocation density adjacent to the grain boundaries compared to the grain interior. Results similar to a composite model are also obtained by considering the plastic deformation of polycrystals according to the Taylor model [48,49]. The value of the H-P constant KH-P was calculated from the plastic deformation kinetics analysis considering both of the above models for the distribution of the additional dislocations due to the grain boundaries. For this, we will write Eq. (7) in the form, g rm b2 nD exp f(DF t bl x )=kTg;
(18) l* /rf 1/2.
Fig. 6. Activation enthalpy DH * and Gibbs free activation energy DG vs. temperature for the intersection of dislocations in Cu. Data from Cao and Conrad [28,29].
where DF * /1.03 eV (from Fig. 7) and Experiments [28,32 /36] indicate that the mobile dislocation density rm :/0.1r and rm :/0.5r (rm /dislocation density responsible for the long-range internal
H. Conrad / Materials Science and Engineering A341 (2003) 216 /228
221
temperatures. Cross slip can influence the dislocation structure (density and arrangement) which develops, and in turn the flow stress, but is not the strain ratecontrolling for the conditions considered above, since both the magnitude of the measured activation volume and its stress dependence (Figs. 4 and 5) are much larger than predicted by the models for cross slip [52 /56]. Cross slip and other recovery mechanisms are included in the flow stress model originally proposed by Kocks [57] and later refined [58 /62]. The inclusion of grain size in these models has yet to be considered. 2.2. Regime II (d /10 8 /10 6 m)
Fig. 7. Flow stress at o /0.8% vs. temperature for vapor deposited Cu foil (d /0.5 mm). Data from Embury and Lahaie [11].
stress). Further, taking r /2g /bd (i.e. b /0.5), rm / pffiffiffi 0.1r , rm /0.5r , t s= 3 and x * /4b , one obtains for both models upon rearranging and solving Eqs. (18) and (6) for o /0.20 at 300 K, pffiffiffi pffiffiffi s 2:1 d and sm;b 2:7 d : (19) b 1=2 gives K /4.8 N Taking sb s s b m;b Kd 3/2 , which is in good accord with the measured mm value KH-P /4.6 N mm 3/2 [8,17]. Moreover, the analysis gives that K is only mildly temperature dependent, in agreement with experimental measurements of KH-P [8]. In summary, the effect of grain size in the range of 8.5 /60 mm on the flow stress of Cu in the temperature range of 77 /300 K and strain rate o : 10 4 s 1 is given by the Hall-Petch equation s /si/KH-Pd 1/2. The experimental data indicate that the rate controlling mechanism which governs both si and KH-P for a constant dislocation structure is the intersection of forest dislocations. The effect of grain size on the plastic deformation kinetics is through the additional dislocations which result from the presence of the grain boundaries. These give an increase in the mobile dislocation density, a decrease in the forest spacing and an increase in the long-range internal stress, all affecting the plastic deformation kinetics in the manner presented above. Observations on the nature of slip lines on the specimen surface and TEM observation of the dislocation structure [8,19,23,50,51] indicate that thermallyactivated cross slip occurs during the straining of bulk single and polycrystalline Cu at low homologous
The data by Embury and Lahaie (E /L) [11] given here in Figs. 7 and 8 are taken to be representative of the plastic deformation kinetics of Cu in grain size Regime II. Their material was 30 mm thick VP Cu foil with a grain size 0.59/0.3 mm [63]. Fig. 7 gives the temperature dependence of the tensile flow stress s for a strain o / 0.8% and strain rate o 6:4 10 5 s 1 in the temperature range 77/473 K. The normalized apparent activation volume (kT @ln o=@s)=b3 vs. temperature in Fig. 8 was obtained from stress relaxation tests following straining to o /0.8%. The magnitude of the flow stress of the E /L material at 300 K is similar to that reported by others for VP [9] and sputter-deposited [10] Cu films with a similar grain size and deformed to the same strain. To be noted in Fig. 8 is that the apparent activation volume shows an unusual temperature dependence in that it decreases with temperature in the range 77/ /
Fig. 8. Normalized apparent activation volume (/kT @ln o=@s)//b3 vs. temperature for VP Cu foil (d /0.5 mm). Data from Embury and Lahaie [11].
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375 K and then increases at higher temperatures. This suggests that different mechanisms are rate-controlling in the two temperature regimes. We will only consider the behavior in the low temperature regime, namely for T B/373 K. Annealing studies [64 /66] on EP Cu with a submicron grain size (Regime II) indicate that the microstructure is relatively stable for T B/373 K. Also, Lu et al. [67] reported that the grain size and microstrain in EP Cu with d /30 nm did not change with annealing up to 370 K, which was also claimed by Cai et al. [6] for their EP Cu with d /30 nm. The anomalous increase in the apparent activation volume with decrease in temperature (increase in stress) which occurs in the low temperature regime (Fig. 8) can be explained by the existence of a stress concentration factor q , so that the effective stress in Eq. (7) becomes, + t qt : e
(20)
Assuming that the stress concentration results from the pile-up of n dislocations at a grain boundary, one obtains [24], pd t n : (21) b m Inserting Eqs. (20) and (21), respectively into Eq. (7) gives DF [pv d(t )2 =mb : (22) g g0 exp kT Taking the logarithm of Eq. (22) and differentiating with respect to t, one obtains, kT @ln g=@t (2pv d=mb)t : (23) pffiffiffi pffiffiffi Taking t s= 3 and g 3o; a plot of kT @ln o=@s vs. s/m should give a straight line with slope 2pv *d/3b and
Fig. 9. /kT @ln o=@s vs. s /m for vapor-deposited Cu foil (d /0.5 mm) tested at T /77 /350 K. Data from Embury and Lahaie [11].
intercept on the ordinate 2pv *dsm/3mb , providing v* is relatively independent of t*. Such a plot is given in Fig. 9 for values of kT @ln o=@s and corresponding s taken every 25 K over the range 77 /350 K from curves drawn through the data points in Figs. 7 and 8. Taking d /0.5 mm, b /2.56 /10 10 m and m /4.21 /1010 Pa, one obtains from the slope of the line in Fig. 9 the activation volume v * /1b3, and from the intercept on the abscissa tm /127 MPa. Further, taking tm ambr1=2 with a / m 14 2 0.5, one obtains rm /5.9 /10 m for the dislocation density responsible for the long-range internal stress tm, which is reasonable. Taking the value te =m 4:4 10 3 at 78 K from Fig. 9 and inserting it into Eq. (22) along with d /0.5 mm one then obtains n /28 dislocations in the pile-up at the maximum stress, which also is reasonable. The values for DF * and g0 can be obtained by taking the logarithm of Eq. (22) and rearranging to give, (t )2 pv d DF0 kTln(g0 =g): mb
(24)
A plot of the left-hand-side of Eq. (24) vs. the temperature should give a straight line with slope k ln(g0 =g) and intercept DF * at T /0 K. Taking d /0.5 mm and v * /b3 such a plot is presented in Fig. 10. The intercept yields DF * /0.70 eV and the slope gives g0 1:4 106 s 1 :/ The values of the activation enthalpy DH / /Tv * @t/ @T are plotted vs. the temperature in Fig. 11. Included are the values of DG /DF * /p (t *)2v*d /mb taken from Fig. 10. The difference between the two lines yield DS / 16.4 J mol 1, which again is reasonable. Also, the slope of the plot of DG vs. temperature gives g0 1:8 106 s 1 and the value of DG at T0 /350 K is 0.70 eV, both in accord with the values obtained from Fig. 10.
Fig. 10. p (t *)2v *d /mb vs. temperature for vapor-deposited Cu foil (d /0.5 mm) tested at T /77 /350 K. Data from Embury and Lahaie [11].
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fideal b=lp ;
223
(26)
where lp is the spacing between the slip planes containing the piled-up groups acting on the grain boundary. An estimate of lp can be obtained as follows. Assuming that all of the dislocations are restricted to piled-up groups of 28 dislocations each (obtained above) on discrete planes, we have, r 28=lp d;
Fig. 11. Activation energy DH and Gibbs free energy DG vs. temperature for vapour-depsoited Cu foil tested at 77 /350 K. Data from Embury and Lahaie [11].
The magnitude of v * /1b3 suggests that the ratecontrolling mechanism is the shearing of individual atoms in the grain boundary, similar to what was proposed for nanocrystalline materials [2]. However, in the present case the grain boundary shear results from the stress concentration due to the pile-up of dislocations at the boundary rather than from the applied stress alone. For grain boundary shear, one expects that the value of DF * will be on the order of that for the migration of vacancies in Cu (/Qvm 0:78 eV [68]) or that for grain boundary diffusion (Qgb /0.64 /1.08 eV [20,69,70]). The magnitude of DF * /0.70 eV indicated in Figs. 10 and 11 is thus in accord with the grain boundary shear mechanism. For grain boundary shearing the pre-exponential in Eq. (7) is given by, g0 NAbn+
fd 3 b nD ; db3
(25)
where N /fd/db3 is the number of places per unit volume where thermal activation can take place. f is the fraction of atoms in the grain boundary which can participate in the process, d :/3b the grain boundary width, d the grain size and b the atomic size. A :/b2 is the area swept out per successful event and n * /nD the vibrational frequency of the participating atom with nD the Debye frequency. The unknown parameter in Eq. (25) is the fraction f of atoms in the grain boundary which at any given moment can participate in the thermally-activated shear event. Inserting into Eq. (25) the values obtained above for the other parameters, one obtains f /1.7 /10 4. Assuming that only those grain boundary atoms situated at the heads of the dislocation pile-ups experience a sufficiently high stress to undergo shear, we obtain,
(27)
Taking r /(t /amb )2 with t/m /6.35 /10 3 (from Fig. 9) and a /0.5, one obtains r /2.5 /1015 m 2. Inserting this value for r into Eq. (27) along with d /0.5 mm, gives lp /2.2 /10 8 m. If each dislocation at the head of a piled-up group made contact with grain boundary atoms along its entire length, the fraction of grain boundary atoms contacted would be fideal /b/lp :/ 10 2. Since the fraction f of grain boundary atoms participating in the shear process determined above is 1.7 /10 4, the calculated value of fideal then indicates that only about one in every 100 grain boundary atoms are favorably positioned with respect to the stress, or other conditions, to undergo thermally-assisted shear. Continuing along these lines it can be shown that the flow stress at 300 K in Regime II is less dependent on grain size than in Regime I. Inserting into Eq. (25), f given by combined Eqs. (26) and (27), one obtains, g0 rdbnD =28:
(28)
Taking r /g/bbd (Eq. (4)), one then obtains g0 gdnD =28bd: Further, considering the possibility of thermally-activated back-jumps, Eq. (22) is given by, 2gdnD (t )2 pv d DF exp : (29) g0 sinh mbkT kT 28bd From Fig. 10 one obtains that the quantity (t*)2pv *d/ mbkT /3.86. Hence, the sinh stress function is intermediate between its argument and an exponential. It then follows that at 300 K the grain size dependence of the flow stress in Regime II is smaller than in Regime I, and becomes independent of grain size when the sinh function reduces to its argument. Finally of interest regarding Regime II is that an analysis [16] of the plastic deformation kinetics of the VP and EP Cu at 373 /473 K gave that the ratecontrolling mechanism at these higher temperatures was the intersection of dislocations. It was concluded that the recovery processes which occurred at these temperatures altered the original as-deposited microstructure and either prevented or relieved the pile-up of dislocations. In summary, the above analysis indicates that the rate-controlling process during the plastic deformation of the VP Cu foil (d /0.5 mm) at 77/350 K is thermally activated grain boundary shear promoted by the pile -up of dislocations at the grain boundaries . The grain size has
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two effects on the process: (a) on the participating dislocation density and in turn on the number of grain boundary atoms acted on by the piled-up groups; and (b) the stress concentration at the head of a pile-up. The former effect is on the pre-exponential g0 which is proportional to 1/d and the latter on the stress component of the activation energy, which is proportional to d. 2.3. Regime III (d B/10 8 m) Taking into consideration the computer simulation results of Shiøtz et al. [25] and Swygenhoven et al. [26], Conrad and Narayan (C /N) [2] proposed that the reversal in the Hall-Petch behavior of materials as the grain size is decreased below /50 nm represents grain boundary shear produced by the applied stress without the assistance of a dislocation pile-up. This model gives for the plastic strain rate [2], 2dnD v te DF g ; (30) sinh exp kT d kT where d is the grain boundary width and DF * is the Helmholtz free energy for the shear of an individual or a small group of atoms residing in the grain boundary. C / N suggested that DF * should be approximately that for the motion of vacancies in the lattice Qvm or grain boundary diffusion Qgb. Employing Eq. (30), C /N obtained from the hardness data by Chokshi et al. [5] on nanocrystalline Cu (compacted EP powder, with d / 7.7 /16 nm) the values v* :/b3 and DF* /0.94 eV and dnD /7.7 /103 m s 1, which are in accord with the grain boundary shear model. Of further interest regarding Regime III are the recent results by Cai et al. [6] on 1.7 mm thick, free-standing EP Cu sheets (d /30 nm) tested in tensile creep at 293/
Fig. 12. Steady state creep rate of electrodeposited Cu (d /30 nm) vs. applied tensile stress. Data from Cai et al. [6].
323 K. A plot of their data is presented in Fig. 12. To be noted is that the strain rate at 293 K is of the order of 10 7 s 1, whereas that by Chokshi et al. [5] is expected to be /10 3 s 1. A graphical analysis by Cai et al. of their data gave that the creep rate is proportional to an effective stress se /s /s0 and that the tensile strain rate, g
Cse kT
exp
Q ; kT
(31)
where C /1.07 /10 22 m3 s 1 and Q /0.729/0.05 eV. They concluded that these values were in accord with the Coble grain boundary diffusion creep mechanism [71]. The existence of the threshold stress s0, which was found to decrease in a linear fashion with temperature, was attributed to the inability of grain boundaries to act as perfect sources and sinks for vacancies. A multivariable linear regression analysis of Cai et al.’s data according to Eq. (30) by the present author gave v* /b3, dnD /6.2 /103 m s 1 and DF * /0.98 eV (94.5 kJ mol 1), which values are in agreement with those from Chokshi et al.’s [5] results. A plot of Cai et al.’s data according to Eq. (30) is presented in Fig. 13. s0 in the C /N model represents a back-stress or threshold stress for shearing atoms in the grain boundary. Thus, the rate-controlling mechanism for both sets of data pertaining to Regime III appears to be thermallyactivated grain boundary shear. Fit of the two sets of data to the Coble grain boundary diffusion creep model [71] or its modifications [72,73] was less satisfactory, when the grain boundary diffusion coefficient given in [20] was employed. A less satisfactory agreement was also found to be the case for grain boundary softening in nanocrystalline Zn [74,75]. It is therefore concluded that
Fig. 13. /o/kTd exp(94.5 kJ mol 1 RT 1) vs. s /s0. Data from Cai et al. [6].
Grain size regime
Reference
d
T (K)
o
III III
[5] [6]
6.9 /13.2 nm 30 nm
298 293 /323
0.10 0.01 /0.10
II
[11]
0.5 mm
77 /350
II
[16]
0.6 mm
I
[8,17,28,29]
8.5 /60 mm
a
G.B., grain boundary.
o (s 1)
/
3
s (MPa)
Mechanism a
Major effect of grain size
467 /767 126 /186
G.B. shear G.B. shear
Number of G.B. sites Number of G.B. sites
0.008
10 2 10 7 /18 10 7 6.4 10 5
326 /578
298 /350
0.01 /0.10
2.8 10 4
520 /670
77 /300
0.01 /0.20
10 4
G.B. shear promoted by dislocation pile-ups G.B. shear promoted by dislocation pile-ups Intersection of dislocations
Number of piled-up dislocations plus number of G.B. sites Number of piled-up dislocations plus number of G.B. sites Mobile and forest dislocation densities
75 /370
H. Conrad / Materials Science and Engineering A341 (2003) 216 /228
Table 1 Summary of the effect of grain size on the rate-controlling mechanism in Cu at low homologous temperatures
225
226
H. Conrad / Materials Science and Engineering A341 (2003) 216 /228
the rate -controlling mechanism in Regime III is grain boundary shear according to the C /N model. The effect of grain size in this mechanism is simply on the number of grain boundary atoms available for participating in the thermally-activated shear event. As pointed out by Raj and Ashby [76] the rate-controlling grain boundary shear may be accommodated by grain boundary diffusion. Of additional interest is the relationship of the data by Chokshi et al. [5] and those by Cai et al. [6] with respect to Regimes II and III as defined in Fig. 2. This is shown in Fig. 14. It is here seen that the data by Chokshi et al. for o :10 3 s 1 and se /s lie partly within and partly outside the given limits for dislocation activity, i.e. they are just borderline for being in Regime III, recognizing that the ratio H /s may not be exactly 3 and that Eq. (3) are based on isotropic elasticity. In contrast, the single data point by Cai et al. for o :10 7 s 1 and se /s /s0 lies completely in Regime III. However, upon increasing the strain rate to 10 5 s 1, the estimated data point for se /s falls well outside of Regime III (as does also se / s /s0) and the rate-controlling mechanism would more likely be that for Regime II, namely grain boundary shear promoted by the pile-up of dislocations. This illustrates the importance of defining the strain rate (and temperature) when considering the transition grain size dc separating one Regime from another. Also to be noted from Fig. 14 is that besides the strain rate, dc depends on the magnitude of the Taylor orientation factor M (i.e. on the crystallographic texture) and on the type of dislocation (edge or screw). The agreement in the values of the parameters for grain boundary shear obtained form the two independent sets of data by Chokshi et al. [5] and by Cai et al. [6] provides support for the existence of a grain size softening phenomenon (inverse H-P effect) in metals. Further support that this is a real phenomenon is
provided by the results [74,75] on laser-deposited, nanocrystalline Zn films, which were free of the defects which might otherwise lead to an apparent grain size softening as discussed by Koch and Narayan [18].
3. Summary Data in the literature on the effect of grain size d in the range of 10 9 /10 3 m on the flow stress of Cu is analyzed with respect to the strain rate-controlling mechanism. Three grain size regimes were identified: Regimes I (d :/10 6 /10 3 m), II (d :/10 8 /10 6 m) and III (d B/ /10 8 m). The dislocation cell structure which occurs in Regime I is concluded to be absent in Regime II; all intragranular dislocation activity is concluded to cease in Regime III. The rate-controlling mechanisms determined from data representing each of the three regimes are summarized in Table 1. Since the plastic deformation in each regime is kinetic in nature, the transition grain size which distinguishes one regime from another will depend on temperature and strain rate. Crystallographic texture is also important.
Acknowledgements The author appreciates the assistance of R. O’Connell and Drs Di Yang and K. Jung in preparation of the manuscript. This study was conducted in collaboration with the NSF Center for Advanced Materials and Smart Structures at NCSU.
Appendix A: List of symbols H s si s0 te te /
/
t* tm tth
Fig. 14. Log s vs. log d illustrating the effects of strain rate, crystallographic orientation (Taylor orientation factor) and dislocation orientation on the grain size separating Regimes II and III.
m b d d
hardness tensile flow stress lattice friction stress critical stress or back stress effective resolved shear stress effective thermal component of the applied shear stress thermal component of the stress long-range internal stress component theoretical shear strength shear modulus Burgers vector grain size grain boundary width
H. Conrad / Materials Science and Engineering A341 (2003) 216 /228
total dislocation density dislocation density due to grain boundaries rI dislocation density in single crystals rm mobile dislocation density rm dislocation density responsible for long-range internal stress rf forest dislocation density w dislocation dissociation width l* dislocation forest spacing lc dislocation cell size ld spacing between dislocations due to elastic interactions lp spacing between parallel slip planes containing piled-up groups of q dislocations on stress concentration factor x* forest dislocation obstacle width A * /l*x * activation area gSF stacking fault energy n number of piled-up dislocations M Taylor orientation factor /g / pre-exponential in ther0 mally-activated strain rate equation nD Debye frequency DG /DH * /T DS * Gibbs free activation energy DF * /DU * /T DS * Helmholtz free activation energy /v MkT @ln o=@s/ shear stress apparent activation volume /v kT @ln g=@t @DG=@t/ shear stress activation volume f fraction of atoms in the grain boundaries which participate in thermallyactivated grain boundary shear k Boltzmann’s constant T temperature a , b, c and C constants r rb
227
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Grain size dependence of the plastic deformation kinetics in Cu Hans Conrad * Department of Materials Science and Engineering, North Carolina State University, Raleigh, NC 27695-7907, USA Received 11 June 2001; received in revised form 4 April 2002
Abstract Data on the effect of grain size d in the range of nm to mm on the plastic deformation kinetics of Cu at 77 /373 K are analyzed to determine the influence of grain size on the strain rate-controlling mechanism. Three grain size regimes were identified: Regimes I (d :/10 6 /10 3 m), II (d :/10 8 /10 6 m) and III (d B/ /10 8 m). A dislocation cell structure characterizes Regime II, which no longer occurs in Regime II. The absence of all intragranular dislocation activity characterizes Regime III. The following mechanisms were concluded to be rate-controlling for o :10 5 10 3 s 1 : (a) Regime I , intersection of dislocations ; (b) Regime I , grain boundary shear promoted by dislocation pile-ups ; and (c) Regime III , grain boundary shear. The major effect of grain size on the intersection mechanism in Regime I is on the mobile and forest dislocation densities; the effect in Regime II is on the number of dislocations and on the number of grain boundary atom sites; the effect in Regime III is on the number of grain boundary atom sites. The transition grain size from one regime to another depends on the strain rate and temperature. Crystallographic texture is also important. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Copper; Dislocations; Grain boundaries; Thermally-activated processes; Hall-Petch constants
1. Introduction There has developed in recent years a renewed interest in the influence of grain size (especially in the submicron range) on the mechanical properties of materials. Two major stimulants have led to this interest: (a) superplasticity; and (b) the desire for ultrahigh strength with reasonable ductility. It is anticipated that these properties may be enhanced by a reduction in grain size. In addition to possible beneficial effects on mechanical properties, novel physical properties can result as the grain size is reduced to the nanometer range, where the number of atoms situated in the grain boundaries comprise a significant fraction of the total [1]. The influence of grain size d on the flow stress s (or hardness H :/3s) of crystalline materials at low homologous temperatures (T B/0.25 TM) is usually considered in terms of the Hall-Petch (H-P) relation, s si KH-P d 1=2 ;
* Tel.: /1-919-515-7443; fax: /1-919-515-7724 E-mail address: [email protected] (H. Conrad).
(1)
where si is the lattice friction stress and KH-P the socalled H-P constant. The H-P relation has been reported to be obeyed down to a grain size of /20 nm, below which there generally occurs a decrease in the constant KH-P, and in some cases ultimately a negative value has been reported, i.e. grain size softening; see Ref. [2]. It has been proposed [3,4] that the mechanism responsible for the H-P relation (i.e. grain size hardening) is the pile-up of dislocations at a grain boundary, giving a stress concentration which activates dislocation generation and/or motion in the adjacent grain. The grain size softening in the low nanocrystalline range (so-called inverse H-P effect) has been attributed by various investigators to: (a) grain boundary diffusion creep (Coble creep) [5,6]); (b) grain boundary sliding [7]; and (c) grain boundary shear [2]. Since in general the flow stress depends on strain rate o and temperature T , the H-P constants si and KH-P (and any departure from the normal H-P behavior) may vary with these test parameters. The objective of the present study was therefore to examine the effects of these parameters on the grain size dependence of the flow stress of Cu over the wide grain size range from nanometers to millimeters and to identify the mechan-
0921-5093/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 0 9 3 ( 0 2 ) 0 0 2 3 8 - 1
H. Conrad / Materials Science and Engineering A341 (2003) 216 /228
isms governing the corresponding plastic deformation kinetics. Cu was chosen because desired data for the analysis are available over the wide grain size range of interest. The analysis was restricted to low homologous temperatures (T B/0.25 TM) to reduce the possible complicating influence of lattice diffusion on recovery and grain growth. A Hall-Petch plot of the effect of grain size on the flow stress of Cu at 300 K and o 10 5 10 3 s 1 is presented in Fig. 1. The data points for the freestanding, vapor-deposited (VP) and electrodeposited (EP) films scatter about the dashed line representing an extrapolation of results by Hansen and Ralph [8] for bulk Cu with d /11/250 mm. The flow stress of the compacted-powder specimens (VP-C and EP-C) tend to fall significantly below the extrapolation for d 5/25 nm (d 1/2 ]/200 mm 1/2) and suggest a grain size softening (i.e. a so-called inverse H-P effect) for d 1/2 ]/200 mm 1/2. As discussed by Koch and Narayan [18] the compacted-powder specimens may contain processing defects (including porosity, non-uniform grain size and contamination), which can lead to a reduction (or increase) in the flow stress. They questioned whether the inverse H-P effect was a real phenomenon or merely reflected the increased difficulty of preparing nanocrystalline materials free of undesirable defects. Another perspective on the effect of grain size on the flow stress of Cu is provided in Fig. 2, which is a plot of log s vs. log d. This plot suggests three grain size regimes: (a) Regime I, d :/10 6 /10 3 m; (b) Regime
Fig. 1. H-P plot of the effect of grain size d on the flow stress s of Cu at 300 K and o 10 5 10 3 s 1 : Symbols " and X are for hardness with s /H /3; the remaining symbols are for the flow stress at o :/0.01 in tension. VP, vapor-deposits; EP, electrodeposits; VP/C, compacted VP powders; EP-C, compacted EP powders. Data from Refs. [5,6,8 / 16].
217
II, d :/10 6 /10 8 m; and (c) Regime III d B/10 8 m. In Regime I, the flow stress increases significantly with decrease in grain size; in Regime II the effect of grain size appears less pronounced; in Regime III a decrease in flow stress with decrease in grain size is suggested. The existence of three regimes is supported by dislocation behavior. In Regime I dislocations form cell structures whose size lc in Cu is given by [19]: (2)
s 21mb=lc ; 10
2
where m /4.21 /10 N m is the shear modulus and b /2.56 /10 10 m the Burgers vector [20]. A plot of Eq. (2) is included in Fig. 2. This indicates that Regime II occurs when the grain size becomes smaller than the cell size, i.e. when d B/ /10 6 m. Hence, no dislocation cells are expected to form during plastic deformation when d B/ /1 mm, thereby terminating Regime I and beginning Regime II. Intragranular dislocation activity however continues in Regime II until, due to elastic interactions, the separation of dislocations becomes greater than the grain size. According to dislocation theory the spacing ld between dislocations due to elastic interactions is given by [24]: t mb=2pld (screw)
and
(3) t mb=2p(1 n)ld (edge); pffiffiffi Plots of Eq. (3) (taking M 3 [20] and M /3 [21 /23]) are also included in Fig. 2. These indicate that Regime III occurs when the spacing between dislocations becomes larger than the grain size, i.e. when d B/ /10 nm. This grain size is in accord with computer simulations [25,26], which gave that intragranular dislocation activity no longer occurred in Cu when d 5/8 nm. The computer simulations [26] also indicated that for d /10 nm there occurs a grain size regime where the flow stress is weakly dependent on the grain size, which is in accord with the behavior in Regime II shown in Fig. 2. Finally to be noted regarding Fig. 2 is that the maximum stress obtained to-date with reduction in grain size in Cu is about 1/6 the theoretical shear strength given by tth /m/ pffiffiffi 16 [27], taking M 3:/ The present paper focuses on three sets of data pertaining to the plastic deformation kinetics in the three regimes indicated in Fig. 2, namely: (a) those by Cao and Conrad [28,29] for Regime I; (b) those by Embury and Lahaie [11] for Regime II; and (c) those by Cai et al. [6] for Regime III. The first set is representative of Regime I and provides kinetic data desired for the analysis. The second two sets are the only data known to the author which provide plastic deformation kinetics data for Regimes II and III, respectively, in Cu. In view of the limited data available for the latter two regimes, the strain rate o to be considered in the present analysis is of the order of 10 5 /10 3 s 1 (except for the data of Cai et al. [6], where o :10 7 10 6 s 1 ) and the strains
218
H. Conrad / Materials Science and Engineering A341 (2003) 216 /228
Fig. 2. Log s vs. log d for Cu at 300 K and o 10 5 10 3 s 1 : Open symbols and * are for o :/0.01; filled symbols and X are for o :/0.2. VP, EP, VP/C and EP/C have the same meaning as in Fig. 1. Data from Refs. [5,6,8 /17].
o are for the most part of the order of /1%, with a maximum of /20%.
2. Plastic deformation kinetics
1=2
tb ambrb ;
where a :/0.5, m (4.21 /1010 N m 2) is the shear modulus and b (2.56 /10 10 m) is the Burgers vector. Inserting rb given by Eq. (4) into Eq. (5), one obtains, tb amb(bg=b)1=2 d 1=2 ;
2.1. Regime I (d /10 6 /10 3 m) This is the grain size regime of most bulk specimens and has received considerable attention in the past. It thus provides a reference framework for the other two regimes. For Cu (and other FCC metals) the flow stress in Regime I obeys the Hall-Petch relation [8,17], with the H-P constant si increasing significantly with strain and with decrease in temperature. The constant KH-P also tends to increase, but is less dependent on these parameters. The H-P constant KH-P has been attributed to the influence of grain size on the average slip distance of dislocations [30] or on the geometrical dislocations needed to accommodate the non-uniform deformation at grain boundaries [31]. In both models the corresponding dislocation density due to grain boundaries rb is given by,
(5)
(6)
giving KH-P /amb (bg /b )1/2. The value of KH-P derived employing Eq. (6) has been found [30] to be in accord with that measured on polycrystalline Cu at 300 K, namely /5 N mm 3/2 [8,17,30]. The flow stress of single and polycrystalline Cu at T B/0.25 TM has been attributed to the intersection of dislocations [28,29,32 /36]. The strain rate for this process is given by [37], g g0 exp [DG(t )=kT]; e
(7)
(4)
where g0 is the pre-exponential and DG the Gibbs free energy of activation, which is a decreasing function of the effective stress t t t e m : t is the applied resolved shear stress and tm the long-range internal stress opposing the motion of the glide dislocations. The preexponential for the intersection process is given by, r b nD rm b2 nD ; g0 m (l )2 b (8) l l
where b /0.25 /2.0, g the resolved shear strain, b the Burgers vector and d the grain size. rb is related to the resolved shear stress tb by the well-known (and experimentally established for Cu [19,23]) relation between shear stress and dislocation density
where rm is the mobile dislocation density, l* the spacing between the forest dislocations, b the Burgers vector and nD :/1013 s 1 the Debye frequency. It is expected that rm :/(0.1 /0.5)r [28,32/36], where r is the total dislocation density, which is given by,
rb bg=bd;
H. Conrad / Materials Science and Engineering A341 (2003) 216 /228
r (t=amb)2 ;
219
(9)
with a :/0.5 [19,23]. Assuming that the effect of t on g0 (through its influence on rm) is small compared to its effect on DG , and thereby that g0/ :/constant, we obtain for intersecting forest dislocations [37,38], DG (Q cvt )=(1c);
(10)
and v @DG=@t bl x (3=2)v;
(11)
where Q * / /Tv (@t */@T ), v kT(@ln g=@t); c / /(T / m) dm/dT , v* the activation volume and x * is the activation distance, i.e. the obstacle width, which is of the order of b . The activation enthalpy DH * pertaining specifically to the short-range stress field of the obstacle 1=2 is given by Q *. Further, since l rf :r 1=2 where rf is the dislocation forest density, one obtains from Eq. (9), l : 0:5mb=t:
(12)
The variation of x * with t , or with t*, will depend on the form of the force f vs. activation distance curve for the intersection process, i.e. on the form of the DG vs. t * curve. An analysis of the plastic deformation kinetics of polycrystalline Cu with d /18 mm (d 1/2 /7.4 mm 1/2) in terms of Eqs. (7) /(12) was performed by Conrad and Cao [28,29] for the temperature range of 77/300 K and a tensile strain rate o 1:7 10 4 s 1 : Plots of the tensile flow stress sf and the long-range internal stress sm (the latter being obtained by decremental unloading tests [39]) vs. the true tensile strain o are presented in Fig. 3. The flow stress values are in accord with those by Hansen and Ralph [8] and Ono and Karashima [17] for
Fig. 3. True tensile stress vs. true strain for Cu (d /18 mm) at 77 /300 K. sf is the flow stress (/o 1:7 10 4 s 1 ) at constant defect structure corresponding to the long-range internal stress sm. Data from Cao and Conrad [28,29].
pffiffiffi Fig. 4. Apparent activation volume v 3kT @ln o=@s vs. temperature for several strains in Cu. Data from Cao and Conrad [28,29].
the same grain size and thus are considered to have the same values for the H-P constants si and KH-P. The variation of the apparent shear stress activation volume v MkT @ln o=@s with temperature for several strains obtained by strain rate change (jump) tests is pffiffiffi shown in Fig. 4. M 3 [20] is the Taylor orientation factor relating the resolved shear stress t to the tensile stress s . To be noted in Fig. 4 is that v decreases with strain, and for a given strain increases with temperature (i.e. with decrease in flow stress), as expected for the dislocation intersection mechanism. The values of the physically-significant activation volume v * / /@DG / @t /(3/2)v range from 420 to 2040b3, which magnitudes are in the range of those expected for the intersection mechanism. Since for this mechanism v * /l*x *b / 1=2 A *b (A * is the activation area and l rf ( /t/
Fig. 5. /kT@ln o=@s vs. 1/s for Cu. Data from Cao and Conrad [28,29].
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amb )), one expects that v* 8/1/t, or the equivalent (kT @ln o=@s)81=s: A plot of kT @ln o=@s vs. 1/s is presented in Fig. 5. The data points lie on straight lines through the origin whose slopes ( /x *mb2/3) increase slightly with temperature. The values of x* obtained from the slopes of the straight lines range from 3.1b at 77 K to 4.2b at 300 K. Considering that some reduction in x * occurs with applied stress, these values are in accord with those (7.19/2.4b ) measured on Cu by TEM using the weak beam technique [40] for the dislocation dissociation width w /mb2/16pgSF at zero stress, where gSF /50 /10 3 J m 2 is the stacking fault energy for Cu [40 /43]. The increase in x * with temperature can then be attributed to the greater temperature dependence of gSF whereby (1/gSF)dgSF/dT / /8 /10 4 K 1 [40 /43] compared to that of the shear modulus m whereby (1/m) dm/dT / /4 /10 4 K 1 [44,45]. Plots of DH* and of DG vs. temperature obtained by employing Eq. (10) and taking DH * /Q * are given in Fig. 6. Both energies increase with increase in test temperature as required by Eq. (7) for a relatively constant g0 ; the difference between the two representing the entropy DS * /3 /10 4 eV K 1, which is a reasonable value. The slope of the straight line for DG gives g0 7:1 106 s 1 : The value of the Helmholtz free energy DF * /1.03 eV ( /0.23mb3) is obtained from an estimate of the temperature T0 at which s* /0, based on the temperature dependence of sf and sm in Fig. 3. The derived values of DF * and DS * are in accord with those for the intersection of dislocations in Cu [32 /34]. The value of the mobile dislocation density rm g0 =b2 nD 1:1 1013 m 2 derived from the pre-exponential g0 is also reasonable. The rate -controlling process in Cu with grain size in Regime I tested at 77 /300 K and o :10 4 s 1 is therefore concluded to be the intersection of forest dislocations by the gliding dislocations.
Of interest here is the relationship between the H-P equation and the rate-controlling mechanism. The above analysis indicates that the same rate-controlling mechanism governs both si and KH-P. As discussed above, the effect of grain size on the flow stress is through an increase in the dislocation density according to Eqs. (4) /(6). The manner in which the additional dislocations due to the presence of grain boundaries contribute to the flow stress is considered in two models. One model [8,30] assumes that the additional dislocations are mostly distributed uniformly throughout the grain interiors, so that the total dislocation density r within the grains is, (13)
r ri rb ;
where ri is essentially that which occurs in single crystals, and sb that due to the presence of grain boundaries. Inserting Eq. (13) for r into the relation t /ambr1/2, one obtains, t amb(ri rb )1=2 amb(ri bg=bd)1=2 ;
(14)
or t2 (amb)2 ri (amb)2 rb t2i t2b t2i (Kd 1=2 )2 :
(15)
The other model [46,47] for the effect of grain size on the flow stress assumes that the additional grain boundary dislocations are concentrated in the vicinity of the grain boundary and therefore the flow stress in this region differs from that in the grain interior. This is essentially a composite model and gives, 1=2
1=2
t ambri ambrb ;
(16)
or t ti tb ti Kd 1=2
(17)
Some support for the composite model is provided by the TEM observations of Hansen and Ralph [8], which reveal a smaller dislocation cell size and a higher dislocation density adjacent to the grain boundaries compared to the grain interior. Results similar to a composite model are also obtained by considering the plastic deformation of polycrystals according to the Taylor model [48,49]. The value of the H-P constant KH-P was calculated from the plastic deformation kinetics analysis considering both of the above models for the distribution of the additional dislocations due to the grain boundaries. For this, we will write Eq. (7) in the form, g rm b2 nD exp f(DF t bl x )=kTg;
(18) l* /rf 1/2.
Fig. 6. Activation enthalpy DH * and Gibbs free activation energy DG vs. temperature for the intersection of dislocations in Cu. Data from Cao and Conrad [28,29].
where DF * /1.03 eV (from Fig. 7) and Experiments [28,32 /36] indicate that the mobile dislocation density rm :/0.1r and rm :/0.5r (rm /dislocation density responsible for the long-range internal
H. Conrad / Materials Science and Engineering A341 (2003) 216 /228
221
temperatures. Cross slip can influence the dislocation structure (density and arrangement) which develops, and in turn the flow stress, but is not the strain ratecontrolling for the conditions considered above, since both the magnitude of the measured activation volume and its stress dependence (Figs. 4 and 5) are much larger than predicted by the models for cross slip [52 /56]. Cross slip and other recovery mechanisms are included in the flow stress model originally proposed by Kocks [57] and later refined [58 /62]. The inclusion of grain size in these models has yet to be considered. 2.2. Regime II (d /10 8 /10 6 m)
Fig. 7. Flow stress at o /0.8% vs. temperature for vapor deposited Cu foil (d /0.5 mm). Data from Embury and Lahaie [11].
stress). Further, taking r /2g /bd (i.e. b /0.5), rm / pffiffiffi 0.1r , rm /0.5r , t s= 3 and x * /4b , one obtains for both models upon rearranging and solving Eqs. (18) and (6) for o /0.20 at 300 K, pffiffiffi pffiffiffi s 2:1 d and sm;b 2:7 d : (19) b 1=2 gives K /4.8 N Taking sb s s b m;b Kd 3/2 , which is in good accord with the measured mm value KH-P /4.6 N mm 3/2 [8,17]. Moreover, the analysis gives that K is only mildly temperature dependent, in agreement with experimental measurements of KH-P [8]. In summary, the effect of grain size in the range of 8.5 /60 mm on the flow stress of Cu in the temperature range of 77 /300 K and strain rate o : 10 4 s 1 is given by the Hall-Petch equation s /si/KH-Pd 1/2. The experimental data indicate that the rate controlling mechanism which governs both si and KH-P for a constant dislocation structure is the intersection of forest dislocations. The effect of grain size on the plastic deformation kinetics is through the additional dislocations which result from the presence of the grain boundaries. These give an increase in the mobile dislocation density, a decrease in the forest spacing and an increase in the long-range internal stress, all affecting the plastic deformation kinetics in the manner presented above. Observations on the nature of slip lines on the specimen surface and TEM observation of the dislocation structure [8,19,23,50,51] indicate that thermallyactivated cross slip occurs during the straining of bulk single and polycrystalline Cu at low homologous
The data by Embury and Lahaie (E /L) [11] given here in Figs. 7 and 8 are taken to be representative of the plastic deformation kinetics of Cu in grain size Regime II. Their material was 30 mm thick VP Cu foil with a grain size 0.59/0.3 mm [63]. Fig. 7 gives the temperature dependence of the tensile flow stress s for a strain o / 0.8% and strain rate o 6:4 10 5 s 1 in the temperature range 77/473 K. The normalized apparent activation volume (kT @ln o=@s)=b3 vs. temperature in Fig. 8 was obtained from stress relaxation tests following straining to o /0.8%. The magnitude of the flow stress of the E /L material at 300 K is similar to that reported by others for VP [9] and sputter-deposited [10] Cu films with a similar grain size and deformed to the same strain. To be noted in Fig. 8 is that the apparent activation volume shows an unusual temperature dependence in that it decreases with temperature in the range 77/ /
Fig. 8. Normalized apparent activation volume (/kT @ln o=@s)//b3 vs. temperature for VP Cu foil (d /0.5 mm). Data from Embury and Lahaie [11].
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375 K and then increases at higher temperatures. This suggests that different mechanisms are rate-controlling in the two temperature regimes. We will only consider the behavior in the low temperature regime, namely for T B/373 K. Annealing studies [64 /66] on EP Cu with a submicron grain size (Regime II) indicate that the microstructure is relatively stable for T B/373 K. Also, Lu et al. [67] reported that the grain size and microstrain in EP Cu with d /30 nm did not change with annealing up to 370 K, which was also claimed by Cai et al. [6] for their EP Cu with d /30 nm. The anomalous increase in the apparent activation volume with decrease in temperature (increase in stress) which occurs in the low temperature regime (Fig. 8) can be explained by the existence of a stress concentration factor q , so that the effective stress in Eq. (7) becomes, + t qt : e
(20)
Assuming that the stress concentration results from the pile-up of n dislocations at a grain boundary, one obtains [24], pd t n : (21) b m Inserting Eqs. (20) and (21), respectively into Eq. (7) gives DF [pv d(t )2 =mb : (22) g g0 exp kT Taking the logarithm of Eq. (22) and differentiating with respect to t, one obtains, kT @ln g=@t (2pv d=mb)t : (23) pffiffiffi pffiffiffi Taking t s= 3 and g 3o; a plot of kT @ln o=@s vs. s/m should give a straight line with slope 2pv *d/3b and
Fig. 9. /kT @ln o=@s vs. s /m for vapor-deposited Cu foil (d /0.5 mm) tested at T /77 /350 K. Data from Embury and Lahaie [11].
intercept on the ordinate 2pv *dsm/3mb , providing v* is relatively independent of t*. Such a plot is given in Fig. 9 for values of kT @ln o=@s and corresponding s taken every 25 K over the range 77 /350 K from curves drawn through the data points in Figs. 7 and 8. Taking d /0.5 mm, b /2.56 /10 10 m and m /4.21 /1010 Pa, one obtains from the slope of the line in Fig. 9 the activation volume v * /1b3, and from the intercept on the abscissa tm /127 MPa. Further, taking tm ambr1=2 with a / m 14 2 0.5, one obtains rm /5.9 /10 m for the dislocation density responsible for the long-range internal stress tm, which is reasonable. Taking the value te =m 4:4 10 3 at 78 K from Fig. 9 and inserting it into Eq. (22) along with d /0.5 mm one then obtains n /28 dislocations in the pile-up at the maximum stress, which also is reasonable. The values for DF * and g0 can be obtained by taking the logarithm of Eq. (22) and rearranging to give, (t )2 pv d DF0 kTln(g0 =g): mb
(24)
A plot of the left-hand-side of Eq. (24) vs. the temperature should give a straight line with slope k ln(g0 =g) and intercept DF * at T /0 K. Taking d /0.5 mm and v * /b3 such a plot is presented in Fig. 10. The intercept yields DF * /0.70 eV and the slope gives g0 1:4 106 s 1 :/ The values of the activation enthalpy DH / /Tv * @t/ @T are plotted vs. the temperature in Fig. 11. Included are the values of DG /DF * /p (t *)2v*d /mb taken from Fig. 10. The difference between the two lines yield DS / 16.4 J mol 1, which again is reasonable. Also, the slope of the plot of DG vs. temperature gives g0 1:8 106 s 1 and the value of DG at T0 /350 K is 0.70 eV, both in accord with the values obtained from Fig. 10.
Fig. 10. p (t *)2v *d /mb vs. temperature for vapor-deposited Cu foil (d /0.5 mm) tested at T /77 /350 K. Data from Embury and Lahaie [11].
H. Conrad / Materials Science and Engineering A341 (2003) 216 /228
fideal b=lp ;
223
(26)
where lp is the spacing between the slip planes containing the piled-up groups acting on the grain boundary. An estimate of lp can be obtained as follows. Assuming that all of the dislocations are restricted to piled-up groups of 28 dislocations each (obtained above) on discrete planes, we have, r 28=lp d;
Fig. 11. Activation energy DH and Gibbs free energy DG vs. temperature for vapour-depsoited Cu foil tested at 77 /350 K. Data from Embury and Lahaie [11].
The magnitude of v * /1b3 suggests that the ratecontrolling mechanism is the shearing of individual atoms in the grain boundary, similar to what was proposed for nanocrystalline materials [2]. However, in the present case the grain boundary shear results from the stress concentration due to the pile-up of dislocations at the boundary rather than from the applied stress alone. For grain boundary shear, one expects that the value of DF * will be on the order of that for the migration of vacancies in Cu (/Qvm 0:78 eV [68]) or that for grain boundary diffusion (Qgb /0.64 /1.08 eV [20,69,70]). The magnitude of DF * /0.70 eV indicated in Figs. 10 and 11 is thus in accord with the grain boundary shear mechanism. For grain boundary shearing the pre-exponential in Eq. (7) is given by, g0 NAbn+
fd 3 b nD ; db3
(25)
where N /fd/db3 is the number of places per unit volume where thermal activation can take place. f is the fraction of atoms in the grain boundary which can participate in the process, d :/3b the grain boundary width, d the grain size and b the atomic size. A :/b2 is the area swept out per successful event and n * /nD the vibrational frequency of the participating atom with nD the Debye frequency. The unknown parameter in Eq. (25) is the fraction f of atoms in the grain boundary which at any given moment can participate in the thermally-activated shear event. Inserting into Eq. (25) the values obtained above for the other parameters, one obtains f /1.7 /10 4. Assuming that only those grain boundary atoms situated at the heads of the dislocation pile-ups experience a sufficiently high stress to undergo shear, we obtain,
(27)
Taking r /(t /amb )2 with t/m /6.35 /10 3 (from Fig. 9) and a /0.5, one obtains r /2.5 /1015 m 2. Inserting this value for r into Eq. (27) along with d /0.5 mm, gives lp /2.2 /10 8 m. If each dislocation at the head of a piled-up group made contact with grain boundary atoms along its entire length, the fraction of grain boundary atoms contacted would be fideal /b/lp :/ 10 2. Since the fraction f of grain boundary atoms participating in the shear process determined above is 1.7 /10 4, the calculated value of fideal then indicates that only about one in every 100 grain boundary atoms are favorably positioned with respect to the stress, or other conditions, to undergo thermally-assisted shear. Continuing along these lines it can be shown that the flow stress at 300 K in Regime II is less dependent on grain size than in Regime I. Inserting into Eq. (25), f given by combined Eqs. (26) and (27), one obtains, g0 rdbnD =28:
(28)
Taking r /g/bbd (Eq. (4)), one then obtains g0 gdnD =28bd: Further, considering the possibility of thermally-activated back-jumps, Eq. (22) is given by, 2gdnD (t )2 pv d DF exp : (29) g0 sinh mbkT kT 28bd From Fig. 10 one obtains that the quantity (t*)2pv *d/ mbkT /3.86. Hence, the sinh stress function is intermediate between its argument and an exponential. It then follows that at 300 K the grain size dependence of the flow stress in Regime II is smaller than in Regime I, and becomes independent of grain size when the sinh function reduces to its argument. Finally of interest regarding Regime II is that an analysis [16] of the plastic deformation kinetics of the VP and EP Cu at 373 /473 K gave that the ratecontrolling mechanism at these higher temperatures was the intersection of dislocations. It was concluded that the recovery processes which occurred at these temperatures altered the original as-deposited microstructure and either prevented or relieved the pile-up of dislocations. In summary, the above analysis indicates that the rate-controlling process during the plastic deformation of the VP Cu foil (d /0.5 mm) at 77/350 K is thermally activated grain boundary shear promoted by the pile -up of dislocations at the grain boundaries . The grain size has
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two effects on the process: (a) on the participating dislocation density and in turn on the number of grain boundary atoms acted on by the piled-up groups; and (b) the stress concentration at the head of a pile-up. The former effect is on the pre-exponential g0 which is proportional to 1/d and the latter on the stress component of the activation energy, which is proportional to d. 2.3. Regime III (d B/10 8 m) Taking into consideration the computer simulation results of Shiøtz et al. [25] and Swygenhoven et al. [26], Conrad and Narayan (C /N) [2] proposed that the reversal in the Hall-Petch behavior of materials as the grain size is decreased below /50 nm represents grain boundary shear produced by the applied stress without the assistance of a dislocation pile-up. This model gives for the plastic strain rate [2], 2dnD v te DF g ; (30) sinh exp kT d kT where d is the grain boundary width and DF * is the Helmholtz free energy for the shear of an individual or a small group of atoms residing in the grain boundary. C / N suggested that DF * should be approximately that for the motion of vacancies in the lattice Qvm or grain boundary diffusion Qgb. Employing Eq. (30), C /N obtained from the hardness data by Chokshi et al. [5] on nanocrystalline Cu (compacted EP powder, with d / 7.7 /16 nm) the values v* :/b3 and DF* /0.94 eV and dnD /7.7 /103 m s 1, which are in accord with the grain boundary shear model. Of further interest regarding Regime III are the recent results by Cai et al. [6] on 1.7 mm thick, free-standing EP Cu sheets (d /30 nm) tested in tensile creep at 293/
Fig. 12. Steady state creep rate of electrodeposited Cu (d /30 nm) vs. applied tensile stress. Data from Cai et al. [6].
323 K. A plot of their data is presented in Fig. 12. To be noted is that the strain rate at 293 K is of the order of 10 7 s 1, whereas that by Chokshi et al. [5] is expected to be /10 3 s 1. A graphical analysis by Cai et al. of their data gave that the creep rate is proportional to an effective stress se /s /s0 and that the tensile strain rate, g
Cse kT
exp
Q ; kT
(31)
where C /1.07 /10 22 m3 s 1 and Q /0.729/0.05 eV. They concluded that these values were in accord with the Coble grain boundary diffusion creep mechanism [71]. The existence of the threshold stress s0, which was found to decrease in a linear fashion with temperature, was attributed to the inability of grain boundaries to act as perfect sources and sinks for vacancies. A multivariable linear regression analysis of Cai et al.’s data according to Eq. (30) by the present author gave v* /b3, dnD /6.2 /103 m s 1 and DF * /0.98 eV (94.5 kJ mol 1), which values are in agreement with those from Chokshi et al.’s [5] results. A plot of Cai et al.’s data according to Eq. (30) is presented in Fig. 13. s0 in the C /N model represents a back-stress or threshold stress for shearing atoms in the grain boundary. Thus, the rate-controlling mechanism for both sets of data pertaining to Regime III appears to be thermallyactivated grain boundary shear. Fit of the two sets of data to the Coble grain boundary diffusion creep model [71] or its modifications [72,73] was less satisfactory, when the grain boundary diffusion coefficient given in [20] was employed. A less satisfactory agreement was also found to be the case for grain boundary softening in nanocrystalline Zn [74,75]. It is therefore concluded that
Fig. 13. /o/kTd exp(94.5 kJ mol 1 RT 1) vs. s /s0. Data from Cai et al. [6].
Grain size regime
Reference
d
T (K)
o
III III
[5] [6]
6.9 /13.2 nm 30 nm
298 293 /323
0.10 0.01 /0.10
II
[11]
0.5 mm
77 /350
II
[16]
0.6 mm
I
[8,17,28,29]
8.5 /60 mm
a
G.B., grain boundary.
o (s 1)
/
3
s (MPa)
Mechanism a
Major effect of grain size
467 /767 126 /186
G.B. shear G.B. shear
Number of G.B. sites Number of G.B. sites
0.008
10 2 10 7 /18 10 7 6.4 10 5
326 /578
298 /350
0.01 /0.10
2.8 10 4
520 /670
77 /300
0.01 /0.20
10 4
G.B. shear promoted by dislocation pile-ups G.B. shear promoted by dislocation pile-ups Intersection of dislocations
Number of piled-up dislocations plus number of G.B. sites Number of piled-up dislocations plus number of G.B. sites Mobile and forest dislocation densities
75 /370
H. Conrad / Materials Science and Engineering A341 (2003) 216 /228
Table 1 Summary of the effect of grain size on the rate-controlling mechanism in Cu at low homologous temperatures
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H. Conrad / Materials Science and Engineering A341 (2003) 216 /228
the rate -controlling mechanism in Regime III is grain boundary shear according to the C /N model. The effect of grain size in this mechanism is simply on the number of grain boundary atoms available for participating in the thermally-activated shear event. As pointed out by Raj and Ashby [76] the rate-controlling grain boundary shear may be accommodated by grain boundary diffusion. Of additional interest is the relationship of the data by Chokshi et al. [5] and those by Cai et al. [6] with respect to Regimes II and III as defined in Fig. 2. This is shown in Fig. 14. It is here seen that the data by Chokshi et al. for o :10 3 s 1 and se /s lie partly within and partly outside the given limits for dislocation activity, i.e. they are just borderline for being in Regime III, recognizing that the ratio H /s may not be exactly 3 and that Eq. (3) are based on isotropic elasticity. In contrast, the single data point by Cai et al. for o :10 7 s 1 and se /s /s0 lies completely in Regime III. However, upon increasing the strain rate to 10 5 s 1, the estimated data point for se /s falls well outside of Regime III (as does also se / s /s0) and the rate-controlling mechanism would more likely be that for Regime II, namely grain boundary shear promoted by the pile-up of dislocations. This illustrates the importance of defining the strain rate (and temperature) when considering the transition grain size dc separating one Regime from another. Also to be noted from Fig. 14 is that besides the strain rate, dc depends on the magnitude of the Taylor orientation factor M (i.e. on the crystallographic texture) and on the type of dislocation (edge or screw). The agreement in the values of the parameters for grain boundary shear obtained form the two independent sets of data by Chokshi et al. [5] and by Cai et al. [6] provides support for the existence of a grain size softening phenomenon (inverse H-P effect) in metals. Further support that this is a real phenomenon is
provided by the results [74,75] on laser-deposited, nanocrystalline Zn films, which were free of the defects which might otherwise lead to an apparent grain size softening as discussed by Koch and Narayan [18].
3. Summary Data in the literature on the effect of grain size d in the range of 10 9 /10 3 m on the flow stress of Cu is analyzed with respect to the strain rate-controlling mechanism. Three grain size regimes were identified: Regimes I (d :/10 6 /10 3 m), II (d :/10 8 /10 6 m) and III (d B/ /10 8 m). The dislocation cell structure which occurs in Regime I is concluded to be absent in Regime II; all intragranular dislocation activity is concluded to cease in Regime III. The rate-controlling mechanisms determined from data representing each of the three regimes are summarized in Table 1. Since the plastic deformation in each regime is kinetic in nature, the transition grain size which distinguishes one regime from another will depend on temperature and strain rate. Crystallographic texture is also important.
Acknowledgements The author appreciates the assistance of R. O’Connell and Drs Di Yang and K. Jung in preparation of the manuscript. This study was conducted in collaboration with the NSF Center for Advanced Materials and Smart Structures at NCSU.
Appendix A: List of symbols H s si s0 te te /
/
t* tm tth
Fig. 14. Log s vs. log d illustrating the effects of strain rate, crystallographic orientation (Taylor orientation factor) and dislocation orientation on the grain size separating Regimes II and III.
m b d d
hardness tensile flow stress lattice friction stress critical stress or back stress effective resolved shear stress effective thermal component of the applied shear stress thermal component of the stress long-range internal stress component theoretical shear strength shear modulus Burgers vector grain size grain boundary width
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total dislocation density dislocation density due to grain boundaries rI dislocation density in single crystals rm mobile dislocation density rm dislocation density responsible for long-range internal stress rf forest dislocation density w dislocation dissociation width l* dislocation forest spacing lc dislocation cell size ld spacing between dislocations due to elastic interactions lp spacing between parallel slip planes containing piled-up groups of q dislocations on stress concentration factor x* forest dislocation obstacle width A * /l*x * activation area gSF stacking fault energy n number of piled-up dislocations M Taylor orientation factor /g / pre-exponential in ther0 mally-activated strain rate equation nD Debye frequency DG /DH * /T DS * Gibbs free activation energy DF * /DU * /T DS * Helmholtz free activation energy /v MkT @ln o=@s/ shear stress apparent activation volume /v kT @ln g=@t @DG=@t/ shear stress activation volume f fraction of atoms in the grain boundaries which participate in thermallyactivated grain boundary shear k Boltzmann’s constant T temperature a , b, c and C constants r rb
227
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