Cent. Eur. J. Phys. • 6(4) • 2008 • 834-842 DOI: 10.2478/s11534-008-0111-4

VERS ITA

Central European Journal of Physics

Onset of local ordering in some copper-based alloys: critical solute concentration vis-a-vis various solution-hardening parameters

Research Article

Muhammad Zakria Butt1*, Mozina Noshi1, Farooq Bashir2

1 Department of Physics, University of Engineering and Technology, Lahore - 54890, Pakistan

2 Central Research Laboratory, Lahore College for Women University, Lahore - 54000, Pakistan

Abstract: The mode of planar distribution of solute atoms in Cu single crystals alloyed with 0.5 to 8.0 at.%Ge has

been investigated via the temperature dependence of the critical resolved shear stress of these alloys. It is found that there exists a critical solute concentration cm x 5 at.%Ge below which the distribution of solute atoms in the crystal is random, and above which some local ordering occurs. This together with such data available in the literature for Cu-Zn, Cu-Al and Cu-Mn alloys, i.e. cm x 27 at. %Zn, 7 at.%Al and 1 at.%Mn, when examined as a function of the size-misfit factor 5 = (1/6)(dfo/dc) of a given binary alloy system, shows that the value of cm strongly depends on 5; the smaller the magnitude of 5, the greater the value of cm and vice versa. Also, the value of cm is found to correlate well with the electron-to-atom ratio (e/a) of the Cu-Zn, Cu-Al, Cu-Ge and Cu-Mn alloys with the solute concentration c = cm. However, no systematic correlation exists between the critical solute concentration cm for the onset of local ordering and the modulus - mismatch parameter n = (1/C)(dC/dc).

PACS (2008): 61.72.Hh; 61.72.Ss; 62.20.Fe; 81.40.Cd

Keywords: copper alloys • yield stress • temperature dependence • composition fluctuations • local order

© Versita Warsaw and Springer-Verlag Berlin Heidelberg.

1. Introduction

Hardening Induced In crystalline materials by foreign atoms dispersed randomly in the host lattice is referred to as solid-solution hardening (SSH). All SSH theories proposed so far involve analysis of the motion of mutually non-interacting dislocations under the action of the

*E-mail: mzbutt49@yahoo.com

Received 21 May 2008; accepted 17 July 2008

applied shear stress piecemeal through random dispersions of similar point-like obstacles over an isolated slip plane. Based on the nature of dislocation-solute interaction, these have been divided into two groups. In the first, typified by the classical paper of Mott and Nabarro [1], the solute concentration is assumed to be high enough to involve several solute atoms in the advance of a dislocation segment. Such models are termed "collective' or "friction" type as named by Suzuki [2]. The second group is made up of the "discrete-obstacle" or "breakaway" type models, exemplified by that of Friedel [3]. In these mod-

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els the low solute-concentration facilitates breakaway of a dislocation segment from individual solute atoms. Butt and Feltham [4] reviewed both types of SSH models highlighting weak and strong points in each case. A rather critical test of the viability of any SSH theory is, inter alia, its ability to account for the observed temperature and concentration dependence of the critical resolved shear stress (CRSS) and of the associated activation volume, the stress-equivalence of SSH, the interdependence of CRSS and activation volume, etc. The kink-pair nucle-ation (KPN) model of plastic flow in solid-solution crystals [5, 6], which is essentially a revised and enlarged version of one initially proposed by Feltham [7], has not only satisfactorily accounted for the SSH features noted above (e.g. [4]), but has also proved to be an effective tool to diagnose the nature of solute distribution in solid-solution crystals, whether metallic [8-11] or non-metallic, [12, 13]. Moreover, Roth et al. [14] in their paper on solute strengthening in nickel-based alloys demonstrated that the mathematical formulation of the unit activation process of yielding envisaged in the KPN model provides a sound base to predict the yield stress as a function of temperature and composition of both binary and ternary solid-solution alloy crystals.

In the present work we shall analyze the wealth of data appertaining to Cu-Ge alloy single crystals (c = 0.5 - 8.0 at.%Ge) deformed by Kamada and Yoshizawa [15] as well as by Traub et al. [16] in the temperature range 4-400 K, within the framework of the KPN model, to characterize the nature of solute distribution in this alloy system. Other objectives were to compare the findings of this study with those available in the literature on some other copper-based alloys, and to offer a plausible explanation of the observed critical solute concentration cm for the onset of local ordering in the crystals vis-à-vis various solution-hardening parameters.

2. The model

The unit activation process of yielding in the KPN model [5-7] involves the stress-assisted, thermally-activated detachment of edge-dislocation segments from short rows of closely spaced solute-atom pinning points. This occurs in a manner similar to the kink-pair mode of escape of screw dislocations from Peierls valleys [17] in nominally pure crystals with high intrinsic lattice friction. The mean spacing between neighbouring solute atoms, denoted by circles in Fig. 1, is taken to be A = b/c^/2, where b is the lattice parameter or the length of the Burgers vector, and c is the solute concentration expressed as an atomic fraction. To facilitate slip under an applied shear stress, the

length of an edge-dislocation segment L(=AB as shown In Fig. 1), after unpinning, must extend to that of the arc ABC with maximum displacement nb (n being a numerical constant) adequate to remove most of it from the short-range stress field of the initial pinning points. The value of the term nb is expected to be of the order of A, so that such a displacement of the arc would also bring it into the vicinity of the new pinning points, leading to renewed stabilization.

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

Now the activation energy W(t) for the formation of the arc ACB (Fig. 1), taking AC and BC to be approximately straight lines, is given by [7]

W(t) = UcV2(L/b) + n2Cb3(b/L) - InTb3(L/b) (1)

where t is the CRSS at the temperature T at which the experiment is carried out, U is the energy expended per solute atom in the initial breakaway of an edge-dislocation segment from an array of solute atoms, and G is the appropriate shear modulus. The terms on the right-hand side of Eq. (1) represent respectively, from left to right, (i) the energy required to overcome the binding energy U(L/A) of the edge-dislocation segment of length L pinned by a "necklace" of L/A solute atoms, (ii) the increase in the line energy due to the extra length, 2n2b2/L, involved in the formation of the "triangle" ACB from the initial length L=AB, taking line tension equal to 2Gb2, and (iii) the work done by the applied force, TbL, for an average displacement, 2nb, in moving the dislocation from AB to ACB configuration. Moreover, if one denotes the radius of curvature of the arc ACB by R (Fig. 1), then t = Gb/2R together with the geometrical relation 2nbR tt L2/4 yields

L/b = (4Gn/T f/2. (2)

If flow is to be observed in a crystal subjected to plastic deformation at constant strain-rate y with the density of moving dislocations held approximately constant, the nucleation of kink-pairs must occur with a frequency v, which is related to the activation energy W(t) through the Boltzmann relation

v = v0 exp[— W(r)/kT], (3)

where k is the Boltzmann constant and the pre-exponential factor v0 is a lattice vibration frequency of the order of 1011 s—1.

In his paper, Feltham [7] assumed that v is of the order of 1 s—1. However, since strain rates involved in usual tensile tests are of the order of 10—5 - 10—3 s—1, it would be more appropriate to consider a range of v-values, e.g. 0.1 - 10 s—1 [4] so that Eq. (3) with these values of v and v0 leads to the yield criterion

W(t) = mkT,m = ln(v0/v) = 25 ± 2.3. (4)

One can readily note that any other plausible values of v and v0 could be taken here without any significant influence on the m-value noted above.

Alternatively, one can re-write Eqs. (3) and (4) in terms of the shear rate y of the crystal, as given below.

y = yo exp[— W(t)/kT], (5)

W(t) = mkT,m = ln(y0/y) = 25 ± 2.3. (6)

Here the shear rate y = y0F(t, T) of the crystal has typical values in the range 10—5-10—3 s—1, and the pre-exponential factor y0 is a constant of the order of 107 s—1 [4].

Butt and Feltham [5] obtained an expression for the temperature and concentration dependence of the CRSS from

Eqs. (1) - (4):

t = T00/[1 + (1 + 9)1/2]2, (7)

where Trj = 4Uc1/2/nb3 is the CRSS as T 0 K, and 9 = 4n2Gb3UcV2/(mkT )2.

It should be noted that T0 « c1/2 provided that neither n nor U vary significantly with c. However, for temperatures high enough to lead to 9-values which are significantly less than unity, the denominator in Eq. (7) is essentially constant and one has t <x T09 <x c as both T0 and 9 are

proportional to c1/2. Thus, If one writes т(Т) ж cr, the rvalues would be expected to lie within the limits of 2 and 1;the larger values occurring at the higher temperatures. An expression for the activation volume, customarily defined as v = kT[d(ln у)/дт]T, associated with the CRSS т (Eq. (7)), in terms of в, is [5]

v = v0[5 + (1+ 52)1/2]2(1 + 52)1/2, (8)

where 5 = в-1/2 and v0 = 4b3n2(Gb3/Uc1/2)1/2 is the activation volume at T — 0 K.

It should be noted that as v — v0 (i.e. when T — 0 K, 5 — 0 K) one has v0 ж c-1/4 assuming that neither n nor U vary significantly with c. However, for relatively high temperatures when в « 1 or 52 » 1, v ж v053 ж c-1. Thus, if one writes v ( T) ж c-q, the g-value would be expected to lie within the limits of 1 and 1;the larger values occurring at the higher temperatures. If, however, one confines attention to the temperatures where diffusional processes are dormant, Eq. (7) can be reduced to an expression suitable for direct application to the experimental т-T data. In their paper, Butt et al. [6] have shown that Eq. (1) for the energy-of-formation W(т) of the arc ABC can also be expressed by the relation:

W(т) = W0(x-1/2 - x1/2), (9)

where W0 = n(Uc1/2Gb3)1/2, and x = т/т0. On defining auxiliary variable 0 by x = exp(-20), they obtained from Eq. (9): W(т) = 2W0 sinh0. For rather low temperatures, where diffusional processes are dormant in the crystal ( i.e. when x — 1,0 — 0 and sinh0 й 0 ), W(т) й 2W00, and hence [6]

W (т )= W01п(т0/т). (10)

Eq. (10), in conjunction with Eq. (4), then finally yields

т = т0 exp(-mkT/W0). (11)

This relation is generally found applicable in practice to both ferrous as well as non-ferrous alloys (see [4, 18]). At a given temperature, the CRSS т(c, T) thus depends on т0 and W0; U and c appear in both parameters only in association as Uc1/2, which is the binding-energy per interatomic spacing along the edge-dislocation segment pinned with solute-atoms. Eq. (11) implies that the slope d(ln^/dT should be constant for a given alloy composition, and equal to - mk/W0. The experimental value of d^n^/dT, thus facilitates the determination of W0, while

the point at which the extrapolated lnT - T line intersects the stress axis (T K), denotes the magnitude of t0. On using the expressions for macroscopic parameters t0 = 4Uc12/nb3 and W0 = n(UcV2Gb3)2, the microscopic parameters n and U of slip envisaged in the KPN model are given by the formulae:

n3 = (Wo/Gb3f(4G/To), (12)

U = W0/(n2Gb3cV2). (13)

0 100 200 300

and (15), one finds that, for a given alloy, the product tv is constant, i.e.

TV = ToVo = Wo. (16)

Also the activation volume v0 in terms of Lo can be shown to be:

Vo = 4nLob . (17)

It should be noted that assuming n, U, G and b are independent of the solute concentration c, t0 « c^/2, Wo « c^/4, Lo « c—^/4 and v0 « c—1/4. However, the four parameters assumed to be constant may depend a little on the solute concentration c leading to some deviation from the predicted c-dependence of t0, Wo, Lo and v0 referred to above. For instance, a "partially flexible" dislocation lying in the glide plane containing a concentration of solute atoms cannot be "fully" pinned at all the individual solute atoms along its length, so that U represents an average value of the binding energy of dislocation per solute atom, lower than the maximum for "full" interaction. So on the basis of the "imperfect" dislocation/solute interaction, the binding may become weaker,on average, as c increases leading to a weak dependence of U on c. Similarly one may expect the microscopic parameter n to decrease slightly with the increase in c, since the maximum displacement nb of the dislocation segment AB to the saddle-point configuration ACB is of the order of A = b/c1/2.

Figure 2. Temperature dependence of the CRSS of Cu-Ge alloy single crystals in semi-logarithmic coordinates. Data points are from Kamada and Yoshizawa [15].

Similarly, on using Eq. (2) one can evaluate the length of the edge-dislocation segment Lo involved in the unit activation process of yielding at T ^ 0 K, i.e.

Lo = (4Gn/To)1/2b. (14)

For the temperature and concentration dependence of activation volume v, customarily defined as (—dW/3t)t or kT[d(1ny)/dT]T, one can readily find from Eq. (11) that [19]:

v = v0 exp(mkT/Wo). (15)

Here Vo = 1 b3n2(Gb3/UcV2)1/2 is the activation volume associated with the CRSS t0 at T ^ 0 K. From Eqs. (11)

3. Data analysis

Reference to Fig. 2 shows the relation between the CRSS t and the temperature T in log-linear coordinates for Cu single crystals alloyed with 0.66 to 8.0 at.%Ge in the temperature range 4-315 K. The data points denote the measurements made by Kamada and Yoshizawa [15] with Cu-Ge single crystals annealed at 1000°C for 48 hours and furnace cooled down to room temperature prior to their deformation at a tensile strain rate of the order of 10—5 s—1. Excepting the anomalous mechanical response observed below 10-50 K in the case of c = 0.66, 0.95 and 2.95 at. %Ge, a straight line can be drawn through the data points for a given solute concentration in accordance with Eq. (11). The slope [= d(lnT)/dT] of the lnT — T straight line facilitates the determination of Wo[= —mk/(d(1n t)/dT)], while the point at which it intersects the stress axis (T ^ 0 K) denotes the magnitude of t0. The values of Wo and t0 determined in this manner have been given in Table 1.

Table 1. The values of various parameters of the KPN model of SSH derived from the experimental t - T data [15, 16] appertaining to Cu-Ge single crystals. G = 4.5 x 104 MPa, b = 0.2556 nm and Gb3 = 4.6 eVwere used in all cases.

c To Wo U n Lo Vo A No t -T

[at.%Ge] [MPa] [meV] [meV] [b] [b3] [b] Data

0.66 6.5 625 13.9 8.0 471 942 12.3 38 [15]

0.95 8.0 715 15.5 8.1 427 865 10.3 41

2.95 16.0 745 15.0 6.7 275 461 5.8 47

4.4 25.0 800 17.9 6.0 208 312 4.8 43

6.0 33.0 780 18.1 5.4 172 232 4.1 42

8.0 37.0 750 17.0 5.0 156 195 3.5 45

0.5 6.0 629 19.0 8.0 490 980 14.1 35 [16]

1.0 9.5 701 19.0 7.5 377 707 10.0 38

2.0 16.5 723 19.0 6.5 266 432 7.1 38

3.8 24.0 783 19.0 6.0 212 318 5.1 42

5.6 31.5 791 19.0 5.5 177 243 4.2 42

7.3 40.0 768 19.0 5.0 150 188 3.7 41

The temperature dependence of the CRSS of Cu - Ge alloys (c = 0.66 - 8.0 at.%Ge) has also been represented In linear coordinates In Fig. 3. The deviations from the monotonlc t - T behaviour are evident in the case of copper single crystals alloyed with 0.66, 0.95 and 2.95 at.%Ge below a certain temperature T0 specific to the alloy. As to the origin of this low-temperature anomaly, Feltham [20, 21] considered it to be a consequence of the deformation-induced enhancement of local stresses, to levels above the applied stress, at barriers to the move-

ment of dislocations at rather low temperatures (T < T0) where dynamic recovery processes will become increasingly inhibited. The applied stress appearing in the kinetic relations of deformation, e.g. t in Eq. (11), must then be multiplied by a stress-concentration factor f(T) for T < T0, whereas for T > T0 one sets f(T) = 1. The anomalous t - T behaviour observed in some copper based solid-solution crystals was found to be well compassed by Eq. (11) in conjunction with the stress-concentration fac-

0.3 1 10

C ( at.% Ge )

Figure 4. Dependence of the parameters t0 and Wo on the solute concentration c in Cu-Ge alloy single crystals in logarithmic coordinates.

tor f(T) of the form (see [4])

f (T) = [(T' + To)/(T' + T)], (18)

where T' = constant, T < To < T', f(T) = 1 at T > To, and f(T) — 1 + (To/T' ) as T — 0 K. Thus the curves drawn through the data points for c = 0.66, 0.95 and 2.95 at.%Ge in Fig. 3(a) comply with Eq. (11) on using the values of t0 and Wo given in Table 1, and after replacing t by Tf(T) below a certain temperature To; the values of To and T' used in f(T) are 50 and 150 K, 50 and 230 K, and 10 and 350 K, respectively. However, Eq. (11) alone was used while drawing the theoretical curves through the data points for relatively concentrated Cu-Ge alloys in Fig. 3(b).

The values of the microscopic parameters of slip, namely n, U, Lo and v0, were also evaluated with the help of Eqs. (12), (13), (14) and (17), in which the values of t0, Wo, G and b given in Table 1 were used. These are of the right order of magnitude for the glide of edge-dislocation segments in solid-solution crystals, as anticipated in the KPN model. The values of t0, n and U used by Butt et al. [22] in Eqs. (7) and (8) to account for the observed temperature dependence of the CRSS and of the associated activation volume appertaining to Cu-Ge alloy single crystals (c = 0.5 - 7.3 at.%Ge) deformed by Traub et al. [16] in tension in the temperature range 4-400 K have also been listed in Table 1 for comparison. It should be noted that the values of n and U chosen by Butt et al. [22] to accomplish agreement with experiment

0.3 1 10

c ( at.% Ge )

Figure 5. Dependence of the parameters n, Lo and v0 on the solute concentration c in Cu-Ge alloy single crystals in logarithmic coordinates.

are unique and cannot be varied independently because n and U appear as a ratio (U/n) in t0 and as a product (n2U) in 9. The values of t0, n and U referred to above were used to obtain other parameters of the KPN model, namely Wo, Lo, v0, A and No. It can be readily seen (Table 1) that the number of solute atoms No = Lo / A from which the dislocation segment Lo gets unpinned in the unit activation process at T — 0 K for solute concentrations 0.5-8.0 at.% Ge ranges from 35 to 47. It should be noted that for an alloy crystal with a given solute concentration c, as the temperature T increases, the CRSS t decreases (Eq. (11)); the length L of the dislocation segment involved in the unit activation process therefore increases with the rise in the temperature T (Eq. (2)), and hence the number of solute atoms N = L/A along L also increases. Figure 4 illustrates, in logarithmic coordinates, the concentration dependence of the parameters t0 and Wo for Cu-Ge alloy single crystals. The circles and squares denote the values (Table 1) derived from the t — T data of Kamada and Yoshizawa [15] and of Traub et al. [16] respectively. One can draw a single straight line through the data points in each case such that t0 k c069 and W0 x c010(c < 5 at.%Ge). On the other hand, the KPN model predicts t0 k c05, Wo x c025, Lo x c-025 and, v0 x c-025 with the stricture that n and U are indepen-

C ( at.% Ge )

Figure 6. Dependence of the parameters W0 and T' on the solute concentration c in Cu-Ge alloy single crystals.

dent of c. This apparent discrepancy can be accounted for as follows. Reference to Fig. 5 shows the concentration dependence of the microscopic parameters n, L0 and v0 (Table 1), in logarithmic coordinates, for Cu-Ge alloy single crystals. Below a critical solute concentration cm & 5 at.%Ge, a straight line fits the data in each case leading to n <x c-016, L0 <x c-042 and v0 <x c-0 55. Now since U is almost independent of c (the mean U-value being 16.2 and 19.0 meV (Table 1) for the data of Kamada and Yoshizawa [15] and of Traub et al. [16] respectively), therefore on using the expressions for various model parameters one finds that t0 « n-1 c05 « c066, W0 « nc025 « c0 09, L0 « nc-025 « c-041 and v0 « n2c-0-25 « c-0 57, i.e. in good agreement with the observations re lated to T0W0, L0 and v0 in Figs. (4) and 5.

To investigate the nature of solute distribution in Cu-Ge alloy crystals, we shall now examine the dependence of W0 on solute concentration c as depicted in Fig. 6. The circles and the squares denote the values of W0 (Table 1) derived from the t -T data of Kamada and Yoshizawa [15] and of Traub et al. [16] respectively. A common curve can be passed through the data points. One can readily note that W0 increases monotonically with c up to about

cm & 5 at.%Ge as predicted in the KPN model (W0 « c025) based on random distribution of solute atoms. However, as c increases beyond cm & 5 at.%Ge, W0decreases till 8.0 at.%Ge, indicating a departure from the random solute distribution. The variation of the values of constant T',denoted by the triangles in Fig. 6, with the solute concentration c in a manner similar to that of W0 shows that the anomalous temperature of the CRSS below a certain temperature T0 specific to the alloy is also influenced by the mode of solute distribution in the crystal. Now we shall examine any possible correlation of the critical solute concentration cm for the onset of local ordering in some copper-based alloy systems with the modulus-mismatch parameter n = (1/C)(dC/dc) and the size-misfit factor 5 = (1/b)(db/dc), which are mainly responsible for the strength of solid-solution crystals. The points in Fig. 7 denote the values of cm for Cu-Zn [8], Cu-Mn [9] and CuAl [10] alloys together with that for Cu-Ge single crystals obtained in the present work, as a function of (a) modulus-mismatch parameter n and (b) size-misfit factor 5 in logarithmic coordinates. One can readily see that no systematic correlation exists between cm and n in Fig. 7(a), while the least-squares fit to the data in Fig. 7(b) can be mathematically represented as

cm =5.8 x 10-75-44 (19)

with correlation factor r = - 0.945. It shows that the critical solute concentration cm corresponding to the maximum in the correlation of W0 with c (which is corroborated by a knee in the c-dependence of the microscopic parameters n, L0, and v0 in Fig. 5) strongly depends on the size-misfit factor 5 of a given binary alloy system;the smaller the magnitude of 5, the greater the value of cm and vice versa. It is worthy of note that in an attempt to account for the loss of stress-equivalence in concentrated solid-solutions, Schwink and Wille [23] opined that the size-misfit factor 5 plays a significant role. Quoting from their paper: "For a given concentration above a critical one to favour solute interactions, a large 5 may enhance deviations from a statistical distribution of solute atoms and thus induce stronger clusterings."

Another parameter of interest in relation to solute hardening of metallic crystals is the electron-to-atom ratio e/a of the alloy. Hibbard [24] found that for binary copper-based alloys with constant lattice parameter, namely Cu - 9.15 at.%Al, Cu - 6.87 at.% Ge, Cu - 7.68 at.%Ga and Cu - 10.83 at.%Zn, the yield stress at room temperature was a function of e/a of the alloys; the greater the value of e/a, the greater the yield strength. He therefore concluded that the models of solute strengthening which depend only on the lattice parameter changes (i.e. 5) are

Figure 7. Relation between the critical solute concentration cm for the onset of local ordering and (a) the modulus-mismatch parameter n, and (b) the size-misfit factor S for some copper-based alloys.

incomplete, and one must also take into account the electrical and chemical factors as well as the geometrical factor based on short-range order. In fact in the KPN model [5-7] the parameter Uc1/2, which is the binding energy per interatomic spacing along the edge-dislocation segment pinned with solute atoms, or the parameter Wo, which is the total binding energy of edge-dislocation segment of length Lo pinned with No solute atoms, is a measure of the overall or cumulative effect of all the factors referred to above. Nevertheless we shall now examine any possible correlation of the critical solute concentration cm with the electron-to-atom ratio e/a for the binary alloys referred to in Fig. 7. The values of cm have been denoted by points as a function of e/a in semi-logarithmic coordinates in Fig. 8. A straight line least-squares fit to the data is encompassed by the relation

cm = 1.3 x 10—9 exp[15.2(e/a)] (20)

with correlation factor r = + 0.971. This shows an excellent linear relationship between the electron-to-atom ratio

e/a of the copper-based binary alloys and the critical solute concentration cm in semi-logarithmic representation. However, the size misfit parameter S = (1 /b)(db/dc) is a constant specific to a given binary solid-solution system for all the solute concentrations until the first solubility limit, whereas the value of e/a varies with the solute concentration c. It therefore appears that the size-misfit parameter S is primarily responsible for the onset of local ordering in the alloy crystals at a critical solute concentrations cm, whereas the dependence of cm on the electron-to-atom ration e/a evaluated for c = cm is of secondary nature.

4. Conclusions

From the results discussed above, one may conclude that:

1. The temperature dependence of the CRSS of Cu-Ge alloy single crystals, whether normal or anomalous, is influenced by the mode of solute distribution.

2. The distribution of solute atoms is random be-

1.0 1.1 1.2 1.3

Figure 8. Relation between the critical solute concentration cm for the onset of local ordering and the electron-to-atom ratio e/a for some copper-based alloys.

low a critical solute concentration cm =5 at.%Ge, whereas for c > cm local ordering (whether clustering or short-range order) occurs in the crystals.

3. The value of the critical solute concentration cm for the binary copper-based alloys, namely Cu-Zn, Cu-Al, Cu-Ge and Cu-Mn, strongly depends on the size-misfit factor S = (1 /b)(db/dc). The smaller the magnitude of S, the greater the value of cm and vice versa.

4. The observed dependence of the critical solute concentration cm on e/a for the discussed copper-based alloys evaluated for c = cm is also good, but is of secondary nature.

5. No systematic correlation exists between the critical solute concentration cm for the onset of local ordering and the modulus-mismatch parameter n = (1/G)(dG/dc).

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