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Non-uniform interfacial impurity segregation
Surface Science 545 (2003) 99–108 www.elsevier.com/locate/susc
Non-uniform interfacial impurity segregation L.I. Stefanovich a, E.P. Feldman a, V.M. Yurchenko a, A.V. Krajnikov D.B. Williams c
b,*
,
a
Donetsk Physics and Technology Institute, 72, R. Luxemburg St., 83114 Donetsk, Ukraine Institute for Problems of Materials Science, 3, Krzhizhanivsky St., 03142 Kiev, Ukraine Department of Materials Science and Engineering, Lehigh University, 5E Packer Avenue, Whitaker Laboratory, Bethlehem, PA 18015-3195, USA b
c
Received 2 March 2003; accepted for publication 28 August 2003
Abstract The mechanism of formation of a non-uniform distribution of segregating atoms along a segregation plane is described as a result of attractive impurity–impurity interactions followed by a two-dimensional spinodal-type decomposition at a given interface. The kinetics of decomposition, which is controlled by the uphill diffusion mechanism, is analysed theoretically. The characteristic size, density and coverage of impurity-rich pre-precipitates are non-monotonic functions of time, even under isothermal conditions. The formulae obtained are applied to the description of possible intergranular decomposition in a Cu–Bi system. 2003 Elsevier B.V. All rights reserved. Keywords: Equilibrium thermodynamics and statistical mechanics; Grain boundaries; Surface segregation; Surface thermodynamics (including phase transitions)
1. Introduction Segregation of impurity atoms to grain boundaries or other interfaces is a well known phenomenon in solids. Many physical and mechanical properties of materials are controlled by impurity segregation. A classical example is embrittlement of grain boundaries in copper-base alloys containing Bi in a ppm range (see, e.g., [1–3]). It is necessary to know both enrichment levels and kinetics of the segregation process for a
*
Corresponding author. Tel.: +380-44-244-57-88/424-02-94; fax: +380-44-244-64-83. E-mail address: [email protected] (A.V. Krajnikov).
correct understanding of segregation–property relationships. The segregation kinetics is traditionally described on the basis of the Langmuir isotherm for a binary system [4,5] or the Guttmann model for multicomponent systems [6, 7]. Further progress in the description of the segregation kinetics is connected to the development of more realistic models, which consider possible interatomic interactions within a segregation layer. For example, the Fowler isotherm, which takes into consideration the limited number of sites within a segregation plane and impurity–impurity interactions, has been successfully used for a description of segregation kinetics in both bulk materials and thin films [8,9].
0039-6028/$ - see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.susc.2003.09.003
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It is traditionally supposed that the segregated atoms are uniformly distributed over the segregation plane. However, such an assumption is not often questioned and contradicts some known facts. For example, a certain accommodation mechanism, suppressing fast growth of the impurity concentration in a segregation layer, becomes operative with increasing surface concentration [10]. As assumed in [10], an island-like mode of impurity accumulation and accommodation, which is operative at early segregation stages, could explain the observed deceleration of the concentration growth. Generally speaking, processes of pre-precipitation, which occur at stages of two-dimensional (2D) spinodal decomposition and are realised by an uphill diffusion mechanism, can be considered as a reason of non-uniform impurity distribution along a segregation plane. Formation of enriched and depleted regions at the grain boundaries has been previously described for ternary alloys with competitive impurities [11]. Weak repulsive impurity interaction has been considered as a main mechanism leading to the formation of a miscibility gap at the grain boundary followed by demixing, occurring at temperatures below a certain critical value. To the authorsÕ knowledge, no special experiments aimed at the detection of non-uniform segregation caused by 2D spinodal decomposition are present in the literature. However, impressive progress in the development of new experimental techniques, e.g. analytical electron microscopy methods [12–14], with a better spatial resolution and higher elemental sensitivity achieved in recent years will provide better possibilities for direct experimental observation of interfacial spinodal decomposition followed by non-uniform segregation. This paper describes a theoretical study of a possible mechanism of 2D decomposition at interfaces in binary alloys, resulting in formation of non-uniform segregation along an interfacial plane. The results of theoretical calculations are used for estimation of the main parameters and conditions necessary for 2D spinodal decomposition in a Cu–Bi system.
2. Results 2.1. Formulation of problem One-dimensional diffusion in a half-infinite medium is considered as a model for study of interfacial segregation. While the model described below is valid for both interstitial and substitutional impurities, substitutional alloys will be considered here for distinctness. The concentration of impurities in the matrix cðz; tÞ is presumed to depend only on z, where z is the coordinate perpendicular to the boundary, and is independent of the coordinates x, y. The impurity concentration in the interface layer cA ðtÞ is also not affected by x and y. In other words, the impurities are supposed to be distributed homogeneously along the xy-plane. Besides, cA ðtÞ does not depend on z due to the extremely small thickness of the segregation layer. We will define the concentration as the ratio of the number of impurity atoms to the total number of available atomic sites. The impurity solution in the matrix is supposed to be dilute, cðz; tÞ 1, and the concentration of impurity in the segregation layer is arbitrary but cA ðtÞ < 1, by definition. Based on the assumption that the chemical potentials of impurity atoms at the interface and in the nearest sub-interface layers equilibrate instantly [4], various isotherms can be written to link the impurity concentrations at the interface, cA , and in the bulk, c. For example, the Fowler isotherm which takes into account the limited site number at the interface and lateral interaction between the neighbouring atoms (see, e.g. [15]) is written as follows: cA Tk Tk c¼ exp c 4 cA ð1Þ ð1 cA Þ T T Two dimensionless parameters are used here, namely the critical temperature Tk and the parameter c, which are defined as follows: ZjuAA j 4 0 4 u c ¼ A Z uAA Tk ¼
ð2Þ ð3Þ
L.I. Stefanovich et al. / Surface Science 545 (2003) 99–108
where u0A is the difference between the energies of impurity atoms in the interface and in the bulk; uAA is the interaction energy of A-type atoms placed in the nearest-neighbour sites; Z is the number of sites nearest to the considered one, i.e. the coordination number of the given grain-boundary structure. Naturally, u0A < 0, otherwise no segregation process occurs. We shall consider only the case of mutual attraction of impurity atoms (uAA < 0). The c ¼ cðcA Þ curves behave in different ways above and below the critical temperature Tk . At higher temperatures ðT > Tk Þ, the curve grows monotonically. At T < Tk , the curve drops down ð1Þ ð2Þ between the points cA ðT Þ and cA ðT Þ which are determined by the following equations: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð1Þ cA ðT Þ ¼ 1 1 ðT =Tk Þ 2 ð4Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2Þ cA ðT Þ ¼ 1 þ 1 ðT =Tk Þ 2 Plotting the curves ciA ¼ ciA ðT Þ ði ¼ 1; 2Þ for both branches of (4), one can obtain a bell-shaped curve (spinodal) on the T –cA diagram. Its maximum corresponds to the critical temperature T ¼ Tk . So, at temperatures below the critical one (T < Tk ) there is an area of compositions: ð1Þ
ð2Þ
cA < cA < cA
ð5Þ
where the impurity distribution in the segregation layer appears to be absolutely unstable with respect to its decomposition to areas with different impurity concentrations. In other words, there is a labile region below the spinodal curve, where a homogeneous impurity distribution is unstable. This instability can, in principle, result in the formation of a non-uniform impurity distribution along a segregation plane. 2.2. Segregation and decomposition rates
impurity concentration cA satisfies condition (5). The quenching time has to be much shorter than the diffusion time for impurities within a segregation plane. Evidently, the subsystem of as-quenched impurities is not in equilibrium and, thus, the distribution of segregated atoms along a segregation plane becomes spatially non-uniform. Such a distribution can be characterized by a random function cA ð~ q; tÞ of a 2D spatial coordinate ~ q and time t. This function will be further considered as a non-equilibrium order parameter and will be used for a description of the time evolution of the impurity distribution along a segregation plane. Important parameters for the description of the impurity redistribution are the characteristics times for the interfacial segregation and spinodal decomposition to occur within a segregation plane. As shown in [8], the characteristic segregation time is on the order of K 2 and is written in dimensional notation as follows: tsegr
K 2 d2 DV
ð6Þ
where KðT Þ ¼ expðju0A j=T Þ is the interfacial enrichment coefficient, d is the thickness of segregation layer; DV is the bulk diffusion coefficient of impurity A. The characteristic time for spinodal decomposition is given by [16]: tSD
R20S ð2Þ
ð7Þ
ð1Þ
c0A DS ½cA cA 2
where DS is the surface diffusion coefficient, R0S is a characteristic spatial scale of the initial inhomogeneity of the function cA ð~ qÞ quenched on the surface at t ¼ 0 and c0A is the mean enrichment level at the initial time. Therefore, the ratio of both characteristic times, which determines the speed of development of the two simultaneous processes (segregation and decomposition), is written as ð2Þ
Evidently, in order to observe non-uniform segregation along a boundary or interface, the subsystem of segregated impurities should be moved in any way to the labile region, e.g., by quenching from a high temperature T0 to a lower temperature T (T < Tk ) at which the interfacial
101
ð1Þ 2
tsegr c0A DS K 2 d2 ðcA cA Þ tSD DV R20S
ð8Þ
For further consideration of possibilities and conditions necessary for the formation of nonuniform segregation within a segregation plane, it is necessary to quantify the relationship of the two
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characteristic times defined by Eq. (8) for a certain practical case. The existence of a labile region on the T –cA diagram at T < Tk is the necessary condition for realisation of the decomposition mechanism under consideration. From the practical viewpoint, the value of Tk should be high enough to cover a field of typical experimental temperatures (e.g. homogenisation and ageing temperatures). From the literature [3,17–19], the Cu–Bi system satisfies the above conditions. Bismuth enriches grain boundaries and causes intergranular embrittlement of Cu-base alloys [1–3]. The Bi segregation is described by the Fowler isotherm [3,17,18]. The main segregation parameters, u0A and uAA , first calculated by the authors of [3] on the basis of their experimental data, are listed in Table 1. These parameters were later determined in [17] as well. The results of [17] agree with those of [3] within 5–8%. A set of the Fowler isotherms is plotted in Fig. 1 for a typical ppm range of Bi bulk concentrations, with the use of the parameters listed in Table 1. Pronounced drooping branches are seen on the c ¼ cðcA Þ curves in a concentration range from 0:1 6 cA 6 0:9, indicating the existence of an extended labile region on the T –cA diagram. As seen from Table 1, the estimated critical temperature is Tk 2000 K. Since homogenization or ageing temperatures for Cu–Bi alloys usually range between 800 and 1300 K, the condition T < Tk is valid for all practical cases. Therefore, the necessary conditions for 2D decomposition are fulfilled under typical experimental conditions in the Cu–Bi system. To estimate the ratio of characteristic times, the thickness of the segregation layer is taken as d 1 nm. As is known from the experiment [3], the equilibrium intergranular concentration of Bi at T > 993 K is rather low. Based on this fact, the
Fig. 1. A set of Fowler isotherms calculated for various quenching temperatures: 1––673 K, 2––773 K, 3––873 K, 4–– 973 K and 5––1123 K. The parameters are c ¼ 2:8 and Tk ¼ 2000 K.
mean Bi concentration is supposed to be c0A 0:1 that corresponds to a lower value of the cðcA Þ curve in a Cu–12 at. ppm Bi alloy at 1123 K (Fig. 1). After quenching to 873 K this concentration does not change but falls into a labile region. Both above-mentioned temperatures and Bi bulk concentration are taken as the typical values used in [19]. Then the enrichment coefficient is approximately KðT Þ 104 . A ratio of the grain boundary and bulk diffusion coefficients is supposed to be fixed for a narrow temperature range of 0.6–0.8 Tmelting . The relationship DS 102 –103 DV is taken as a typical value for diffusion of large metalloid atoms in FCC metals at considered temperatures ð2Þ ð1Þ [20,21]. Assuming ðcA cA Þ 1 and R0S 106
Table 1 Parameters used for calculation of the characteristic segregation and spinodal decomposition times and the main decomposition parameters Parameter
u0A , kJ/mol
uAA , kJ/mol
Tk , K
c
D0V , m2 /s
Q, kJ/mole
Value Source
)58.6 [3]
20.9 [3]
2000 Eq. (2)
2.8 Eq. (3)
7.66 · 105 [31]
178.1 [31]
L.I. Stefanovich et al. / Surface Science 545 (2003) 99–108
m, the ratio of characteristic times is tseqr =tSD 105 106 . In other words, the characteristic time of segregation is much larger than that of spinodal decomposition ðtseqr tSD Þ. Therefore, the spinodal decomposition can be considered as a fast process, occurring against a background of much slower impurity segregation from the bulk to the interface, which obeys the square-root law [8] 2Dc pffiffiffi cA ðsÞ cA ð0Þ þ pffiffiffi s p
ð9Þ
where s is the dimensionless segregation time expressed in t ¼ d2 =DV units, Dc ¼ c0 ðzÞ c0 ð0Þ is the difference in the bulk impurity concentration between the bulk and the nearest subsurface layer at t ¼ 0, and c0A is the mean interfacial enrichment level at t ¼ 0. 2.3. Basic equations for spinodal decomposition at interfaces For the case of bulk (3D) spinodal decomposition, relaxation of the non-equilibrium order parameter (bulk concentration) obeys the conservation law of local composition in the form of a continuity equation [16]. In contrast to 3D decomposition, interfacial spinodal decomposition occurs under continuous replenishment of the segregation layer by impurity atoms from the bulk. Therefore, evolution of the atomic concentration within a segregation layer should follow the matter-balance equation: ocA þ div~ jA ¼ rA ð~ q; tÞ ot
ð10Þ
q; tÞ is the local density of the source of where rA ð~ component A at the point ~ q at time t. On the supposition that the impurity element A comes to the interface from the bulk on average uniformly, one can note: dcA ðtÞ rA ð~ q; tÞ ¼ dt
ð11Þ
where rA > 0 and rA < 0 for segregation and desegregation, respectively. The flux density of segregated atoms along an interfacial plane is written as [16]:
qÞ ¼ MS r jA ð~
dF dcA
103
ð12Þ
where MS is the kinetics coefficient. It is generally defined as MS ¼ bS cA ð1 cA Þ
ð13Þ
where bS is the surface mobility of segregated atoms. By analogy with the bulk [16], the free energy function, F , for an impurity subsystem in the segregation layer, is written as Z 2 F ¼ ½f ðcA Þ þ kðrcA Þ nS dS ð14Þ where nS is the number of impurity atom sites per unit area at the interface and f ðcA Þ is the specific free energy of impurities distributed uniformly over the segregation layer. The second term in Eq. (14) describes the contribution of spatial correlative factors to the free energy. In the area of labile compositions (5), the expression for the specific bulk free energy obtained in [16] with an additional term linear in cA , which is caused by a difference between the surface and bulk energies of an impurity atom, can be used to express f ðcA Þ: 2 f ðcA Þ ¼ f0 Tk ðcA cA Þ þ ðcA cA Þ
2 4 ðcA cA Þ ð15Þ 3 where f0 is an addition to the specific free energy, ð1Þ ð2Þ independent on cA and T , and cA ¼ ðcA þ cA Þ=2. For simplicity, the temperature is expressed in units of energy. Since only the initial segregation stage, (the socalled root stage) is considered further, the value of the kinetics coefficient MS ðcA Þ from (13) can be approximated by a constant which is expressed as MS ðcA Þ ¼ bS cA ð1 cA Þ bSc0A ð1 c0A Þ bSc0A
ð16Þ
As already mentioned, the concentration of impurities within a segregation layer is a random function of the 2D coordinate ~ q along the surface. It is reasonable to introduce two new notations, namely cA ðtÞ and vA ð~ q; tÞ, which are the mean
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concentration and the deviation from the mean value, respectively, and are connected as q; tÞ cA ð~ q; tÞ cA ðtÞ vA ð~
ð17Þ
The value of cA ðtÞ is a known function defined in (9). Using Eqs. (10) and (12) and taking into account expressions (11) and (14)–(17), the following non-linear differential equation, expressed in a dimensionless form, can be written to describe the evolution of the concentration fluctuation vA ð~ q; tÞ with time. ov ð1Þ ð2Þ 0 ¼ 8cA D ðcA ðsÞ cA ÞðcA ðsÞ cA Þv ot
1 3 1 2 þ ðcA ðsÞ cA Þv þ v D2 v ð18Þ 3 2 where D is the Laplacian. The spatial and time coordinates were reduced to their dimensionless forms ffiffiffiffiffiffiffiffiffiffiffiffiby normalizingl2 the coordinates by l ¼ p k=ju0A j and by t ¼ b ju 0 j ¼ l2 =D0 , respectively. S
A
If the bulk concentrations of both components in a binary system are close to each other, phase decomposition often leads to the formation of lamellar or quasi-one-dimensional structures [22]. If the concentrations are essentially different, then the surface distribution of elements becomes statistically isotropic and, thus, the decomposition problem becomes effectively one-dimensional. Drawing an analogy between the film growth on a free surface and the formation of segregation at the interface, the mean impurity concentration can be considered as a 1D function of the coordinate, for example, of x: cA ð~ q; tÞ cA ðx; tÞ
ð19Þ
Then Eq. (18) can be transformed to 1D analogue of the Cahn–Hilliard diffusion equation [23] in dimensionless coordinates s and n: 2 ov ð1Þ ð2Þ o v ¼ 8c0A ðcA ðsÞ cA ÞðcA ðsÞ cA Þ 2 os on
2 2 2 3 1ov 1 o4 v o v þ ðcA ðsÞ cA Þ 2 þ ð20Þ 3 on2 4 on4 on Eq. (20) is similar to that describing evolution of the concentration fluctuation in the bulk [16] with only one difference. Here, the mean enrichment value cA is the known time-dependent function (9)
rather than a constant, as in [16]. The latter term in Eq. (20) is determined by the presence of a gradient-energy term in the free-energy function (14). This gradient-energy term starts to play an essential role during the later stages of spinodal decomposition only. Its potential contribution will be estimated later. For the early decomposition stages, Eq. (20) can be truncated to the following form: 2 ov ð1Þ ð2Þ o v 0 ¼ 8cA ðcA ðsÞ cA ÞðcA ðsÞ cA Þ 2 os on
2 2 2 3 ov 1ov þ ðcA ðsÞ cA Þ 2 þ 3 on2 on
ð21Þ
where the time dependence of the mean surface enrichment obeys Eq. (9). The enrichment fluctuation in the as-quenched state, i.e. at t ¼ 0, is taken as an initial condition: vA ðn; sÞjs¼0 ¼ vA ðn; 0Þ
ð22Þ
Since vðn; 0Þ is a random function of the coordinate within a segregation plane, the determinate equation with random initial conditions can therefore be solved using probabilistic terms. In particular, it is necessary to find the correlation characteristics of the random function v. Eq. (21) allows the writing of a set of engaging equations for the correlators of various orders. A statistical method developed in [16] can be used to do this under the assumption that the segregation layer is statistically uniform with respect to the impurity distribution. As a result, Eq. (21) can be transformed to the following set of equations: 8 dr 2aðsÞ 2 > > < ds ¼ r þ r2 þ 2bðsÞg ð23Þ > dg 3aðsÞ 6 12bðsÞ > : ¼ þ 3 g ds r2 r r4 where rðsÞ ¼ RðsÞ=R0S is the mean dimensionless spatial period of the structure of the impurity atoms at the segregation layer at t ¼ 0. A normalized correlation function of the third order, gðsÞ, is introduced here to characterize the possible distribution asymmetry of the random function cA ð~ q; tÞ. The newly introduced function is written as
L.I. Stefanovich et al. / Surface Science 545 (2003) 99–108 3=2
gðsÞ hv3 ðsÞi=K0
ð24Þ
where K0 ¼ hv2 ð0Þi is the correlation function of the second order (dispersion distribution) at initial time ðs ¼ 0Þ. Eqs. (23) can be reduced to their dimensionless forms by the following substitutions: s ¼ t=t0 ;
t0 ¼ R20S =ð8c0S DS K0 Þ
ð25Þ
3. Discussion 3.1. Analysis of equations Parameters a and b from Eq. (26) can be rewritten as functions of real time t: pffi ð28Þ aðtÞ ¼ b2 ðtÞ d 2 ðT Þ; bðtÞ ¼ k þ kðT Þ t; where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 T =Tk pffiffiffiffiffi dðT Þ ¼ ; K0 c0 hðT Þ kðT Þ ¼ Apffiffiffiffiffi K0 0:5
Initial conditions for Eqs. (23) are rð0Þ ¼ 1 and gð0Þ ¼ g0 . The outward appearance of Eqs. (23) looks generally similar to those obtained in [16] for the case of bulk spinodal decomposition. However, the present equations have time-dependent coefficients aðsÞ and bðsÞ that essentially complicates their analytical solution. The evolution of the above coefficients with time is rather slow as compared with the interfacial spinodal decomposition because it is controlled by the inflow of the segregating atoms from the bulk. Using dimensionless forms, the coefficients aðsÞ and bðsÞ can be written as pffiffiffi pffiffiffi ð1Þ ð2Þ ½c0 ð1 þ h sÞ cA ½c0A ð1 þ h sÞ cA
aðsÞ ¼ A K0 pffiffiffi c0 ð1 þ h sÞ cA pffiffiffiffiffi bðsÞ ¼ A ð26Þ K0 where parameter h is given by rffiffiffiffiffiffi 2DcR0S DV h ¼ pffiffiffiffiffiffiffiffi 0 pK0 dcA DS
ð27Þ
Evidently, the solution of Eqs. (23) gives a complete description of the decomposition processes at the interfaces under conditions of continuous inflow of the segregating atoms. However, it is a very difficult task to solve these equations analytically. As was estimated by (8), the segregation processes are much slower than the decomposition ones. This allowed the authors to use adiabatic approximation in the next section and analyse evolution of quasi-steady structures, which form at the interfaces as a result of spinodal decomposition at some critical points, instead of the exact solution of Eqs. (23).
105
c0 c k ¼ Apffiffiffiffiffi A ; K0
Let us suppose that impurities have already enriched the interface before quenching and the mean interfacial concentration at t ¼ 0, cA ð0Þ, is within the spinodal area, satisfying condition (5). Evidently, the parameters a and b are then negative, at least at the initial stage of the quenching. Therefore, the considered Eqs. (23) have two critical points, such as the saddle and stable junctions. In contrast to the bulk spinodal decomposition [16], the coordinates of such points are not fixed at a phase plane and they change with time in accordance with the following equations: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4b2 ðtÞ 3aðtÞ þ ð4b2 ðtÞ 3aðtÞÞ 8a2 ðtÞ r1 ðtÞ ¼ 2a2 ðtÞ ð29aÞ g1 ðtÞ ¼
r2 ðtÞ ¼
aðtÞr1 ðtÞ þ 1 bðtÞr12 ðtÞ
4b2 ðtÞ 3aðtÞ
ð29bÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð4b2 ðtÞ 3aðtÞÞ 8a2 ðtÞ 2a2 ðtÞ ð30aÞ
g2 ðtÞ ¼
aðtÞr2 ðtÞ þ 1 bðtÞr22 ðtÞ
ð30bÞ
The stable junction point represented by the r2 ðtÞ and g2 ðtÞ functions is responsible for the formation and evolution of quasi-steady non-uniform impurity structures at the interfaces. The first function determines the time dependence of the mean period of segregation non-uniformity, appearing as a
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result of 2D spinodal decomposition, and the second describes evolution of the asymmetry of the distribution function. To clarify the physical meaning of the results obtained, let us consider a single-point function of the impurity distribution in the segregation layer, constructed as a combination of two sharp d-like maxima instead of the functions r2 ðtÞ and g2 ðtÞ: pðvÞ ¼ pe dðvA ve Þ þ pd dðvA vd Þ
ð31Þ
where ve and vd are the deviations from the mean concentrations in impurity-enriched and depleted regions of the surface, respectively, and pe and pd are the areas (coverages) of the enriched and depleted regions. Evidently, the distribution function (31) satisfies the following normalization condition: Z 1 pðvÞ dv ¼ 1 ð32Þ 1
Using correlations of the hv2 i and hv3 i correlators with the distribution parameters described in (31), on the one hand, and functional expressions of the above correlators through the r2 ðtÞ and g2 ðtÞ functions, on the other hand, the values of pe ðr2 ; g2 Þ and pd ðr2 ; g2 Þ are written as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 1 g22 r23 1þ ; pe ¼ 2 4 þ g22 r23 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 1 g22 r23 1 pd ¼ ð33Þ 2 4 þ g22 r23 Since the r2 ðtÞ and g2 ðtÞ functions are known from Eq. (30), then the time evolution of the coverages of both enriched and depleted surface parts can be calculated by (33). Based on the known function r2 ðtÞ, the density of interfacial pre-precipitates can be calculated as a function of time by the following equation: nðtÞ ¼ ðR0S r2 ðtÞÞ1
ð34Þ
Furthermore, the time dependencies of the mean size of the enriched pre-precipitates (or enriched islands), Re ðtÞ, and depleted islands, Rd ðtÞ, can be calculated using Eqs. (30) and (33): Re ðtÞ ¼ R0S pe ðtÞr2 ðtÞ;
Rd ðtÞ ¼ R0S pd ðtÞr2 ðtÞ
ð35Þ
3.2. Numerical estimations for Cu–Bi system There are many reported experimental observations of concentration inhomogeneities along a segregation surface. However, typical reasons for such behaviour are • segregation anisotropy, often enhanced by impurity interaction [24], which results in variation of the impurity concentration between different grains of polycrystalline materials (see e.g. [25] for general information and [18] for a particular case of Cu–Bi alloys), • preferential enrichment of local defects on the surface, such as dislocation lines [26], deformation twins [27], steps [28], etc., and • experimental artefacts, such as e.g. composition variations caused by zigzag crack propagation along the in-situ fractured grain boundary in Auger electron spectroscopy experiments [27, 29]. Indications of inhomogeneities in Bi segregation along some grain boundaries were already reported in transmission electron microscopy (TEM) experiments [19,30]. However, the origin of such Bi-rich regions is not understood yet. As was mentioned before, all necessary conditions for 2D spinodal decomposition at grain boundaries in Cu–Bi alloys are fulfilled for a ppm range of Bi bulk concentrations and a wide temperature range. The main parameters for possible non-uniform segregation caused by 2D spinodal decomposition can be estimated for a Cu–Bi system on the basis of the formulae derived above. For distinctness, we consider the Bi bulk concentration c ¼ 12 at. ppm that corresponds to the bulk composition of one of two alloys studied by TEM in [19]. The supposed heat treatment route generally corresponds to that used in [19]. The alloy is first homogenized at T0 ¼ 1123 K. The equilibrium intergranular Bi segregation at 1123 K is c0A ¼ 0:1, as follows from the Fowler isotherm (Fig. 1). Then the alloy is quenched to T ¼ 873 K and annealed at this temperature for a few hours. All other parameters necessary for the calculations, such as segregation and impurity interaction energies, and
L.I. Stefanovich et al. / Surface Science 545 (2003) 99–108
Fig. 2. Mean size of Bi-rich islands, Ri ðtÞ, in a segregation layer as a function of annealing time at 873 K calculated by (35).
bulk diffusion data for Bi in Cu alloys, are taken from the literature and are listed in Table 1. Fig. 2 shows the calculated characteristic mean size of the Bi-rich islands, Ri ðtÞ as a function of annealing time at 873 K. The mean island size shows a non-monotonic time dependence, changing from Re max 34 nm to Re min 20 nm. The estimated values of Bi islands generally correlate with the dimensions of Bi inhomogeneities experimentally observed at grain boundaries in a Cu–12 at. ppm Bi alloy, which vary typically from 15 to 40 nm [19]. While the origin of the observed Bi regions is not clear and could perhaps be caused by other than 2D decomposition reasons (e.g. by artefacts of sample preparation as discussed in [19]), the proposed mechanism of spinodal decomposition can, in principle, be realised under the considered experimental conditions. Undoubtedly, more precise experiments on a nanometre scale are necessary to generate conclusive experimental evidences for 2D impurity decomposition. In accordance with theoretical predictions (Fig. 2), a minimum size of Bi-rich pre-precipitates is reached at a border of the labile area (5), when ð2Þ cA ðtÞ ¼ cA , after annealing for tc 7 103 s at 873 K. Simultaneously the island density, nðtÞ, reaches a maximum and amounts to nmax 4 107 cm1 (Fig. 3). The Bi-rich island density reaches its minimum value nmin 2 107 cm1 (Fig. 3) when the mean surface enrichment cA ðtÞ reaches the centre of the
107
Fig. 3. Density of Bi pre-precipitates, nðtÞ, as a function of annealing time at 873 K calculated by (34).
labile region, i.e. at cA ¼ cA ¼ 0:5 (Fig. 1), and the coverages of the Bi-enriched and the Bi-depleted islands become equal, pdmax ¼ pemin ¼ 0:5 (Fig. 4). When cA ðtÞ approaches the right border of the labile region (5), the coverage of Bi-rich islands, pe , increases with time (Fig. 4). Since the subsystem of impurities gets the isothermal conditions (T ¼ const) after the quenching and since a half-infinite medium is considered, then the bulk impurity concentration does not practically change during segregation ðc constÞ and the mean impurity enrichment cA ðtÞ is the only parameter, controlling 2D impurity decomposition. The value of cA ðtÞ, evolving during segregation, crosses the area under the spinodal curve
Fig. 4. Surface coverage of Bi pre-precipitates, pe ðtÞ, as a function of annealing time at 873 K calculated by (33).
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from a quenching point to a point at the right border of the labile region. The thermodynamic force, driving 2D spinodal decomposition, changes non-monotonically. As a result, the main decomposition parameters, such as RðtÞ and nðtÞ also evolve non-monotonically, as seen from Figs. 2 and 3. 4. Conclusions 1. In contrast to bulk spinodal decomposition, when the order parameter (average bulk concentration) remains constant, the proposed mechanism of interfacial spinodal decomposition is controlled by the order parameter (average surface concentration) cA ðtÞ which changes with time and is a function of segregation rate. 2. Even under isothermal conditions, the density and mean size of the impurity-rich islands are non-monotonic functions of time. 3. The impurity-rich island density reaches a minimum, when half of the surface sites are occupied with impurity atoms ðcA ¼ cA ¼ 0:5Þ, and ð2Þ attains its maximum value at cA ¼ cA . Acknowledgements This work was supported in part by Award #UE2-2072 of the U.S. Civilian Research & Development Foundation for the Independent States of the Former Soviet Union (CRDF). One of the authors, DBW wishes to acknowledge the support of the National Science Foundation through grant DMR 99-72670. References [1] A. Joshi, D.F. Stein, J. Inst. Metals 99 (1971) 178. [2] S.F. Baumann, D.B. Williams, J. Microsc. 123 (1984) 299. [3] M. Menyhard, B. Blum, C.J. Mahon Jr., Acta Metall. 37 (1989) 549.
[4] D. McLean, Grain Boundaries in Metals, Clarendon Press, Oxford, 1957. [5] C. Lea, M.P. Seah, Philos. Mag. 35 (1977) 213. [6] M. Guttmann, Surf. Sci. 53 (1975) 213. [7] M. Militzer, J. Wieting, Acta Metall. 34 (1986) 1229. [8] T.N. Melnik, E.P. Feldman, V.M. Yurchenko, Metal Phys. Adv. Technol. (in Russian) 21 (4) (1999) 28. [9] A.V. Krajnikov, V.M. Yurchenko, E.P. Feldman, D.B. Williams, Surf. Sci. 515 (2002) 36. [10] M.J. van Staden, J.P. Roux, Appl. Surf. Sci. 44 (1990) 271. [11] M. Militzer, J. Wieting, Acta Metall. 37 (1989) 2585. [12] D.B. Williams, A.D. Romig Jr., Ultramicroscopy 30 (1989) 38. [13] V.J. Keast, D.B. Williams, Curr. Opinion Solid State Mater. Sci. 5 (2001) 23. [14] D.B. Williams, D.E. Newbury, Acta Mater. 48 (2000) 323– 346. [15] R.H. Fowler, E.A. Guggenheim, Statistical Thermodynamics, University Press, Cambridge, 1960. [16] E.P. Feldman, L.I. Stefanofich, Sov. Phys. JETP 71 (1990) 951. [17] L.-S. Chang, E. Rabkin, B.B. Straumal, B. Baretzky, W. Gust, Acta Mater. 47 (1999) 4041. [18] U. Alber, H. Muelejans, M. Ruele, Acta Mater. 47 (1999) 4047. [19] V.J. Keast, D.B. Williams, Acta Mater. 47 (1999) 3999. [20] B. Aufray, F. Cabane-Brouty, J. Cabane, Acta Metall. 27 (1979) 1849. [21] F. Moya, G.E. Moya-Gontier, F. Cabane-Brouty, Phys. Status Solidi 35 (1969) 893. [22] M. Atzmon, D.A. Kessler, D.J. Srolovitz, J. Appl. Phys. 72 (1992) 442. [23] J.W. Cahn, A. Hilliard, J. Chem. Phys. 28 (1958) 258. [24] A.V. Krajnikov, M. Militzer, J. Wieting, Mater. Sci. Technol. 40 (1997) 877. [25] P. Lejcek, S. Hoffmann, Crit. Rev. Sol. State Mater. Sci. 20 (1995) 1. [26] L.-S. Chang, E. Rabkin, B. Baretzky, W. Gust, Scripta Mater. 38 (1998) 1033. [27] M. Menyhard, B. Rothman, C.J. McMahon Jr., P. Lejcek, V. Paidar, Acta Metal. Mater. 39 (1991) 1289. [28] P. Lejcek, M. Koutnik, J. Bradler, V. Paidar, A. Potmesilova, Appl. Surf. Sci. 44 (1990) 75. [29] M. Menyhard, C.J. McMahon Jr., Scripta Met. Mater. 25 (1991) 935. [30] V.J. Keast, D.B. Williams, in: H.A. Calder on Benavides, M.J. Yacaman (Eds.), Proc. Electron Microscopy, Institute of Physics, Bristol, 1998, p. 569. [31] V.A. Gorbachev, Ja.A. Rabovsky, S.M. Klotsman, Phys. Metall. Metallograph. (in Russian) 44 (1977) 214.
Non-uniform interfacial impurity segregation L.I. Stefanovich a, E.P. Feldman a, V.M. Yurchenko a, A.V. Krajnikov D.B. Williams c
b,*
,
a
Donetsk Physics and Technology Institute, 72, R. Luxemburg St., 83114 Donetsk, Ukraine Institute for Problems of Materials Science, 3, Krzhizhanivsky St., 03142 Kiev, Ukraine Department of Materials Science and Engineering, Lehigh University, 5E Packer Avenue, Whitaker Laboratory, Bethlehem, PA 18015-3195, USA b
c
Received 2 March 2003; accepted for publication 28 August 2003
Abstract The mechanism of formation of a non-uniform distribution of segregating atoms along a segregation plane is described as a result of attractive impurity–impurity interactions followed by a two-dimensional spinodal-type decomposition at a given interface. The kinetics of decomposition, which is controlled by the uphill diffusion mechanism, is analysed theoretically. The characteristic size, density and coverage of impurity-rich pre-precipitates are non-monotonic functions of time, even under isothermal conditions. The formulae obtained are applied to the description of possible intergranular decomposition in a Cu–Bi system. 2003 Elsevier B.V. All rights reserved. Keywords: Equilibrium thermodynamics and statistical mechanics; Grain boundaries; Surface segregation; Surface thermodynamics (including phase transitions)
1. Introduction Segregation of impurity atoms to grain boundaries or other interfaces is a well known phenomenon in solids. Many physical and mechanical properties of materials are controlled by impurity segregation. A classical example is embrittlement of grain boundaries in copper-base alloys containing Bi in a ppm range (see, e.g., [1–3]). It is necessary to know both enrichment levels and kinetics of the segregation process for a
*
Corresponding author. Tel.: +380-44-244-57-88/424-02-94; fax: +380-44-244-64-83. E-mail address: [email protected] (A.V. Krajnikov).
correct understanding of segregation–property relationships. The segregation kinetics is traditionally described on the basis of the Langmuir isotherm for a binary system [4,5] or the Guttmann model for multicomponent systems [6, 7]. Further progress in the description of the segregation kinetics is connected to the development of more realistic models, which consider possible interatomic interactions within a segregation layer. For example, the Fowler isotherm, which takes into consideration the limited number of sites within a segregation plane and impurity–impurity interactions, has been successfully used for a description of segregation kinetics in both bulk materials and thin films [8,9].
0039-6028/$ - see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.susc.2003.09.003
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It is traditionally supposed that the segregated atoms are uniformly distributed over the segregation plane. However, such an assumption is not often questioned and contradicts some known facts. For example, a certain accommodation mechanism, suppressing fast growth of the impurity concentration in a segregation layer, becomes operative with increasing surface concentration [10]. As assumed in [10], an island-like mode of impurity accumulation and accommodation, which is operative at early segregation stages, could explain the observed deceleration of the concentration growth. Generally speaking, processes of pre-precipitation, which occur at stages of two-dimensional (2D) spinodal decomposition and are realised by an uphill diffusion mechanism, can be considered as a reason of non-uniform impurity distribution along a segregation plane. Formation of enriched and depleted regions at the grain boundaries has been previously described for ternary alloys with competitive impurities [11]. Weak repulsive impurity interaction has been considered as a main mechanism leading to the formation of a miscibility gap at the grain boundary followed by demixing, occurring at temperatures below a certain critical value. To the authorsÕ knowledge, no special experiments aimed at the detection of non-uniform segregation caused by 2D spinodal decomposition are present in the literature. However, impressive progress in the development of new experimental techniques, e.g. analytical electron microscopy methods [12–14], with a better spatial resolution and higher elemental sensitivity achieved in recent years will provide better possibilities for direct experimental observation of interfacial spinodal decomposition followed by non-uniform segregation. This paper describes a theoretical study of a possible mechanism of 2D decomposition at interfaces in binary alloys, resulting in formation of non-uniform segregation along an interfacial plane. The results of theoretical calculations are used for estimation of the main parameters and conditions necessary for 2D spinodal decomposition in a Cu–Bi system.
2. Results 2.1. Formulation of problem One-dimensional diffusion in a half-infinite medium is considered as a model for study of interfacial segregation. While the model described below is valid for both interstitial and substitutional impurities, substitutional alloys will be considered here for distinctness. The concentration of impurities in the matrix cðz; tÞ is presumed to depend only on z, where z is the coordinate perpendicular to the boundary, and is independent of the coordinates x, y. The impurity concentration in the interface layer cA ðtÞ is also not affected by x and y. In other words, the impurities are supposed to be distributed homogeneously along the xy-plane. Besides, cA ðtÞ does not depend on z due to the extremely small thickness of the segregation layer. We will define the concentration as the ratio of the number of impurity atoms to the total number of available atomic sites. The impurity solution in the matrix is supposed to be dilute, cðz; tÞ 1, and the concentration of impurity in the segregation layer is arbitrary but cA ðtÞ < 1, by definition. Based on the assumption that the chemical potentials of impurity atoms at the interface and in the nearest sub-interface layers equilibrate instantly [4], various isotherms can be written to link the impurity concentrations at the interface, cA , and in the bulk, c. For example, the Fowler isotherm which takes into account the limited site number at the interface and lateral interaction between the neighbouring atoms (see, e.g. [15]) is written as follows: cA Tk Tk c¼ exp c 4 cA ð1Þ ð1 cA Þ T T Two dimensionless parameters are used here, namely the critical temperature Tk and the parameter c, which are defined as follows: ZjuAA j 4 0 4 u c ¼ A Z uAA Tk ¼
ð2Þ ð3Þ
L.I. Stefanovich et al. / Surface Science 545 (2003) 99–108
where u0A is the difference between the energies of impurity atoms in the interface and in the bulk; uAA is the interaction energy of A-type atoms placed in the nearest-neighbour sites; Z is the number of sites nearest to the considered one, i.e. the coordination number of the given grain-boundary structure. Naturally, u0A < 0, otherwise no segregation process occurs. We shall consider only the case of mutual attraction of impurity atoms (uAA < 0). The c ¼ cðcA Þ curves behave in different ways above and below the critical temperature Tk . At higher temperatures ðT > Tk Þ, the curve grows monotonically. At T < Tk , the curve drops down ð1Þ ð2Þ between the points cA ðT Þ and cA ðT Þ which are determined by the following equations: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð1Þ cA ðT Þ ¼ 1 1 ðT =Tk Þ 2 ð4Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð2Þ cA ðT Þ ¼ 1 þ 1 ðT =Tk Þ 2 Plotting the curves ciA ¼ ciA ðT Þ ði ¼ 1; 2Þ for both branches of (4), one can obtain a bell-shaped curve (spinodal) on the T –cA diagram. Its maximum corresponds to the critical temperature T ¼ Tk . So, at temperatures below the critical one (T < Tk ) there is an area of compositions: ð1Þ
ð2Þ
cA < cA < cA
ð5Þ
where the impurity distribution in the segregation layer appears to be absolutely unstable with respect to its decomposition to areas with different impurity concentrations. In other words, there is a labile region below the spinodal curve, where a homogeneous impurity distribution is unstable. This instability can, in principle, result in the formation of a non-uniform impurity distribution along a segregation plane. 2.2. Segregation and decomposition rates
impurity concentration cA satisfies condition (5). The quenching time has to be much shorter than the diffusion time for impurities within a segregation plane. Evidently, the subsystem of as-quenched impurities is not in equilibrium and, thus, the distribution of segregated atoms along a segregation plane becomes spatially non-uniform. Such a distribution can be characterized by a random function cA ð~ q; tÞ of a 2D spatial coordinate ~ q and time t. This function will be further considered as a non-equilibrium order parameter and will be used for a description of the time evolution of the impurity distribution along a segregation plane. Important parameters for the description of the impurity redistribution are the characteristics times for the interfacial segregation and spinodal decomposition to occur within a segregation plane. As shown in [8], the characteristic segregation time is on the order of K 2 and is written in dimensional notation as follows: tsegr
K 2 d2 DV
ð6Þ
where KðT Þ ¼ expðju0A j=T Þ is the interfacial enrichment coefficient, d is the thickness of segregation layer; DV is the bulk diffusion coefficient of impurity A. The characteristic time for spinodal decomposition is given by [16]: tSD
R20S ð2Þ
ð7Þ
ð1Þ
c0A DS ½cA cA 2
where DS is the surface diffusion coefficient, R0S is a characteristic spatial scale of the initial inhomogeneity of the function cA ð~ qÞ quenched on the surface at t ¼ 0 and c0A is the mean enrichment level at the initial time. Therefore, the ratio of both characteristic times, which determines the speed of development of the two simultaneous processes (segregation and decomposition), is written as ð2Þ
Evidently, in order to observe non-uniform segregation along a boundary or interface, the subsystem of segregated impurities should be moved in any way to the labile region, e.g., by quenching from a high temperature T0 to a lower temperature T (T < Tk ) at which the interfacial
101
ð1Þ 2
tsegr c0A DS K 2 d2 ðcA cA Þ tSD DV R20S
ð8Þ
For further consideration of possibilities and conditions necessary for the formation of nonuniform segregation within a segregation plane, it is necessary to quantify the relationship of the two
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characteristic times defined by Eq. (8) for a certain practical case. The existence of a labile region on the T –cA diagram at T < Tk is the necessary condition for realisation of the decomposition mechanism under consideration. From the practical viewpoint, the value of Tk should be high enough to cover a field of typical experimental temperatures (e.g. homogenisation and ageing temperatures). From the literature [3,17–19], the Cu–Bi system satisfies the above conditions. Bismuth enriches grain boundaries and causes intergranular embrittlement of Cu-base alloys [1–3]. The Bi segregation is described by the Fowler isotherm [3,17,18]. The main segregation parameters, u0A and uAA , first calculated by the authors of [3] on the basis of their experimental data, are listed in Table 1. These parameters were later determined in [17] as well. The results of [17] agree with those of [3] within 5–8%. A set of the Fowler isotherms is plotted in Fig. 1 for a typical ppm range of Bi bulk concentrations, with the use of the parameters listed in Table 1. Pronounced drooping branches are seen on the c ¼ cðcA Þ curves in a concentration range from 0:1 6 cA 6 0:9, indicating the existence of an extended labile region on the T –cA diagram. As seen from Table 1, the estimated critical temperature is Tk 2000 K. Since homogenization or ageing temperatures for Cu–Bi alloys usually range between 800 and 1300 K, the condition T < Tk is valid for all practical cases. Therefore, the necessary conditions for 2D decomposition are fulfilled under typical experimental conditions in the Cu–Bi system. To estimate the ratio of characteristic times, the thickness of the segregation layer is taken as d 1 nm. As is known from the experiment [3], the equilibrium intergranular concentration of Bi at T > 993 K is rather low. Based on this fact, the
Fig. 1. A set of Fowler isotherms calculated for various quenching temperatures: 1––673 K, 2––773 K, 3––873 K, 4–– 973 K and 5––1123 K. The parameters are c ¼ 2:8 and Tk ¼ 2000 K.
mean Bi concentration is supposed to be c0A 0:1 that corresponds to a lower value of the cðcA Þ curve in a Cu–12 at. ppm Bi alloy at 1123 K (Fig. 1). After quenching to 873 K this concentration does not change but falls into a labile region. Both above-mentioned temperatures and Bi bulk concentration are taken as the typical values used in [19]. Then the enrichment coefficient is approximately KðT Þ 104 . A ratio of the grain boundary and bulk diffusion coefficients is supposed to be fixed for a narrow temperature range of 0.6–0.8 Tmelting . The relationship DS 102 –103 DV is taken as a typical value for diffusion of large metalloid atoms in FCC metals at considered temperatures ð2Þ ð1Þ [20,21]. Assuming ðcA cA Þ 1 and R0S 106
Table 1 Parameters used for calculation of the characteristic segregation and spinodal decomposition times and the main decomposition parameters Parameter
u0A , kJ/mol
uAA , kJ/mol
Tk , K
c
D0V , m2 /s
Q, kJ/mole
Value Source
)58.6 [3]
20.9 [3]
2000 Eq. (2)
2.8 Eq. (3)
7.66 · 105 [31]
178.1 [31]
L.I. Stefanovich et al. / Surface Science 545 (2003) 99–108
m, the ratio of characteristic times is tseqr =tSD 105 106 . In other words, the characteristic time of segregation is much larger than that of spinodal decomposition ðtseqr tSD Þ. Therefore, the spinodal decomposition can be considered as a fast process, occurring against a background of much slower impurity segregation from the bulk to the interface, which obeys the square-root law [8] 2Dc pffiffiffi cA ðsÞ cA ð0Þ þ pffiffiffi s p
ð9Þ
where s is the dimensionless segregation time expressed in t ¼ d2 =DV units, Dc ¼ c0 ðzÞ c0 ð0Þ is the difference in the bulk impurity concentration between the bulk and the nearest subsurface layer at t ¼ 0, and c0A is the mean interfacial enrichment level at t ¼ 0. 2.3. Basic equations for spinodal decomposition at interfaces For the case of bulk (3D) spinodal decomposition, relaxation of the non-equilibrium order parameter (bulk concentration) obeys the conservation law of local composition in the form of a continuity equation [16]. In contrast to 3D decomposition, interfacial spinodal decomposition occurs under continuous replenishment of the segregation layer by impurity atoms from the bulk. Therefore, evolution of the atomic concentration within a segregation layer should follow the matter-balance equation: ocA þ div~ jA ¼ rA ð~ q; tÞ ot
ð10Þ
q; tÞ is the local density of the source of where rA ð~ component A at the point ~ q at time t. On the supposition that the impurity element A comes to the interface from the bulk on average uniformly, one can note: dcA ðtÞ rA ð~ q; tÞ ¼ dt
ð11Þ
where rA > 0 and rA < 0 for segregation and desegregation, respectively. The flux density of segregated atoms along an interfacial plane is written as [16]:
qÞ ¼ MS r jA ð~
dF dcA
103
ð12Þ
where MS is the kinetics coefficient. It is generally defined as MS ¼ bS cA ð1 cA Þ
ð13Þ
where bS is the surface mobility of segregated atoms. By analogy with the bulk [16], the free energy function, F , for an impurity subsystem in the segregation layer, is written as Z 2 F ¼ ½f ðcA Þ þ kðrcA Þ nS dS ð14Þ where nS is the number of impurity atom sites per unit area at the interface and f ðcA Þ is the specific free energy of impurities distributed uniformly over the segregation layer. The second term in Eq. (14) describes the contribution of spatial correlative factors to the free energy. In the area of labile compositions (5), the expression for the specific bulk free energy obtained in [16] with an additional term linear in cA , which is caused by a difference between the surface and bulk energies of an impurity atom, can be used to express f ðcA Þ: 2 f ðcA Þ ¼ f0 Tk ðcA cA Þ þ ðcA cA Þ
2 4 ðcA cA Þ ð15Þ 3 where f0 is an addition to the specific free energy, ð1Þ ð2Þ independent on cA and T , and cA ¼ ðcA þ cA Þ=2. For simplicity, the temperature is expressed in units of energy. Since only the initial segregation stage, (the socalled root stage) is considered further, the value of the kinetics coefficient MS ðcA Þ from (13) can be approximated by a constant which is expressed as MS ðcA Þ ¼ bS cA ð1 cA Þ bSc0A ð1 c0A Þ bSc0A
ð16Þ
As already mentioned, the concentration of impurities within a segregation layer is a random function of the 2D coordinate ~ q along the surface. It is reasonable to introduce two new notations, namely cA ðtÞ and vA ð~ q; tÞ, which are the mean
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concentration and the deviation from the mean value, respectively, and are connected as q; tÞ cA ð~ q; tÞ cA ðtÞ vA ð~
ð17Þ
The value of cA ðtÞ is a known function defined in (9). Using Eqs. (10) and (12) and taking into account expressions (11) and (14)–(17), the following non-linear differential equation, expressed in a dimensionless form, can be written to describe the evolution of the concentration fluctuation vA ð~ q; tÞ with time. ov ð1Þ ð2Þ 0 ¼ 8cA D ðcA ðsÞ cA ÞðcA ðsÞ cA Þv ot
1 3 1 2 þ ðcA ðsÞ cA Þv þ v D2 v ð18Þ 3 2 where D is the Laplacian. The spatial and time coordinates were reduced to their dimensionless forms ffiffiffiffiffiffiffiffiffiffiffiffiby normalizingl2 the coordinates by l ¼ p k=ju0A j and by t ¼ b ju 0 j ¼ l2 =D0 , respectively. S
A
If the bulk concentrations of both components in a binary system are close to each other, phase decomposition often leads to the formation of lamellar or quasi-one-dimensional structures [22]. If the concentrations are essentially different, then the surface distribution of elements becomes statistically isotropic and, thus, the decomposition problem becomes effectively one-dimensional. Drawing an analogy between the film growth on a free surface and the formation of segregation at the interface, the mean impurity concentration can be considered as a 1D function of the coordinate, for example, of x: cA ð~ q; tÞ cA ðx; tÞ
ð19Þ
Then Eq. (18) can be transformed to 1D analogue of the Cahn–Hilliard diffusion equation [23] in dimensionless coordinates s and n: 2 ov ð1Þ ð2Þ o v ¼ 8c0A ðcA ðsÞ cA ÞðcA ðsÞ cA Þ 2 os on
2 2 2 3 1ov 1 o4 v o v þ ðcA ðsÞ cA Þ 2 þ ð20Þ 3 on2 4 on4 on Eq. (20) is similar to that describing evolution of the concentration fluctuation in the bulk [16] with only one difference. Here, the mean enrichment value cA is the known time-dependent function (9)
rather than a constant, as in [16]. The latter term in Eq. (20) is determined by the presence of a gradient-energy term in the free-energy function (14). This gradient-energy term starts to play an essential role during the later stages of spinodal decomposition only. Its potential contribution will be estimated later. For the early decomposition stages, Eq. (20) can be truncated to the following form: 2 ov ð1Þ ð2Þ o v 0 ¼ 8cA ðcA ðsÞ cA ÞðcA ðsÞ cA Þ 2 os on
2 2 2 3 ov 1ov þ ðcA ðsÞ cA Þ 2 þ 3 on2 on
ð21Þ
where the time dependence of the mean surface enrichment obeys Eq. (9). The enrichment fluctuation in the as-quenched state, i.e. at t ¼ 0, is taken as an initial condition: vA ðn; sÞjs¼0 ¼ vA ðn; 0Þ
ð22Þ
Since vðn; 0Þ is a random function of the coordinate within a segregation plane, the determinate equation with random initial conditions can therefore be solved using probabilistic terms. In particular, it is necessary to find the correlation characteristics of the random function v. Eq. (21) allows the writing of a set of engaging equations for the correlators of various orders. A statistical method developed in [16] can be used to do this under the assumption that the segregation layer is statistically uniform with respect to the impurity distribution. As a result, Eq. (21) can be transformed to the following set of equations: 8 dr 2aðsÞ 2 > > < ds ¼ r þ r2 þ 2bðsÞg ð23Þ > dg 3aðsÞ 6 12bðsÞ > : ¼ þ 3 g ds r2 r r4 where rðsÞ ¼ RðsÞ=R0S is the mean dimensionless spatial period of the structure of the impurity atoms at the segregation layer at t ¼ 0. A normalized correlation function of the third order, gðsÞ, is introduced here to characterize the possible distribution asymmetry of the random function cA ð~ q; tÞ. The newly introduced function is written as
L.I. Stefanovich et al. / Surface Science 545 (2003) 99–108 3=2
gðsÞ hv3 ðsÞi=K0
ð24Þ
where K0 ¼ hv2 ð0Þi is the correlation function of the second order (dispersion distribution) at initial time ðs ¼ 0Þ. Eqs. (23) can be reduced to their dimensionless forms by the following substitutions: s ¼ t=t0 ;
t0 ¼ R20S =ð8c0S DS K0 Þ
ð25Þ
3. Discussion 3.1. Analysis of equations Parameters a and b from Eq. (26) can be rewritten as functions of real time t: pffi ð28Þ aðtÞ ¼ b2 ðtÞ d 2 ðT Þ; bðtÞ ¼ k þ kðT Þ t; where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 T =Tk pffiffiffiffiffi dðT Þ ¼ ; K0 c0 hðT Þ kðT Þ ¼ Apffiffiffiffiffi K0 0:5
Initial conditions for Eqs. (23) are rð0Þ ¼ 1 and gð0Þ ¼ g0 . The outward appearance of Eqs. (23) looks generally similar to those obtained in [16] for the case of bulk spinodal decomposition. However, the present equations have time-dependent coefficients aðsÞ and bðsÞ that essentially complicates their analytical solution. The evolution of the above coefficients with time is rather slow as compared with the interfacial spinodal decomposition because it is controlled by the inflow of the segregating atoms from the bulk. Using dimensionless forms, the coefficients aðsÞ and bðsÞ can be written as pffiffiffi pffiffiffi ð1Þ ð2Þ ½c0 ð1 þ h sÞ cA ½c0A ð1 þ h sÞ cA
aðsÞ ¼ A K0 pffiffiffi c0 ð1 þ h sÞ cA pffiffiffiffiffi bðsÞ ¼ A ð26Þ K0 where parameter h is given by rffiffiffiffiffiffi 2DcR0S DV h ¼ pffiffiffiffiffiffiffiffi 0 pK0 dcA DS
ð27Þ
Evidently, the solution of Eqs. (23) gives a complete description of the decomposition processes at the interfaces under conditions of continuous inflow of the segregating atoms. However, it is a very difficult task to solve these equations analytically. As was estimated by (8), the segregation processes are much slower than the decomposition ones. This allowed the authors to use adiabatic approximation in the next section and analyse evolution of quasi-steady structures, which form at the interfaces as a result of spinodal decomposition at some critical points, instead of the exact solution of Eqs. (23).
105
c0 c k ¼ Apffiffiffiffiffi A ; K0
Let us suppose that impurities have already enriched the interface before quenching and the mean interfacial concentration at t ¼ 0, cA ð0Þ, is within the spinodal area, satisfying condition (5). Evidently, the parameters a and b are then negative, at least at the initial stage of the quenching. Therefore, the considered Eqs. (23) have two critical points, such as the saddle and stable junctions. In contrast to the bulk spinodal decomposition [16], the coordinates of such points are not fixed at a phase plane and they change with time in accordance with the following equations: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4b2 ðtÞ 3aðtÞ þ ð4b2 ðtÞ 3aðtÞÞ 8a2 ðtÞ r1 ðtÞ ¼ 2a2 ðtÞ ð29aÞ g1 ðtÞ ¼
r2 ðtÞ ¼
aðtÞr1 ðtÞ þ 1 bðtÞr12 ðtÞ
4b2 ðtÞ 3aðtÞ
ð29bÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð4b2 ðtÞ 3aðtÞÞ 8a2 ðtÞ 2a2 ðtÞ ð30aÞ
g2 ðtÞ ¼
aðtÞr2 ðtÞ þ 1 bðtÞr22 ðtÞ
ð30bÞ
The stable junction point represented by the r2 ðtÞ and g2 ðtÞ functions is responsible for the formation and evolution of quasi-steady non-uniform impurity structures at the interfaces. The first function determines the time dependence of the mean period of segregation non-uniformity, appearing as a
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result of 2D spinodal decomposition, and the second describes evolution of the asymmetry of the distribution function. To clarify the physical meaning of the results obtained, let us consider a single-point function of the impurity distribution in the segregation layer, constructed as a combination of two sharp d-like maxima instead of the functions r2 ðtÞ and g2 ðtÞ: pðvÞ ¼ pe dðvA ve Þ þ pd dðvA vd Þ
ð31Þ
where ve and vd are the deviations from the mean concentrations in impurity-enriched and depleted regions of the surface, respectively, and pe and pd are the areas (coverages) of the enriched and depleted regions. Evidently, the distribution function (31) satisfies the following normalization condition: Z 1 pðvÞ dv ¼ 1 ð32Þ 1
Using correlations of the hv2 i and hv3 i correlators with the distribution parameters described in (31), on the one hand, and functional expressions of the above correlators through the r2 ðtÞ and g2 ðtÞ functions, on the other hand, the values of pe ðr2 ; g2 Þ and pd ðr2 ; g2 Þ are written as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 1 g22 r23 1þ ; pe ¼ 2 4 þ g22 r23 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 1 g22 r23 1 pd ¼ ð33Þ 2 4 þ g22 r23 Since the r2 ðtÞ and g2 ðtÞ functions are known from Eq. (30), then the time evolution of the coverages of both enriched and depleted surface parts can be calculated by (33). Based on the known function r2 ðtÞ, the density of interfacial pre-precipitates can be calculated as a function of time by the following equation: nðtÞ ¼ ðR0S r2 ðtÞÞ1
ð34Þ
Furthermore, the time dependencies of the mean size of the enriched pre-precipitates (or enriched islands), Re ðtÞ, and depleted islands, Rd ðtÞ, can be calculated using Eqs. (30) and (33): Re ðtÞ ¼ R0S pe ðtÞr2 ðtÞ;
Rd ðtÞ ¼ R0S pd ðtÞr2 ðtÞ
ð35Þ
3.2. Numerical estimations for Cu–Bi system There are many reported experimental observations of concentration inhomogeneities along a segregation surface. However, typical reasons for such behaviour are • segregation anisotropy, often enhanced by impurity interaction [24], which results in variation of the impurity concentration between different grains of polycrystalline materials (see e.g. [25] for general information and [18] for a particular case of Cu–Bi alloys), • preferential enrichment of local defects on the surface, such as dislocation lines [26], deformation twins [27], steps [28], etc., and • experimental artefacts, such as e.g. composition variations caused by zigzag crack propagation along the in-situ fractured grain boundary in Auger electron spectroscopy experiments [27, 29]. Indications of inhomogeneities in Bi segregation along some grain boundaries were already reported in transmission electron microscopy (TEM) experiments [19,30]. However, the origin of such Bi-rich regions is not understood yet. As was mentioned before, all necessary conditions for 2D spinodal decomposition at grain boundaries in Cu–Bi alloys are fulfilled for a ppm range of Bi bulk concentrations and a wide temperature range. The main parameters for possible non-uniform segregation caused by 2D spinodal decomposition can be estimated for a Cu–Bi system on the basis of the formulae derived above. For distinctness, we consider the Bi bulk concentration c ¼ 12 at. ppm that corresponds to the bulk composition of one of two alloys studied by TEM in [19]. The supposed heat treatment route generally corresponds to that used in [19]. The alloy is first homogenized at T0 ¼ 1123 K. The equilibrium intergranular Bi segregation at 1123 K is c0A ¼ 0:1, as follows from the Fowler isotherm (Fig. 1). Then the alloy is quenched to T ¼ 873 K and annealed at this temperature for a few hours. All other parameters necessary for the calculations, such as segregation and impurity interaction energies, and
L.I. Stefanovich et al. / Surface Science 545 (2003) 99–108
Fig. 2. Mean size of Bi-rich islands, Ri ðtÞ, in a segregation layer as a function of annealing time at 873 K calculated by (35).
bulk diffusion data for Bi in Cu alloys, are taken from the literature and are listed in Table 1. Fig. 2 shows the calculated characteristic mean size of the Bi-rich islands, Ri ðtÞ as a function of annealing time at 873 K. The mean island size shows a non-monotonic time dependence, changing from Re max 34 nm to Re min 20 nm. The estimated values of Bi islands generally correlate with the dimensions of Bi inhomogeneities experimentally observed at grain boundaries in a Cu–12 at. ppm Bi alloy, which vary typically from 15 to 40 nm [19]. While the origin of the observed Bi regions is not clear and could perhaps be caused by other than 2D decomposition reasons (e.g. by artefacts of sample preparation as discussed in [19]), the proposed mechanism of spinodal decomposition can, in principle, be realised under the considered experimental conditions. Undoubtedly, more precise experiments on a nanometre scale are necessary to generate conclusive experimental evidences for 2D impurity decomposition. In accordance with theoretical predictions (Fig. 2), a minimum size of Bi-rich pre-precipitates is reached at a border of the labile area (5), when ð2Þ cA ðtÞ ¼ cA , after annealing for tc 7 103 s at 873 K. Simultaneously the island density, nðtÞ, reaches a maximum and amounts to nmax 4 107 cm1 (Fig. 3). The Bi-rich island density reaches its minimum value nmin 2 107 cm1 (Fig. 3) when the mean surface enrichment cA ðtÞ reaches the centre of the
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Fig. 3. Density of Bi pre-precipitates, nðtÞ, as a function of annealing time at 873 K calculated by (34).
labile region, i.e. at cA ¼ cA ¼ 0:5 (Fig. 1), and the coverages of the Bi-enriched and the Bi-depleted islands become equal, pdmax ¼ pemin ¼ 0:5 (Fig. 4). When cA ðtÞ approaches the right border of the labile region (5), the coverage of Bi-rich islands, pe , increases with time (Fig. 4). Since the subsystem of impurities gets the isothermal conditions (T ¼ const) after the quenching and since a half-infinite medium is considered, then the bulk impurity concentration does not practically change during segregation ðc constÞ and the mean impurity enrichment cA ðtÞ is the only parameter, controlling 2D impurity decomposition. The value of cA ðtÞ, evolving during segregation, crosses the area under the spinodal curve
Fig. 4. Surface coverage of Bi pre-precipitates, pe ðtÞ, as a function of annealing time at 873 K calculated by (33).
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from a quenching point to a point at the right border of the labile region. The thermodynamic force, driving 2D spinodal decomposition, changes non-monotonically. As a result, the main decomposition parameters, such as RðtÞ and nðtÞ also evolve non-monotonically, as seen from Figs. 2 and 3. 4. Conclusions 1. In contrast to bulk spinodal decomposition, when the order parameter (average bulk concentration) remains constant, the proposed mechanism of interfacial spinodal decomposition is controlled by the order parameter (average surface concentration) cA ðtÞ which changes with time and is a function of segregation rate. 2. Even under isothermal conditions, the density and mean size of the impurity-rich islands are non-monotonic functions of time. 3. The impurity-rich island density reaches a minimum, when half of the surface sites are occupied with impurity atoms ðcA ¼ cA ¼ 0:5Þ, and ð2Þ attains its maximum value at cA ¼ cA . Acknowledgements This work was supported in part by Award #UE2-2072 of the U.S. Civilian Research & Development Foundation for the Independent States of the Former Soviet Union (CRDF). One of the authors, DBW wishes to acknowledge the support of the National Science Foundation through grant DMR 99-72670. References [1] A. Joshi, D.F. Stein, J. Inst. Metals 99 (1971) 178. [2] S.F. Baumann, D.B. Williams, J. Microsc. 123 (1984) 299. [3] M. Menyhard, B. Blum, C.J. Mahon Jr., Acta Metall. 37 (1989) 549.
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