Radiation-induced precipitation at the alloy surface during ion bombardment

Nuclear Instruments and Methods in Physics Research B71 (1992) 148-154 North-Holland

Beam Interactions with Materials&Atoms

Radiation-induced precipitation at the alloy surface during ion bombardment A.M. Yacout and N .Q . Lam

Argonne National Laboraton-, Argonne, IL 60439, USA

J .F. Stabbins

Unirersit.v of Illinols, Urban«, IL 61801, USA Received 21 February 1992 and in revised form 16 March 1992 A kinetic approach to model radiation-induced formation of a precipitate layer on the surface of a binary alloy during ion bombardment is described. Sample calculations were performed for the case of Ni ;Si coating on a Ni-Si alloy surface . The strong coupling between Si atoms and radiation-generated defect fluxes causes a significant Si enrichment at the surface, which gives rise to the formation of a precipitate layer when it exceeds the Si solubility limit . The stability of this layer depends on the competition between the rates of precipitation and sputtering . Both the receding surface and the moving precipitate /matrix interface were accounted for by means of a mathematical scheme of boundary immobilization . The dependences of the precipitation kinetics and the development of solute concentration profiles in the alloy matrix on bombardment temperature, ion flux and alloy composition were examined. 1. Introduction Radiation-induced segregation of alloying elements toward extended defect sinks such as dislocations, voids, grain boundaries and external surfaces is commonly observed in alloys during irradiation at elevated temperatures [1,2] . In the particular case of Ni-Si alloys, it is well established that preferential coupling between Si-atoms and radiation-generated interstitial fluxes gives rise to significant enrichment of Si at defect sinks. Whenever the local Si enrichment surpasses the solubility limit, precipitation of the ordered N' 3S' (y') phase occurs. As additional Si segregates from the bulk toward the sink, this new phase grows. N',S' coatings on dislocation loops and external surfaces have been observed [3,4]. The growth of these coatings on the surface of Ni-12 .7 at .% Si alloys during I-McV proton irradiation at various temperatures was measured by Rehn et al . [5] . The kinetics of precipitate growth under uniform irradiation conditions was theoretically modeled by Lam et al. [6] . A comparison of model calculations with the experimental measurements has * Work supported by the US Department of Energy, BESMaterials Sciences, under Contract W-31-109-Eng-38, and the National Science Foundation, under Contract ECS85157722. Correspondence to. Dr. N .Q. Lam, Argonne National Laboratory, MSD-212, Argonne IL 60439, USA . 0168-583X/92/$05.00

n

provided useful information about the defect properties and the defect production efficiency in Ni-Si alloys. In the present paper, our theoretical approach [6] was extended to model the kinetics of radiation-induced precipitation at the alloy surface during ion bombardment, taking into account the effects of sputtering and spatially-nonuniform damage rate. Nonequilibrium solute segregation leading to phase instability in the near-surface region was modeled, based on a kinetic theory of radiation-induced segregation in concentrated alloys [7]. Both the receding surface (owing to sputtering) and the moving precipitate /matrix interface (due to the growth of the precipitate layer) were accommodated by a mathematical scheme that transforms the spatial coordinates into a reference frame in which the boundaries are immobile . Using this model, the effects of ion flux, target temperature and alloy composition on the evolution of the precipitate layer examined systematically. 2 . Model for radiation-induced surface precipitation during ion bombardment 2.1 . Radiation-induced segregation

When a material is exposed to irradiation by energetic particles, its microstructure experiences a quick

1992 - Elsevier Science Publishers B.V. All rights reserved

A . Yacout et ai. / Radiation-induced precipitation

buildup of nonequilibrium point defects. At sufficiently high temperatures, point defects become mobile, and tend to annihilate by mutual recombination and/or diffusion to extended sinks . Since the motion of defects is associated with the motion of atoms, atom fluxes are always coupled to point-defect fluxes . For a multicomponent alloy, preferential association of one alloy component with the defect fluxes leads to the transport of this component in an amount that is not proportional to its initial concentration in the alloy, producing local changes in the alloy composition. These changes can lead to the formation of new phases within the alloy because the local alloy composition may be shifted into a different part of the phase diagram. The radiation-induced segregation phenomenon in concentrated binary alloys A-B was previously modeled by Wiedersich et al . [7]. The model describes the local changes in the point-defect and solute concentrations with time using the following set of equations : ac, =- V-J +F_ (1) at

ac,

at

=

-V . J. + F,

(2)

and

aCA =- V . JA +FA , (3) at the concentrations of vacanwhere C, C; and CA are cies, interstitials and solute A-atoms, respectively, and Jv, J, and JA are the corresponding currents . P F, and FA are source/sink terms. The species currents are given by (4) J = (dA - d a )C«VCA - DvVC  , J,= - (dA;-dn;)C;aVCA -D,VC;,

(5)

and JA = - DAaVCA - CA(dAYC, -dAvVC,) ,

(b) where a is a thermodynamic factor which is related to the activity coefficients of the alloy components, dAi, dß;, dA and de, are the diffusivity coefficients of Aand B-atoms via interstitials and vacancies, respectively, and are defined as ~ z dkl -_ fiÀ 1Z1Vki,

(7)

where k = A or B, j = i or v, A, and Z, are the elementary jump distance and the coordination number of defect j, respectively, and vki is the effective exchange-jump frequency of a k -j pair. The total local diffusion coefficients of defect j and atom k are respectively given by D, = F_dkjCk

149

and Dk = Y_dk,C, 1

Detailed description of these equations and the different coefficients involved are given in ref. [7]. The kinetic equations (1)-(3) can be written in a general form : ack r 1 at

=-V-{ F_G,VC, } +Fk, 1

(8)

with j and k = v, i, A. Here, G's are functions of the partial and total diffusion coefficients as well as the species concentrations. 2.2. Radiation-induced precipitation and sputtering

If radiation-induced segregation gives rise to enrichment of solute A-atoms at the surface, a precipitate layer forms there whenever the solute concentration exceeds the solubility limit . This precipitate film grows with irradiation time as additional A-atoms are preferentially transported toward the surface. Concurrently, the precipitate layer is eroded by ion sputtering. As a result, both the bombarded surface and the precipitate/matrix interface recede into the alloy interior during irradiation. The sputtered surface recedes at a velocity (sputteringrate) S s , defined as = 00J SACÁrec + SBCBrecl ,

bti

where 0 is the ion flux (ions/cm' s), JZ is the average atomic volume, SA and SB are the sputtering coefficients of the alloy components A and B, respectively, and CÁ"` and CBrcc are the concentrations of these components in the precipitate layer. It is assumed that the sputtering coefficients are the same for both components (SA = SB) and all sputtered atoms originate from the first atom layer. The former assumption may lead to a change in the stoichiometry of the precipitate surface, because CAp,ec and Cpprec are not necessarily equal to each other. However, for the sake of simplicity, the precipitate composition is assumed to remain uniform during bombardment. The precipitate /matrix interface, on the other hand, moves deeper into the alloy with a velocity S p. To calculate this velocity, we consider the conservation of mass during bombardment. If after an irradiation time t, the sputtered surface and the precipitate/matrix interface are located at S. and Sp, respectively, (fig . 1) then the conservation of the solute A-atoms is given by CAL=

0

ODSACAedt+(SP-8,)CAn+

f t-CA(x, t) dx, ,5n

(10)

A . Yacout et al. / Radiation-induced precipitation

150 SURFACE

C1

At the surface of the alloy (x = 8,), and far in the bulk (x = L), the point-defect concentrations are maintained at their thermal equilibrium values, i.e .,

INTERFACE

C yr~e

C (S, t)=C,,(L, t)=C~"

A

C i (8, t)=C,(L, i)=C," .

(16)

The solute concentration at x = L is unperturbed by irradiation, i .e .,

$oam C A

CA (L, t) =C'A' .

2 Fig. l . Schematic description of the solute concentration profile from the bombarded surface .

where CA" ils the initial concentration of A-atoms in the alloy and L is the initial thickness of the sample. In order to simulate a semi-infinite target, L is chosen to be sufficiently large, so that all concentration gradients at L are zero . The right-hand side of eq. (10) contains three terms : the first represents the quantity of solute atoms lost by sputtering, the second is the amount of solute in the precipitate layer of thickness (S P - S,), and the third is the solute amount remaining in the bulk, below the interface. Differentiating eq. (10) with respect to time t and rearranging the terms, we obtain the precipitation rate (or interface velocity) 1 __ Pree SP CAprec - CA-j- (CA (SS -O d1SA)

- ni,(x=a') },

(15)

and

SP

(11)

where CA"" is the solubility limit of A-atoms at the irradiation temperature .

(17)

The solute concentration at the surface, however, depends on irradiation time . Before the onset of precipitation, the concentration of A-atoms at the surface varies with time, because of the combined effects of radiation-induced segregation and sputtering. Whenever the surface concentration of A atoms exceeds the solubility limit, CÄ"ß, precipitation of a new phase occurs . At this stage, the concentration of A atoms within the surface precipitate layer becomes Cppr- , while the concentration at the precipitate/matrix interface is maintained at CÄ,"m. Thus, for S P > 0, CÁree (18) CA(S ., t) = and

C .SA"hm (19) I) = where Sp denotes the matrix side of the interface . For simplicity, it is assumed that the precipitate /matrix interface is not a sink for point defects. This assumption can be too severe, because in most cases the interface is not perfectly coherent. The effect of the interface nature (i .e ., coherent versus incoherent) on the precipitation kinetics is indeed a subject for further investigation . CA S

3. Numerical procedure 3.1. Immobilization of the moving surface and precipitate /matrix interface

C,,(x,0)=C, "= exp(Sf/k)exp ( -Hc/kT),

(12)

Before the method of lines [8] is applied, the receding surface and the moving precipitate /matrix interface have to be immobilized by means of a domain transformation . Before the occurrence of precipitation (i .e., when SP = tl), the immobilization of the moving planar surface can be accomplished by the introduction of a reduced space z [9,10] given by

C(x,0)=C "= exp ( S,/k)exp ( -H,r/kT),

(13)

z= I-

(14)

where /3 is a scaling factor. In this domain transformation, the physical space S, -< x < w is mapped into a fixed region O :g z -< 1 . Eq . (20) is used when the solute concentration at the surface is still below the solubility limit .

2.3. Initial and boundary conditions Initially, at I = 0, the alloy is maintained at thermal equilibrium ; the concentrations of point defects and solute atoms in the alloy are given by

and CA(x, 0) = CA'

Here, S ft,.;, and H,'., arc the effective defect formation entropies and enthalpies, respectively.

ell(-10,

(20)

A. Yacout et at. / Radiation-induced precipitation Whenever precipitation occurs at the surface, the moving precipitate/matrix interface must also be accounted for . Both the receding surface and interface can be immobilized in the mathematical formalism by the following transformation : for S, _< x < Sp,

z=

2 _ e-f,-s  i

(21)

for S P
With this transformation, the sputtered precipitate layer (S, < x <_ S r,) and the remaining matrix (S p < x < L) can be mapped into a fixed region 0 < z _< 2 with the interface located at z = 1 (see fig . 1). 3.2. Kinetic equations in reduced space In the reduced domain z = f(x, S), defined by eqs. (20) and (21), the kinetic equation (8) can be re-written as [6,9-11]: í1C

1 1C 8 + í1z dtR - ~ W A Oz~ 1

, aC 8zß

K'

+ fA

,

(22)

with C(z, t)=C(x, t),

(23)

f(C, z, t)=F(C,x, l),

(24)

W*~

dz

K~

(25)

- (8x)dz(dx)G~, -(

0

az

GR 1x) az ( 1x)

Oz ílt '

(26)

and W

, R- (8x) Gt

(27)

where k = v, i, or A, j = v, i, and A, and W's and K's must be appropriately defined for different stages of irradiation, due to different space-transformation formalisms [eqs . (20) and (21)] . Before precipitation (S P = 0): W R =ß 2 (1 -z)GJ +ß(1 - z)S}ó Aj

(28)

and =ß2(1 _z)2Gi , (29) where S R , is the Kronecker delta function . On the other hand, after the onset of precipitation (S p > 0): K,

Wi =~

1 (SP_SS) [S'+z(SP-S,)]Sk, 13 2 (2-z)G, +PS p(2-z)S R,

for0_
and G'

for0<_z<1,

(2-z) 2 G,

for 1
~SP

02

-

S,)

2

(31)

The system of eqs . (22) can be solved numerically [9-11] with the aid of the LSODE package of subroutines [8]. 4 . Application to Ni-Si alloys The formation of a y'-Ni 3 Si film on the irradiated surface of Ni-Si alloys is a typical example of radia;ion-induced precipitation [3,4] . In this case, the Si enrichment at the surface, brought about by a strong coupling between Si atoms and the interstitial flux toward defect sinks, exceeds the Si solubility limit, resulting in the precipitation of the new phase . The growth of the precipitate layer during high-energy proton irradiation (i .e . no sputtering) was already investigated theoretically by Lam et al . [6]. Here, we examine the effects of sputtering and spatially-dependent damage rate on the evolution of the precipitate film and on the development of the solute concentration profile in the alloy matrix during 100-keV Ne + ion irradiation . The basic physical parameters used in the previous calculations [6] were also employed in the present work. Since the sputtering coefficient S N , for 100-keV Ne + on Ni is about 1 atom/ion [12], and since SN; and Ss, calculated for a Ni-12 at .% Si alloy using the TRIM code [13] were only slightly larger than 1, a value of 1 atom/ion was taken for both elements . The peak damage occurs at 43 nm and the total damage range is - 154 nm . A maximum damage rate of 5.7 X 10 - ' dpa/s was calculated for a flux of 6.25 X 1012 ions/cm 2 s. Various ion fluxes between 6 .25 X 10"° and 6 .25 X 10'° ions/cm 2 s and temperatures between 200 and 800°C were used in the calculations. The temperature dependence of the Si solubility limit in Ni is shown in table 1 . Table 1 Solubility limit of Si in Ni-Si solution Temperature (°C) 2011 300 400 500 600 700 800 9110

Solubility limit (at .%) 4 .4 6.0 7.6 9.0 10.2 11 .3 12.0 12.7

152

A . Yacout et al. / Radiation-induced p,-,,,ipit .ti.,, 0.30 F

030 r 025

O

020

-

0.15

-

0 10 3

O 10~ O 10 5

ó 0.10 E

025x x IO s O 10 2

0.20

-

015

-

0 10

x

10

0

W

0

ó t 0 05 E0

x

0.05

O

200

400 600 DISTANCE ("m)

800

20

30

40

50

60

1000

Fig . 2 . Si concentration in a Ni-6 at .% Si alloy calculated as a function of depth from the surface (a) and from the precipi tate/matrix interface (b) for various irradiation times at 500°C Ion flux 4!1 = 6 .25 x 10 12 ions/cm 2 s.

4.1 . Effect of initial solute concentration The time evolution of Si concentration profiles of Ni-6 at % Si and Ni-12 at.% Si alloys during Ne + bombardment with a flux of 6.25 x 1012 ions/cm 2 s at 500°C is shown in figs. 2 and 3, respectively. The vertical lines indicate the positions of the precipitate/matrix interface. For the case of Ni-6 at .% Si (fig. 2), the initial Si concentration is less than the solubility limit of the alloy at 500°C (Cs; 1 "" =9 at.%) . The concentration of Si at the surface increases rapidly to values larger than Cs'," within the first few seconds of irradiation, and precipitation of a y'-N' 3 S' layer occurs at the surface, at the expense of a Si depletion beneath the precipitate/matrix interface . As irradiation continues, this depletion extends deeper into the alloy. The precipitate thickness increases with time when the transport of Si atoms from the alloy interior by radiation-induced segregation is dominant . In this time regime, a pileup of Si ini the beyond-range region is also observed, due to Si segregation out of the peak-damage zone [141. However, after long irradiation times, sputtering of the

Fig. 3 . Si concentration in a Ni-12 at . /c. Si alloy calculated as a function of depth from the surface (a) and from the precipi tate/matrix interface (b) for various irradiation times at 500°C . oh = 6.25 x 101 ` ions/cmz s. precipitate layer combined with Si back-diffusion outweighs the effect of radiation-induced segregation, and the thickness of the layer starts to decrease . The balance between radiation-induced segregation and sputtering determines the thickness of the Ni 3Si film at steady state (see curves for 200, 300 and 400°C in fig. 4).

Fig. 4 . Time evolution of the precipitate layer thickness in a Ni-12 at .%r Si alloy during irradiation at various temperatures . 0 = 6.25 x 10 12 ions/cm= s .

A. Yacout et at. / Radiatiorrinducedprecipitation

In the case of Ni-12 at .% Si irradiation (fig. 3), since the initial concentration is higher than Cs; I '"' at 500°C, the formation of the y' phase instantly takes place at the surface as a result of radiation-induced segregation . The time evolution of the concentration profile below the precipitate/ matrix interface is similar to the previous case; however, for a given dose, the thickness of the precipitate layer is larger, simply because of the larger Si supply. 4.2. Effect of temperature The time evolution of the thickness of the y' precipitate layer at the surface of a Ni-12 at .% Si alloy is shown in fig . 4 for different bombardment temperatures. After a relatively short incubation period, the precipitate starts to grow. The kinetics of precipitate growth largely depends on temperature . At 200°C, radiation-induced segregation is weak due to dominant defect recombination, and the growth of the precipitate layer is slow. In this case, the precipitate thickness increases monotonically to a steady-state value . At higher temperatures, radiation-induced segregation becomes effective, the N' ;Si layer first grows rapidly to a maximum thickness, and then becomes thinner and thinner as a result of sputtering . A steady-state thickness will eventually be achieved after a long irradiation time. The higher the temperature, the faster the initial growth of the layer and the thinner the steady-state thickness. The values of the maximum thicknesses are determined by the competition between radiation-induced segregation and sputtering, i .e., by (8 P In fact, as shown in section 2.2, the sputtering rate S, is constant (because sputtering is assumed to be nonprefcrential and the composition of the precipitate layer does not change with time) and the precipitation rate S P is strongly dependent on the flux of Si atoms across

153

U Z O U

zoo

400 600 DISTANCE (nm)

800

Fig. 6. Si concentration in a Ni-12 at.% Si alloy calculated as a function of depth from the precipitate/ matrix interface after 10 4 s of bombardment with various ion fluxes at 500°C.

the precipitate/ matrix interface, Js ,(x = 8 P ). Initially, Sp increases with time as more and more Si atoms are transported to the interface by radiation-induced segregation . After a certain time, a Si-depleted zone develops below the interface, back-diffusion becomes effective in reducing Js ,(x = S p ), and S P starts to decrease after going through a maximum. Maximum precipitate thickness is achieved at a time when (S P - S,) = 0 . Beyond this time, S p becomes smaller than S,, and the thickness of the precipitate layer begins to decrease. Steady-state is reached when S P increases again to the value of 8,. In some cases, depending on the initial Si concentration and irradiation temperature, S p decreases monotonically from its maximum value to S, and, hence, the Ni ;Si layer grows slowly to a steady-state thickness (see, e .g ., the 200°C curve in fig. 4). Above 600°C, radiation-induced segregation becomes small, due to fast back-diffusion, and the precipitate layer is quickly formed and sputtered away . No precipitation is possible at very high temperatures, above - 900°C. 4.3. Effect of ion flux

Fig. 5. Time evolution of the precipitate layer thickness in a Ni-l2 at.%o Si alloy during irradiation at 500°C with different ion fluxes.

Fig . 5 shows the effect of ion flux on the development of the Ni ;Si precipitate layer on a Ni-12 at .% Si alloy at 500°C. The maximum precipitate thickness achieved during irradiation is nearly the same for a four-orders-of-magnitude change in the ion flux . However, the time period for maximum precipitate thickness is shifted to longer irradiation times. This behavior is expected since the incubation period for precipitation increases with decreasing ion flux. Fig. 6 illustrates the Si concentration profiles obtained after 104 s of bombardment with various ion fluxes . The extent of Si depletion in the region below the interface, which reflects the total amount of Si available for the forma-

154

A. Yacout et al. / Radiation-btdacedprecipitation

tion of the Ni ;Si layer, is significantly smaller for a flux of 6 .25 x 10 1° than for 6.25 x 10 1° Nc +/cm = s . 5. Conclusions Irradiation of binary alloys at elevated temperatures can lead to the enrichment of solute atoms at the surface, as a result of radiation-induced segregation . If the solute concentration exceeds the solubility limit, a new surface phase is formed. A kinetic approach to model the time evolution of this precipitate layer during ion bombardment, when sputtering cannot be neglected, was proposed in the present study . Model calculations were performed for Ni-Si alloys undergoing 100-keV Ne+ bombardment in the temperature range 200-900°C. In general, the precipitate layer grew to a maximum thickness before being eroded away by sputtering. The magnitude and the occurrence time of the maximum thickness, and whether or not a layer of steady-state thickness could be obtained depended on the initial Si concentration, ion flux, and irradiation temperature. For bombardment with a flux of 6 .25 x 10 1- Ne + /cm= s, for example, a steady-state precipitate layer was observed on a Ni-12 at.% Si alloy only at temperatures below - 400°C . The present kinetic approach may be of importance in the area of ion beam-induced compositional modifications oil alloy surfaces at elevated temperatures .

References [l] J .R. Holland, L.K . Mansur and D.I . Potter (eds .), Phase Stability during Irradiation (The Metallurgical Society of AIMS, New York, 1981). [21 F .V. Nolfi, Jr . (ed .), Phase Transformation during Irradiation (Applied Science Publishers, Barking, Essex, 1983). [3] P .R . Okamoto and L.E. Rehn, J . Nucl. Mater. 83 (1979) 2. [4] D .I . Potter, P .R. Okamoto, H. Wiedersich, J.R. Wallace and A .W. McCormick, Acta Met . 27 (1979) 1175. [5] L.E. Rehn, P.R . Okamoto and R .S. Averback, Phys. Rev . B30 (1984) 3073. [6] N .Q . Lam, T . Nguyen, G .K. Leaf and S . Yip, Nucl. Instr, and Meth. B31 (1988) 415 . [71 H . Wiedersich, P.R. Okamoto and N .Q . Lam, J. Nucl . Mater. 83 (1979) 98. [8] A.C. Hindmarsh, in Scientific Computing, eds . R.S . Stepleman, M . Carver, R . Peskin, M .F . Ames and R . Vichnevetsky (North-Holland, Amsterdam, 1983) p. 55 . [91 N .Q . Lam and G.K. Leaf, J . Mater. Res. 1 (1986) 251 . [101 N.Q . Lam, G .K. Leaf and H . Wiedersich, J . Nucl. Mater. 88 (1980) 289. [111 A.M . Yacout, N.Q. Lam and J .F . Stubbins, Nucl. Insir. and Meth . B42 (1989) 49. [12] H.H. Andersen and H .L. Bay, in : Sputtering by Particle Bombardment, vol . I, ed . R . Behrisch (Springer, Heidelberg, 1981) p . 145 . [13] J .P . Biersack and W . Eckstein, Appl. Phys. 34 (1984) 73 . [14] N .Q. Lam, P.R . Okamoto and R .A. Johnson, J . Nucl . Mater. 78 (1978) 408 .