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Compositional patterning in irradiated immiscible alloys
Surface Science 514 (2002) 68–73 www.elsevier.com/locate/susc
Compositional patterning in irradiated immiscible alloys Yoshihisa Enomoto *, Masataka Sawa Department of Environmental Technology, Nagoya Institute of Technology, Gokiso, Nagoya 466-8555, Japan Received 1 October 2001; accepted for publication 23 January 2002
Abstract The spontaneous formation of surface compositional patterning of quenched binary alloys driven by irradiation is numerically studied by means of a phenomenologically modified Cahn–Hilliard type equation for the local concentration field. In the present model the irradiation effect is modeled as a ballistic mixing with an average distance of atomic relocation R and its frequency C. From several two-dimensional computer simulations, it is shown that competitive effects between phase separation and irradiation-induced mixing might provide a novel way to stabilize and tune the steady-state nanostructures on the surface of phase separating materials under some irradiation conditions (i.e., in some region on (R,C) space). Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Alloys; Surface structure, morphology, roughness, and topography; Computer simulations; Surface thermodynamics (including phase transitions)
1. Introduction Whenever a system is driven from thermodynamic equilibrium by irradiation with energetic particles (e.g., ions), a variety of different phenomena, such as collision cascades, cascade damage, and radiation-enhanced diffusion, occur depending sensitively on the actual experimental condition [1]. For metals and miscible alloy systems, experimental and theoretical study on irradiation effects have been done extensively. For instance, spontaneous formation of metallic nanostructures under irradiation has been recently observed in Au/SiO2 and Pt/SiO2 [2]. On the other hand, immiscible
*
Corresponding author. Tel.: +81-52-7355153; fax: +81-527355158. E-mail address: [email protected] (Y. Enomoto).
alloys under irradiation have, however, been less studied, since more complicated situations can be encountered through the interplay between irradiation and the internal kinetics in these systems [1]. In particular, interesting but less studied case is ion-beam mixing of immiscible quenched alloys, where irradiation-induced ballistic mixing acts in opposition to thermodynamically driven kinetics like a spinodal decomposition. Each time an external particle collides with the solid, a local atomic rearrangement is produced. This rearrangement has a ballistic component that mixes the atoms regardless of their chemical identity, leading to bring the system to a random solid solution. During this ballistic mixing, a number of exchanges of atomic positions occur with an average relocation distance R and the frequency of these forced exchanges C, whose typical values are R 10 nm
0039-6028/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 9 - 6 0 2 8 ( 0 2 ) 0 1 6 0 9 - 6
Y. Enomoto, M. Sawa / Surface Science 514 (2002) 68–73
and C 104 s1 [1]. To explore such competitive effects, the Cahn–Hilliard type dynamical equation for the local concentration field [3] is known to provide a powerful tool, which has proposed by Enrique and Bellon [4]. As far, their continuum model has been used only to discuss the linear stability analysis for the limiting case of spinodal decomposition from the unstable state under ballistic mixing. In this work, we report on the long-time behavior of surface compositional patterns and their structure factor obtained by two-dimensional computer simulations of Enrique–Bellon model. We will study the following effects on the morphology: (1) the average exchange range R, (2) the exchange frequency C, and (3) the initial homogeneous concentration Si of the alloy. Despite the importance of controlling the surface compositional structures, these have been less studied previously, except for the mean field treatment and kinetic Monte Carlo simulation only for Si ¼ 0.
2. The model We study the spontaneous formation of surface compositional patterns in a quenched binary alloy with a positive heat of ballistic mixing under irradiation. The equation governing the time evolution of the system is [4] o Sðr; tÞ ot
h i ¼ Mr2 2ASðr; tÞ þ 4BðSðr; tÞÞ3 2Cr2 Sðr; tÞ C½Sðr; tÞ hSiR ðr; tÞ ;
ð1Þ
where Sðr; tÞ is a local composition difference of the two constituents of the alloy at time t and position r on the material surface. Here, we have chosen the system free energy density as AS 2 þ 2 BS 4 þ CðrSÞ with positive coefficients A, B, and C, which are assumed to be constant as well as a mobility M in Eq. (1). Estimating these parameters would require microscopic information about a specific system. Since we would understand generic properties of irradiated materials in the most general manner possible, we do not discuss them any more. The first term in the above equation repre-
69
sents the usual phase separation dynamics given by Cahn and Hilliard [3], and the second term describes the forced mixing term due to irradiation with the frequency of atomic relocation C. In the second term, hSiR represents a finite average of the concentration profile around r, given by Z Z 0 0 0 hSiR ðr; tÞ Sðr ; tÞ wR ðr rÞ dr = wR ðr0 rÞ dr0 ; ð2Þ
where the weight function wR ðxÞ expðjxj=RÞ denotes the distribution of atomic relocation distance with the average exchange distance R. In the model, for the sake of simplicity, the concentration of irradiation-induced point defects is assumed to have reached a steady value, and also the contribution of interstitials, as well as defect sinks and clusters [5], is neglected. Moreover, non-ballistic mixing effects and also any coupling between the degree of order and the concentration field [6] are not considered. Under these assumptions, the above model excludes phenomena related to the order–disorder transition, and macroscopic solute migration such as irradiation-induced heterogeneous precipitation. We also focus on the surface compositional patterning on the irradiated materials, neglecting the bulk diffusion and irradiationinduced phenomena occurring in the interior of the sample. These situations can be easily realized in thin film experiments with not too energetic cascades and some temperature region depending on the irradiated material properties [1] (e.g., 1 MeV Kr ion irradiation in Ag–Cu thin films at temperature ranging from 80 to 473 K [7]). Now, to discuss the early stage dynamics and the long-time steady-state behavior, we apply the linear stability analysis and the variational analysis to the model. Similar analyses have been done for the case Si ¼ 0 [4]. Linearizing Eq. (1) about Sðr; tÞ ¼ Si in Fourier space, one obtains R the equation for the fluctuations, S~ðk; tÞ dr½Sðr; tÞ Si expðik rÞ, as o ~ S ðk; tÞ ot
¼ 2MðA 6Si2 BÞk 2 2MCk 4 C
R2 k 2 S~ðk; tÞ 1 þ R2 k 2
ð3Þ
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Y. Enomoto, M. Sawa / Surface Science 514 (2002) 68–73
with k ¼ jkj. The linear stability analyses of Eq. (3) have shown that there are two characteristic values of C, denoted by C1 ðR; Si Þ and C2 ðR; Si Þ. For C > C2 ðR; Si Þ the system is homogeneously mixed (solid solution), while for C < C1 ðR; Si Þ the homogeneous state is linearly unstable to grow, leading to macroscopic phase separation. Moreover, for C1 < C < C2 , certain modes k with k2 < k 2 < kþ2 are linearly unstable, suggesting a wavelength selection at long times (i.e., the steady-state patterning). An interval of the exchange frequency window ½C1 ; C2 decreases with R, reaching a zero value at a critical value Rc , when C1 ðRc ; Si Þ ¼ C2 ðRc ; Si Þ. For ballistic mixing with exchange distance smaller than Rc , the steady-state patterning is not possible. The variational analysis for the system energy ffi results in that for R Rc ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p C=ðA 6Si2 BÞ one has qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C1 ¼ 2R3 M ðA 6Si2 BÞC ; 2
C2 ¼ MðA 6Si2 B þ CR2 Þ =ð2CÞ; k ¼ 1=ð2RÞ; and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 kþ ¼ 0:5R2 c 0:5R : These analytic predictions have been partially confirmed by the kinetic Monte Carlo simulations only for the case Si ¼ 0 [4]. The question is whether the similar results are obtained for the present continuum approach including the non-zero Si case as well.
3. Simulation results We carry out two-dimensional computer simulations of the surface phase separation dynamics under irradiation by changing values of Si , C, and R. The system considered here is a two-dimensional plane, which corresponds to irradiated material surface. In the following simulations, we use the dimensionless variables such as re R=L and p0ffiffiffiffiffiffiffiffiffi g t0 C, by using the units of length L0 ¼ C=A, time t0 ¼ffi C=ð2MA2 Þ, and concentration field S0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi A=ð2BÞ. The resulting dimensionless equation (1) is solved numerically on a 256 256 square lattice with periodic boundary conditions, using a standard finite-difference scheme with mesh size
being 0:5 and time step 0:01 [8]. The initial state of the system consists of a Gaussian random distribution of the concentration with the mean value Si and standard deviation 0.1. First of all, the time evolution of the concentration field and its amplitude along the x-axis at re ¼ 8 are shown in Figs. 1 and 2 for Si ¼ 0 and 0.3, respectively. In these figures, lattice points with Sðr; tÞ < 0 are plotted by dots. For re ¼ 8, one has g1 t0 C1 ¼ 0:0019 and g2 t0 C2 ¼ 0:258 for Si ¼ 0, and g1 ¼ 0:0016 and g2 ¼ 0:139 for Si ¼ 0:3. For g ¼ 0:0005 (g2 ). We should remark that the pattern data in Fig. 1(c) represent extremely small concentration fluctuation, but not obvious structure, originating from the small randomness of the initial configuration. Then, the amplitude data in Fig. 1(c) are almost zero in this scale. The same behavior has been obtained for Si ¼ 0:3 and g ¼ 0:5, but this result is not shown here. For intermediate value of g at long times (here, g ¼ 0:05 and t P 500), we can see the steady-state labyrinthine pattern at Si ¼ 0 in Fig. 1(b) and the steady-state droplet patterns at Si ¼ 0:3 in Fig. 2(b). These results are in agreement with theoretical analyses mentioned in the preceding section. To analyze both the time evolution and longtime behavior of compositional patterns, we examine the structure factor of the system, Iðk; tÞ 2 hjS~ðk; tÞj i, where h i denotes the average over 10 independent simulation runs. In the following discussion, we use the spherically averaged structure factor Iðk; tÞ and its characteristicR wave number hki, which are defined by Iðk; tÞ ¼ Iðk; tÞ dX with kR ¼ jkj and solid angle X in k-space, and hki ¼ R kIðk; tÞ dk= Iðk; tÞ dk, respectively. We show the averaged structure factor Iðk; tÞ at long time t ¼ 500 in Fig. 3(a), and the time evolution of the characteristic wave number hki in Fig. 3(b), respectively, for three values of g at re ¼ 8 and Si ¼ 0. In Fig. 3(b) the result at g ¼ 0:5 is not shown, since hki 1 in this case. From Fig. 3 and
Y. Enomoto, M. Sawa / Surface Science 514 (2002) 68–73
71
Fig. 1. Time evolution of compositional patterns (left column) and its amplitude, Sðx; y ¼ 64Þ (right column) at re ¼ 8 and Si ¼ 0 for g ¼ 0:0005 (a), 0.05 (b), and 0.5 (c). Dots are plotted at lattice points with Sðr; tÞ < 0.
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Y. Enomoto, M. Sawa / Surface Science 514 (2002) 68–73
Fig. 3. (a) Spherically averaged structure factor Iðk; t ¼ 500Þ at Si ¼ 0 and re ¼ 8 for g ¼ 0:0005 ( ), 0.05 ( ), and 0.5 ( ). (b) Time evolution of the characteristic wave number hki at Si ¼ 0 and re ¼ 8 for g ¼ 0:0005 ( ), and 0.05 ( ). The slope of a line indicates the value of temporal power law exponent of hki.
Fig. 2. Time evolution of compositional patterns (left column) and its amplitude, Sðx; y ¼ 64Þ (right column) at re ¼ 8 and Si ¼ 0:3 for g ¼ 0:0005 (a) and 0.05 (b). Dots are plotted at lattice points with Sðr; tÞ < 0.
other simulations with different parameter values, asymptotic behavior of structure factors is classified into three regimes, corresponding to respective pattern morphology: (1) for phase separation regime (e.g., g ¼ 0:0005 in Fig. 3), Iðk; tÞ has a single peak at a time-decreasing peak position, and the characteristic wave number decreases with time as hki t1=3 (the well-known 1=3 growth law exponent); (2) for steady-state regime (e.g., g ¼ 0:05 in Fig. 3), Iðk; tÞ has a single peak, and the peak position and hki gradually become constant, which correspond to the characteristic length of
Y. Enomoto, M. Sawa / Surface Science 514 (2002) 68–73
steady-state patterns; (3) for solid solution regime (e.g., g ¼ 0:5 in Fig. 3), Iðk; tÞ ’ 0 which corresponds to a homogeneous state. For the steadystate regime, hki is found to be an increasing function of g, as is shown in Fig. 4(a). From Fig. 4(b) with further simulations, we also find a scaling relation between hki and re for re P 8 as hkire ¼ F ðgre3 Þ with a scaling function F ðxÞ. Such scaling behavior has been predicted theoretically for large re [4], but this is the first evident confirmation.
73
Finally, we note that the similar results have been obtained for the non-zero Si case as well. Since we have obtained hki 0:6 in the present simulation at g ¼ 0:005, re ¼ 8, and Si ¼ 0, and also Rc 0:1 nm in most cases [4], the characteristic wavelength of the steady-state pattern in this case is estimated as 2p=hki 1 nm. Thus, the irradiation might be used as a processing tool to synthesize nanostructures in a self-organized manner. 4. Conclusion In summary, we have numerically studied the two-dimensional phase separation dynamics of quenched alloys driven by irradiation, where the surface phase separation dynamics competes with the irradiation-induced ballistic mixing. This is the first simulation study of the continuum Enrique– Bellon model. From several simulation results it has been found that the irradiation-induced ballistic mixing can affect the dynamical process of phase separation in various manners, and it might provide a novel way to control the steady-state surface morphology of phase separating materials under some irradiation conditions. In the present work we have concentrated on rather simplified situation. In order to compare simulation results with real experimental data, both of further simulation study including other important factors neglected here (as was mentioned in Section 2) and further theoretical analysis of numerical results are needed, as well as estimations of phenomenological parameters in the model. These are left for future study. References
Fig. 4. (a) Characteristic wave number hki for the steady-state patterns at Si ¼ 0 as a function of g for re ¼ 4 ( ), 8 ( ), and 16 ( ). (b) Scaling relation between hki and g for the steadystate patterns at Si ¼ 0 for re ¼ 4 ( ), 8 ( ), and 16 ( ).
[1] K.C. Rusell, Prog. Mater. Sci. 28 (1984) 229. [2] X. Hu, D.G. Cahill, R.S. Averback, Appl. Phys. Lett. 76 (2000) 3215. [3] J.W. Cahn, J.E. Hilliard, J. Chem. Phys. 28 (1958) 258. [4] R.A. Enrique, P. Bellon, Phys. Rev. Lett. 84 (2000) 2885; Phys. Rev. B 63 (2001) 134111-1. [5] S.L. Dudarev, Phys. Rev. B 62 (2000) 9325. [6] G. Schmitz, J.C. Ewert, F. Harbsmeier, M. Uhrmacher, F. Haider, Phys. Rev. B 63 (2001) 224113-1. [7] L.C. Wei, R.S. Averback, J. Appl. Phys. 81 (1997) 613. [8] Y. Enomoto, Thin Solid Films 369 (2000) 21.
Compositional patterning in irradiated immiscible alloys Yoshihisa Enomoto *, Masataka Sawa Department of Environmental Technology, Nagoya Institute of Technology, Gokiso, Nagoya 466-8555, Japan Received 1 October 2001; accepted for publication 23 January 2002
Abstract The spontaneous formation of surface compositional patterning of quenched binary alloys driven by irradiation is numerically studied by means of a phenomenologically modified Cahn–Hilliard type equation for the local concentration field. In the present model the irradiation effect is modeled as a ballistic mixing with an average distance of atomic relocation R and its frequency C. From several two-dimensional computer simulations, it is shown that competitive effects between phase separation and irradiation-induced mixing might provide a novel way to stabilize and tune the steady-state nanostructures on the surface of phase separating materials under some irradiation conditions (i.e., in some region on (R,C) space). Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Alloys; Surface structure, morphology, roughness, and topography; Computer simulations; Surface thermodynamics (including phase transitions)
1. Introduction Whenever a system is driven from thermodynamic equilibrium by irradiation with energetic particles (e.g., ions), a variety of different phenomena, such as collision cascades, cascade damage, and radiation-enhanced diffusion, occur depending sensitively on the actual experimental condition [1]. For metals and miscible alloy systems, experimental and theoretical study on irradiation effects have been done extensively. For instance, spontaneous formation of metallic nanostructures under irradiation has been recently observed in Au/SiO2 and Pt/SiO2 [2]. On the other hand, immiscible
*
Corresponding author. Tel.: +81-52-7355153; fax: +81-527355158. E-mail address: [email protected] (Y. Enomoto).
alloys under irradiation have, however, been less studied, since more complicated situations can be encountered through the interplay between irradiation and the internal kinetics in these systems [1]. In particular, interesting but less studied case is ion-beam mixing of immiscible quenched alloys, where irradiation-induced ballistic mixing acts in opposition to thermodynamically driven kinetics like a spinodal decomposition. Each time an external particle collides with the solid, a local atomic rearrangement is produced. This rearrangement has a ballistic component that mixes the atoms regardless of their chemical identity, leading to bring the system to a random solid solution. During this ballistic mixing, a number of exchanges of atomic positions occur with an average relocation distance R and the frequency of these forced exchanges C, whose typical values are R 10 nm
0039-6028/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 9 - 6 0 2 8 ( 0 2 ) 0 1 6 0 9 - 6
Y. Enomoto, M. Sawa / Surface Science 514 (2002) 68–73
and C 104 s1 [1]. To explore such competitive effects, the Cahn–Hilliard type dynamical equation for the local concentration field [3] is known to provide a powerful tool, which has proposed by Enrique and Bellon [4]. As far, their continuum model has been used only to discuss the linear stability analysis for the limiting case of spinodal decomposition from the unstable state under ballistic mixing. In this work, we report on the long-time behavior of surface compositional patterns and their structure factor obtained by two-dimensional computer simulations of Enrique–Bellon model. We will study the following effects on the morphology: (1) the average exchange range R, (2) the exchange frequency C, and (3) the initial homogeneous concentration Si of the alloy. Despite the importance of controlling the surface compositional structures, these have been less studied previously, except for the mean field treatment and kinetic Monte Carlo simulation only for Si ¼ 0.
2. The model We study the spontaneous formation of surface compositional patterns in a quenched binary alloy with a positive heat of ballistic mixing under irradiation. The equation governing the time evolution of the system is [4] o Sðr; tÞ ot
h i ¼ Mr2 2ASðr; tÞ þ 4BðSðr; tÞÞ3 2Cr2 Sðr; tÞ C½Sðr; tÞ hSiR ðr; tÞ ;
ð1Þ
where Sðr; tÞ is a local composition difference of the two constituents of the alloy at time t and position r on the material surface. Here, we have chosen the system free energy density as AS 2 þ 2 BS 4 þ CðrSÞ with positive coefficients A, B, and C, which are assumed to be constant as well as a mobility M in Eq. (1). Estimating these parameters would require microscopic information about a specific system. Since we would understand generic properties of irradiated materials in the most general manner possible, we do not discuss them any more. The first term in the above equation repre-
69
sents the usual phase separation dynamics given by Cahn and Hilliard [3], and the second term describes the forced mixing term due to irradiation with the frequency of atomic relocation C. In the second term, hSiR represents a finite average of the concentration profile around r, given by Z Z 0 0 0 hSiR ðr; tÞ Sðr ; tÞ wR ðr rÞ dr = wR ðr0 rÞ dr0 ; ð2Þ
where the weight function wR ðxÞ expðjxj=RÞ denotes the distribution of atomic relocation distance with the average exchange distance R. In the model, for the sake of simplicity, the concentration of irradiation-induced point defects is assumed to have reached a steady value, and also the contribution of interstitials, as well as defect sinks and clusters [5], is neglected. Moreover, non-ballistic mixing effects and also any coupling between the degree of order and the concentration field [6] are not considered. Under these assumptions, the above model excludes phenomena related to the order–disorder transition, and macroscopic solute migration such as irradiation-induced heterogeneous precipitation. We also focus on the surface compositional patterning on the irradiated materials, neglecting the bulk diffusion and irradiationinduced phenomena occurring in the interior of the sample. These situations can be easily realized in thin film experiments with not too energetic cascades and some temperature region depending on the irradiated material properties [1] (e.g., 1 MeV Kr ion irradiation in Ag–Cu thin films at temperature ranging from 80 to 473 K [7]). Now, to discuss the early stage dynamics and the long-time steady-state behavior, we apply the linear stability analysis and the variational analysis to the model. Similar analyses have been done for the case Si ¼ 0 [4]. Linearizing Eq. (1) about Sðr; tÞ ¼ Si in Fourier space, one obtains R the equation for the fluctuations, S~ðk; tÞ dr½Sðr; tÞ Si expðik rÞ, as o ~ S ðk; tÞ ot
¼ 2MðA 6Si2 BÞk 2 2MCk 4 C
R2 k 2 S~ðk; tÞ 1 þ R2 k 2
ð3Þ
70
Y. Enomoto, M. Sawa / Surface Science 514 (2002) 68–73
with k ¼ jkj. The linear stability analyses of Eq. (3) have shown that there are two characteristic values of C, denoted by C1 ðR; Si Þ and C2 ðR; Si Þ. For C > C2 ðR; Si Þ the system is homogeneously mixed (solid solution), while for C < C1 ðR; Si Þ the homogeneous state is linearly unstable to grow, leading to macroscopic phase separation. Moreover, for C1 < C < C2 , certain modes k with k2 < k 2 < kþ2 are linearly unstable, suggesting a wavelength selection at long times (i.e., the steady-state patterning). An interval of the exchange frequency window ½C1 ; C2 decreases with R, reaching a zero value at a critical value Rc , when C1 ðRc ; Si Þ ¼ C2 ðRc ; Si Þ. For ballistic mixing with exchange distance smaller than Rc , the steady-state patterning is not possible. The variational analysis for the system energy ffi results in that for R Rc ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p C=ðA 6Si2 BÞ one has qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C1 ¼ 2R3 M ðA 6Si2 BÞC ; 2
C2 ¼ MðA 6Si2 B þ CR2 Þ =ð2CÞ; k ¼ 1=ð2RÞ; and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 kþ ¼ 0:5R2 c 0:5R : These analytic predictions have been partially confirmed by the kinetic Monte Carlo simulations only for the case Si ¼ 0 [4]. The question is whether the similar results are obtained for the present continuum approach including the non-zero Si case as well.
3. Simulation results We carry out two-dimensional computer simulations of the surface phase separation dynamics under irradiation by changing values of Si , C, and R. The system considered here is a two-dimensional plane, which corresponds to irradiated material surface. In the following simulations, we use the dimensionless variables such as re R=L and p0ffiffiffiffiffiffiffiffiffi g t0 C, by using the units of length L0 ¼ C=A, time t0 ¼ffi C=ð2MA2 Þ, and concentration field S0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi A=ð2BÞ. The resulting dimensionless equation (1) is solved numerically on a 256 256 square lattice with periodic boundary conditions, using a standard finite-difference scheme with mesh size
being 0:5 and time step 0:01 [8]. The initial state of the system consists of a Gaussian random distribution of the concentration with the mean value Si and standard deviation 0.1. First of all, the time evolution of the concentration field and its amplitude along the x-axis at re ¼ 8 are shown in Figs. 1 and 2 for Si ¼ 0 and 0.3, respectively. In these figures, lattice points with Sðr; tÞ < 0 are plotted by dots. For re ¼ 8, one has g1 t0 C1 ¼ 0:0019 and g2 t0 C2 ¼ 0:258 for Si ¼ 0, and g1 ¼ 0:0016 and g2 ¼ 0:139 for Si ¼ 0:3. For g ¼ 0:0005 (
Y. Enomoto, M. Sawa / Surface Science 514 (2002) 68–73
71
Fig. 1. Time evolution of compositional patterns (left column) and its amplitude, Sðx; y ¼ 64Þ (right column) at re ¼ 8 and Si ¼ 0 for g ¼ 0:0005 (a), 0.05 (b), and 0.5 (c). Dots are plotted at lattice points with Sðr; tÞ < 0.
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Y. Enomoto, M. Sawa / Surface Science 514 (2002) 68–73
Fig. 3. (a) Spherically averaged structure factor Iðk; t ¼ 500Þ at Si ¼ 0 and re ¼ 8 for g ¼ 0:0005 ( ), 0.05 ( ), and 0.5 ( ). (b) Time evolution of the characteristic wave number hki at Si ¼ 0 and re ¼ 8 for g ¼ 0:0005 ( ), and 0.05 ( ). The slope of a line indicates the value of temporal power law exponent of hki.
Fig. 2. Time evolution of compositional patterns (left column) and its amplitude, Sðx; y ¼ 64Þ (right column) at re ¼ 8 and Si ¼ 0:3 for g ¼ 0:0005 (a) and 0.05 (b). Dots are plotted at lattice points with Sðr; tÞ < 0.
other simulations with different parameter values, asymptotic behavior of structure factors is classified into three regimes, corresponding to respective pattern morphology: (1) for phase separation regime (e.g., g ¼ 0:0005 in Fig. 3), Iðk; tÞ has a single peak at a time-decreasing peak position, and the characteristic wave number decreases with time as hki t1=3 (the well-known 1=3 growth law exponent); (2) for steady-state regime (e.g., g ¼ 0:05 in Fig. 3), Iðk; tÞ has a single peak, and the peak position and hki gradually become constant, which correspond to the characteristic length of
Y. Enomoto, M. Sawa / Surface Science 514 (2002) 68–73
steady-state patterns; (3) for solid solution regime (e.g., g ¼ 0:5 in Fig. 3), Iðk; tÞ ’ 0 which corresponds to a homogeneous state. For the steadystate regime, hki is found to be an increasing function of g, as is shown in Fig. 4(a). From Fig. 4(b) with further simulations, we also find a scaling relation between hki and re for re P 8 as hkire ¼ F ðgre3 Þ with a scaling function F ðxÞ. Such scaling behavior has been predicted theoretically for large re [4], but this is the first evident confirmation.
73
Finally, we note that the similar results have been obtained for the non-zero Si case as well. Since we have obtained hki 0:6 in the present simulation at g ¼ 0:005, re ¼ 8, and Si ¼ 0, and also Rc 0:1 nm in most cases [4], the characteristic wavelength of the steady-state pattern in this case is estimated as 2p=hki 1 nm. Thus, the irradiation might be used as a processing tool to synthesize nanostructures in a self-organized manner. 4. Conclusion In summary, we have numerically studied the two-dimensional phase separation dynamics of quenched alloys driven by irradiation, where the surface phase separation dynamics competes with the irradiation-induced ballistic mixing. This is the first simulation study of the continuum Enrique– Bellon model. From several simulation results it has been found that the irradiation-induced ballistic mixing can affect the dynamical process of phase separation in various manners, and it might provide a novel way to control the steady-state surface morphology of phase separating materials under some irradiation conditions. In the present work we have concentrated on rather simplified situation. In order to compare simulation results with real experimental data, both of further simulation study including other important factors neglected here (as was mentioned in Section 2) and further theoretical analysis of numerical results are needed, as well as estimations of phenomenological parameters in the model. These are left for future study. References
Fig. 4. (a) Characteristic wave number hki for the steady-state patterns at Si ¼ 0 as a function of g for re ¼ 4 ( ), 8 ( ), and 16 ( ). (b) Scaling relation between hki and g for the steadystate patterns at Si ¼ 0 for re ¼ 4 ( ), 8 ( ), and 16 ( ).
[1] K.C. Rusell, Prog. Mater. Sci. 28 (1984) 229. [2] X. Hu, D.G. Cahill, R.S. Averback, Appl. Phys. Lett. 76 (2000) 3215. [3] J.W. Cahn, J.E. Hilliard, J. Chem. Phys. 28 (1958) 258. [4] R.A. Enrique, P. Bellon, Phys. Rev. Lett. 84 (2000) 2885; Phys. Rev. B 63 (2001) 134111-1. [5] S.L. Dudarev, Phys. Rev. B 62 (2000) 9325. [6] G. Schmitz, J.C. Ewert, F. Harbsmeier, M. Uhrmacher, F. Haider, Phys. Rev. B 63 (2001) 224113-1. [7] L.C. Wei, R.S. Averback, J. Appl. Phys. 81 (1997) 613. [8] Y. Enomoto, Thin Solid Films 369 (2000) 21.