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Molecular dynamics simulation of structural anisotropy in glassy metals
Materials Science and Engineering, A 134 ( 1991 ) 931- 934
931
Molecular dynamics simulation of structural anisotropy in glassy metals T. Tomida and T. Egami Department of Materials Science and Engineering, Universityof lYnnsylvania, Philadelphia, PA 19104-6272 (U.S.A.
Abstract Molecular dynamics simulations have been carried out to study the creep-induced structural amsotropy in amorphous alloys. A model binary glass structure of 2048 particles interacting via Lennard-Jones potentials was creep-deformed below its glass transition temperature. Creep-deformation resulted in the bond-orientational anisotropy (BOA) within the first nearest-neighbor shell with a large sixth-order spherical harmonic component. The BOA caused an anisotropy characterized by uniform elongation along the tensile principal axis of the creep-deformation beyond the third nearest-neighbor shell.
1. Introduction
function) using spherical harmonics [4],
Ideally a glassy metal should be macroscopically isotropic. -However, a number of experiments have shown, mainly through magnetic anisotropy measurements, that structural anisotropy can be induced during various anisotropic processing of glassy metals [1-4]. In particular creep-deformation is well known to result in uniaxial magnetic anisotropy [1-3]. The structural anisotropy within a creep-deformed glassy alloy was observed recently by means of X-ray diffraction by Suzuki et al. [4]. The authors reported that the observed anisotropy in the scattering intensity is best explained by the bondorientational anisotropy (BOA). The BOA is the anisotropic existence probability of the nearestneighbor atoms; the existence probability of the nearest-neighbor atoms along the tensile principal axis of the creep-deformation is reported to be less than that along the compressive axis [4]. However, the detailed structure of such anisotropy has not been clarified. For this purpose a molecular dynamics (MD) simulation of the creep-deformation of a glassy alloy was carried out using 2048 particles interacting via Lennard-Jones potentials.
g(r, i / r ) = Z p,m(r) Y/"(r/r)
2. Directional and anisotropic pair distribution functions In order to describe the directional short-range order, the directional RDF (radial distribution 0921-5093/91/$3.50
Im
where Y/" is a spherical harmonic and
pro(r) = 4:r/N
~ d(r - r,j) Yg(i'ijrij), i]
and likewise the directional PDF (pair distribution function), p(r, f/r) = 1/(4:rr2po)g(r, f /r),
where P0 is the average density, are utilized. In this paper the spherical harmonics with a nonzero value of m and odd l were ignored because of the uniaxial symmetry of the creep-deformation to be described in the next section. Furthermore, the term anisotropic PDF will be used to express the difference in the directional PDF along the expansive principal axis from that along the compressive principal axis. 3. Computer model of the creep deformation An assembly consisting of 2048 particles interacting via Lennard-Jones potentials (4e((o/r) 12(o/r)6)) has been employed. The mixture of potentials with different length parameters OAA and o ~ was introduced to simulate a binary alloy. The length parameter for unlike pairs of particles was given by OA~e = (OAA2 + OBB2)/2. The ratio of OAA2/O~B 2 and the atomic concentration were © Elsevier Sequoia/Printed in The Netherlands
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80'88
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Time, t/to
Fig. 1. Typicaltime dependence of the elongation along the expansiveprincipal axis of the simulated creep-deformation. chosen to be 0.8 and 0.5, respectively. The depths of these potentials were chosen for the ratio between them to be eAA:eAB: eBB= 1 : 1/0.9:1/0.8. The integration time step was taken to be 0.02t 0 (to=(ma2/eAA)l/2).The force due to all the potentials was truncated at the particle distance of 1.55 OAA. Periodic boundary conditions were maintained on all six faces of the cubic assembly. The shear strains, as described below, were applied to the model structure at the temperature of 0.275 in the unit of eAA/K every 100th time step so that the expansive stress along the Z-axis, az, is kept constant, simulating creep-deformation.
°°0)
0
0
2e
Each deformation was performed by displacing every atom in accordance with the linear elastic displacement field corresponding with the strain. The time-strain curves shown in Fig. 1 were thus obtained. The directional RDFs during creepdeformation were calculated every tenth time step after the 2500th step to the 10 000th step, and averaged. Furthermore, to explore the anelastic part of the anisotropy, the creep-deformed .assembly was quenched to the temperature of 0.1, and then the remaining stress was released. The directional RDFs for these relaxed states were then computed and averaged.
4. The anisotropy of the coordination number (existence of BOA) To explore the anisotropy of the coordination number, t h e directional coordination number
\
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\,\j~
•~ -11.8 O
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Stress.~z Fig. 2. Stress dependence of the Y2°, Y4° and Y6° spherical harmonic components in the total D C N F calculated during creep-deformation.
functions (DCNF) were defined integrating the partial directional RDFs in regard to the particle distance from zero to the cut-off lengths. Its expression, using the polar angle from the Z-axis, is as follows: 1.5 aAA
DCNFAA(O) =
f gAA(r,O)dr 0
for A - A pairs 1,42oAA
DCNFAB(O ) =
f
gAB(r,O ) dr
0
for A-B pairs or 1.35oAA
DCNFBB(O) =
f gBB(r,O)dr 0
for B - B pairs The stress dependence of Y2°, Y4° and Y6° spherical harmonic components in the total DCNF, including all the above-mentioned partial DCNFs, during the creep-deformation shown in Fig. 2 was calculated. It was found that the Y2° component increases linearly with increasing stress while the Y6° component shows a minimum at a stress of around 0.2. In contrast with these components, the Y4° component stays at a slightly negative value, which is considered to be the anisotropy quenched in from the liquid state of the
933
13.1,
12.8.0.05Y2* I
,// -0.06 y o z~ ~ 7 1 3 . ' 0 -1.27Y~ 126
/
~
, '/
1
II
t 12.2
08
0.8 0.6-'~'---~ ~ 0.4 ~ 0.2
..... ];
dP(r) ro01 1 ,,
,
1
r ~_g 0" I
<
N o
0.2
30 60 Angle, O/degree
90
Fig. 3. Total DCNF calculated for the stress released state after creep-deformation with the averaged stress o z=0.19 and the difference between the partial DCNFs for the same stress released state and the corresponding partial DCNFs calculated without deformation. The broken line shows the function consisting of the Y2 °, Y4° and Y6° spherical harmonic components in the total D C N E
model structure, and is independent of a z. These trends imply that the I12° component is elastic and the anelastic part of the anisotropy, i.e. the BOA, induced by the creep-deformation, is the }16° component. This point was confirmed by calculating the DCNF after stress release, as shown in Fig. 3. Almost the same amount of the Y6° component as before stress release was found; however, the remaining II2° component was negligible and was within statistical error. Furthermore the preference of the B O A to certain kinds of atomic pairs was investigated. For this purpose, the partial DCNFs calculated without deformation were subtracted from the corresponding partial DCNF's after stress release to eliminate the effect of the anisotropy quenched in from the liquid state. The results are shown in Fig. 3. No such clear preference of the B O A were found. Therefore the atomic pair ordering [1, 3] was not induced by this simulation.
5. Anisotropy in the medium-range distance The important features of the medium-range anisotropy were again found in the ]/2 0 and }16° components in the anisotropic PDF calculated after stress release, as shown in Fig. 4. The }12° component is negligible within the first nearestneighbor shell; however, this component tends to increase with increasing inter-particle distance. In
-0./,L0
1 2 3 /, 5 Particle Distance. r/o~A
J
6
Fig. 4. Non-directional total PDF, a derivative of the PDF, and the Yz°, 1/4° and y 0 spherical harmonic components of the total anisotropic PDF calculated for the stress released state after the creep-deformation with the averaged stress a~=0.27; the non-directional PDF is equal to the Y," component of the directional PDE
contrast to the Y2° component, the Y6° component is large within the distance of about 3 aAA, while it tends to die out beyond it. The behavior of the Y2° component in the medium-range distance is well characterized by the uniform elongation of about l% strain along the Z-axis, which is consistent with the experiment [4]. The Y2 ° component beyond the distance of 30AA is very close to the derivative of the PDF of this assembly - d p ( r ) / d r" re (s = 0.01 ) shown in Fig. 4.
6. Discussion According to Suzuki e t al. [4], the directional imbalance of the probability of bond annihilation and creation caused induces the BOA. The probability difference between bond annihilation and creation in a direction may be proportional to sinh(of2/2KT), where a is the stress and f2 is the volume strain element, or an activation volume [6, 7]; oQ expresses the activation energy due to an applied stress. If the stress is small, the probability difference is linear to the stress, therefore the anisotropy of the difference, i.e. the BOA, for small stress creep-deformation should be dominated by the }12° term. In the case of our simulation, the activation energy for the stress o z = 0.2 is estimated to be comparable to CAn, using the measured volume strain element for metallic glasses below the glass transition temperature of around
934
.--.. Induced /l/...~ D C N F
[ -~'y
v Neorest " ~ " neighbor shell
\])"
Z-oxis
Fig. 5. Schematic representation of the proposed mechanism to explain the large Y6° component in the DCNF.
10 times the atomic volume [7]. The activation energy is then about twice 2 K T (~0.55EAA)for the simulation.~' Hence the above-mentioned probability difference is no longer linear to the stress. This energetic condition leads to a strong directional preference of the bond annihilation and creation. Consequently such a bond rearrangement is activated only at a part of a glass which has a nearest-neighbor configuration, as shown in Fig. 5 with full circles. The nearestneighbor atoms along the expansive direction are excluded from the second nearest-neighbor shell and thus along the compressive direction, resulting in the new configuration shown as open circles in Fig. 5. The preference causes a six-fold anlsotropy in the DCNF, as shown in Fig. 5 as a solid wavy line. The anisotropy should have a
dominant higher spherical harmonics Y6° component rather than a Y4° component. Hence the fact that the dominant component of the BOA was a Y6° spherical harmonic is considered to be one of the consequences of accelerated creep-simulations with a large stress in MD calculation. Furthermore such a large stress can be responsible for the stress dependence minimum of the BOA shown in Fig. 2, through an inhomogeneous flow. Since the inhomogeneous flow causes thin shear bands and leaves the rest of the material plastically undeformed, the bond annihilation and creation are localized in the bands. Therefore, the structural anisotropy averaged over the entire material is reduced. Consequently the BOA is suppressed and shows a minimum. References 1 H. Fujimori, in E E. Luborsky (ed.), Amorphous Metallic Alloys, Butterworth & Co. Ltd., London, 1983, p. 300. 2 0 . V . Nielsen and H. J. V. Nielsen, J. Magn. Magn. Mater., 22(1980) 21. 3 H. R. Hilzinger, in T. Masumoto and K. Suzuki (eds.), Proc. 4th Int. Conf. on Rapidly Quenched Metals, Vol. 2, Japan Institute of Metals, Sendal, 1982, p. 791. 4 Y. Suzuki, J. Haimovieh and T. Egami, Phys. Rev. B35 (1987) 2162. 5 S. Chikazumi and C. D. Graham, in A. Berkowitz and E. Kneller (eds.), Magnetism and Metallurgy, Academic Press, New York, 1969, p. 577. 6 E Spaepen, Acta Metall., 25 (1977) 407. 7 F. Spaepen and A. I. Taub, in F. E. Luborsky (ed.), Amorphous Metallic Alloys, Butterworth & Co. Ltd., London, 1983, p. 231.
931
Molecular dynamics simulation of structural anisotropy in glassy metals T. Tomida and T. Egami Department of Materials Science and Engineering, Universityof lYnnsylvania, Philadelphia, PA 19104-6272 (U.S.A.
Abstract Molecular dynamics simulations have been carried out to study the creep-induced structural amsotropy in amorphous alloys. A model binary glass structure of 2048 particles interacting via Lennard-Jones potentials was creep-deformed below its glass transition temperature. Creep-deformation resulted in the bond-orientational anisotropy (BOA) within the first nearest-neighbor shell with a large sixth-order spherical harmonic component. The BOA caused an anisotropy characterized by uniform elongation along the tensile principal axis of the creep-deformation beyond the third nearest-neighbor shell.
1. Introduction
function) using spherical harmonics [4],
Ideally a glassy metal should be macroscopically isotropic. -However, a number of experiments have shown, mainly through magnetic anisotropy measurements, that structural anisotropy can be induced during various anisotropic processing of glassy metals [1-4]. In particular creep-deformation is well known to result in uniaxial magnetic anisotropy [1-3]. The structural anisotropy within a creep-deformed glassy alloy was observed recently by means of X-ray diffraction by Suzuki et al. [4]. The authors reported that the observed anisotropy in the scattering intensity is best explained by the bondorientational anisotropy (BOA). The BOA is the anisotropic existence probability of the nearestneighbor atoms; the existence probability of the nearest-neighbor atoms along the tensile principal axis of the creep-deformation is reported to be less than that along the compressive axis [4]. However, the detailed structure of such anisotropy has not been clarified. For this purpose a molecular dynamics (MD) simulation of the creep-deformation of a glassy alloy was carried out using 2048 particles interacting via Lennard-Jones potentials.
g(r, i / r ) = Z p,m(r) Y/"(r/r)
2. Directional and anisotropic pair distribution functions In order to describe the directional short-range order, the directional RDF (radial distribution 0921-5093/91/$3.50
Im
where Y/" is a spherical harmonic and
pro(r) = 4:r/N
~ d(r - r,j) Yg(i'ijrij), i]
and likewise the directional PDF (pair distribution function), p(r, f/r) = 1/(4:rr2po)g(r, f /r),
where P0 is the average density, are utilized. In this paper the spherical harmonics with a nonzero value of m and odd l were ignored because of the uniaxial symmetry of the creep-deformation to be described in the next section. Furthermore, the term anisotropic PDF will be used to express the difference in the directional PDF along the expansive principal axis from that along the compressive principal axis. 3. Computer model of the creep deformation An assembly consisting of 2048 particles interacting via Lennard-Jones potentials (4e((o/r) 12(o/r)6)) has been employed. The mixture of potentials with different length parameters OAA and o ~ was introduced to simulate a binary alloy. The length parameter for unlike pairs of particles was given by OA~e = (OAA2 + OBB2)/2. The ratio of OAA2/O~B 2 and the atomic concentration were © Elsevier Sequoia/Printed in The Netherlands
932 ,
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,
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i
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£::
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.ca "0
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E f,,_ ,-lo e-
~
°'27_,E7-~
\, •
'
2&o
'
48'00
'
,~00
'
80'88
'
10'
Time, t/to
Fig. 1. Typicaltime dependence of the elongation along the expansiveprincipal axis of the simulated creep-deformation. chosen to be 0.8 and 0.5, respectively. The depths of these potentials were chosen for the ratio between them to be eAA:eAB: eBB= 1 : 1/0.9:1/0.8. The integration time step was taken to be 0.02t 0 (to=(ma2/eAA)l/2).The force due to all the potentials was truncated at the particle distance of 1.55 OAA. Periodic boundary conditions were maintained on all six faces of the cubic assembly. The shear strains, as described below, were applied to the model structure at the temperature of 0.275 in the unit of eAA/K every 100th time step so that the expansive stress along the Z-axis, az, is kept constant, simulating creep-deformation.
°°0)
0
0
2e
Each deformation was performed by displacing every atom in accordance with the linear elastic displacement field corresponding with the strain. The time-strain curves shown in Fig. 1 were thus obtained. The directional RDFs during creepdeformation were calculated every tenth time step after the 2500th step to the 10 000th step, and averaged. Furthermore, to explore the anelastic part of the anisotropy, the creep-deformed .assembly was quenched to the temperature of 0.1, and then the remaining stress was released. The directional RDFs for these relaxed states were then computed and averaged.
4. The anisotropy of the coordination number (existence of BOA) To explore the anisotropy of the coordination number, t h e directional coordination number
\
/y~ /'
'\
'\
O
E
\,\j~
•~ -11.8 O
- e
\
03 ..,.. - 0 . 6
~
0
\
..£:: t:)..
~---~-----'---~ ' "~-"--"--'-"~--
o
1/3
,
";- - 0 . 4 ~
Y4°
,
O (.J
5
LLJ
0
"1" - 0 . 2
-1.0
.I
0
0.1
0.2
0.3
0.4
Stress.~z Fig. 2. Stress dependence of the Y2°, Y4° and Y6° spherical harmonic components in the total D C N F calculated during creep-deformation.
functions (DCNF) were defined integrating the partial directional RDFs in regard to the particle distance from zero to the cut-off lengths. Its expression, using the polar angle from the Z-axis, is as follows: 1.5 aAA
DCNFAA(O) =
f gAA(r,O)dr 0
for A - A pairs 1,42oAA
DCNFAB(O ) =
f
gAB(r,O ) dr
0
for A-B pairs or 1.35oAA
DCNFBB(O) =
f gBB(r,O)dr 0
for B - B pairs The stress dependence of Y2°, Y4° and Y6° spherical harmonic components in the total DCNF, including all the above-mentioned partial DCNFs, during the creep-deformation shown in Fig. 2 was calculated. It was found that the Y2° component increases linearly with increasing stress while the Y6° component shows a minimum at a stress of around 0.2. In contrast with these components, the Y4° component stays at a slightly negative value, which is considered to be the anisotropy quenched in from the liquid state of the
933
13.1,
12.8.0.05Y2* I
,// -0.06 y o z~ ~ 7 1 3 . ' 0 -1.27Y~ 126
/
~
, '/
1
II
t 12.2
08
0.8 0.6-'~'---~ ~ 0.4 ~ 0.2
..... ];
dP(r) ro01 1 ,,
,
1
r ~_g 0" I
<
N o
0.2
30 60 Angle, O/degree
90
Fig. 3. Total DCNF calculated for the stress released state after creep-deformation with the averaged stress o z=0.19 and the difference between the partial DCNFs for the same stress released state and the corresponding partial DCNFs calculated without deformation. The broken line shows the function consisting of the Y2 °, Y4° and Y6° spherical harmonic components in the total D C N E
model structure, and is independent of a z. These trends imply that the I12° component is elastic and the anelastic part of the anisotropy, i.e. the BOA, induced by the creep-deformation, is the }16° component. This point was confirmed by calculating the DCNF after stress release, as shown in Fig. 3. Almost the same amount of the Y6° component as before stress release was found; however, the remaining II2° component was negligible and was within statistical error. Furthermore the preference of the B O A to certain kinds of atomic pairs was investigated. For this purpose, the partial DCNFs calculated without deformation were subtracted from the corresponding partial DCNF's after stress release to eliminate the effect of the anisotropy quenched in from the liquid state. The results are shown in Fig. 3. No such clear preference of the B O A were found. Therefore the atomic pair ordering [1, 3] was not induced by this simulation.
5. Anisotropy in the medium-range distance The important features of the medium-range anisotropy were again found in the ]/2 0 and }16° components in the anisotropic PDF calculated after stress release, as shown in Fig. 4. The }12° component is negligible within the first nearestneighbor shell; however, this component tends to increase with increasing inter-particle distance. In
-0./,L0
1 2 3 /, 5 Particle Distance. r/o~A
J
6
Fig. 4. Non-directional total PDF, a derivative of the PDF, and the Yz°, 1/4° and y 0 spherical harmonic components of the total anisotropic PDF calculated for the stress released state after the creep-deformation with the averaged stress a~=0.27; the non-directional PDF is equal to the Y," component of the directional PDE
contrast to the Y2° component, the Y6° component is large within the distance of about 3 aAA, while it tends to die out beyond it. The behavior of the Y2° component in the medium-range distance is well characterized by the uniform elongation of about l% strain along the Z-axis, which is consistent with the experiment [4]. The Y2 ° component beyond the distance of 30AA is very close to the derivative of the PDF of this assembly - d p ( r ) / d r" re (s = 0.01 ) shown in Fig. 4.
6. Discussion According to Suzuki e t al. [4], the directional imbalance of the probability of bond annihilation and creation caused induces the BOA. The probability difference between bond annihilation and creation in a direction may be proportional to sinh(of2/2KT), where a is the stress and f2 is the volume strain element, or an activation volume [6, 7]; oQ expresses the activation energy due to an applied stress. If the stress is small, the probability difference is linear to the stress, therefore the anisotropy of the difference, i.e. the BOA, for small stress creep-deformation should be dominated by the }12° term. In the case of our simulation, the activation energy for the stress o z = 0.2 is estimated to be comparable to CAn, using the measured volume strain element for metallic glasses below the glass transition temperature of around
934
.--.. Induced /l/...~ D C N F
[ -~'y
v Neorest " ~ " neighbor shell
\])"
Z-oxis
Fig. 5. Schematic representation of the proposed mechanism to explain the large Y6° component in the DCNF.
10 times the atomic volume [7]. The activation energy is then about twice 2 K T (~0.55EAA)for the simulation.~' Hence the above-mentioned probability difference is no longer linear to the stress. This energetic condition leads to a strong directional preference of the bond annihilation and creation. Consequently such a bond rearrangement is activated only at a part of a glass which has a nearest-neighbor configuration, as shown in Fig. 5 with full circles. The nearestneighbor atoms along the expansive direction are excluded from the second nearest-neighbor shell and thus along the compressive direction, resulting in the new configuration shown as open circles in Fig. 5. The preference causes a six-fold anlsotropy in the DCNF, as shown in Fig. 5 as a solid wavy line. The anisotropy should have a
dominant higher spherical harmonics Y6° component rather than a Y4° component. Hence the fact that the dominant component of the BOA was a Y6° spherical harmonic is considered to be one of the consequences of accelerated creep-simulations with a large stress in MD calculation. Furthermore such a large stress can be responsible for the stress dependence minimum of the BOA shown in Fig. 2, through an inhomogeneous flow. Since the inhomogeneous flow causes thin shear bands and leaves the rest of the material plastically undeformed, the bond annihilation and creation are localized in the bands. Therefore, the structural anisotropy averaged over the entire material is reduced. Consequently the BOA is suppressed and shows a minimum. References 1 H. Fujimori, in E E. Luborsky (ed.), Amorphous Metallic Alloys, Butterworth & Co. Ltd., London, 1983, p. 300. 2 0 . V . Nielsen and H. J. V. Nielsen, J. Magn. Magn. Mater., 22(1980) 21. 3 H. R. Hilzinger, in T. Masumoto and K. Suzuki (eds.), Proc. 4th Int. Conf. on Rapidly Quenched Metals, Vol. 2, Japan Institute of Metals, Sendal, 1982, p. 791. 4 Y. Suzuki, J. Haimovieh and T. Egami, Phys. Rev. B35 (1987) 2162. 5 S. Chikazumi and C. D. Graham, in A. Berkowitz and E. Kneller (eds.), Magnetism and Metallurgy, Academic Press, New York, 1969, p. 577. 6 E Spaepen, Acta Metall., 25 (1977) 407. 7 F. Spaepen and A. I. Taub, in F. E. Luborsky (ed.), Amorphous Metallic Alloys, Butterworth & Co. Ltd., London, 1983, p. 231.