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Suspension effect of nanocrystalline grain growth under electropulsing
NantShudurcdMataials.Vol. 10,No. 1.pp. 71-76.1998 Ekevia ScienceLtd 0 1998AI% Mctallurpica Inc. RintedinthousA. Allrights-al 09654773’98$19.00+ .OO
Pergamon
PII 809659773(98)00030-0
SUSPENSION EFFECT OF NANOCRYSTALLINE GRAIN GROWTH UNDER ELECTROPULSING R.S. Qid2,
S.X. Sd, J.D. Guo’, G.H. I-Ie’J, and B.L. Zhou’J
‘Institute of Metal Research, Chinese Academy of Science, Shenyang 110015, P.R. China 21nstitute of Corrosion and Protection for Metals, Chinese Academy of Science, Shenyang 110015, P.R. China 31ntemational Center for Materials Physics, Chinese Academy of Science, Shenyang 110015, P.R. China (AcceptedFebruary 11,1998)
Abstract - The e#ect of electtopulsingon nanocrystallinegrain growth in amorphous material is considered. A size-dependentgrowthrate of a nanoscale grain is revealed, which is due to the size-dependentelectrical resistivity.A suspensione$ect isfound when the densityof electric current is at a critical valuej,. Theformed grain may shrinkwhen the densityof applied electric current is higher thanjC. Numerical calculationshowsthatj, is about 2.59 x IdAl& for nanocryst#alline Nis&o alloy withgrain size of 13 nm. 01998 Acta MetallurgicaInc.
I. INTRODUCTION Electropulsing with current density of ldA/mm2 is used frequently in synthesizing nanocrystalline alloy from amorphous alloy (1). The advantages of using electropulsing over annealing are that lower temperature and less time are needed in the experiment. It is known from both experimental and theoretical studies that the side effects of electropulsing (i.e. skin, Joule heat, pinch force) are less important However, the effects of electropulsing on crystallization are far from understood. Our previous work came to the conclusion that electropulsing can increase the nucleatioln rate and undercooling (2,3). The aim of this paper is to the study of effect of electropulsing on the growth rate of a nanocrystalline grain. The investigation is along the standard line of the first-order phase transition theory. The paper is organized as follows. In Section II we present our theoretical study for the effect of electropulsing on the growtbrate of ananoscale-grain in amorphous alloy by which the suspension effect is found. In Section III we analyze the suspension effect and calculate it numerically. In Section IV some concluding remarks are given. 71
72
Fl!3QIN, SX Su, JD Guo, GH HEANDBL ZMOU
IL THEORETICAL
STUDY
For simplicity, we consider a system which only one grain is formed in the amorphous material. The grain is supposed to grow with the velocity (4) U=U0exp(-f)[l-exp(+)]
111
The preexponential factor Uo is typically of the order of 16 cm/s (roughly estimated as aov, ae is the characteristic interatomic length in a solid, and v is the characteristic atomic frequency inasolid). Vandnareusuallytermedasthekineticandthermodynamicbarriers tocrystallization, respectively, k is Boltzmann’s constant, and T is temperature. The multiplier in square brackets on the right-hand side in eq. [l] reflects the possibility of a backward process when the energy n gainedbyamoleculejoiningagrainisnotlargeenough. Aslongasitisgivenbyeq. [l],thegrowth rate is determinated by both kinetic and thermodynamic barriers. There are many experimental investigations on the change of kinetic barrier under electropulsing during crystallization in amorphous alloys (5.6). Here, one would like topay more attention to the thermodynamic barrier. In a current-carrying system, f2 consists of two parts Q=szo+f&
PI
where f& is the free-energy change due to grain growth in a current-free system, which is given by f2, = -AVji+AS.a,, where AV is the volume of a molecule, ji(>O) is the free energy difference between the nanocrystalline phase and amorphous phase per unit volume. AS is the increase of the surface area of a grain when a molecule is joined to it, and cranis the interfacial energy per unit area. Anegative value of Sz implies that a grain has more probability to grow, otherwise the grain may shrink. f& is the energy change in the grain growing process caused by electropulsing, which can be determined from the following equation (7)
where p is the magnetic susceptibility. In the temperature region of our studies l.t = po, ~MIis the magnetic susceptibility in vacuum, ;I (r) and 1; (r) are current densities when the grain sixes are at and ax, respectively. The change of the volume of the grain is denoted by AV. The relationship between al and a2 is given by: -a:)=AV
SUSPENSON EFFECT OFNAWCRYSTALLF-JE GRAIN GROWTH UNDER ELECTROPULWG
73
Electric current distribution in the adiabatic approximation can be determined from the following equ.ations: divji = 0
rOt(ji/@ = 0,
161
where i=O,l,Z!represent in amorphous, grain with size al and size az, respectively. oi is the conductivity of corresponding medium. In the calculation of current density distribution, the medium can be assumed infinite (because the grain size is in nanoscale and the amorphous size is above microscale) and the solution of eq. [6] is as follows:
[71
where @I
-sineq
i’o = jo(cosef
191
where crois the conductivity of amorphous, 3 and 6 are the unit vector in spherical coordinate. Bringing eq. 1173 to eq. [4] one obtains the f& with an accuracy of order a; by using the method of physical replacement:
where b is the size of amorphous alloy. It is obvious that in the region of
&, Cl
In b a: 0 =1 ln b
a;
0 =2 f& will be positive, otherwise f& will be negative.
(4, > 0,41>
0)
WI
74
RS QIN, SX Su, JD Guo, GH HE ANDBL ZHOU
4
- 650 - 600 - 550 - 500
/\”’ 350,
. 0
I 2
.
, 4
.
/_:I1 I 6
.
I 0
- 450
- 400
I . t . I . I . I . 1350 14 16 18 20 10 12
Grain size
nm
Figure 1. A schematic diagram of the relationship between electrical resistivity and grain size.
ThepositivevalueofR,impliesthattheelectropulsingmayreducethegrowthrateofagrain. Intheconditionofn,=-~(i.e.thecurrentdensityisinacriticalvalue)thegrainwillstopgrowing. The zero growth rate of a grain is termed suspension effect.
III. DISCUSSION AND NUMERICAL CALCULATION Eq. [ 1l] is only valid for nanoscale grain. The electrical conductivity of a metallic material with coarse grain is higher than that of a corresponding amorphous material generally. According to eq. [8], & will be negative, but the electrical conductivity of some nanocrystalline materials is lower than that of the corresponding amorphous materials. For example, the electrical conductivity ratio between nanocrystalline FeMoSiB alloy with grain size of 28 nm and the amorphous FeMoSiB alloy is 1.02 at room temperature (8), and it is 1.26 of nanocrystalline FeCuSiB alloy with grain size of 25 nm to its amorphous alloy at room temperature (9). So & will be positive in this case. Theoretical investigation shows that the electrical conductivity of the nanocrystalline materials depends on the grain size (lo), which is due to the size-dependent lattice distortion and interface excess volume (11). So the growth rate of a nauoscale grain will depend on the grain size under electropulsing. In order to know the effect of electropulsing on grain growth quantitatively, one would like to perform some numerical calculation. Nis&u is a well studied material. Amorphous Nis&o alloy is synthesized by a single rollerrapidlyquenching equipment. Nanocrystalline Nis&o alloy can be obtained by the method of heat treatment. A nanocrystalline Nis&o alloy with the mean grain size of 13 nm is synthesized in a heat treatment procedure of rapidly heating (3OOK/min) the amorphous sample to 598 K, anuealing isothermally for 10 min and cooling to room temperature
75
SUSPENSIONEFFECTOF NAN~CXYSTALLINE GRAINGFOWTHIJNDERELECTROPULSING
3 & 2
.b G s 0 .r( Ei R
S!OOO
. __--_ 680 - _
Linear Saddle
shape shape
-640 -1960 -
-3280 $ t
-4600 0
1
I 2
Current density
II II II II
I 3
4
103Almm*
Figure 2. Thermodynamic barrier vs. current density, where j, is the critical value.
( 12).The electrical resistivity of the nanocrystallineNi8&u alloy with the mean grain size of 13 nm is 585.0 + O.l23T(@. cm), and the electrical resistivity of the amorphous b&&u alloy is 210.7 + 0.505T Q&l - cm) (13). The Gibbs fret energy change in amorphous-nanocrystline transition is A.6 J/ mol in 598 K (14). The aim of the present calculation is to know the critical value j, by which the grain cannot grow yet. The problem here is that one does not know the relationshipbetween electrical conductivity and grain size. In order to overcome this difficulty two cases are assumed as illustrated in Figure 1: (a) the electrical resistivity is proportional to grain size linearly in this size range; (b) the relationshipbetween electrical resistivity and grain sizeis a saddle-shape.The results of numerical calculations are shown in Figure 2. The critical current densities for above-mentioned two cases are 2.59 x @A/mm2 and 2.56 x ldA/mm2, respectively. The grain may have more possibility to shrink when the current density is above the critical value. The present investigation is also valid for the solidification of a melt. Experiment reveals a size-dependent electrical resistivity behavior in the soliditication of Pb (15). The relationship between electrical resistivity and grain size is a saddle-shapein the early stage of crystallization. There will be a suspension effect when the current is carried in a certain size range, which will be useful for the fabricating bulk nanocrystallineby electric rheological technique. IV. CONCLUSION The nanocrystalline grain takes an unusual correspondence to the applied electropulsing. The effect of electropulsing on the growth rate of a nanoscale grain is size dependent. When the electrical conductivity of a grain is largerthan that of amorphous,theelectropulsing would prevent
76
=
GN,
Sx
.%,
JD
Guo, GH HE AND BL Ztiou
the grain from growing. In the extra condition of applied electric current density larger than the critical value, the grain may shrink.
ACKNOWLEDGMENT This work is supported by tbe Chinese National Science Foundation -59431040, and Liaoning Doctoral Foundation - 97 1106.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
Teng,G., Chao,Y.,Lai, Z. and Dong,L., ChineseScience Bulletin, 1994,39,974. Qin, R.S., Yan, H.C., He, G.H. aud Z~OU,B.L., Chinese Journal ofMateriols Research, 1995,9, 219. Qiu, R.S. and Zbou, B.L., ChineseJournal ofMaterials Research, 1997, 11.69. Fine, M.F., Phase Trun&ormation in Condensed Systems, MacMillan, New York, 1974. Takemoto, R. and Mizubayasbi,H., Actu Metallurgica et Materiulia, 1995,43, 1495. Mizubayaabi,H. aud Okuda, S., Physical Review B, 1989,40,8057. Landau, L.D. aud Lifabitz, E.M., Electrodynamics of Continuous Media. Pergamon Press, England, 1984. Wang, Y., Qiao, G., Liu, X, Ding, B. aud Hu, Z., MaterialsLetters, 1993, 17, 152. Liu, X., Ding, B., Hu, Z., Lu. K., aud Wang, Y., Physica, 1993, 192B, 345. Qin, R.S. aud Zbou, B.L., NatwstructuredMateri& 1996,7,741. Qiu, R.S. and Zbou, B.L., Actu Metullutgica Sinicu. 1996.32, 1093. Lu, K., Wand, LT. audWei, W.D., Journal of Applied Physics, 1991,69,522. Lu, K., Wand, J.T. audWei, W.D., Journal of Physics D, 1992,25,508. Lu, K., Luck, R. and Predel, B., Journal of Non-Crystalline Solidr, 1993.156,589. Liu, S.H., Poirer, D.R. aud Qausry,RN., Metallurgical Transuctions, 1995,26A, 741.
Pergamon
PII 809659773(98)00030-0
SUSPENSION EFFECT OF NANOCRYSTALLINE GRAIN GROWTH UNDER ELECTROPULSING R.S. Qid2,
S.X. Sd, J.D. Guo’, G.H. I-Ie’J, and B.L. Zhou’J
‘Institute of Metal Research, Chinese Academy of Science, Shenyang 110015, P.R. China 21nstitute of Corrosion and Protection for Metals, Chinese Academy of Science, Shenyang 110015, P.R. China 31ntemational Center for Materials Physics, Chinese Academy of Science, Shenyang 110015, P.R. China (AcceptedFebruary 11,1998)
Abstract - The e#ect of electtopulsingon nanocrystallinegrain growth in amorphous material is considered. A size-dependentgrowthrate of a nanoscale grain is revealed, which is due to the size-dependentelectrical resistivity.A suspensione$ect isfound when the densityof electric current is at a critical valuej,. Theformed grain may shrinkwhen the densityof applied electric current is higher thanjC. Numerical calculationshowsthatj, is about 2.59 x IdAl& for nanocryst#alline Nis&o alloy withgrain size of 13 nm. 01998 Acta MetallurgicaInc.
I. INTRODUCTION Electropulsing with current density of ldA/mm2 is used frequently in synthesizing nanocrystalline alloy from amorphous alloy (1). The advantages of using electropulsing over annealing are that lower temperature and less time are needed in the experiment. It is known from both experimental and theoretical studies that the side effects of electropulsing (i.e. skin, Joule heat, pinch force) are less important However, the effects of electropulsing on crystallization are far from understood. Our previous work came to the conclusion that electropulsing can increase the nucleatioln rate and undercooling (2,3). The aim of this paper is to the study of effect of electropulsing on the growth rate of a nanocrystalline grain. The investigation is along the standard line of the first-order phase transition theory. The paper is organized as follows. In Section II we present our theoretical study for the effect of electropulsing on the growtbrate of ananoscale-grain in amorphous alloy by which the suspension effect is found. In Section III we analyze the suspension effect and calculate it numerically. In Section IV some concluding remarks are given. 71
72
Fl!3QIN, SX Su, JD Guo, GH HEANDBL ZMOU
IL THEORETICAL
STUDY
For simplicity, we consider a system which only one grain is formed in the amorphous material. The grain is supposed to grow with the velocity (4) U=U0exp(-f)[l-exp(+)]
111
The preexponential factor Uo is typically of the order of 16 cm/s (roughly estimated as aov, ae is the characteristic interatomic length in a solid, and v is the characteristic atomic frequency inasolid). Vandnareusuallytermedasthekineticandthermodynamicbarriers tocrystallization, respectively, k is Boltzmann’s constant, and T is temperature. The multiplier in square brackets on the right-hand side in eq. [l] reflects the possibility of a backward process when the energy n gainedbyamoleculejoiningagrainisnotlargeenough. Aslongasitisgivenbyeq. [l],thegrowth rate is determinated by both kinetic and thermodynamic barriers. There are many experimental investigations on the change of kinetic barrier under electropulsing during crystallization in amorphous alloys (5.6). Here, one would like topay more attention to the thermodynamic barrier. In a current-carrying system, f2 consists of two parts Q=szo+f&
PI
where f& is the free-energy change due to grain growth in a current-free system, which is given by f2, = -AVji+AS.a,, where AV is the volume of a molecule, ji(>O) is the free energy difference between the nanocrystalline phase and amorphous phase per unit volume. AS is the increase of the surface area of a grain when a molecule is joined to it, and cranis the interfacial energy per unit area. Anegative value of Sz implies that a grain has more probability to grow, otherwise the grain may shrink. f& is the energy change in the grain growing process caused by electropulsing, which can be determined from the following equation (7)
where p is the magnetic susceptibility. In the temperature region of our studies l.t = po, ~MIis the magnetic susceptibility in vacuum, ;I (r) and 1; (r) are current densities when the grain sixes are at and ax, respectively. The change of the volume of the grain is denoted by AV. The relationship between al and a2 is given by: -a:)=AV
SUSPENSON EFFECT OFNAWCRYSTALLF-JE GRAIN GROWTH UNDER ELECTROPULWG
73
Electric current distribution in the adiabatic approximation can be determined from the following equ.ations: divji = 0
rOt(ji/@ = 0,
161
where i=O,l,Z!represent in amorphous, grain with size al and size az, respectively. oi is the conductivity of corresponding medium. In the calculation of current density distribution, the medium can be assumed infinite (because the grain size is in nanoscale and the amorphous size is above microscale) and the solution of eq. [6] is as follows:
[71
where @I
-sineq
i’o = jo(cosef
191
where crois the conductivity of amorphous, 3 and 6 are the unit vector in spherical coordinate. Bringing eq. 1173 to eq. [4] one obtains the f& with an accuracy of order a; by using the method of physical replacement:
where b is the size of amorphous alloy. It is obvious that in the region of
&, Cl
In b a: 0 =1 ln b
a;
0 =2 f& will be positive, otherwise f& will be negative.
(4, > 0,41>
0)
WI
74
RS QIN, SX Su, JD Guo, GH HE ANDBL ZHOU
4
- 650 - 600 - 550 - 500
/\”’ 350,
. 0
I 2
.
, 4
.
/_:I1 I 6
.
I 0
- 450
- 400
I . t . I . I . I . 1350 14 16 18 20 10 12
Grain size
nm
Figure 1. A schematic diagram of the relationship between electrical resistivity and grain size.
ThepositivevalueofR,impliesthattheelectropulsingmayreducethegrowthrateofagrain. Intheconditionofn,=-~(i.e.thecurrentdensityisinacriticalvalue)thegrainwillstopgrowing. The zero growth rate of a grain is termed suspension effect.
III. DISCUSSION AND NUMERICAL CALCULATION Eq. [ 1l] is only valid for nanoscale grain. The electrical conductivity of a metallic material with coarse grain is higher than that of a corresponding amorphous material generally. According to eq. [8], & will be negative, but the electrical conductivity of some nanocrystalline materials is lower than that of the corresponding amorphous materials. For example, the electrical conductivity ratio between nanocrystalline FeMoSiB alloy with grain size of 28 nm and the amorphous FeMoSiB alloy is 1.02 at room temperature (8), and it is 1.26 of nanocrystalline FeCuSiB alloy with grain size of 25 nm to its amorphous alloy at room temperature (9). So & will be positive in this case. Theoretical investigation shows that the electrical conductivity of the nanocrystalline materials depends on the grain size (lo), which is due to the size-dependent lattice distortion and interface excess volume (11). So the growth rate of a nauoscale grain will depend on the grain size under electropulsing. In order to know the effect of electropulsing on grain growth quantitatively, one would like to perform some numerical calculation. Nis&u is a well studied material. Amorphous Nis&o alloy is synthesized by a single rollerrapidlyquenching equipment. Nanocrystalline Nis&o alloy can be obtained by the method of heat treatment. A nanocrystalline Nis&o alloy with the mean grain size of 13 nm is synthesized in a heat treatment procedure of rapidly heating (3OOK/min) the amorphous sample to 598 K, anuealing isothermally for 10 min and cooling to room temperature
75
SUSPENSIONEFFECTOF NAN~CXYSTALLINE GRAINGFOWTHIJNDERELECTROPULSING
3 & 2
.b G s 0 .r( Ei R
S!OOO
. __--_ 680 - _
Linear Saddle
shape shape
-640 -1960 -
-3280 $ t
-4600 0
1
I 2
Current density
II II II II
I 3
4
103Almm*
Figure 2. Thermodynamic barrier vs. current density, where j, is the critical value.
( 12).The electrical resistivity of the nanocrystallineNi8&u alloy with the mean grain size of 13 nm is 585.0 + O.l23T(@. cm), and the electrical resistivity of the amorphous b&&u alloy is 210.7 + 0.505T Q&l - cm) (13). The Gibbs fret energy change in amorphous-nanocrystline transition is A.6 J/ mol in 598 K (14). The aim of the present calculation is to know the critical value j, by which the grain cannot grow yet. The problem here is that one does not know the relationshipbetween electrical conductivity and grain size. In order to overcome this difficulty two cases are assumed as illustrated in Figure 1: (a) the electrical resistivity is proportional to grain size linearly in this size range; (b) the relationshipbetween electrical resistivity and grain sizeis a saddle-shape.The results of numerical calculations are shown in Figure 2. The critical current densities for above-mentioned two cases are 2.59 x @A/mm2 and 2.56 x ldA/mm2, respectively. The grain may have more possibility to shrink when the current density is above the critical value. The present investigation is also valid for the solidification of a melt. Experiment reveals a size-dependent electrical resistivity behavior in the soliditication of Pb (15). The relationship between electrical resistivity and grain size is a saddle-shapein the early stage of crystallization. There will be a suspension effect when the current is carried in a certain size range, which will be useful for the fabricating bulk nanocrystallineby electric rheological technique. IV. CONCLUSION The nanocrystalline grain takes an unusual correspondence to the applied electropulsing. The effect of electropulsing on the growth rate of a nanoscale grain is size dependent. When the electrical conductivity of a grain is largerthan that of amorphous,theelectropulsing would prevent
76
=
GN,
Sx
.%,
JD
Guo, GH HE AND BL Ztiou
the grain from growing. In the extra condition of applied electric current density larger than the critical value, the grain may shrink.
ACKNOWLEDGMENT This work is supported by tbe Chinese National Science Foundation -59431040, and Liaoning Doctoral Foundation - 97 1106.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
Teng,G., Chao,Y.,Lai, Z. and Dong,L., ChineseScience Bulletin, 1994,39,974. Qin, R.S., Yan, H.C., He, G.H. aud Z~OU,B.L., Chinese Journal ofMateriols Research, 1995,9, 219. Qiu, R.S. and Zbou, B.L., ChineseJournal ofMaterials Research, 1997, 11.69. Fine, M.F., Phase Trun&ormation in Condensed Systems, MacMillan, New York, 1974. Takemoto, R. and Mizubayasbi,H., Actu Metallurgica et Materiulia, 1995,43, 1495. Mizubayaabi,H. aud Okuda, S., Physical Review B, 1989,40,8057. Landau, L.D. aud Lifabitz, E.M., Electrodynamics of Continuous Media. Pergamon Press, England, 1984. Wang, Y., Qiao, G., Liu, X, Ding, B. aud Hu, Z., MaterialsLetters, 1993, 17, 152. Liu, X., Ding, B., Hu, Z., Lu. K., aud Wang, Y., Physica, 1993, 192B, 345. Qin, R.S. aud Zbou, B.L., NatwstructuredMateri& 1996,7,741. Qiu, R.S. and Zbou, B.L., Actu Metullutgica Sinicu. 1996.32, 1093. Lu, K., Wand, LT. audWei, W.D., Journal of Applied Physics, 1991,69,522. Lu, K., Wand, J.T. audWei, W.D., Journal of Physics D, 1992,25,508. Lu, K., Luck, R. and Predel, B., Journal of Non-Crystalline Solidr, 1993.156,589. Liu, S.H., Poirer, D.R. aud Qausry,RN., Metallurgical Transuctions, 1995,26A, 741.