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Nucleation in nanostructures: Effects of embryo loss
NANoSTRUCTURED MATERIALS VoL. 2, PP. 295-300, 1993 COPYRIGHT©1993 PERGAMONPRESSLTD, ALL RIGHTSRESERVED.
NUCLEATION
IN NANOSTRUCTURES: EMBRYO LOSS
0965-9773/93 $6.00 + .00 PRINTEDIN THE USA
EFFECTS
OF
K. C. Russell Department of Materials Science and Engineering Department of Nuclear Engineering Massachusetts Institute of Technology, Cambridge, MA 02139
(Accepted May 1993) Abstract---Conventional nucleation theory considers only the growth and decay of embryos by single atom addition or subtraction. The numerous processes involved in mechanical alloying, such as creation and movement of dislocations and point defects, and passage of thermal and stress waves, could readily destroy a subcritical embryo which typically contains at most only a few tens of atoms. This paper modifies the nucleation equation to accountfor such embryo loss, and solves the resulting equation analytically in the steady state approximation. The effects of embryo loss are described by several dimensionless groupings of kinetic and thermodynamic parameters. Embryo loss isfound significant only if the probability of destruction is comparable to or greater than the probability of growth by single atom capture. The effects of embryo loss are greatest for phases which form with large critical nuclei or complex unit cells. INTRODUCTION Formation of nanostructures by mechanically alloying puts the system into an unusual physical and chemical regime (1,2). Intense, continual deformation dramatically deforms the metal particles, giving a very fine scale lamellar structure with a very high ratio of surface to volume. The deformation also produces large numbers of mobile point and line defects. In addition the comminution process produces thermal and stress pulses of significant magnitude. Phase selection during mechanical alloying is an extremely complex process, and is certainly not explainable on the basis of formation of the minimum free energy phase (3,4). For example, solid solution phases which are thermally unstable are formed, as are amorphous phases and intermetallics. The energy of the milling process is clearly sometimes being utilized in a way to cause the behavior of the system to deviate sharply from what would be observed thermally. Phase selection is most easily altered in the nucleation step, and a number of interesting nucleation problems have surfaced in nanostructured materials. This paper attacks the problem of destruction of subcritical nuclei before they can reach critical nucleus size. No specific destruction mechanism is assumed, though moving dislocations and thermal and stress waves are likely candidates. This paper outlines the analysis; a more complete discussion will be published later (5). 295
296
KC RUSSELL
CLASSICAL THEORY We will very briefly outline the classical theory of nucleation to provide a point of reference for the inclusion of cluster destruction. The development of nucleation theory is discussed in detail by Feder, et al. (6). The flux of clusters, Jx, along the size coordinate, x, is given by:
_
o(~Cx/C~,~
)
(1)
where:
Cx = concentration of clusters of x atoms (x-mer); 13x= rate of capture of solute atoms by x-mer; C ° -- concentration of n-mer in equilibrium with the bulk material in the system. Application of the divergence theorem to Equation 1 gives:
(2) Equation (2) is deceptively difficult. It cannot be solved in closed form for the transient case and even numerical evaluation is difficult. Equation (2) may readily be integrated at steady state, 0c where ~ - = 0, to give the steady state nucleation rate of critical nuclei, J*.
J* = Zfl* Cx, o
(3)
I~* and C°, are evaluated at critical nucleus size. From Equations (1,3):
z
(4)
The Zeldovich factor, Z, accounts for the concentration of critical nuclei being half the equilibrium and for the probability of supercritical nuclei decaying to subcritical size. Typically, Z ~- 0.1. By far the most important quantity is J* is C°,, which is related exponentially to the free energy of forming a critical nucleus. EFFECTS OF CLUSTER LOSS We assume that suberitical clusters are are destroyed at a frequency, 1¢. The continuity equation then becomes:
297
NUCLEATIONIN NANOSTRUCTURES; EFFECTSOF EMBRYOLOSS
-
~L
x x~.
~
(5)
) j - ~Cx
Equation 5 is usually intractable analytically, even at steady state. Solution is possible in terms of Bessel functions or Legendre polynomials only for certain specific functional forms of the coefficients of the derivatives. Equation 5 may be rendered tractable if we restrict our attention to steady state and do the following: • •
Define y = Cx / C °. Define K2 = ~Jl~xand assume that ~ has the same weak dependence on x as ~x.
•
Define G2=
d l n C ~ = 1--L(dAG°x~andlinearize the activation barder as shown in dx kT ~. dx )
Figure 1. Then (32 = AG*/(x*kT) (for x
(6)
where y, - - -dy . - The ODE is of standard form and may be solved directly by the method of the dx Laplace Transform (7) or from Handbooks (8). Details of the solution will be published later (5). 90
CLASSICAL .......
LINEARIZED
Iv,
L9
A G*/kT
60
30
0 0
30
6O
90
# ATOMS
Figure 1. Comparison of classical and linearized nucleation barriers. The height of the barrier (AG*/kT) and critical nucleus size (x*) are the same for the two cases.
298
KC RUSSELL
RESULTS AND DISCUSSION
We assume boundary conditions such that very small clusters are present at the equilibrium concentration and all clusters reaching critical size are removed from the system. Thus, y(0) = 1, y(x*) = 0. The solution to Equation 6 is: erl xer2 x*_er2 xerl x*
y=
er2X._erlX,
2
where
2 1/2
GZIl+(l+4KZ) rl= 2 L [, G 4 J
], J
(7)
G2I 1 ('1+4K2~)1/2] r2=-'2"[-~ "~-~ J
Equation 7 gives for the first time the steady state cluster concentrations in nucleation with cluster loss. The cluster flux equation (1) does not depend explicitly on the rate of cluster loss, so Equation 3 is still valid, with:
Cluster loss thus affects J* through a modified Zeldovich factor, given by: dy] - ~
(rl - r2)erlX*e r2x* x*
er2X*-e rlx*
(8)
Equation 8 is too complicated to analyze directly. Instead, we consider some limiting cases, which may be classified according to the value of 4K2/G4 which appears in both rl and r2. We first check the solution to Equation 6 by eliminating cluster destruction to see if the classical nucleation rate equation is recovered. We set K 2 = 0 to obtain Z = G2
(9)
We thus recover the classical expression, though Z is increased from the usual value of--0.1 to G2, which in general is expected to lie between about 1 and 10. The differenceis believed due to errors introduced in replacing a difference equation by a derivative in applying the divergence theorem in obtaining Equation (2). Some fine structure is lost by going from differences to derivatives, and perhaps in the linearization of AG°, but recovery of the main features of the classical nucleation rate equation shows these effects to be minor. We now assume that the destruction rate is small but finite, so that 0<4K2/G4 <<1. The inequality is then valid if r < 13". The effect of this low level of nucleus destruction is to modify the Zeldovich factor to:
NUCLEATIONIN NANOSTRUCTURES; EFFECTSOF EMBRYOLoss
Z = G 2 exp -
299
(lO)
Equations (9,10) may be more easily interpreted by substituting AG*/(x*kT) for G2. Examination of Equation 10 shows that if x* is only a few atoms, Z is near unity and cluster destruction has little effecton the nucleation rate. If, however, x* is large, nucleus destruction gives Z<
D.R.Maurice and T.H. Courmey,Metall. Trans. 21A, 289 (1990). A.K.Bhattacharya and E. Arzt, Scripta Metall. et Mater. 27, 749 (1992).
300
3. 4. 5. 6. 7. 8.
KC RUSSELL
L. Schultz, Phil. Mag. B 61,453 (1990). Q.J. Meng, C.W. Nieh, and W.L. Johnson, Appl. Phys. Lett. 51, 1693 (1987). K.C. Russell, Acta Metallurgica (In Preparation). J. Feder, K.C. Russell, J. Lothe, and G.M. Pound, Advan. in Phys. 15, 111 (1966). R.V. Churchill, Modem Operational Mathematics in Engineering, p. 1, McGraw-Hill, New York (1944). D.R. Lide, ed., Handbook of Chemistry and Physics, 71st ed., p. A-65, CRC Press, Boca Raton, Florida (1990).
NUCLEATION
IN NANOSTRUCTURES: EMBRYO LOSS
0965-9773/93 $6.00 + .00 PRINTEDIN THE USA
EFFECTS
OF
K. C. Russell Department of Materials Science and Engineering Department of Nuclear Engineering Massachusetts Institute of Technology, Cambridge, MA 02139
(Accepted May 1993) Abstract---Conventional nucleation theory considers only the growth and decay of embryos by single atom addition or subtraction. The numerous processes involved in mechanical alloying, such as creation and movement of dislocations and point defects, and passage of thermal and stress waves, could readily destroy a subcritical embryo which typically contains at most only a few tens of atoms. This paper modifies the nucleation equation to accountfor such embryo loss, and solves the resulting equation analytically in the steady state approximation. The effects of embryo loss are described by several dimensionless groupings of kinetic and thermodynamic parameters. Embryo loss isfound significant only if the probability of destruction is comparable to or greater than the probability of growth by single atom capture. The effects of embryo loss are greatest for phases which form with large critical nuclei or complex unit cells. INTRODUCTION Formation of nanostructures by mechanically alloying puts the system into an unusual physical and chemical regime (1,2). Intense, continual deformation dramatically deforms the metal particles, giving a very fine scale lamellar structure with a very high ratio of surface to volume. The deformation also produces large numbers of mobile point and line defects. In addition the comminution process produces thermal and stress pulses of significant magnitude. Phase selection during mechanical alloying is an extremely complex process, and is certainly not explainable on the basis of formation of the minimum free energy phase (3,4). For example, solid solution phases which are thermally unstable are formed, as are amorphous phases and intermetallics. The energy of the milling process is clearly sometimes being utilized in a way to cause the behavior of the system to deviate sharply from what would be observed thermally. Phase selection is most easily altered in the nucleation step, and a number of interesting nucleation problems have surfaced in nanostructured materials. This paper attacks the problem of destruction of subcritical nuclei before they can reach critical nucleus size. No specific destruction mechanism is assumed, though moving dislocations and thermal and stress waves are likely candidates. This paper outlines the analysis; a more complete discussion will be published later (5). 295
296
KC RUSSELL
CLASSICAL THEORY We will very briefly outline the classical theory of nucleation to provide a point of reference for the inclusion of cluster destruction. The development of nucleation theory is discussed in detail by Feder, et al. (6). The flux of clusters, Jx, along the size coordinate, x, is given by:
_
o(~Cx/C~,~
)
(1)
where:
Cx = concentration of clusters of x atoms (x-mer); 13x= rate of capture of solute atoms by x-mer; C ° -- concentration of n-mer in equilibrium with the bulk material in the system. Application of the divergence theorem to Equation 1 gives:
(2) Equation (2) is deceptively difficult. It cannot be solved in closed form for the transient case and even numerical evaluation is difficult. Equation (2) may readily be integrated at steady state, 0c where ~ - = 0, to give the steady state nucleation rate of critical nuclei, J*.
J* = Zfl* Cx, o
(3)
I~* and C°, are evaluated at critical nucleus size. From Equations (1,3):
z
(4)
The Zeldovich factor, Z, accounts for the concentration of critical nuclei being half the equilibrium and for the probability of supercritical nuclei decaying to subcritical size. Typically, Z ~- 0.1. By far the most important quantity is J* is C°,, which is related exponentially to the free energy of forming a critical nucleus. EFFECTS OF CLUSTER LOSS We assume that suberitical clusters are are destroyed at a frequency, 1¢. The continuity equation then becomes:
297
NUCLEATIONIN NANOSTRUCTURES; EFFECTSOF EMBRYOLOSS
-
~L
x x~.
~
(5)
) j - ~Cx
Equation 5 is usually intractable analytically, even at steady state. Solution is possible in terms of Bessel functions or Legendre polynomials only for certain specific functional forms of the coefficients of the derivatives. Equation 5 may be rendered tractable if we restrict our attention to steady state and do the following: • •
Define y = Cx / C °. Define K2 = ~Jl~xand assume that ~ has the same weak dependence on x as ~x.
•
Define G2=
d l n C ~ = 1--L(dAG°x~andlinearize the activation barder as shown in dx kT ~. dx )
Figure 1. Then (32 = AG*/(x*kT) (for x
(6)
where y, - - -dy . - The ODE is of standard form and may be solved directly by the method of the dx Laplace Transform (7) or from Handbooks (8). Details of the solution will be published later (5). 90
CLASSICAL .......
LINEARIZED
Iv,
L9
A G*/kT
60
30
0 0
30
6O
90
# ATOMS
Figure 1. Comparison of classical and linearized nucleation barriers. The height of the barrier (AG*/kT) and critical nucleus size (x*) are the same for the two cases.
298
KC RUSSELL
RESULTS AND DISCUSSION
We assume boundary conditions such that very small clusters are present at the equilibrium concentration and all clusters reaching critical size are removed from the system. Thus, y(0) = 1, y(x*) = 0. The solution to Equation 6 is: erl xer2 x*_er2 xerl x*
y=
er2X._erlX,
2
where
2 1/2
GZIl+(l+4KZ) rl= 2 L [, G 4 J
], J
(7)
G2I 1 ('1+4K2~)1/2] r2=-'2"[-~ "~-~ J
Equation 7 gives for the first time the steady state cluster concentrations in nucleation with cluster loss. The cluster flux equation (1) does not depend explicitly on the rate of cluster loss, so Equation 3 is still valid, with:
Cluster loss thus affects J* through a modified Zeldovich factor, given by: dy] - ~
(rl - r2)erlX*e r2x* x*
er2X*-e rlx*
(8)
Equation 8 is too complicated to analyze directly. Instead, we consider some limiting cases, which may be classified according to the value of 4K2/G4 which appears in both rl and r2. We first check the solution to Equation 6 by eliminating cluster destruction to see if the classical nucleation rate equation is recovered. We set K 2 = 0 to obtain Z = G2
(9)
We thus recover the classical expression, though Z is increased from the usual value of--0.1 to G2, which in general is expected to lie between about 1 and 10. The differenceis believed due to errors introduced in replacing a difference equation by a derivative in applying the divergence theorem in obtaining Equation (2). Some fine structure is lost by going from differences to derivatives, and perhaps in the linearization of AG°, but recovery of the main features of the classical nucleation rate equation shows these effects to be minor. We now assume that the destruction rate is small but finite, so that 0<4K2/G4 <<1. The inequality is then valid if r < 13". The effect of this low level of nucleus destruction is to modify the Zeldovich factor to:
NUCLEATIONIN NANOSTRUCTURES; EFFECTSOF EMBRYOLoss
Z = G 2 exp -
299
(lO)
Equations (9,10) may be more easily interpreted by substituting AG*/(x*kT) for G2. Examination of Equation 10 shows that if x* is only a few atoms, Z is near unity and cluster destruction has little effecton the nucleation rate. If, however, x* is large, nucleus destruction gives Z<
D.R.Maurice and T.H. Courmey,Metall. Trans. 21A, 289 (1990). A.K.Bhattacharya and E. Arzt, Scripta Metall. et Mater. 27, 749 (1992).
300
3. 4. 5. 6. 7. 8.
KC RUSSELL
L. Schultz, Phil. Mag. B 61,453 (1990). Q.J. Meng, C.W. Nieh, and W.L. Johnson, Appl. Phys. Lett. 51, 1693 (1987). K.C. Russell, Acta Metallurgica (In Preparation). J. Feder, K.C. Russell, J. Lothe, and G.M. Pound, Advan. in Phys. 15, 111 (1966). R.V. Churchill, Modem Operational Mathematics in Engineering, p. 1, McGraw-Hill, New York (1944). D.R. Lide, ed., Handbook of Chemistry and Physics, 71st ed., p. A-65, CRC Press, Boca Raton, Florida (1990).