Physical metallurgy

Nucleation and growth kinetics (異質成核推導補充) Materials Science and EngineeringPhase Transformations in Metals

Homogeneous Nucleation  Gibbs free energy (G) - Gibbs free energy is a function of the internal energy of the system (enthalpy, H) and a measurement of the randomness or disorder of the atoms or molecules (entropy, S). - A transformation will occur spontaneously only when the change in free energy G has a negative value.  Case of homogeneous nucleation - For the sake of simplicity, let us first consider the solidification of a pure material, and that nuclei of the solid phase form in the interior of the liquid as atoms cluster together so as to form a packing arrangement similar to that found in the solid phase. - There are two contributions to the total free energy change that accompany a solidification transformation. The first is the free energy difference between the solid and liquid phases, or the volume

free energy, Gv. The second energy contribution results from the formation of the solid–liquid phase boundary during the solidification transformation, or the surface energy, . FIGURE 12.1 Schematic diagram showing the nucleation of a spherical solid particle in a liquid. 2 Materials Science and EngineeringPhase Transformations in Metals Homogeneous Nucleation 4 3 2 V: volume of spherical nucleus, G  VGv  A  r Gv  4r  3 G : volume free energy, d(G) v When r = r*   4r 2G  8r  0 A: surface area of spherical nucleus, dr v : surface free energy, 2  r*   r: radius of spherical nucleus, Gv 3 r*: critical radius of spherical nucleus, 4 2 3 2 2 16  G*   ( ) Gv  4 ( )   2 G*: critical free energy, activation energy 3 Gv Gv 3(Gv )

FIGURE 12. 2 (a) Schematic curves for volume free energy and surface free energy contributions to the total free energy change attending the formation of a spherical embryo/nucleus during Solidification, (b) Schematic plot of free energy versus embryo/nucleus radius, on which is 3 shown the critical free energy change (G*) and the critical nucleus radius (r*). Materials Science and EngineeringPhase Transformations in Metals Homogeneous Nucleation

The volume free energy change Gv is the driving force for the solidification transformation, and its magnitude is a function of temperature. At the equilibrium

solidification temperature Tm, the value of Gv is zero, and with diminishing temperature its value becomes increasingly more negative. It can be shown that

Gv is a function of temperature as

H f (Tm T) Gv  Tm 2 2  2T   1   r*       m    H (T T)     Gv f m  H f  Tm T 

Tm 16 3 16 3 16 3T 2  1  G*     m   3(G ) 2 2  3H 2  (T  T ) 2 v H f (Tm  T )  f  m 3   Tm 

Hf: latent heat of fusion,

Tm: equilibrium melting temperature (K), T: real solidification temperature (K), 4 Materials Science and EngineeringPhase Transformations in Metals Homogeneous Nucleation Both the critical radius r* and the activation free energy G* decrease as temperature T decreases. With a lowering of temperature at temperatures

below the equilibrium solidification temperature (Tm), nucleation occurs more readily.

 2T   1  16 3T 2  1 r*   m    G*   m    H  T T   2  2  f   m   3H f  (Tm T)

<

FIGURE 12.3 Schematic free energy- versus-embryo/nucleus radius curves for two different temperatures. The critical free energy change (G*) and critical nucleus radius (r*) are Indicated for each temperature. 5 Materials Science and EngineeringPhase Transformations in Metals Nucleation rate The number of stable nuclei n* (having radii greater than r*) is a function of temperature as

Where K1 is related to the total number of nuclei of the solid phase Another temperature-dependent step influences nucleation: the clusting of atoms by short-range diffusion during the formation of nuclei. The diffusion effect is related to

the frequency at which atoms from the liquid attach themselves to the solid nucleus, d.

Where Qd is a temperature-independent parameter-the activation energy for diffusion,

and K2 is a temperature-dependenct constant. A decrease of temperature results in a

reduction in d.

The nucleation rate is simply proportional to the product of n* and d, that is

6 Materials Science and EngineeringPhase Transformations in Metals

FIGURE 12.4 For Solidification: (a) number of stable nuclei v.s. temperature, (b) frequency of atomic attachment v.s. temperature and (c) nucleation rate v.s. temperature.

7 Materials Science and EngineeringPhase Transformations in Metals

Heterogeneous Nucleation

In practical supercoolings are often on the order of only several degrees Celsius.The reason for this is that the activation energy for nucleation is lowered when nuclei form on preexisting surfaces or interfaces, since the surface free energy is reduced. In other words, it is easier for nucleation to occur at surfaces and interfaces than at other sites. Again, this type of nucleation is termed heterogeneous. Let us consider the nucleation, on a flat surface, of a solid particle from a liquid phase. It is assumed that both the liquid and solid phases “wet” this flat surface, that is, both of these phases spread out and cover the surface.

 IL   SI   SL cos

FIGURE 12.5 Heterogeneous nucleation of

a solid from a liquid. The solid–surface (SI),

solid–liquid (SL), and liquid–surface (IL) interfacial energies are represented by vectors. The wetting angle () is also shown.

8 Materials Science and EngineeringPhase Transformations in Metals

Heterogeneous Nucleation

2 ASI   (r sin )

2 ASL  2r (1 cos ) r 3 V  (2  3cos  cos3 ) r sin r sin S 3      cos r r IL SI SL   Ghet  VS Gv  ASL SL  ASI  SI  ASI  IL

 Ghet  G  S()

3 4r 2 VS: volume of heterogeneous nucleus, G  G  4r  3 v SL Gv: volume free energy, (2  cos )(1 cos ) 2 ASL: area of solid-liquid interface, S()  4 ASI: area of solid-surface interface, 2 SL SL: surface free energy of solid-liquid interface,  r*   Gv SI: surface free energy of solid-surface interface, 3 r: radius of spherical nucleus, * 16 SL  Ghet  2  S() r*: critical radius of spherical nucleus, 3(Gv ) G *: critical free energy, activation energy * * het  Ghet  Ghom  S( ) 9 Materials Science and EngineeringPhase Transformations in Metals

Heterogeneous Nucleation

 G * Q N  exp( )exp( d ) kT kT G *  G *  S( ) 16 3T 2  1 het hom G*   m    3H 2  (T T)2 2  3cos  cos3   f  m S()  4 FIGURE 12.6 Schematic free energy-versus- FIGURE 12.7 Nucleation rate versus embryo/nucleus radius plot on which is presented temperature for both homogeneous and curves for both homogeneous and heterogeneous heterogeneous nucleation. Degree of nucleation. Critical free energies and the critical supercooling (T) for each is also shown. radius are also shown. 10 Materials Science and EngineeringPhase Transformations in Metals

Heterogeneous versus Homogeneous Nucleation  Critical radius The critical radius for heterogeneous nucleation is the same as for homogeneous.  Activation energy The activation energy barrier for heterogeneous nucleation is smaller than the homogeneous barrier.  Supercooling A much smaller degree of supercooling is required for heterogeneous nucleation.

11 Materials Science and EngineeringPhase Transformations in Metals

Growth The growth step in a phase transformation begins once an embryo has exceeded the critical size, r*, and becomes a stable nucleus. The growth process will cease in any region where particles of the new phase meet, since here the transformation will have reached completion. Particle growth occurs by long-range atomic diffusion, which normally involves several steps- for example, diffusion through the parent phase, across a phase boundary, and then into the nucleus. Consequently, the growth rate is determined by the rate of diffusion, and its temperature dependence is the same as for the diffusion coefficient,   Q  G  C  exp   kT  Q: activation energy, independent of temperature; C: constant, independent of temperature. At some temperature, the overall transformation rate   is equal to some product of N and G. The third curve for the total rate represents this combined effect. The general shape of this curve is the same as for the nucleation rate, in that it has a peak or maximum that has been shifted upward relative to the curve. FIGURE 12.8 Schematic plot showing curves for   nucleation rate (N), growth rate (G), and overall12 transformation versus temperature. Materials Science and EngineeringPhase Transformations in Metals

Growth As we shall see below, the rate of transformation and the time required for the transformation to proceed to some degree of completion are inversely proportional to one another. First, the size of the product phase particles will depend on transformation temperature. Secondly, when a material is cooled very rapidly through the temperature range encompassed by the transformation rate curve to a relatively low temperature where the rate is extremely low, it is possible to produce nonequilibrium phase structures.

FIGURE 12.9 Schematic plots of (a) transformation rate versus temperature, and (b) logarithm time [to some degree (e.g., 0.5 fraction) of transformation] versus temperature. The curves in both (a) and (b) are generated from the same set of data—i.e., for horizontal axes, the time 13 [scaled logarithmically in the (b) plot] is just the reciprocal of the rate from plot (a). 相變態 Heterogeneous Nucleation

 Driving force and critical size of heterogeneous nucleation

- Balance among tensions SLsin Consider a solid embryo forming in contact with a perfectly flat mold wall as depicted

in Fig. 4.7. Assuming SL is isotropic,

the total interfacial energy of the system SLcos is minimized if the embryo has the shape of a spherical cap. Fig. 4.7 Heterogeneous nucleation of spherical  ML   SM   SL cos    (4.14) cap on a flat mold wall. cos  ML SM  SL

SL: solid/liquid interfacial tension, ML: mold/liquid interfacial tension,

SM: solid/mold interfacial tension, : a wetting angle.

Note that the vertical component of SL remains unbalanced. Given time, this force would pull the mold surface upwards until the surface tension forces balance in all direction.

14 相變態 Heterogeneous Nucleation

- Driving force The formation of such an embryo will be associated with an excess free energy

Ghet VS GV  ASL SL  ASM  SM  ASM ML (4.15)

VS: volume of the spherical cap, ASL: areas of solid/liquid interface,

ASM: areas of solid/mold interface, SL: solid/liquid interfacial tension,

ML: mold/liquid interfacial tension, SM: solid/mold interfacial tension.

 2  2 2  2 ASL    r sind rd   2r sind  2r cos  2r 1 cos  0 0 0 0 2 2 2 ASM   r sin   r sin   2 r 1 2 VS     dr r sind rd   r sin  r cos 0 0 0 3  2 1 3 1 3 2  VS    r sind d  r 1 cos  cos 0 0 3 3  rsin  2 3 1 3 3  VS   r sind  r cos  cos   0 3 3 2 3 1 3 3  V  r 1 cos  r cos  cos   r S 3 3 3 r 3   VS  2  3cos  cos   3 15 相變態 Heterogeneous Nucleation

 Ghet VS GV  ASL SL  ASM  SM  ASM  ML (4.15)

 Ghet VS GV  ASL SL  ASM  SM  ML 

  ML   SM  SL cos (4.14)

 Ghet VS GV  ASL SL  ASM  SL cos 

 Ghet VS GV  SL ASL  ASM cos 

2 2 2 ASL  ASM cos  2r 1 cos  r sin  cos  r 2 2  2cos  sin2  cos   r 2 2  2cos  1 cos2  cos   r 2 2  3cos  cos3  

Ghet  VS GV   SL ASL  ASM cos  3 r 3  2 3   2  3cos  cos  GV   SL r 2  3cos  cos    3   2  3cos  cos3    4r 3     2     GV  4r  SL   4   3  16 相變態 Heterogeneous Nucleation

 4r 3   2  Ghet   GV  4r  SL  S  (4.16)  3 

2  3cos  cos3  2  cos 1 cos 2 S    (4.17) 4 4

Note that except for factor S(), the expression for heterogeneous nucleation is the same as that obtained for homogeneous nucleation. S() has a numerical value  1 dependent only on .

1

0.8

) 0.6 

S( 0.4

0.2

0 0 20 40 60 80 100 120 140 160 180  () 17 相變態 Heterogeneous Nucleation

- Critical nucleus size

dGhet/dr = 0 at r = r* d  4     r 3G  4r 2  S   0 dr  3 V SL     rr* 2  4r * GV  8r * SL  S   0 2  4r * GV  8r * SL  0

 2 r*  SL (4.18) GV

 4r 3   2   Ghet   GV  4r  SL  S   3  Fig. 4.8 The excess free energy of  3  solid clusters for homogeneous and  2 SL   4   2  heterogeneous nucleation. Note r* *   GV   2 SL    Ghet  GV  4    SL  S  is independent of the nucleation site.  3  G     V    

16 3  G*   SL  S  het  2    (4.19)  3GV 

18