Nucleation and Growth
• Goal Understand the basic thermodynamics behind the nucleation and growth processes
• References Handout Ch. 8, Intro. to Ceramics by Kingery, Bowen and Uhlmann Most books on glasses and glass-ceramics
• Homework None
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1 Phase Transformations
• Considered as a transformation of a homogeneous solution to a mixture of two phases ∆ • For a stable solution, Gmix is less than zero. In other words, the solution is more stable than the individual components ∆ ∆ ∆ • Gmix is composed of entropic (-T Smix) and enthalpic ( Hmix) parts • Consider ∆ 1. Hmix less than zero: stable solution ∆ 2. Hmix = zero (ideal solution), stable solution due to entropic ∆ 3. Hmix slightly greater than zero: stable solution entropy dominates ∆ 4. Hmix >> 0: enthalpy dominates, phase separation occurs • Note: in all cases as T increases, entropy becomes more important, so at very high temperatures, solutions are usually favored
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2 Phase separation
∆ ∆ • If Hmix is greater than zero, the overall Gmix can be greater than zero meaning that phase separation is favored
• As T increases, homogeneous solution is favored
• Tc, the consulate temperature is the point above which solution is favored
• Behavior described by a series of G vs. composition curves at different temperatures Inflection points and minima plotted on T vs. comp. Diagram
• Spinodals from inflection points
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3 Spinodal Decomposition
• A continuous phase transformation Initially, small composition changes that are wide-ranging Give interpenetrating microstructure (2 continuous phases)
• No thermodynamic barrier to phase separation One phase separates into two Infinitesimal composition changes lower the system free energy
• Very important in glass and liquids Vycor Liquid-liquid phase separation
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4 Nucleation and Growth
• Important for: Phase transitions, precipitation, crystallization of glasses Many other phenomena • Nucleation has thermodynamic barrier β π V = 34 Iutv • Initially, large compositional change V 3 Small in size Nucleation and growth 1 • Volume transformations α to β phase transformation 0.8
Avrami equation 0.6 Vβ is the volume of second phase V is system volume 0.4
Iv is the nucleation rate 0.2 u is the growth rate Volume Fraction Transformed 0 t is time 0 0.2 0.4 0.6 0.8 1 Sigmoidal transformation curves Normalized Time of Reaction
• Infinitesimal changes raise system free energy
5 Volume Energy
∆ ∆ • Gv is Grxn (energy/volume) times the new phase volume • Spherical clusters have the minimum surface area/volume ratio • So: the volume term can be:
∆ ()volume Gv or 4 πrG3∆ 3 v
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6 Surface Energy
• The LaPlace equation shows the importance of surface energy 2γ ∆P = r Where: ∆P is the pressure drop across a curved surface γ is the surface energy LaPlace Equation/Kelvin Effect r is particle radius 15000
• Surface energy is important for small particles 10000 • Nuclei are on the order of 100 molecules P (atm) ∆ • More generally, surface energy is given by: 5000
∂G γ = ∂A 0 T,, P composition 0.001 0.01 0.1 1 10 Radius (µm)
WS2002 Where: A is the surface area of the particle, bubble, etc. 7
7 Nucleation
• Consider the nucleation of a new phase at a temperature T The transition temperature (T) is below that predicted by thermodynamics when surface or volume are not considered
• We can estimate the free energy change as a function of the radius of the nuclei Nucleation
-13 2 from the volume and surface terms 4 10 Surface Term (~x )
3 Volume Term (~x ) Sum of Surface and Volume • When r is small, surface dominates 2 10-13 ∆ G* 0 100 G (J) * • When r is large, volume dominates ∆ r
-13 • r* is the inflection point -2 10
-4 10-13 T To
0 2 10-8 4 10-8 6 10-8 8 10-8 1 10-7 ∆= − β phase stable α phase stable TT0 T Radius (m) ∆T
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8 Nucleation
• r* is the critical size nucleus and inflection point on the curve ∂()∆G At r*: r = 0 ∂r
∆ • We can use this to calculate r* and Gr* 216γπγ3 r**=− ∆=G ∆G ∆ 2 v 3()G v
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9 Critical Nuclei
• The number of molecules in the critical nucleus, n*, can be calculated by equating the volume of the critical nucleus, 4/3π(r*)3, with the volume of each molecule, V, times the number of molecules per nucleus 4 π(*)rnV3 = * 3
• Substituting the previous equations and solving gives
32πγ 3 n* =− ∆ 3 3VG()v
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10 Nucleus Formation
• The number of nuclei can be calculated using statistical entropy
N N N N ∆=∆+GNGkT r ln r + ln nrr NN+ NN+ NN+ NN+ r rrr ∆ Where: Gn is the free energy for cluster formation Nr is the number of clusters of radius r per unit volume N is the number of molecules per unit volume
• At equilibrium, Nr < ∆G* NN=−exp r* kT WS2002 11 11 Nucleation Rate • The nucleation rate, I, is then the product of a thermodynamic barrier described by Nr* and a kinetic barrier given by the rate of atomic attachment ∆G* kT ∆G IN=−exp N exp− m S kT h kT • As the degree of undercooling increases, the thermodynamic driving force increases, but atomic mobility decreases Thermo Kinetic Driving Limitation Force Nucleation Rate ∆T To ∆T increasing T increasing kT ∆G 16πγ 32T IN=−exp moN exp− s h kT 3()(∆∆THkT22 ) WS2002 rxn 12 12 Heterogeneous Nucleation • In many cases (some argue all cases), nucleation occurs at a surface, interface, impurity, or other heterogeneities in the system • The energy required for nucleation is reduced by a factor related to the contact angle of the nucleus on the foreign surface ∆=∆* * θ GGfhethom o () (21+− cosθθ )( cos )2 f()θ = 4 WS2002 13 13 Growth • Compared to nucleation, growth is relatively simple Assume that stable nuclei exist prior to growth Add molecules to a stable cluster Driven by free energy decrease of phase change Kinetically limited ∆G ua=−−ν 1 exp m o kT Where: u = growth rate per unit area of interface α β ao = distance across the - interface (~ 1 atomic dia.) ∆ Gm = activation energy for mobility or diffusion ν = frequency factor ν = kT πη3 3 ao Where: η is atomic mobility of viscosity WS2002 14 14 Summary • The thermodynamic driving force for both nucleation and growth increases as undercooling increases, but both become limited by atomic mobility Growth Nucleation II III I IV I and u Rate ∆ To ∆T increasing T T increasing • As we cool from the reaction temperature To we find 4 regions: Region I, α is metastable, no β grows since no nuclei have formed Region II, mixed nucleation and growth Region III, nucleation only Region IV, no nucleation or growth due to atomic mobility • Implications for tailoring microstructure WS2002 15 15