Cer103 WS00 Notes Shelby Chapter 2 2-1 Kinetic Theory of Glass Formation Since most glasses are formed from melts, it is appropriate to first discuss glass formation in terms of crystallization kinetics. Any melt will form a glass if cooled fast enough. Kinetic theory of glass formation is an attempt to define 'fast enough'. Most Glasses are Formed by Melt Techniques · Oxides (SiO2, B2O3, P2O5, etc.) · Halides (F, Cl, Br, I) · Chalcogenides (S, Se, Te, etc.) · Metals · Salts (see N. J. Kreidl, “Glass Forming Systems,” Glass Science and Technology, Vol. 1 (1983).) All Bond Types: · Covalent (SiO2, chalcogenides) · Ionic (KNO3•Ca(NO3)2) · Metallic (Au4Si, Pd4Si, Fe-Ni-P-B) · van der Waal’s (toluene) · Hydrogen (water) · and all combinations (homodesmic and heterodesmic). So-- What is 'fast enough'? Why don't all melts form glasses? Classical Nucleation/Crystallization Theory § Applied to any solution®solid transformation § Metals, rock candy, snow, etc. § See D. R. Uhlmann, "Kinetics of glass formation and devitrification behavior," J. de Physique, C9 43 175 (1982). 'Crystallization' is the opposite of 'Glass Formation'. § Avoid crystallization to form a glass. 'Crystallization' refers to two processes: 1. Nucleation: formation of crystal embryos- critical size 2. Crystal growth: beyond the critical size Cer103 WS00 Notes Shelby Chapter 2 2-2 Step 1: Nucleation Homogeneous: spontaneous formation from the melt; composition of the melt and crystal are the same. Heterogeneous: nuclei form on some 'foreign' surface: impurities, crucible wall, bubbles, seeds, etc. If you do not first form a nuclei (and there are no heterogeneities), crystallization can be avoided. Two contributions to the energetics of nucleation § Thermodynamics: free energy changes associated with crystal Energy formation (W*) Barrier § Surface creation barrier § Volume creation driving force Energy Meta- Driving § Kinetics: energy barrier associated with stable Force moving material to create/grow nuclei glass (DGD, where 'D' refers to diffusion.) Stable nuclei Nucleation is an activated process- that is, the nucleation rate (I) increases with temperature, and can be described by an Arrhenius equation: I = Aexp[ -(W * +DGD ) kT ] I is the homogeneous nucleation rate A is the pre-exponential factor k is Boltzmann's constant Notes about Activated Processes Property ln Property -1 Temperature 1/Temp. (K) Typical behavior for chemical Property = A exp(-Ea/kT) reactions: rates (slope of property A = pre-exponential factor curve) increase with temperature. (intercept) E = activation energy Arrhenius (1889) recognized that this a (slope) could be described by an exponential k = Boltzmann’s constant function (sometimes ‘R’) Cer103 WS00 Notes Shelby Chapter 2 2-3 Thermodynamic Considerations: Consider the Free Energy changes associated with nuclei formation glass DG supercooled v liquid solid Free Energy (G) liquid Tmelt Temperature 1. Volume Energy Gain Above the melting point, the liquid is the thermodynamically favored state (has the lower free energy). Upon cooling below Tm, the solid (crystal) state is favored, but can be avoided during supercooling. The free energy difference (per volume) between a glass and crystal (or nucleus) is DGV. § Spherical nuclei, radius 'r' § Free energy reduction for a volume of material to transform from a glass to the thermodynamically favored crystalline nucleus is given by: DGV ´ volume of sphere ([4/3]pr3) 2. Surface Energy Penalty (Barrier) Nucleation opposed by energy needed to create a new surface (blowing bubbles, surface energy minimized with spheres): Surface tension (g) ´ surface area (4pr2) Nucleation dependent on the balance between surface & volume energies. 3 2 W = (4/3)pr DGv + 4pr g (eq. 2-4) Cer103 WS00 Notes Shelby Chapter 2 2-4 Surface Energy (+r2) W* 3 2 W = (4/3)pr DGv + 4pr g r* r (nucleus radius) W (energy) Volume energy (-r3) § Small nuclei (r<r*) are dominated by surface energy term and will tend to re- dissolve because an increase in r causes an increase in W. § If the nucleus reaches a critical size (r*), that nucleus will grow because the dominance of the volume term causes W to decrease with increasing r. § Thermodynamically stable particles when r³r* Find r* from the first derivative of eq. 2.4: dW eq. 2.5: = 0 = 4pr 2 DG +8prg dr V - 2g eq. 2-6: Solve for r: r * = DGV § Note: blowing bubbles? Replace DGV with P. § In practice, stable nuclei are about 100 molecules in size. Definition for thermodynamic energy barrier to nucleation (substitute 2-6 into 2-4) 16 g 3 eq. 2-7: W * = p 2 3 DGV g & DGV are experimentally obtainable. Large g ® better glass forming tendency reduce g ® promote nucleation Cer103 WS00 Notes Shelby Chapter 2 2-5 Heterogeneous Nucleation: effective reduction of surface energy by the presence of 'impurity' surfaces- seeds, bubbles, crucible walls, etc. (Don't need to expend energy to initially create 'homogeneous' surfaces. From eq. 2-7: reducing geff reduces W* and so lowers the energy barrier for nucleation. Homogeneous Heterogeneous Nucleation Nucleation Meta- stable Surface glass Energy Barrier Energy Volume Energy Energy Gain Stable nucleus Surfaces that are readily wet by the liquid (defined by the contact angle, q, between the growing nucleus and the heterogeneous surface) provide better nucleation sites. § Smaller contact angles ® lower surface energies® greater nucleation rates q q Small contact Poor nucleation angle (q): good nucleation Cer103 WS00 Notes Shelby Chapter 2 2-6 Recall: glass Thermodynamic estimate for driving energy: DG supercooled v liquid eq. 2-7: DGV » -DHV (Tm -T)/VmTm (DHV is the solid heat of fusion) Free Energy (G) The further below Tm, the greater the driving liquid Tmelt force for nucleation becomes. Temperature So: if DGV becomes greater and greater with decreasing temperature, why don't all melts crystallize upon cooling? What keeps the thermodynamics from controlling glass formation? I. Kinetic Considerations DGD: related to how fast material moves to the surface of a growing nuclei: expressed as a diffusion coefficient: kTl2 -G eq. 2-13: D = ( )exp( D ), h kT l is atomic jump distance, and D is related to viscosity by Stokes-Einstein: kT eq. 2-14: D = ( ) . Note, high h, low diffusion coefficient. 3plh Plug these equations into eq. 2-1 to obtain description of nucleation rate (I): Ah -W * eq. 2-15: I = ( )exp( ) 3pl3h kT -1 Note: I ~ h : increasing h decreases I (counterbalances the increasing DGV). So: what does eq. 2-15 look like? Cer103 WS00 Notes Shelby Chapter 2 2-7 ® Increasing I: DGV»(Tm-T) Decreasing I: kinetics (h-1) Metastable zone of undercooling Nucleation Rate (I) Tmelt Tg ¬Temperature Metastable zone of undercooling: no stable nuclei form at these temperatures just below Tm: high surface energies of small particles decreases melting points (small particles melt at T<Tm). So, nucleation must begin at the melting point of particles with r*, not at the macroscopic Tm shown above. Step 2: Crystal Growth Growing particles with r>r*. § Surface tension less important; barrier to r* already overcome § Same thermodynamic (DG) and kinetic (DE) considerations. § Balance crystallization energies against diffusional barriers Consider atoms jumping back and forth across the liquid/solid interface: u ~ atomic jump frequency ulx =u exp(-DE/kT) Crystal Growth uxl =u exp((-DE-DG)/kT) (more energy needed for x®l jump) the net crystal growth rate is Diffusional liquid Energy U=a0(ulx - uxl), a0 is interatomic distance Barrier (DE) And so: Energy Xtal. eq. 2.20: U = a0u exp(- DE kT )[1- exp(DG / kt) ] Energy (diffusion barrier) (Xtal. Driving force) Gain ( DG) Crystal Cer103 WS00 Notes Shelby Chapter 2 2-8 again, using Stokes-Einstein (eq. 2.13 & 2.14): kT - DG eq. 2.21: U = ( 2 )[1-exp( )] 3pa0h kT Shelby Figure 2.1: Shelby, Figure 2.1 a. no metastable undercooling zone for U; particles are large enough to melt at Tm. (a) b. extent of U-I overlap is an indication of glass forming tendency. Large overlap, poor glass forming melt (nucleation and growth occurring T (b) contemporaneously). 1 T2 What happens if we rapidly cool from above Tm to T1 and hold? § Nothing, no nuclei can form in metastable undercooling zone What happens if we rapidly cool from above Tm to T2 and hold? § Nuclei will form, but will not grow. Reheat to T(Umax) to crystallize sample (process cycle for glass-ceramics). So: to avoid crystallization, must cool below Tg faster than nucleation and crystallization rates. Consider the modified Avrami equation from classical transformation kinetics. The volume fraction of a melt that has crystallized: V t t eq. 2.22: x 1 exp[ I ( Ud )3 dt'] = - -ò V ò t V 0 t' (I is volume, U is linear (so U3). Under isothermal conditions, (2.22) becomes: V p eq. 2.23: x =1- exp[- I U 3t 4 ] V 3 V where t is the time the sample is held at experimental temperature. By knowing I, U at various temperatures, one can calculate (Vx/V) in time/temperature space to produce Time-Temperature-Transformation (TTT) curves: Cer103 WS00 Notes Shelby Chapter 2 2-9 -8 Tm 10 -6 (Vx/V)=10 Tn (Crystal) Temperature (Glass) tn time Note: shape of curves a consequence of the balance between thermodynamic and kinetic processes: § Crystallization driving force increases with undercooling § Viscosity increases with undercooling. Most favorable conditions for crystallization (and so least favorable conditions for glass formation) when temperature is held at nose temperature (Tn) where the shortest amount of time (tn) is needed for transformation to occur. (Large viscosity at Tn means greater tn). Note: for smaller transformation volumes, tn is shorter. Thus, to avoid crystallization, the melt must be cooled at a critical cooling rate: · eq. 2.24: (dT dt) » » (T -T ) t C TC m n n If thermal diffusivity (DT) is known, the maximum sample thickness is: eq.