Nucleation and growth kinetics

Nucleation and growth kinetics Hoogeneous nucleation Critical radius, nucleation rate Heterogeneous nucleation Nucleation in elting and boiling Growth echaniss Rate of a phase transforation Reading: Chapters 4.1and 4.2 of Porter and Easterling,

Nucleation and growth - the ain echanis of phase transforations in aterials α X B 2 0 X B α X 1 B Batos α X B 2 0 X B α X 1 B Batos α X B 2 0 X B α X 1 B spatial coordinate P liquid solid T

Nucleation can be Nucleation Heterogeneous the new phase appears on the walls of the container, at ipurity particles, etc. Hoogeneous solid nuclei spontaneously appear within the undercooled phase. Let s consider solidification of a liquid phase undercooled below the elting teperature as a siple exaple of a phase transforation. supercooled liquid liquid liquid solid hoogeneous nucleation solid heterogeneous nucleation

Hoogeneous nucleation 1 2 supercooled liquid liquid solid Is the transition fro undercooled liquid to a solid spherical particle in the liquid a spontaneous one? That is, does the Gibbs free energy decreases? The foration of a solid nucleus leads to a Gibbs free energy change of = G 2 -G 1 = -V S (G L G S ) + A SL γ SL V S olue of the solid sphere A SL solid/liquid interfacial area γ SL solid/liquid interfacial energy negatie below T always positie = G L G S is the difference between free energies per unit olue of solid and liquid at T < T, G S < G L solid is the equilibriu phase

Reinder: Driing force for solidification ( ) When a liquid is cooled below the elting teperature, there is a driing force for solidification, = G L -G S At teperature T L L L G = H - T S S S S G = H - T S = ΔH - T ΔS At teperature T 0 G ΔT G S G L = ΔH - TΔS = ΔS ΔH = T For sall undercooling ΔT we can assue that ΔH and ΔS are independent of teperature (neglect the difference in C p between liquid and solid) At any teperature below T there is a driing force for solidification. The liquid solidify at T < T. If energy is added/reoed quickly, the syste can be significantly undercooled or (supercooled). As we will see, the contribution of interfacial energy (γ SL ) results in a kinetic barrier for the phase transforation. T T ΔH ΔH ΔT ΔH T = T T

Origin of the interfacial energy (γ SL ) Consider a solid-liquid interface. Depending on the type of aterial and crystallographic orientation of the interface, the interface can be atoically flat (sooth, faceted) or rough (diffuse). liquid solid liquid solid free energies of liquid and solid per unit olue: H S H L H G = H L G = H S L S - TS - TS L S - TS S - T S interface L - T S G S G γ SL L G spatial coordinate

Hoogeneous nucleation = G 2 -G 1 = -V S Δ G + A SL γ SL For a spherical nucleus with radius r: 4 r π r + 3 4π r 4 V S = π r 3 SL A = 4π r 3 2 SL = - 2 γ 3 interfacial energy ~ r 2 r r olue energy ~ r 3 For nucleus with a radius r > r, the Gibbs free energy will decrease if the nucleus grows. r is the critical nucleus size, is the nucleation barrier.

Hoogeneous nucleation At r = r dδ G 2 SL = -4 π r + 8π r γ dr = 0 SL 2 γ ( SL ) 3 16π γ r = = 3 ( ) 2 G G S = 2 γ r SL r = r ΔT G L T T Teperature of unstable equilibriu of a solid cluster of radius r with undercooled liquid.

Hoogeneous nucleation SL 2 γ ( SL ) 3 16π γ r = = 3 ( ) 2 The difference between the Gibbs free energy of liquid and solid (also called driing force for the phase transforation) is proportional to the undercooling below the elting teperature, ΔT = T T: ΔHΔT = T where H is the latent heat of elting (or fusion) Therefore: r = 2 γ ΔH SL T 1 ΔT = 16 π 3 ( SL ) 3 2 γ T 1 ( ΔH ) 2 ( ΔT ) 2 Both r and G decrease with increasing undercooling

Hoogeneous nucleation 2 1 r 2 r 1 T < 2 < T1 T r r = 2 γ ΔH SL T 1 ΔT = 16 π 3 ( SL ) 3 2 γ T 1 ( ΔH ) 2 ( ΔT ) 2 Both r and G decrease with increasing undercooling

MD siulation of laser elting of Au fils laser pulse 20 n Au fils irradiated by 200 fs laser pulse 6000 T e 1.2 1.15 20 ps 500 ps two elting fronts propagate fro free surfaces, teperature drops (energy goes into ΔH V l ). Melting stops when T approaches T. Teperature (K) 5000 4000 3000 2000 1000 T l T/T 1.1 1.05 1 T = 963 K 0.95 0 100 200 300 400 500 Tie (ps) F abs =45J/ 2 0 100 200 300 400 500 Tie (ps) 1.1 nanocrystalline saple with aerage grain diaeter of ~8 n T/T 1.05 1 T =963K 0.95 20 ps 100 ps 0.9 0 100 200 300 400 500 Tie (ps) elting starts at grain boundaries and continues een after T drops below T at 30 ps. The last crystalline region disappears at ~250 ps Lin et al., J. Phys. Che. C 114, 5686, 2010

MD siulation of laser elting of Au fils The continuation of the elting process below T can be explained based on the nucleation theory 2γ SL 2γ SLT 1 4 r = = 3 2 r = πr + 4πr γ SL G H Δ ΔT 3 Δ critical radius at ΔT T 2γ 1 ΔH 1 r SL = T - teperature of the equilibriu between the cluster of size r and the surrounding liquid 6000 Nuber of atos in the crystalline cluster 5000 4000 3000 2000 850 K 820 K 847 K 835 K 843 K 841 K T 845 K 845 K 844 K 840 K 1000 0 50 100 150 200 250 300 Tie (ps) Critical undercooling teperature of a crystalline cluster surrounded by undercooled liquid Lin et al., J. Phys. Che. C 114, 5686, 2010

Hoogeneous nucleation There is an energy barrier of for foration of a solid nucleus of critical size r. The probability of energy fluctuation of size is gien by the Arrhenius equation and the rate of hoogeneous nucleation is N& ~ ν d exp kt nuclei per 3 per s where ν d is the frequency with which atos fro liquid attach to the solid nucleus. The rearrangeent of atos needed for joining the solid nucleus typically follows the sae teperature dependence as the diffusion coefficient: ν d ~ exp Q d kt Therefore: N& ~ exp Q d kt exp kt

Rate of hoogeneous nucleation N& ~ exp Q d kt exp kt exp Q d kt N & exp kt Teperature T > Qd exp ( kt) << exp( Q kt) d is too high - nucleation is suppressed Q ( ) d exp kt > exp( Qd kt) ~ 1/ΔT 2 decreases with T sharp rise of hoogeneous nucleation (diffusion is still actie) exp ( Q kt) d too sall low atoic obility suppresses the nucleation rate

Rate of Hoogeneous Nucleation In any phase transforations, it is difficult to achiee the leel of undercooling that would suppress nucleation due to the drop in the atoic obility (regie 3 in the preious slide). The nucleation typically happens in regie 2 and is defined by the probability of energy fluctuation sufficient to oercoe the actiation barrier r : Using N& r = ~ exp 16 π 3 kt ( SL ) 3 2 γ T 1 ( ΔH ) 2 ( ΔT ) 2 N & exp A = I0 2 ( ΔT) ery strong teperature dependence! where A has a relatiely weak dependence on teperature (as copared to ΔT 2 ) N & There is critical undercooling for hoogeneous nucleation ΔT cr there are irtually no nuclei until ΔT cr is reached, and there is an explosie nucleation at ΔT cr. 0 ΔT cr ΔT

Coputer siulation of laser oerheating and explosie boiling rate of hoogeneous nucleation (explosie boiling) increases sharply in a ery narrow teperature range at ~0.9T spin sharp threshold for laser ablation Phys. Re. E 68, 041501 (2003) oerheating and rapid decoposition into a ixture of gas phase olecules and liquid droplets http://www.faculty.irginia.edu/copmat/ablation/aniations/ 50 ps 250 ps Appl. Phys. A 114, 11, 2014

Heterogeneous nucleation supercooled liquid liquid solid the new phase appears on the walls of the container, at ipurity particles, grain boundaries, etc. Let s consider a siple exaple of heterogeneous nucleation of a nucleus of the shape of a spherical cap on a wall of a container. Three interfacial energies: γ LC liquid container interface, γ LS liquid-solid interface, γ SC solid-container interface. liquid γ LC θ γ SC solid nucleus γls Balancing the interfacial tensions in the plane of the container wall gies γ LC = γ SC + γ LS cos(θ) and the wetting angle θ is defined by cos(θ) = (γ LC - γ SC )/ γ LS

How about the out-of-plane coponent of the liquidapor surface tension? γ LS = γ SV + γ LV cos(θ) θ = 90º γ LS = γ SV The out-of-plane coponent of the liquid-apor surface tension is expected to be balanced by the elastic response of the solid, but theoretical analysis is not straightforward due to an apparent diergence of stress at the contact line. See [Physical Reiew Letters 106, 186103, 2011] for extracurricular reading

Heterogeneous Nucleation liquid γ LC θ γ SC solid nucleus γls The foration of the nucleus leads to a Gibbs free energy change of r het = -V S + A SL γ SL + A SC γ SC -A SC γ LC V S = π r 3 (2 + cos(θ)) (1 cos(θ)) 2 /3 A SL = 2π r 2 (1 cos(θ)) and A SC = π r 2 sin 2 (θ) One can show that het 4 3 2 SL r = - π r + 4π r γ S = 3 ho () θ S() θ r S 2 () θ = ( 2 + cos θ )( 1 cos θ ) /4

where Heterogeneous nucleation het 4 3 2 SL r = - π r + 4π r γ S = 3 dδ G dr ho () θ S() θ At r = r r ( 2 SL = - 4π r + 8π r γ ) S() θ = 0 r = S 2 γ 2 () θ = ( 2 + cos θ )( 1 cos θ ) /4 1 SL - sae as for hoogeneous nucleation het ( SL 16 π γ ) () θ = S = 2 3 ( ) 3 S r () θ ho S ( θ) can be sall if θ = het S 10 ho o 2-4 ( θ ) = ( 2 + cos θ )( 1 cos θ ) / 4 10 Actie nucleation starts cr ΔT het cr ΔT ho ΔT

Heterogeneous Nucleation SL 2 γ r = het = S( θ ) ho ho het r heterogeneous nucleation starts at a lower undercooling N& ho ~ exp kt ho N & N & het N & ho N& het ~ exp kt het & het >> N N& ho ΔT

Pre-elting For solid/liquid/apor interfaces, often γ Solid-Vapor > γ Solid-Liquid + γ Liquid-Vapor in this case, no superheating is needed for nucleation of liquid and surface elting can take place below T pre-elting cross-section of an atoic cluster close to T (siulations by J. Sethna, Cornell Uniersity) Why ice is slippery? Physics Today, Dec. 2005, pp. 50-55

Two-step nucleation in solid solid phase transitions (extracurricular - not tested) Fortini & Dijkstra, J. Phys. Condens. Matter 18, L371, 2006 icrogel colloidal spheres with σ ~ 0.7 μ packing fraction phase transforations in a syste of hard spheres (colloidal particles) confined between parallel hard plates: 1 2 2 3... n (n + 1) (n + 1)..., where where n is the nuber of crystal layers, and the phases hae triangular ( ) or square ( ) syetry single-particle resolution ideo icroscopy of a transition fro square to triangular lattice in a colloidal fil Yi Peng et al. Nature Mater. 14, 101, 2015

Two-step nucleation in solid solid phase transitions (extracurricular - not tested) two-step nucleation with an interediate liquid phase when γs-l < γs1-s2 Yi Peng et al. Nature Mater. 14, 101, 2015 single-particle resolution ideo icroscopy of a transition fro square to triangular lattice in a colloidal fil

γ LV (T, C) Marangoni effect (extracurricular - not tested) The decrease of surface energy of a liquid with increasing T can be described by a seiepirical equation: γ γ LV alue of γ 0 LV γ LV γ 0 when 0 LV (1 T / T ) extrapolat ed T LV T c n 1.2 for etals c n to 0 K liquid becoes indistinguishable fro gas gradient of cheical coposition or teperature T + ΔT gradient of surface tension Marangoni effect - flow along the gradient of surface tension alcohol eaporation high γ low γ tears of wine: flow fro regions with higher concentration of alcohol (lower γ LV ) to regions where concentration of alcohol decreased due to eaporation (higher γ LV )

γ LV (T, C) Marangoni effect (extracurricular - not tested) Marangoni effect in laser synthesis of TiN: surface-tension-drien conection has to be accounted for to explain experiental obserations Höche & Schaaf, Heat and Mass Transfer 47, 519, 2011 γ T LV P 0.24 J 2-1 - K strong dependence of T on X N N concentration gradient resolidification starts at surface

Growth echaniss The next step after the nucleation is growth. Atoically rough (diffuse) interfaces igrate by continuous growth, whereas atoically flat interfaces igrate by ledge foration and lateral growth. liquid liquid solid solid The rate of the continuous growth (typical for etals) is typically controlled by heat transfer to the interfacial region for pure aterials and by solute diffusion for alloys. Growth in the case of atoically flat interfaces can proceed fro existing interfacial steps (e.g. due to the screw dislocations or twin boundaries) or by surface nucleation and lateral growth of 2D islands.

Growth echaniss Sooth solid-liquid interfaces typically adance by the lateral growth of ledges. Ledges can result fro surface nucleation or fro dislocations that is intersecting the interface. Spiral growth on dislocations AFM iages of growing crystal of KDP (potassiu dihydrogen phosphate) by De Yoreo and Land, LLNL and Malkin and Kuznetso, Uniersity of California

Growth: Teperature dependence In ulti-coponent systes, non-congruent phase transforations typically inole long-range diffusion of coponents necessary for achieing the equilibriu phase coposition. D E = d D 0exp RT The atoic rearrangeents necessary for growth of a onecoponent phase or growth in a congruent phase transforation also inole therally-actiated eleentary processes (diffusion). When growth is diffusion controlled, it slow down with decreasing teperature.

Rate of phase transforations Total rate of a phase transforation induced by cooling is a product of the nucleation rate (driing force increases with undercooling but diffusion needed for atoic rearrangeent slows down with T decrease) and growth rate (diffusion controlled - slows down with T decrease). oerall transforation rate growth rate nucleation rate teperature T high T (close to T ): low nucleation and high growth rates coarse icrostructure with large grains low T (strong undercooling): high nucleation and low growth rates fine structure with sall grains

Surface energy of crystals, γ SV directional dependence of the surface energy in crystals: the nuber of broken bonds in an fcc crystal increases fro {111} to {100} to {110} faces and, in general, H s can be expected to be higher for high {hkl} index of the crystal face. {111} {100} {110} θ a face at an angle θ to a low-energy close-packed plane a E SV = energy per unit surface area: (cos θ + sin θ ) ε / a 2 E SV where ε is the surface energy of close-packed plane per area of a 2 0 θ although surface entropy contribution ay sooth the energy cusps, the cusps are still present in γ(θ)

Growth echaniss directional dependence The shape of a growing crystal can be affected by the fact that different crystal faces hae different growth rates. Close-packed low-energy faces tend to grow slower and, as a result, they are the ones that are ostly present in a growing crystallite. For exaple, water ice I(h) has hexagonal crystal syetry that is reflected in the syetry of snow crystals. The growth rate is fast parallel to the basal {0001} and pris {1010} faces. As a result, ery sall snow crystals hae shape of hexagonal priss. As they grow, growth instabilities result in ore coplex shapes of larger snow crystals. MSE 3050, Phase Diagras and Kinetics, P. V. Leonid Hobbs, Zhigilei Ice Physics, Oxford Uni. Press, Oxford, 1974.

Growth instabilities, dendrites Material and heat diffusion liits the rate at which a crystal can grow, often greatly affecting the shape of the growing crystals. An exaple is the Mullins-Sekerka instability. Consider a flat solid surface growing into a supersaturated apor. If a sall bup appears on the surface, then the bup sticks out farther into the supersaturated ediu, and hence tends to grow faster than the surrounding surface. crystal supersaturated apor As a result the flat growth is unstable, and a crystal tend to grow into ore coplex shapes, e.g. snowflakes MSE 3050, Phase Diagras and Kinetics, http://www.its.caltech.edu/~atoic/snowcrystals/ Leonid Zhigilei

Patterns of snow crystals: A letter fro the sky Nakaya diagra: The shape of snow crystals depends on the teperature and huidity of the atosphere in which they hae grown. Vertical axis shows the density of water apor in excess of saturation with respect to ice. The black cure shows the saturation with respect to liquid water as a function of teperature. Physics Today, Dec. 2007, pp. 70-71, 2007 http://www.lowte.hokudai.ac.jp/ptdice/english/aletter.htl

Two-diensional ice dendrites on windows by Harry Bhadeshia by Nadezhda Bulgakoa

Water dendrites in ice http://www.s.ca.ac.uk/

Crystal growth and heat flow during solidification Theral dendrites Bragard et al., Interfacial Science 10, 121, 2002 icrostructure of a etal ingot http://www.s.ca.ac.uk/phase-trans/

Crystal growth and heat flow during solidification Theral dendrites liquid etal heat flow What are the processes leading to the foration of the icrostructure scheatically shown in this figure? cold ould

Suary Make sure you understand language and concepts: Hoogeneous nucleation Interfacial energy Critical radius, nucleation rate Heterogeneous nucleation Teperature dependence hoogeneous and heterogeneous nucleation rates Nucleation in elting and boiling Nucleation in solidification Growth echaniss Rate of a phase transforation Growth instabilities, dendrites H. Iai and Y. Oaki, MRS Bulletin 35, February issue, 138-144 (2010)