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UVA MSE 3050 - LECTURE NOTES

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MSE 3050, Phase Diagrams and Kinetics, Leonid ZhigileiNucleation and growth kinetics¾ Homogeneous nucleation ¾ Critical radius, nucleation rate ¾ Heterogeneous nucleation¾ Nucleation in melting and boiling¾ Growth mechanisms¾ Rate of a phase transformation Reading: Chapters 4.1and 4.2 of Porter and Easterling,MSE 3050, Phase Diagrams and Kinetics, Leonid ZhigileiNucleation and growth - the main mechanism of phase transformations in materials αB1X αB2X 0BXcoordinate spatial αB1X αB2X 0BX αB1X αB2X 0BXatomsBatomsBTPsolidliquidMSE 3050, Phase Diagrams and Kinetics, Leonid ZhigileiNucleationNucleation can be Heterogeneous – the new phase appears on the walls of the container, at impurity particles, etc.Homogeneous – solid nuclei spontaneously appear within the undercooled phase.Let’s consider solidification of a liquid phase undercooled below the melting temperature as a simple example of a phase transformation.solidsolidliquid liquidhomogeneousnucleationheterogeneousnucleationsupercooledliquidMSE 3050, Phase Diagrams and Kinetics, Leonid ZhigileiHomogeneous nucleationsolidliquidIs the transition from undercooled liquid to a solid spherical particle in the liquid a spontaneous one?That is, does the Gibbs free energy decreases?supercooledliquidThe formation of a solid nucleus leads to a Gibbs free energy change of ΔG = G2-G1= -VS (GvL–GvS) + ASLγSLnegative below Tmalways positive12VS– volume of the solid sphereASL– solid/liquid interfacial areaγSL– solid/liquid interfacial energyΔGv= GvL–GvSis the difference between free energies per unit volume of solid and liquidat T < Tm, GvS< GvL– solid is the equilibrium phaseMSE 3050, Phase Diagrams and Kinetics, Leonid ZhigileiWhen a liquid is cooled below the melting temperature, there is a driving force for solidification, ΔGv= GvL-GvSGT*ΔGvGvSGvLTmΔTReminder: Driving force for solidification (ΔGv)At any temperature below Tmthere is a driving force for solidification. The liquid solidify at T < Tm. If energy is added/removed quickly, the system can be significantly undercooled or (supercooled).As we will see, the contribution of interfacial energy (γSL) results in a kinetic barrier for the phase transformation.At temperature T*Lv*LvLvST-HG =Sv*SvSvST-HG =v*vvST-HG ΔΔ=ΔAt temperature Tm0ST-HGmvmmvv=ΔΔ=ΔmmvmvTHSΔ=Δmmvmmv*mvvTΔTΔHTΔHTΔHΔG =−≈For small undercooling ΔT we can assume that ΔHvand ΔSvare independent of temperature (neglect the difference in Cpbetween liquid and solid)MSE 3050, Phase Diagrams and Kinetics, Leonid ZhigileiOrigin of the interfacial energy (γSL)Consider a solid-liquid interface. Depending on the type of material and crystallographic orientation of the interface, the interface can be atomically flat (smooth, faceted) or rough (diffuse).liquidsolidliquidsolidHSvHLvHTS-SvmST-GLvmST-spatial coordinateSvGLvGγSLinterfaceLvLvLvTS-HG =SvSvSvTS-HG =free energies of liquid and solid per unit volume:MSE 3050, Phase Diagrams and Kinetics, Leonid ZhigileiHomogeneous nucleationΔG = G2-G1= -VS Δ Gv+ ASLγSLFor a spherical nucleus with radius r:3Sr π34V =SL2v3rγr 4πΔGr π34-ΔG +=2SLr 4πA =rΔG*ΔG*rΔGinterfacial energy ~ r2volume energy ~ r3For nucleus with a radius r > r*, the Gibbs free energy will decrease if the nucleus grows. r* is the critical nucleus size, ΔG*is the nucleation barrier.MSE 3050, Phase Diagrams and Kinetics, Leonid ZhigileiHomogeneous nucleationAt r = r*0r γ 8πΔGr -4 πdrG dΔSLv2=+=vSL*ΔG γ2r =()()2v3SL*ΔG3γ 16πΔG =GT*GvSGvLTmΔT*rr =*SLvr γ2ΔG =Temperature of unstable equilibrium of a solid cluster of radius r* with undercooled liquid.MSE 3050, Phase Diagrams and Kinetics, Leonid ZhigileiHomogeneous nucleationvSL*ΔG γ2r =()()2v3SL*ΔG3γ 16πΔG =mmvTΔTΔHΔG =ΔT1ΔHT γ2rmmSL*⎟⎟⎠⎞⎜⎜⎝⎛=()()()22m2m3SL*ΔT1ΔH3Tγ 16ΔG⎟⎟⎠⎞⎜⎜⎝⎛π=Both r* and G* decrease with increasing undercooling The difference between the Gibbs free energy of liquid and solid (also called “driving force” for the phase transformation) is proportional to the undercooling below the melting temperature, ΔT = Tm–T:where Hmis the latent heat of melting (or fusion) Therefore:MSE 3050, Phase Diagrams and Kinetics, Leonid ZhigileiHomogeneous nucleationΔT1ΔHT γ2rmmSL*⎟⎟⎠⎞⎜⎜⎝⎛=()()()22m2m3SL*ΔT1ΔH3Tγ 16ΔG⎟⎟⎠⎞⎜⎜⎝⎛π=Both r* and G* decrease with increasing undercooling rΔG*ΔG1*1rΔG*2r*ΔG2m12TTT<<MSE 3050, Phase Diagrams and Kinetics, Leonid ZhigileiMD simulation of laser melting of Au filmsLin, Leveugle, Bringa, Zhigilei, J. Phys. Chem. C 114, 5686, 2010laser pulse20 psTime (ps)Temperature (K)0 100 200 300 400 500100020003000400050006000TlFabs=45J/m2TeTime (ps)T/Tm0 100 200 300 400 5000.9511.051.11.151.2Tm= 963 K500 pstwo melting fronts propagate from free surfaces, temperature drops (energy goes into ΔHm×Vl). Melting stops when T approaches Tm.nanocrystalline sample with average grain diameter of ~8 nm20 nm Au films irradiated by 200 fs laser pulseTime (ps)T/Tm0 100 200 300 400 5000.90.9511.051.1Tm=963Kmelting starts at grain boundaries and continues even after T drops below Tmat 30 ps. The last crystalline region disappears at ~250 ps20 ps 100 psMSE 3050, Phase Diagrams and Kinetics, Leonid ZhigileiMD simulation of laser melting of Au filmsThe continuation of the melting process below Tmcan be explained based on the nucleation theorySLvrrGrG γπ+Δπ−=Δ23434THTGrmmSLvSLΔ⎟⎟⎠⎞⎜⎜⎝⎛Δγ=Δγ=122*⎥⎦⎤⎢⎣⎡Δγ−=rHTTmSLm121*- temperature of the equilibrium between the cluster of size r and the surrounding liquidcritical radius at ΔTTime (ps)Number of atoms in the crystalline clust er0 50 100 150 200 250 300100020003000400050006000T* ≈ 845 K820 K840 K835 K850 K841 K844 K843 K845 K847 KLin, Leveugle, Bringa, Zhigilei, J. Phys. Chem. C114, 5686, 2010Critical undercooling temperature of a crystalline cluster surrounded by undercooled liquidMSE 3050, Phase Diagrams and Kinetics, Leonid ZhigileiHomogeneous nucleationThere is an energy barrier of ΔG* for formation of a solid nucleus of critical size r*. The probability of energy fluctuation of size ΔG* is given by the Arrhenius equation and the rate of homogeneous nucleation is⎟⎟⎠⎞⎜⎜⎝⎛−νkTΔGexpN*d~&nuclei per m3per swhere νdis the frequency with which atoms from liquid attach to the solid nucleus. The rearrangement of atoms needed for joining the solid nucleus typically follows the


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