结晶动力学和结晶热力学补充材料

1、徐军地址：英士楼218电话：62784740 手机mail：jun-Process of Polymer CrystallizationTwo steps: 1. Nucleation (成核成核) processSizeD DG0embryosuper critic. sizestablecritical nucleusD DG = Gcrystal Gmelt , D DG = D DH TD DSGcrystal = Gbulk + S SAs sD DG = Gbulk - Gmelt + S SAs s =D DGf + S SAs sA is the su

2、face area and D DGf is the bulk free energy change.abcSchematic representation of the change in free energy as a function of size illustrating the nucleation processTypes of crystal nuclei. (a) primary (初次初次), (b) secondary (二次二次), (c) tertiary (三次三次) nucleus. Homogeneous (均相均相) and heterogeneous (异

3、相异相) nucleationEstimate the critical nucleus size*44eemffTlghTssDDDefaalglaGss2224DD0fmfmfgThTsDDD For primary nucleus For secondary nucleus*22eemffTlghTssDDDefalgalaGss442DDsagalGf42DDfgaDs4*feglDs4*mffThsDD22feGabl gblabssD D2*mfTahTsDD4*mfTahTsDD0GaD0GlD ffmfcfcffmhsTTgThTshTDDDD DD DGibbs-Thomso

4、n方程：Hoffman-Weeks方程：Hoffman JD, Weeks JJ. J. Res. Natl. Bur. Std. (U. S.) 1962, 66A, 13-28.l10mT0熔融线熔融线cTmTcmTT 0mTfepgllDs4*cTmTcmTT zeroTcL10mTzeroL1zeroT熔融线熔融线结晶线结晶线Hoffman-Weeks 和和Gibbs-Thomson方程的联系方程的联系acT1),(czeromzeroTTTTcmmLKTT101),11(zerocczeroLLKTTzerozeromLKTT10Thermodynamics of Crystalli

5、zation: 2. Melting point of Polymer mixtureChemical Potential of Polymer in amorphous phase2022212ln11RTxxConsider a system of Polymer/Impurity mixture:Chemical Potential of one monomer unit of polymer in amorphous phasePolymer is compatible with impurity but cant co-crystallize with impurity00uCuuu

6、I20212ln111uuRTxxEffects of “Impurities” on TmCondition of Phase Equilibrium00uICuuuSuperscript of C: Crystalline Phase0uCuuuumGHTS DDD00ummmmuHSTHTSDDDD02212ln111IuuRTxx22120ln1111mumRTTHxxTD02212ln11111mmuTTHRxxDSuperscript of I: Amorphous Phase01mumTHT DEffects of “Impurities” on Tm22120ln11111mm

7、uRTTHxx DFor x2, x1=1211011mmuRTTH DFor end-group effects: 1=2/N, =00112mmuRTTHNDFor copolymer011lnmmuRPTTH Dalternating copolymer: PXA 1P: Probability of crystalline unit with sequential connection 2221111211ln22uuRRHH DD211121ln 1ln2Note:For polymer blends220111mmuRTTH D22120212ln11111mmuRTTHxxx D

8、polymer/small moleculepolymer/polymer2. Linear Growth of Polymer Crystallization Temperature dependence of linear growth rate (线生长速度线生长速度)Temperature dependence of the radial growth rate u of spherulites in isotactic polystyrene (left), polyamid 6 (center) and poly(tetramethl-p-silpheylene siloxane)

9、 (right). Tf : equilibrium melting temperature. Surface nucleation on substrate with length L with a rate i Linear growth rate G: the growth rate of crystal perpendicular to the substratel: fold length; a: width of stem; b: thickness of the stem; g: substrate completion rateAvrami指数指数n生长类型均相成核n=生长维数

10、生长维数+1异相成核n=生长维数生长维数三维生长三维生长(球状晶体球状晶体)n=3+1=4n=3+0=3二维生长二维生长(片状晶体片状晶体)n=2+1=3n=2+0=2一维生长一维生长(针状晶体针状晶体)n=1+1=2n=1+0=1= 空间维数空间维数 + 时间维数时间维数Kinetics models of overall crystallizationn1. Free growth modelRvt3crystaltotal43icVVvNvtVVb. new nucleus produced - homogeneousa. constant numbers of nucleus- het

11、erogeneoust tDDt tt tD Dt t3*3 403*043334d/tctvIv tIIvvtttttttDt3* 3 4IttvtttDI* nucleation rate*33*340430 +4.3.3iIv tItv ttIvtttttDDDtPoissons raindrop problemIf raindrops fall randomly on a pond creating expanding circular waves, what is the chance that the number of waves created by different rai

12、ndrops which pass over a representative point P up to time t is exactly n?Kinetics models of overall crystallization-Avramin2.Impingement modelP solution: / !EnnP teEnPoisson distributionWhat is the chance that no wave reaches a given point P in space ?1cv 01EcPev iV tE tVheterogeneoushomogeneous 34

13、/31Nvtcve*3 4/31cIv tveThe Avrami equation for 3DE(t): wave spreading area during tWhile t or v is very small cvE t1xex *3 4/3Iv t343Nvtor1. Johnson-Mehl方程（方程（I、u不随时间而变）不随时间而变）设新相设新相在母相在母相中成核，并经过一段孕育期中成核，并经过一段孕育期以后长大成球状以后长大成球状新相半径新相半径一个新相体积一个新相体积d时间间隔内成核的时间间隔内成核的相数目为：相数目为：()ru tt33344()33ru ttVIdt则这些新相长大到时间则这些新相长大到时间t时的转变体积：时的转变体积：不同时间内形核的不同时间内形核的相在时间相在时间t的转变总体积的转变总体积：exdVIVdt 0texexVdV为扩张体积，重复计算为扩张体积，重复计算已转变的体积不能再成核已转变的体积不能再成核新相长大到相互接触时，不能继续长大新相长大到相互接触时，不能继续长大（真正的转变体积）在在d时间内有时间内