Lecture27 - EMA 6165 - Polymer Physics Crystallinity...

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Unformatted text preview: EMA 6165 - Polymer Physics Crystallinity Lecture 27 Professor Anthony Brennan Department of Materials Science & Engineering University of Florida UF - EMA 6165 Polymer Physics - Lecture 27 1 Rate of growth e E px = x! px probability function E size, x is no. of sites -E x "Like raindrops on a puddle, these are the days of our lives" Terms: Avrami probability that fronts cross average number of fronts of all p fronts of growing spherulites UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan 2 Rate of growth and po = e po = 1 - xt UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan -E for amorphous where xt = volume fraction of crystals 3 Rate of growth Thus 1- xt = e xt E -E which at low degrees of crystallinity yields UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan 4 Rate of growth And in bulk crystallization xt E Vt thus Volume 1- xt = e -Vt 5 UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan Rate of growth Consider 2 cases to evaluate Vt Spontaneous Nucleation Spinodal decomposition process Multiple sites initiating Broad distribution of sizes Binodal process Site specific Size specific II. Sporadic Nucleation UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan 6 Avrami Spontaneous kinetics Case I: L: nuclei g: growth rate 4 3 Vt = r L 3 2 dVt = 4 r L dv Where r = gt at time t, dr/dt = g, and dr = g*dt UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan 7 Avrami Dimensional Analysis Rate of growth Thus... ( Vt = 4 t 0 t 2 g 2 ) L g dt 4 3 3 t = t g L V 3 UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan Spontaneous 8 Avrami: Sporadic Growth Kinetics Rate of growth Case II: dVt = 4g ( t - ti ) L tgdt 2 2 Thus 2 3 4 Vt = g Lt 3 Sporadic UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan 9 Avrami: Power Law Analysis Spontaneous Rate of growth Substituted into q Spontaneous 1 - xt = e - t V 1- xt = =e ( e 4 g 3 Lt 3 - 3 ) 10 - Zt n UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan Avrami: Power Law Analysis Sporadic Rate of growth Sporadic 1- xt ( =e - Zt =e 2 g 3 Lt 4 - 3 n ) UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan 11 Rate of growth Avrami General Form ln (1 - xt ) = - Zt n ln - ln( 1 - xt ) = ln Z + n ln t UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan 12 [ OR ln - ln X c ( t ) = ln Z + n ln t ln[-lnXc(t)] =n [ Dimensionality Constant Z ln(t) UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan 13 Dimensionality Constant n is the dimensionality constant which describes morphology of crystallite in terms of: 1 dimensional n=2 n=3 n=4 2 dimensional 3 dimensional UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan 14 Nucleation Factor Nucleation Factor Gn C Tm Tm = kTc Tc Tc T UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan n -1 n -1 15 Nucleation Factor q Nucleation Factor Gn C Tm Tc T kTc n -1 Tc - Crystallization Tm - Melting n - Dimensionality UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan 16 Crystallization Rate Overall rate of crystallization Avrami Equation ( t ) = 1 - exp( - kt n ) 17 UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan Crystallization Rate - t - t ( t ) = - a - a a t density at time amorphous crystallite density at time = t UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan 18 Rate of growth Keith and Padden How does the Avrami describe the texture of spherulites? i.e. Branching, entrapped impurities, configurations that do not crystallize, etc? UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan 19 Rate of growth Where D is the diffusion coefficient of an impurity and G is the growth rate. D = = length G varies from a maximum for radial growth of a lamellae to a minimum when non-crystallographic branching occurs. UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan 20 Keith & Padden: Rate of growth is a measure of the internal structure of a spherulite. i.e. its Texture This can be expressed as 1 d 1 d D 1 d G = - d T D d T G d T UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan 21 Rate of growth dD >1 dT as whereas dG <0 dT dG 0 dT or dG <0 dT Coarseness or branching increases 22 UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan Rate of growth G = Goe E RT * e - F * RT E - free energy of activation for chain to diffuse onto crystal surface F* - free energy of formation of a stable nuclei UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan 23 Keith & Padden: Rate of growth The arguments are based upon the recognized limitation of both nucleation and diffusion Linear Growth Rate Diffusion Controlled Nucleation Controlled 80 60 40 20 120 PET Tm~280C Tg~67C 140 160 180 200 220 T(C) 24 UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan Rate of growth Hoffman Chain folding is kinetically controlled e l x e: fold surface x interfacial energy : lateral surface energy 25 UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan Rate of growth formation = 4xl + 2 2 2x e - x l f Free energy formation of a single chain folded crystal ( ) 26 f = bulk free energy of fusion UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan Keith & Padden Free Energy of Formation f Energy = H f H f - TS f = - TH f Tf = 27 UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan Rate of growth H f ( T ) Tf which assumes Hf (Enthalpy of Fusion) is independent of temperature UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan 28 At Tm: cryst = 0 for x>> l This one has the melting point depression expressed in fundamental quantities 2 e o 1- Tf = Tf H f l 29 UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan Hoffman Kinetics: Regime I Crystallization Model Regime I Ilium g G r bo x lg* One surface nucleus results in complete formation of substrate length L. Many chains can participate but all growth occurs from the one nucleus. UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan 30 Regime Growth G Ilium Loop bo r x g lg* Vacancy UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan 31 Reptation Rate in Regime I x: one chain G: growth rate axially g: substrate completion rate ao: molecular width bo: layer thickness l*g: initial fold thickness Stem is portion of chain occupying l*g 32 l g r = g ao * Reptation rate UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan Theories on Diffusion in crystallization Yoon and Flory believe that crystallization proceeds after elimination of entanglements. q Whereas Emil Fischer suggests the <rg> does not change upon crystallization. q Hoffman considers both the thermodynamics and kinetics of crystallization. UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan 33 Thermodynamics of Crystallization Free energy of Fusion Thermodynamics cryst cryst = Gibbs Free Energy = 4xl + 2 x e - x l ( f 2 2 ) l = thickness x = length of crystal e = fold surface energy f = bulk free energy of fusion 34 = lateral surface energy UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan Energy of Fusion Equation Approximate f f = h f - T S f hf = hf - T o Tm ( ) 35 UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan Energy of Fusion Equation (cont'd) f = hf ( T) Tm o Tmo = melting point of crystal with very large l T = undercooling = Tmo - Tc at Tmo and x >> l UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan 36 Energy of Fusion Equation (cont'd) e o Tm = Tm 1 - 2 h f UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan Gibbs Free Energy then = cryst = 0 l 37 Energy of Fusion Equation (cont'd) Thus a plot of Tm versus 1 / l produces a slope of Tm - 2Tm o e (degrees) hf with an intercept of Tm o 1/l (reciprocal dimensions) UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan 38 Surface Energy of Crystal If hf is known one can calculate e Huseby, Bair, Salovey measured P.E. that was gamma irradiated to prevent thickening e = 93 8 ergs/cm2 Tmo = 146 o C UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan 39 Summary Avrami Kinetics are geometric based Limitations include the size and the volume fraction of crystals Keith & Padden are energy based kinetics - better predictor of growth UF - EMA 6165 Polymer Physics - Lecture 27EMA 6165 Polymer Physics AB Brennan 40 ...
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This note was uploaded on 07/20/2011 for the course EMA 6165 taught by Professor Brennan during the Spring '08 term at University of Florida.

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