Chain_Dimensions_Review

Chain_Dimensions_Review - Polymer Processing Chain...

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Unformatted text preview: Polymer Processing Chain Conformations A Review Dr. Anthony Brennan University of Florida Department of Materials Science & Engineering 07/22/11 EMA 4666C Chain Dimensions Review Copyright Protected 1 Agenda Rotational States q Bond Rotational Energetics q Spatial Relationships q Characteristic Dimensions q Freely Jointed Model q Freely Rotating Model q Hindered Rotation q Random Flight Statistical Model q 07/22/11 07/22/11 EMA 4666C Chain Dimensions Review Copyright Protected 2 Polymer Structure -Property Polymer Behavior Behavior q q q q q 07/22/11 07/22/11 chain chain thermodynamics thermodynamics chain dimensions chain environment chain topography mechanical behavior EMA 4666C Chain Dimensions Review Copyright Protected 3 Chain Dimensions Chain Rotational Isomers Rotational 07/22/11 07/22/11 EMA 4666C Chain Dimensions Review Copyright Protected 4 Chain Dimensions Chain Ethanes Rotational Isomer Energies States Ethanes 07/22/11 07/22/11 EMA 4666C Chain Dimensions Review Copyright Protected 5 Chain Dimensions Chain n-Butane Conformational States n-Butane 07/22/11 07/22/11 EMA 4666C Chain Dimensions Review Copyright Protected 6 Chain Dimensions Chain n-Butane Conformational States n-Butane 07/22/11 07/22/11 EMA 4666C Chain Dimensions Review Copyright Protected 7 Chain Dimensions Chain Conformational States Conformational Number of Conformation States is: N = RIS n −1 RIS = Number of Rotational Isomeric States (Gauche+, Trans, Gauche-) n = number of sigma bonds in chain backbone N = 1 x 104770 for PE where Xn ~ 10,000 07/22/11 07/22/11 EMA 4666C Chain Dimensions Review Copyright Protected 8 Chain Dimensions Chain Local Steric Repulsion Local 07/22/11 07/22/11 EMA 4666C Chain Dimensions Review Copyright Protected 9 Chain Dimensions Gaussian Chain q q q Random Random Chain Conformation Conformation End-to-End End-to-End Distance Distance Radius of Radius Gyration Gyration 07/22/11 07/22/11 EMA 4666C Chain Dimensions Review Copyright Protected 10 Chain Dimensions q Gaussian Chain or Gaussian random chain: random – Gaussian Gaussian distribution of RIS, I.e., trans and gauche states gauche 07/22/11 07/22/11 q End - to - end End distance and radius of gyration : radius – characteristic characteristic dimensions based upon molar mass, chain structure and temperature temperature EMA 4666C Chain Dimensions Review Copyright Protected 11 Chain Statistics Chain Random Walk Random q Maxwellian Demon Maxwellian – Random walk – No restrictions on No direction direction – Fixed segment q Begin by Coin toss 07/22/11 07/22/11 EMA 4666C Chain Dimensions Review Copyright Protected 12 q 10 Steps – H = +1 step – T = -1 step 07/22/11 07/22/11 Chain Statistics Chain Random Ten Step Random EMA 4666C Chain Dimensions Review Copyright Protected 13 Chain Dimensions Chain Definitions Definitions q q q q Projections of Projections vectors vectors Mean dimensions Bond angles Statistical Statistical Segments Segments 07/22/11 07/22/11 EMA 4666C Chain Dimensions Review Copyright Protected 14 Hindered Rotating Chain Model Assumptions q Freely rotating model: – high temperature – solvated xi+1 i+2 yi i+1 i zi+1 θ φ xi zi i-1 yi+1 07/22/11 07/22/11 EMA 4666C Chain Dimensions Review Copyright Protected 15 Hindered Rotating Chain Model Assumptions q Hindered rotating model: –torsion (valence) angle dependence –first order interactions xi+1 –high level interactions i+2 yi i+1 i zi+1 θ φ xi zi i-1 yi+1 07/22/11 07/22/11 EMA 4666C Chain Dimensions Review Copyright Protected 16 Freely Rotating Chain Vector Analysis Mean Square End to End Distance r 2 = nl 2 As shown previously: ri rj 07/22/11 07/22/11 = l cosθij 2 EMA 4666C Chain Dimensions Review Copyright Protected 17 Chain Dimensions Chain n r 2 = ∑mr i =1 n 2 ii ∑m i =1 i End to End Distance s 2 = r 2 6 Radius of Gyration m - mass of repeat unit (segment) r - vector form center of gravity to atom i 07/22/11 07/22/11 EMA 4666C Chain Dimensions Review Copyright Protected 18 Freely Rotating Chain Model Freely Vector Analysis Vector r 2 r 07/22/11 07/22/11 = nl 2 2 1 + cos(180 − τ ) 1 − cos(180 − τ ) = 2nl 2 (Fixed angle = (Fixed 109.5°) 109.5°) EMA 4666C Chain Dimensions Review Copyright Protected 19 Freely Jointed Chain Vector Analysis r 2 = nl 2 1 + cosθ 1 − cosθ For the Freely Jointed Model there are For no restrictions on the valence bond angle, thus 07/22/11 07/22/11 EMA 4666C Chain Dimensions Review Copyright Protected 20 Freely Rotating Chain Model Freely Vector Analysis Vector r 2 r 07/22/11 07/22/11 = nl 2 2 1 + cos(180 − τ ) 1 − cos(180 − τ ) = 2nl 2 (Fixed angle = (Fixed 109.5°) 109.5°) EMA 4666C Chain Dimensions Review Copyright Protected 21 Freely Rotating Chain Vector Analysis r q q q 2 = 2 Ml 2 Freely rotating model: – high temperature – solvated Fixed bond angle of 109.5° expands by 2 Ignores bond rotational energy barriers (RIS) 07/22/11 07/22/11 EMA 4666C Chain Dimensions Review Copyright Protected 22 Hindered Rotating Chain Model Ilustration of RIS C C C H C H H φ C H C H H φ H H Equal probability for conformations 07/22/11 07/22/11 EMA 4666C Chain Dimensions Review Copyright Protected 23 Hindered Rotating Chain Model Ilustration of RIS CH C H H C H φ H H C H C φ C H Periodic fluctuations as 2π /3 07/22/11 07/22/11 EMA 4666C Chain Dimensions Review Copyright Protected 24 Hindered Rotating Chain Model r 2 = nl 07/22/11 07/22/11 (180 − τ ) 1 + cos φ 1 − cos(180 − τ ) 1 − cos φ 2 1 + cos EMA 4666C Chain Dimensions Review Copyright Protected 25 Hindered Rotating Chain Model PE Torsional Energy Potentials Torsional −Eg σ = e RT one can define f ( φ ) whereby all states are averaged : 07/22/11 07/22/11 EMA 4666C Chain Dimensions Review Copyright Protected 26 Hindered Rotating Chain Model Hindered PE Torsional Energy Potentials PE Torsional r 2 r 2 1 + cos( 180 − τ ) 2 + σ 2 = nl 1 − cos( 180 − τ ) 3σ = 3.4 nl 2 Experimental value ~ (6.7+/- 0.1)nl2 q Explain discrepancies q 07/22/11 07/22/11 EMA 4666C Chain Dimensions Review Copyright Protected 27 Chain Dimensions Characteristic Dimensions r 2 0 = Cnl 2 where: <r2>o - chain dimensions at Theta conditions conditions C θ- Characteristic Chain Ratio n - number of segments l - segment dimension (length) 07/22/11 07/22/11 EMA 4666C Chain Dimensions Review Copyright Protected 28 Chain Dimensions Chain Expansion Factor θ α − α = Cψ n 1 − T 5 q q q 3 α - chain expansion factor expansion C - characteristic characteristic chain ratio chain ψ - interaction entropy entropy 07/22/11 07/22/11 q q θ - Theta condition temperature temperature Τ − temperature temperature EMA 4666C Chain Dimensions Review Copyright Protected 29 Chain Dimensions Chain Experimental Determination of Hydrodynamic Volume Volume 07/22/11 07/22/11 EMA 4666C Chain Dimensions Review Copyright Protected 30 Chain Dimensions Chain Expansion Factor θ α − α = Cψ n 1 − T 5 3 α = 1: when T = θ 1/10 qα − > n 1/10; T > θ q q <r2> = <r2>o at T = q q Chains behave as Chains phantoms phantoms θ 12 α − α ∝ ψ 1 − M T 5 07/22/11 07/22/11 3 EMA 4666C Chain Dimensions Review Copyright Protected 31 Chain Dimensions Scattering Function Kc 1 1 16 2 r 2θ = + π + 2 A2 c 2 sin Rθ MW MW 3 6λ 2 2 where: K - contrast factor c - concentration of scattering species Rθ - Rayleigh ratio (reduced intensity) r - distance from scattering center to detector. detector. I0 - incident wave intensity A2 - 2nd virial coefficient 07/22/11 07/22/11 EMA 4666C Chain Dimensions Review Copyright Protected 32 Chain Dimensions Generation of a Zimm Plot limθ →0 Kc 1 = + 2 A2 c Rθ MW Kc 1 1 16 2 r 2θ = + π 2 sin Rθ MW MW 3 6λ 2 2 limc→0 07/22/11 07/22/11 EMA 4666C Chain Dimensions Review Copyright Protected 33 Chain Dimensions Chain Zimm Plot Zimm 07/22/11 07/22/11 EMA 4666C Chain Dimensions Review Copyright Protected 34 Chain Dimensions Viscosity Relationships η η r= −1 ηo η η r= ηo ηred 1 η = − 1 c 2 ηo [ η] = lim 07/22/11 07/22/11 c →0 ηinh 1 c2 1 η = ln c2 ηo η − 1 ηo EMA 4666C Chain Dimensions Review Copyright Protected 35 Three Dimensional Random Walk Three Calculated Dimensions Out q – Molar mass = 106 g/mol – Mo = 104 g/mol Solvent In – <rr2>o = 73.5 nm q q q q q q 07/22/11 07/22/11 Assume: n = {106/104} x 2 = 1.92 x 104 l = 0.154 nm <r2> = 455 nm2 Theta Restriction; <r2> = 910 910 nm2 nm <r2>calc = 30.2 nm2 <r2>calc/ <rr2>o = 2.44 EMA 4666C Chain Dimensions Review Copyright Protected 36 Summary q Chain bond rotational potential energies Chain dictate properties dictate q Chain dimensions can be calculated Chain statistically or by vector analysis statistically q Zimm Plots from scattering measurements Zimm give dimensions Mw and rg separately, along give with A2 with q Characteristic dimensions defined by Characteristic structure and environment structure 07/22/11 07/22/11 EMA 4666C Chain Dimensions Review Copyright Protected 37 References q Introduction to Physical Polymer Science, 2nd Introduction Edition, Lesley H. Sperling, Wiley Interscience (1992) ISBN 0-471-53035-2 ISBN q Principles of Polymer Chemistry, P.J. Flory (1953) Principles Cornell University Press, Inc., New York. Cornell q The Physics of Polymers, Gert Strobl (1996) The Springer-Verlag, New York. Springer-Verlag, q Figures were reproduced from Polymer Physics, Figures (1996) Ulf Gedde, Chapman & Hall, New York. (1996) 07/22/11 07/22/11 EMA 4666C Chain Dimensions Review Copyright Protected 38 ...
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