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# Lecture4 - Polymeric Materials Models for Calculating Chain...

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Unformatted text preview: Polymeric Materials Models for Calculating Chain Models Dimensions Dimensions Dr. Anthony Brennan University of Florida Department of Materials Science & Engineering EMA 6165 Polymer Physics – AB Brennan EMA 1 Agenda • • • • • • • • Rotational States Bond Rotational Energetics Spatial Relationships Characteristic Dimensions Freely Jointed Model Freely Rotating Model Hindered Rotation Random Flight Statistical Model EMA 6165 Polymer Physics – AB Brennan EMA 2 Chain Dimensions Chain Definitions Definitions • Projections of Projections vectors vectors • Mean Mean dimensions dimensions • Bond angles • Statistical Statistical Segments Segments EMA 6165 Polymer Physics – AB Brennan EMA 3 Chain Dimensions Chain Definitions Definitions • Projections of Projections vectors vectors • Mean Mean dimensions dimensions • Bond angles • Statistical Statistical Segments Segments EMA 6165 Polymer Physics – AB Brennan EMA 4 Freely Jointed Chain Conditions • Gaussian Chain or random Gaussian chain: chain: – N: total number of segments N: must be sufficiently large (n>50, z-distribution) z-distribution) – free rotation about all bonds – bonds are invisible bonds EMA 6165 Polymer Physics – AB Brennan EMA 5 Freely Jointed Chain Conditions • End - to - end distance and End radius of gyration : radius – all valence angles are allowed – No excluded volume EMA 6165 Polymer Physics – AB Brennan EMA 6 Freely Jointed Chain Vector Analysis r = ∑ ri r : end to end vector project the vector r; r • r = r = ∑ ri ∑ rj = ∑ ri + 2∑ 2 2 i =1 j =1 i =1 i =1 n n n n −1 n i =1 i = j +1 ∑rr n ij •Valid for any polymer chain •Evaluate dimensions as an ensemble Evaluate of N chains of EMA 6165 Polymer Physics – AB Brennan EMA 7 Freely Jointed Chain Vector Analysis Mean Square End to End Distance N n n −1 n 1 2 2 2 r = ∑ rk * = ∑ ri + 2∑ ∑ ri rj N k =1 i =1 i =1 j =i +1 ∑ i =1 n ri 2 Sum of the diagonal elements Sum of a square array (matrix) of n 2 ∑∑ ri rj i= j= + 1 i1 n− 1 Sum of the elements Sum below/above the diagonal below/above 8 EMA 6165 Polymer Physics – AB Brennan EMA Freely Jointed Chain Vector Analysis r 2 2 = nl + 2nl 2 2 2 θ + cos3 θ + ... cos n − 2 θ ) (1 + cosθ + cos + 2l cos θ (1 + 2 cosθ + 3 cos θ + ...n cos 2 n −3 θ) which by combining terms and use of which a McLauren Expansion one can McLauren show that: EMA 6165 Polymer Physics – AB Brennan EMA 9 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 EMA 6165 Polymer Physics – AB Brennan EMA 10 Freely Jointed Chain Vector Analysis cosθij = 0; for i ≠ j Thus, the FREELY JOINTED CHAIN is defined as: r 2 = nl 2 Consider how <r2> scales with Molar scales Mass Mass EMA 6165 Polymer Physics – AB Brennan EMA 11 Freely Rotating Chain Vector Analysis Mean Square End to End Distance r 2 = nl 2 2 As shown previously: ri rj = l cosθij 12 EMA 6165 Polymer Physics – AB Brennan EMA Freely Rotating Chain Model Freely Vector Analysis Vector r 2 = nl 2 2 1 + cos(180 − τ ) 1 − cos(180 − τ ) 2 (Fixed angle = (Fixed 109.5°) 109.5°) 13 r = 2nl EMA 6165 Polymer Physics – AB Brennan EMA Freely Rotating Chain Vector Analysis r 2 = 2 Ml 2 • Freely rotating model: – high temperature – solvated • Fixed bond angle of 109.5° expands by 2 • Ignores bond rotational energy barriers Ignores (RIS) (RIS) EMA 6165 Polymer Physics – AB Brennan EMA 14 Hindered Rotating Chain Model Ilustration of RIS C H C H C H CH φ H H H C φ C H Periodic fluctuations as 2π /3 EMA 6165 Polymer Physics – AB Brennan EMA 15 Hindered Rotating Chain Model Assumptions • Freely rotating model: – high temperature – solvated θ = fixed value fixed – 0< φ <360 <360 i+1 i i-1 i+2 φ θ EMA 6165 Polymer Physics – AB Brennan EMA 16 Hindered Rotating Chain Model Assumptions • Hindered rotating model: –torsion (valence) angle dependence –first order interactions –high level interactions – θ <= 109.5 for a C-C-C bond angle <= –0< φ <360 <360 i+1 i i-1 EMA 6165 Polymer Physics – AB Brennan EMA 17 i+2 φ θ Freely Rotating Chain Vector Analysis •However, now the bond angle is restricted to a constant value. •Thus, one must restrict the chains projection back upon itself. EMA 6165 Polymer Physics – AB Brennan EMA 18 Freely Rotating Chain Vector Analysis 2 k r = n l 1 + ∑ (n − k ) α n k =1 where α = cos (180 − τ ) 2 2 n −1 For the Freely Rotating Model, PE For the equation is further simplified to give: give: EMA 6165 Polymer Physics – AB Brennan EMA 19 Freely Rotating Chain Vector Analysis r 2 = 2nl 2 •Which clearly demonstrates the increase in the chain dimensions by the restriction of bond angle. •This is thus critical in terms of relative measurements such as viscosity for complex polymer and copolymer structures EMA 6165 Polymer Physics – AB Brennan EMA 20 Chain Dimensions Characteristic Dimensions r where: 2 0 = Cnl 2 <r2>o - chain dimensions at Theta conditions C - Characteristic Chain Ratio n θ - number of segments l - segment dimension (length) EMA 6165 Polymer Physics – AB Brennan EMA 21 Chain Dimensions Chain Expansion Factor ∀ α - chain expansion factor expansion • C - characteristic characteristic chain ratio chain θ α − α = Cψ n 1 − T 5 3 ∀ ψ - interaction entropy entropy ∀ θ - Theta condition temperature temperature ∀ Τ − temperature temperature EMA 6165 Polymer Physics – AB Brennan EMA 22 Chain Dimensions Chain Expansion Factor ∀ α = 1: when T = θ 1/10 ∀ α −> n 1/10; T > θ • • θ α − α = Cψ n 1 − T 5 3 <r2> = <r2>o at T = q Chains behave as phantoms θ 12 α − α ∝ ψ 1 − M T 5 3 EMA 6165 Polymer Physics – AB Brennan EMA 23 Chain Dimensions Viscosity Relationships η ηr = η o η red η η r= −1 η o 1 η 1 η inh = −1 η = c ln η o c2 η 2 o [ η] = lim c →0 1 c2 η − 1 ηo 24 EMA 6165 Polymer Physics – AB Brennan EMA Summary • Freely Jointed Chain dimensions Freely scale with Molar Mass scale • Freely Rotating Model increases Freely chain dimensions of simple polyolefins by a factor of 2 compared to the freely jointed model. to EMA 6165 Polymer Physics – AB Brennan EMA 25 References • Introduction to Physical Polymer Science, Introduction 4th Edition, Lesley H. Sperling, Wiley Interscience (2006) ISBN 13-978-0-471-70606Interscience 9 • Principles of Polymer Chemistry, P.J. Flory Principles (1953) Cornell University Press, Inc., New York. York. • The Physics of Polymers, Gert Strobl (1996) The Springer-Verlag, New York. Springer-Verlag, • Figures were reproduced from Polymer Figures Physics, (1996) Ulf Gedde, Chapman & Hall, New York. New EMA 6165 Polymer Physics – AB Brennan EMA 26 ...
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