Physical Picture for Diffusion of Polymers Low Molecular Weight (M < M e ) chains shown moving past one another. Z X Y Rouse chains, unentangled Figure by MIT OCW. High Molecular weight (M > M e ) Entanglements in a polymer melt (a short portion of one chain is outlined in bold). Lateral chain motion is severely restricted by the presence of neighboring chains. Portion of an effective constraining tube defined by Reptating chains, entanglements about a given chain entangled Figure by MIT OCW. Figure by MIT OCW.
Rouse Chain A fleible string comprised of connected Brownian particles that interact with a featureless viscous medium. Number of repeat units is less than the entanglement limit, small N, where N < N e Viscosity depends on monomeric friction factor ξ M and η ~ ξ ~ ξ M ~ M Diffusivity depends on friction factor via Einstein equation: D = kt/ξ ~ kt/m
(Self) Diffusivity Regimes (Two more scaling laws!) Small N or M Rouse Regime (follows directly from Einstein definition of D) D ~ N -1 Large N or M Reptation Regime (see derivation in net section) D ~ N -
Reptating Chains Key idea: Consider a Rouse chain moving inside a tube defined by the surrounding topological constraints. The number of repeat units in the chain is much larger than the entanglement limit, i.e. large N, where N > N e Basics In this regime, zero shear rate viscosity η 0 ~ M 3.4 < > = D t Fundamental equation of Brownian notion D = kt ξ F = ξ v Einstein relation for diffusion of Brownian particle with a friction factor ξ in host medium Frictional force for particle moving with velocity v in host medium
Reptation (P. de Gennes, S. Edwards, 1971) 1) Identify set of topological constraints to chain motion. ) Construct a tube of diameter a tube about the contour of the test chain which meanders through these constraints. (Edwards, 1967). N e is the average number of segments between entanglements. There are Z pieces of the tube such that the tube contour length L is the same as that of the diffusing chain Nb = L Za = Nb (tube itself is a random walk; treat entangled chain as physicist s universal chain) 3) Test chain moves by continuous creation/destruction of its tube. As the chain moves, one end comes out of the tube and a new part of the tube is created. The end of the tube from which the chain has withdrawn is considered destroyed since it no longer has any influence on the mobility of the test chain. reptating string inside tube Figure by MIT OCW.
Reptation cont d 4) The diffusion coefficient of the Rouse-like chain within the tube (not entangled) is given by: D tube = where N is the number of statistical segments of the chain and ξ M is the friction factor per statistical segment. 5) As a result of the local wiggling of the chain, the tube migrates. The time τ for the chain to completely escape its old tube of total length L is approimately: τ = where L is the length of the tube which is also the contour length of the chain, L = N b. τ is called the longest relaation time, since it characterizes the motion of the entire chain. The relaation time for the chain to eit its old tube thus scales with N 3 since Nb = L. τ ~ N 3 Recall viscosity scales with flow time, so we predict that for an entangled melt η ~ M 3 kt Nξ M L D tube
D rept N τ 1 N Note D rept =D self Eperimentally, for well entangled polymer melts, the self-diffusion coefficient is found to scale as 1/(molecular weight) in accord with the reptation model. Reptation cont d 6) Diffusion coefficient for entangled chains: Center of mass motion of the chain defines a reptation diffusion coefficient, D rept. The distance diffused for a time t is given by < > = D rept t For t = τ, the chain center of mass can be estimated to have moved a distance of approimately the radius of gyration. R g = D rept τ Therefore D rept scales with chain length as
Polymer-Polymer Diffusion Entropic Effects - consider diffusion of chains with different molecular weights: scenarios: 1. Fied obstacles : diffusion of short chain in matri of large, entangled chains. Constraint release : diffusion of long chain in matri of short chains Enthalpic Effects Self diffusion (same species) χ = 0 χ > 0 χ < 0 No effect Mutual diffusion (different species) Slows or stops inter-diffusion Thermodynamic acceleration of inter-diffusion of species in addition to Brownian motion Ass t Prof ELT DeGennes Prediction: PVC/PCL
Enthalpic Influence on Diffusion cont d (Mutual Diffusion) Recall F-H free energy of miing per lattice site: ΔG M N o kt = φ 1 1 lnφ 1 + φ lnφ + φ 1 φ χ 1 In the case of the mutual interdiffusion of two (miscible) high MW polymers with 1 = >> 1, the free energy of miing for a negative χ is dominated by the enthalpic term. For chains comprised of N monomers, ΔG M ~ Nχ 1 φ 1 φ This enthalpic effect influences the scaling dependence of the diffusion coefficients by increasing the diffusion scaling law by a factor of N for compatible, negative chi blends.
(self) Diffusion and Viscosity Scaling Laws Property N < N e N > N e η D N N -1 N 3 N - Where N e is the number of statistical segments between entanglements. Note M e = N e M o, where M o is the molecular weight per monomer.
Rubber Elasticity Assumptions: 1. Gaussian subchains between permanent crosslinks. Temperature is above T g 3. Fleible chains with relatively low backbone bond rotation potentials 4. Deformation occurs by conformational changes only 5. No relaation (permanent network is fied on time scale of eperiment) 6. Affine deformation (microscopic deformation : macroscopic deformation) 7. No crystallization at large strains 8. No change in volume (density) with deformation
Ideal Rubber Elasticity cont d 3D Gaussian Subchain Network Relaed subchains M = Average Molecular weight between Xlinks crosslink Gaussian subchain (n segments, segment step size l)
Relaed vs Etended Subchain Conformations Relaed r 0 1/ Etended r 1/ F F
l 0 Entropic Elastic Force F = T S l T,P State 1: Relaed F = 0 l 0 l 0 State 1 Ω 1 r 0 1/,y,z F l y F State : Etended F > 0 l z l State Ω r 1/,y,z
Etension Ratio, i,, y Local coordinates z are the 3 orthogonal etension ratios which depend on the type of loading geometry. ' = y'= y y z'= z z Assumption #6: Affine Deformation Macro coordinates l = l y l y = 0 l 0 z = l z l 0 Assumption #8: ΔV = 0 the rubber is incompressible during deformation y z = 1
Entropic Elastic Force cont d The entropic elastic force is derived by considering the change in the conformational entropy of the etended vs. the relaed state + + = Ω Ω = Δ z y 1 nl 1)] ( z 1) ( y 1) ( 3[ klnep kln S 1 1 ln Ω Ω Δ k S S S = = = 0 1 3 ep nl r const Ω + + = ) 3( ep nl z y const z y Ω r It s all about entropy again 0 r z y = + +
Entropic Elastic Force cont d + y + z = r 0 and n steps of size l in subchain = y = z = nl = r 0 r 0 3 (for the relaed subchain) ΔS( i ) = k ( + y + z 3) per sub-chain (general relationship) ΔF i = T ΔS( i) l i T,P ΔS ) ( i depends on details of initial sample shape and applied loads and this in turn controls ΔF i
Eample: Uniaial Deformation y With Ai-symmetric sample cross-section l dl l 0 l, d Deform along, = = l0 l0 1 =, y = z = Poisson contraction in lateral directions since y z = 1 Rewriting ΔS ) eplicitly in terms of F ( i k 1 1 ΔS( i ) = + + 3 = T l ΔS( ) ( ) T,P = kt l z + 3
Uniaial Deformation cont d F = kt l 0 l 0 = kt At small etensions, the stress behavior is Hookean (F ~ const ) Stress-Etension Relationship σ A l = V 0 0 F ( ) A zkt 1 = for z subchains in volume V A 0l0 z = # of subchains per unit volume = N links V = total 0 Usually the entropic stress of an elastomeric network is written in terms of M where M = avg. molecular weight of subchain between -links mass/vol molesofcrosslinks/volume = M = ρn AkT 1 σ = ( ) M ρ 1 N /N A uniaial
Young s Modulus of a Rubber E dσ ρrt lim = 1 + 1 3 d M E = 3ρRT/M Notice that Young s Modulus of a rubber is : 1. Directly proportional to temperature. Indirectly proportional to M Can measure modulus of crosslinked rubber to derive M In an analogous fashion, the entanglement network of a melt, gives rise to entropic restoring elastic force provided the time scale of the measurement is sufficiently short so the chains do not slip out of their entanglements. In this case, we can measure modulus of non-crosslinked rubber melt to derive M e!
Stress-Uniaial Etension Ratio Behavior of Elastomers Uniaial tensile 5? σ() 4 Image removed due to copyright restrictions. Please see Fig. 5 in Treloar, L. R. G. Stress- Strain Data for Vulcanised Rubber Under Various Types of Deformation. Transactions of the Faraday Society 40 (1944): 59-70 compressive σ() Stress / MN m - 3 Theoretical 1 0 1 3 4 5 6 7 8 Small deformations High tensile elongations Figures by MIT OCW.