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Lecture4Notes - Lecture 4: Polymer Conformation Real Chains...

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Unformatted text preview: Lecture 4: Polymer Conformation Real Chains 1 Monday, October 4, 2010 Recall from last time … A polymer can adopt many conformations Polymers with no interactions between monomers separated by many bonds along the chain are called IDEAL CHAINS The mean square end to end distance for an ideal (long, linear) chain is: R 2 ≅ C∞ nl 2 C∞ is a characteristic ratio that describes the stiffness of the chain We have defined the Kuhn segment, b, for a polymer chain, which is related to the number of monomers N. From the Kuhn segment, we obtain an expression for the mean square end to end distance for ideal chains Several models of ideal chain conformation exist: freely rotating chain, freely jointed chain, worm-like chain, hindered rotation and rotational isomeric state The mean square radius of gyration, <Rg2>, is defined as the average square distance from all monomers to the center of mass of the polymer General scaling for IDEAL CHAIN: RN Monday, October 4, 2010 1 2 2 Polymer Conformation • The conformation of a polymer chain will be strongly affect by the intraand intermolecular interactions of the chain with itself and its environment • In order to better understand and predict the 3D shape of a polymer chain based on these interactions, we need a starting point (we did this last time) • Models of conformation often treat polymers as ideal chains, but the real world (and our experiments) is more complicated DEFINITIONS Ideal chain: no interactions between repeat units far apart along the chain, even if they approach each other in space (do not interact with solvent or itself) Real chain: attractive and/or repulsive interactions between/within repeat units as well as with solvent (interacts with solvent and itself) 3 Monday, October 4, 2010 Today’s Concepts Real chains do not exist in a vacuum, but rather in a solution with either solvent, or other chains surrounding them Real chains have interactions with the things surrounding them, and between monomers along the chain These interactions can be described in terms of excluded volume, which describes the net two-body interaction between monomers Temperature can affect excluded volume When concentration is high, net three body interactions become important A polymer “solution” can be described as good, theta, and poor; these can change with concentration 4 Monday, October 4, 2010 Potential Energy versus Distance Too crowded repulsion Potential Energy Potential due to repulsive forces Potential due to attractive forces Perfect balance of attraction and repulsion Monday, October 4, 2010 r Too far away 5 Possible interactions of beads on the chain • • • • Dispersion forces (van der Waals) Dipole/dipole interactions Hydrogen bonding Coulombic Interactions 6 Monday, October 4, 2010 van der Waals Forces, dipole dipole interactions • • weak interactions between dipoles (δ+ and δ -) alignment of liquid crystals, gecko adhesion ⎡⎛ σ ⎞12 ⎛ σ ⎞ 6 ⎤ φ LJ ( r) = 4ε⎢⎜ ⎟ − ⎜ ⎟ ⎥ ⎣⎝ r ⎠ ⎝ r ⎠ ⎦ ϕ = Lennard-Jones Potential (energy) σ = hard shell radius r = nuclear separation repulsive, exchange energy ε attractive, dispersion energy 7 Monday, October 4, 2010 Hydrogen Bonding • hydrogen is δ+ from bonding to O, N, F, (electronegative atoms, forms polar covalent bond) • δ+ H associates with lone pairs of electrons on heteroatom, typically O, N, sometimes (weaker) S, P, B, Cl, Br, I H O H O N N O O O N O R O H O O N H O R O N H H R O N O R O O O O H N N H N N H O O N O H H O R O O N O R H 8 Monday, October 4, 2010 Ionic Bonds • usually metallic and non-metallic atoms involved • metal gives up valence electrons to non-metallic atom(s) • two oppositely charged IONS come together to form an ionic bond + Ca2+ - 9 Monday, October 4, 2010 Summary of Interaction Strengths Characteristics Approx. Strength (kcal/mole) Examples Dispersion forces Short Range, varies as 1/r6 0.2 - 0.5 PE, PSty Dipole/Dipole Interactions Short range, varies as 1/r6 0.5 - 2 PAN, PVC Strong Polar Interactions Complex form, and Hydrogen Bonds but also short range 1 -10 Nylons, Polyurethanes Coulombic Interactions (Ionomers) Increasing interaction strength Type of Interaction 10 - 20 alginate Long range, varies as 1/r 10 Monday, October 4, 2010 Excluded Volume • • • • Excluded volume describes the volume around a monomer from which another monomer is excluded A positive excluded volume means that repulsion occurs between monomers A negative excluded volume means that the monomers are attracted to each other When excluded volume equals zero, this is called the theta state and it describes an ideal chain Monomers have some repulsion, resulting in a stretched chain ν>0 R~N3/5 Monday, October 4, 2010 Remember, this is our starting point Monomer-monomer repulsion is perfectly balanced by attraction ν=0 R~N1/2 Monomer-monomer attraction results in smaller coil v<0 R~N1/3 Excluded Volume • • • U(r) (effective interaction potential) is the energy cost of bringing two monomers from infinite separation to within distance r of each other. Hard-core repulsion when r is small Often, attractive at intermediate distances U(r) includes all interactions, including those mediated by solvent 12 Monday, October 4, 2010 Excluded Volume • • • Boltzmann factor describes the probability of finding two monomers separated by a distance r in a solvent at temperature T The Mayer f-function is defined as the difference between the Boltzmann factor for two monomers at distance r and that for the case of no interaction (or infinite distance) The excluded volume v summarizes the net two-body interaction between monomers Hard-core repulsion e − U (r ) kB T f (r ) = e −U ( r ) kB T −1 What is the area under this function? The area describes the probability of finding a monomer close to another monomer Attractive interactions 13 Monday, October 4, 2010 Excluded Volume • The EXCLUDED VOLUME v is minus the integral of the Mayer f-function over the whole space: f (r ) = e • • • −U ( r ) kB T −1 U (r ) − ⎛ ⎞ v = − ∫ f (r )d 3r = ∫ ⎜ 1 − e kB T ⎟ d 3r ⎝ ⎠ EXCLUDED VOLUME is the net two-body interaction between monomers. Hard core repulsion (r<1) makes a negative contribution to the integration of the Mayer f-function, and a positive contribution to excluded volume. Attractive forces between monomer (r>1) makes a positive contribution to the Mayer f-function and a negative contribution to excluded volume A net attraction has excluded volume less than zero, and a net repulsion has excluded volume greater than 0. In practical terms, excluded volume is the volume around a segment (monomer) that another segment (monomer) cannot occupy, determined by energetics of the interaction between monomers. 14 Monday, October 4, 2010 Flory theory, cont. • • • • • • Flory theory overestimates repulsion energy (correlations between monomers along the chain are omitted) The number of contacts per chain, in Flory theory, scales with N1/5, but in computer simulations of random walks , the number of contacts between monomers that are far apart does not scale with N. The result is that the attractive forces are also overestimated The two overestimations cancel each other out. The elastic energy of the chain is also overestimated, because the ideal chain conformational entropy is assumed Still, Flory theory is simple, and the predictions are in good agreement with experiment, and more sophisticated experiments. Flory theory leads to a UNIVERSAL POWER LAW dependence of the size of the polymer on the number of monomers N R~N υ 15 Monday, October 4, 2010 R~N υ • The Flory approximation of the scaling exponent is ν =3/5 • • For an ideal chain, the exponent is ν= 0.5 If we again think of a chain as a fractal object, the fractal dimension of an ideal chain is D=1/ν=2 • For a swollen chain, the fractal dimension is D=1/ν=5/3 • More sophisticated theories lead to a better estimate of the scaling exponent ν for a swollen linear chain in 3D: υ ≅ 0.588 16 Monday, October 4, 2010 Flory Theory of a polymer in a poor solvent • The Flory free energy for polymer chain is given by: ⎛ R2 N2 ⎞ F ≈ kT⎜ 2 + v 3 ⎟ R⎠ ⎝ Nb • • • • • In a poor solvent, the excluded volume is negative, indicating a net attraction Both entropic and enthalpic contributions decrease with decreasing R Strong collapse of a polymer into a point is not physically possible; need stabilizing term in the free energy There is an entropic cost to confining a chain, so we can add in a term to account for that: ⎛ R 2 Nb 2 N2 ⎞ F ≈ kT⎜ 2 + 2 + v 3 ⎟ R R⎠ ⎝ Nb The free energy still has a minimum at R=0; the confinement entropy term is still not enough 17 Monday, October 4, 2010 What now? • • • Stabilization of the coil comes from other terms of the interaction part of free energy The interaction energy per unit volume is expressed a virial expansion in powers of the number density of monomers cn In a pervaded coil volume R3, the excluded volume term is the first term in the virial series and counts two-body interactions as vcn2 ( ) Fint 2 3 ≈ kT vcn + wcn + ... 3 R • The next term in the expansion counts three-body interactions as wcn3, where w is the three-body interaction coefficient ⎛ N2 ⎞ N3 Fint ≈ kT ⎜ v 3 + w 6 + ...⎟ R ⎝R ⎠ • At low concentration, the two-body term dominates the interaction, but the three-body term becomes important at higher concentrations and can stabilize the collapse of the globule 18 Monday, October 4, 2010 Excluded Volume Allows us to understand Chain Behavior • • • If the attraction between monomers just balances the effect of the hard core repulsion the net excluded volume is zero and the chain will adopt a nearly ideal conformation (θ-condition) If the attraction between monomers is weaker than the hard core repulsion, the excluded volume is positive and the chain swells; the coil size is larger than the ideal size If there are attractive forces between monomers stronger than the hard core repulsion, excluded volume is negative and the chain collapses; the size of the globule is smaller than ideal 19 Monday, October 4, 2010 What happens when we deform the chains? tension compression 20 Monday, October 4, 2010 Temperature Dependence of Excluded Volume T − Tθ 3 v≈ b T • For T<Tθ, the excluded volume is negative (coil collapse, poor solvent) • For T> Tθ, the excluded volume is positive and the chain is swollen • For T>> Tθ, excluded volume becomes independent of temperature • At T= Tθ, the net excluded volume is zero and the chain adopts an ideal conformation 21 Monday, October 4, 2010 How does concentration affect chain conformation? 22 Monday, October 4, 2010 Polymer Chains in Solution • • • We were treating (ideal) polymer chains as if they existed in space; in reality, they are surrounded by something Since polymers are never in the gas phase, they have to be surrounded by something – solvent or other polymers Polymer solution behavior is affected by concentration c cv mon N Av φ= = ρ M mon Volume fraction, φ: the ratio of the occupied volume of polymer in the solution and the total volume of the solution c: polymer mass concentration M mon vmon:occupied volume of a single monomer ρ= v mon N Av ρ: polymer density Mmon: molar mass of the monomer 23 Monday, October 4, 2010 Pervaded Volume, V • The pervaded volume V is the volume of solution spanned by the polymer chain: V ≈ R3 This volume is typically orders of magnitude larger than that occupied by the polymer chain. The fractal nature of the polymer chain N~RD with typical fractal dimensions of D<3 means that most of V is occupied by solvent or other chains. 24 Monday, October 4, 2010 Solution Regimes for Flexible Polymers The volume fraction φ, of a single molecule inside its pervaded volume V is called the overlap volume fraction, φ* or the corresponding overlap concentration c*: Nv φ* = mon V Nv mon M c* = ρ = V VN Av Polymer coils in dilute solutions (where the average distance between chains is larger than their size) are far away from each other. In dilute solutions, the solution properties are similar to that of the solvent, with slight modifications due to the presence of the polymer. 25 Monday, October 4, 2010 Solution Regimes for Flexible Polymers: Semidilute Solutions are called semidilute at polymer volume fractions above overlap. While most of the polymer volume fractions in semi-dilute solutions is still very low (φ<<1), the polymer coils overlap and dominate most of the physical properties of the solution. In semidilute solutions both solvent and other chains are found in the pervaded volume of a given coil. The overlap parameter P is the average number of chains in a pervaded volume randomly placed in solution: φV P= Nv mon 26 Monday, October 4, 2010 Solution Regimes for Flexible Polymers: Bulk In the absence of solvent, polymers can form a bulk liquid state, called a polymer melt. Polymer melts are neat polymeric liquids above their glass transition temperature and melting temperatures A macroscopic piece of a polymer melt remembers its shape and has elasticity on short time scales, but exhibits liquid flow (with high viscosity) at long times. In the polymer melt the overlap parameter is large (P>>1) and the strong overlap with neighboring chains leads to entanglement that greatly slows the motion of polymers. However, individual chains in a polymer melt do move over large distances on long time scales, a property characteristic of fluids. 27 Monday, October 4, 2010 Solution Regimes for Flexible Polymers 28 Monday, October 4, 2010 Summary of Real Chains • • • • Real chains have interactions with their environment If the attraction between monomers just balances the effect of the hard core repulsion the net excluded volume is zero and the chain will adopt a nearly ideal conformation (θ-condition) If the attraction between monomers is weaker than the hard core repulsion the excluded volume is positive and the chain swells; the coil size is larger than the ideal size; the chain is a self-avoiding walk If there are attractive forces between monomers stronger than the hard core repulsion, excluded volume is negative and the chain collapses; the size of the globule is smaller than ideal (below Tθ) • Most chains in a poor solvent collapse into a globule, agglomerate with other chains and precipitate from solution; this occurs far below Tθ • This is the non-solvent limit and an individual chain in that solvent has a fully collapsed chain; most chains therefore precipitate from solution into a melt and then adopt ideal conformation to maximize entropy Stretching a real linear chain in a good solvent is easier than an ideal chain Compressing an ideal chain is easier than compressing a real chain Excluded volume changes with temperature in the vicinity of Tθ 29 • • • Monday, October 4, 2010 References • Rubenstein and Colby Chpt 3 • Strobl Chpt 2-3 (Chpt 3 has more info than we’ve covered so far) • Painter and Coleman Fundamentals Chpt 9.D. • Topics in Polymer Physics by Richard Stein and Joseph Powers Chpt 2 30 Monday, October 4, 2010 ...
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This note was uploaded on 11/29/2010 for the course BME 104 taught by Professor Kasko during the Fall '10 term at UCLA.

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