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Examples of different random coil conformations Conformation = stereostructure of the molecule defined by its sequence of bonds and torsion angles Conformation of polymers

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1) Average of end-to-end distance (mean square end-to-end distance) Note: if the Contour lenght of the chain is L maximum of r max = L. denotes the average calculated from a large number of conformations 2) Radius of gyration R g Root mean square distance of the collection of atoms from their common centre of gravity Gaussian coil is the vector from the centre of the gravity to atom i Mean square end-to-end distance and Radius of gyration R g are related Polymer coil size

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The essential message … a) Polymer chains tend to retract to coils (unless there exist opposing factors..) and the chain conformation can be characterized as a random coil (solution, polymer melt and also glassy amorphous state) The polymer chain size can be evaluated based on average end-to- end distance r and radius of gyration R g. b) Individual polymer chains can be studied as dissolved in solvents. Unperturbed state is characterized by the absence of long- range interactions ( solvent) c) Models for polymer chain dimensions are developed. Mean square of end-to-end distance scales like C = 1 freely jointed chain C 2 freely rotating chain C 3.4 hindered rotation C 6.7 Advanced models (statistical weight matrices) Experimental value for polyethylene 6.7 C 4 – 10 Experiments for different polymers

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Different conformations Example 1: individual C-C bonds, simplest case ethane CH 3 -CH 3 Both end form tetrahedrons. What is the relative orientation of these two tetrahedrions ?

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Example 2: n-butane 4 carbons Trans-conformation All carbon atoms in the same plane Gauche-conformation G G’

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9 potential combination: but only six are independent TG = GT, TG ’= G’T, and GG’ = G’G Conformations … if the polymer have carbons thusIn general (number of distingguishable conformations is less due to symmetry and also some comformations have low probability e.g. GG’) Statistical treatment: Flory: Rotation isomeric state approximation. Each repeat unit can only be in T, G, or G’. Fluctuations around minima are ignores Example 3: pentane 5 carbons

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We have demostrated that the allowed conformations of the consequtive C-C bonds are T, G, G’. We can start constructing chains just by taking random orders of T, G, G’ {TTGGTG’T….} = polymer chain conformation Random coil model ideal ”phantom” chain But long range interactions.. Polymer chain tends to revisit a spatial point that is already occupied. We have to take also account the chain segments that are ”far apart” - long range interactions. Self avoiding walks – more open coils Models for polymers

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Individual polymer chains We want to eliminate interactions between the polymer chains Analogy with the idal gas model: ideal gas = no interactions between the gas molecules - real gas (interactions: repulsion and attractions) Gas: Low concentration: No interactions – like ideal gas High concentration: Interaction between the molecules ”real gas” Polymer solutions: Low concentration – no interaction between the different chains High concentration: Entanglemets etc.. But: In low concentration there is still internal interactions within the single polymer chain and interactions with the solvent molecules... Dilute solutions

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Example Good solvent (T > 35 o C) ”open coil” R g large But if solvent is changed to ethanol, which is ”bad” solvent for polystyrene ”compact coil’’ R g small Good solvent ”open coil” ”extended coil” ”Bad” solvent ”compact coil” In general: solvent Polymer-polymer and polymer-solvent interactions compensate each others... polymer-polymer 1) interaction >> polymer-solvent 2) interaction polymer-polymer 1) interaction << polymer-solvent 2) interaction Low concentration 1% polystyre 99% cyklohexane

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T > Temperature ”open coil” ”extended coil” T < Temperature ”compact coil” One can define for polymer-solvent system Temperature T = Temperature ”in between” where Polymer-polymer and polymer-solvent interactions compensate each others Polymer solubility depents typically also on Temperature.. In general solubility is better in high Temperatures (typical case but not always..) Compare ideal gas Boyle-Temperature where attractive interaction between the gas molecules are equal to repulsive ”hard core” interactions. In this temperature gas molecules behave like ideal gas Temperature

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General model: Assume that the each mainchain bond (or repeat unit) in the polymer is a vectore segment r i r i Chain consist of these segments: n = total number of segments l = lenght of each segment End-to-end vector:vector: magnitude and direction Vector magnitude Remember: where is the angle between vectors Models for statistical chains

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… Take large ensemble of different conformations mean square end-to-end distance and end-to-end distance This result is still a general formulation and is valid for any continuos polymer chain. (Unperturbed phantom chains, - solvents and also polymer melt) Average value of

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1) The freely jointed chain The freely jointed chain consist of a chain of bonds: the orientation of the different bonds is completely uncorrelated and no direction is preferred. i.e. bond angle [0, 180] and torsion angle [0, 360] can have any value There is no correlation between the segments: angle between the two bond vector ij (= 180- ) can have all values mean square end-to-end distance average end-to-end distance now C = 1 C 4 – 10 Experiments for different polymers …

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i.e. bond angle is fixed but torsion angle [0, 360] can have any value. Carbon-carbon bond angle 110 o summation can be performed over single variable (k) by 2) The freely rotating chain

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On the other hnd Differentiate previous Multibly by x Geometrical series sum

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infinitely long chains (n= ) if =110 o freely rotating model gives C 2, for polyethylene chain good approximation at high T when T,G, and G’states are almost ”equally” populated. At low T trans is more populated and chain conformation is more extended

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i.e. bond angle is fixed and also torsion angle can only have three values T, G, G’. Relative population of T, G and G’ states depends on the energy levels (figure) and they are temperature dependent Again we can use same equation: but this case it means a lot of work … see Example Gedde pages result First term is the freely rotation result and second term is the correction due to hindered rotation is the average value for cos , where is the torsion angle. And it is temperature dependend 3) The hindered rotation model

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How to calculate average values? i = 1i =2i = 3 Now we have a system which have a fixed number of states (3: G, T, and G’). System partition function Z measures the number of different states the system can adopt at the tremperature of interest At T = 0 this equals 1 and it increases with increasing temperature Take any parameter f, which takes a value f i for the state i. Average Now for polyethylene 3 states: i = 1 (Gauche) 1 = -120, E 1 = E g i = 2 (Trans) 2 = 0 E 2 = E t =0 i = 3 (Gauhe’) 3 = 120, E 3 = E g Averages in statistical mechanics

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For hindered rotation square end-to-end average For example polyethylene at 140 o C, E t =0 and E g 2.1, Note that in this case the constant C is higher than freely rotating model- hindered rotation includes some rigidity to the chain.. but still less than experimentally found C = 6.7 for polyethylene where Average for cos( ), polyethylene

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i.e. bond angle is fixed and also torsion angle can only have three values T, G, G’. And potential energy of a given bond i depends on the states of the adjacent bonds i-1 and i+1 The conformation of n bond each having three possible torsion angles may specified by n-2 componenets. For example heptane 7 bonds: one possible conformation TGTGG’ Total conformation energy 4) The chains with interdependent bonds: statistical weight matrices

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The chains with interdependent bonds …average square end-to-end distance …

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Statistical problem - What is the propability that the chain displacement vector reaches from origin to the point r and lies within the volume element dV=dxdydz ? Similar random process problems - Brownian particles Previous models for chain conformation predicts quite well the polymer chain end-to-end distance scaling But they does not lead any further analysis – for example what is the distribution of end-to-end distances r ? Random flight analyzis – The Gaussian Chain

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Also similar random process problem – rifle shots on a target

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The function which models random processes is the Gaussian function 1D The probability that polymer end-to-end lenght lying between x and x+dx is the product of p(x) and dx correspondingly in 3D The probability that chain end-to-end lenght lies between r and r+dr is the product of P(r) and dV (4 r 2 is the area of the sphere)

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Mean square value of r ? definite integral from mathematics tables now n = 2 and a = 1/ 2 And from random freely jointed chain

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Crystallization: High density polyethylene HDPE Crystallization Liquid crystals or stiff chains helices -Chains with preferred conformations

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If hydrogen atoms (from PE) are replaced by fluorine atoms with van der Waals diameter 270 pm – to accommodate the fluorine atoms a rotation around each C-C bond of about 20 o is induced. This is accompanied by a slight opening of the C-C chain bond angle to about 116 o poly(tetrafluoroethylene) (PTFE) 13:1 helix

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Vinyl polymers: example Isotactic PP (i-PP)

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Dense Polymer Systems: Entanglements How can we sketch a melt of polymers ? Given the red polymer with N monomers, we call N’ the number of monomers in between two entanglements In a polymer melt where N>>N’, a chain will see N/N’ obstacles or entanglements. It is the number of entanglements (or N/N’) which rules motion of macromolecules in melt. We now aim at describing a theory which captures sufficient details of a polymer motion, but with a picture as simple as possible: Problems should be made as simple as possible….but not any simpler !!! (A. Einstein) From Mezzenga lecture

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Reptation Model (in simple linear chains) Suppose to Sketch a chain in a melt of chains constrained within obstacles O 1, O 2. These obstacles represent the entanglements, and the chain is not allowed to cross them. How can the chain move within this network ? The obstacles are fixed. The chain can only move tangentially to its countour, since transversal motion is impeeded. The chain moves as a snake, a reptile and the movement is called reptation (after DeGennes) From Mezzenga lecture

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From Rubinstein talk

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Diffusion of Branched Chains We study now the diffusion of branched chains. Lets start with a chain with 1 side group having N s monomers. If N 2 monomers now carry a side chain with having N s monomers (and N 1 are free) we can work out the total force f to apply to the chain to have a contour velocity v: f=v(N 1 μ N 2 μ 2 -1 ) In order to be mobile a branch unit with N s monomers needs to refold back. If z is the coordination of the lattice, this may happen with z Ns possibilities. The probability for the branch chain to get back to the branch starting point is then P≈e -αNs with α≈1. Thus we can write μ 2 =μ 1 e -αNs and f=vμ 1 -1 (N 1 + N 2 e αNs ). The effect of branching is exponential. Suppose N 2 =1. τis dominated by branching as soon as N s >1/α*ln(N 1 ) The motion of the main red chains is identical as before. However, the mobility of J, the monomer to which the side chain is attached is not longer μ 1 but μ 2 From Mezzenga lecture

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From Rubinstein talk

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