Lecture notes Polymer Chemistry

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Published Date:21-07-2017
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GENERAL POLYMER CHEMISTRY (KJM 5500) Part II-Macromolecules in solution Lecture notes By Bo Nyström Institute of Chemistry, University of Oslo Translation by Anna-Lena Kjøniksen 1MACROMOLECULES IN SOLUTION • Macromolecules size, conformation and statistics in dilute solutions • The thermodynamics of polymer solutions • Characterization of polymer molecules in dilute polymer solutions a) End-group analysis b) Osmotic pressure c) Light scattering (static) d) Ultra centrifugation (equilibrium - and velocity sedimentation) e) Diffusion f) Viscosity g) Gel permeation chromatography (GPC) 2• Introduction of the scaling notion The size, conformation and statistics of random coils In order to describe the conformation of random coils two parameters are used: • End-to-end Distance • Radius of Gyration With experimental measurements one may measure the radius of gyration, but not the end-to-end distance. The end-to-end distance is though of theoretical interest in connection with polymer statistics. 3End-to end distance, r, for a conformation of a random coil. Radius of gyration, R : The distance from the center G of gravity that all the mass can be gathered into without changing the moment of inertia of the molecule 2 Moment of inertia = mass⋅ R G Centre of gravity 2 mr ∑ ii 2 i R= (1a) G m ∑ i i 42 mr ∑ ii 2 i R= (1b) G m ∑ i i 12 / 2  mr ∑ ii 12 / 2 i RR== () (1c)  GG m ∑ i  i  If all mass points have an identical mass, M : 0 2 2 mr=M r and mn=⋅M ∑∑∑ ii 0 i i 0 i i i (n = number of monomer units) From equ. (1c): 12 / 2  r ∑ i  i R= (2a)  G n   512 /  2 r ∑ i   i R= (2b) G 12 / n 2 r ∑ i 2 i R= (2c) G n The molecular weight dependency of the radius of gyration Sphere: 2 rm ∑ ii 2 i R= G m ∑ i i R 22 rm⋅= rdm ∑ ∫ i i i 0 6m 2 2 dm= 4⋅πr⋅ρ⋅dr (ρ= ); (V=4πr dr) V 5 R 4⋅π⋅ρ⋅ R 2 4 r⋅ m= 4⋅π⋅ r⋅ρ⋅dr= ∑ ∫ i i 5 i 0 (n+1) x n 〈 x dx=+const.〉 ∫ n+1 3 R 4⋅π⋅ρ⋅ R 2 m= 4⋅π⋅ρ⋅ r⋅ dr= ∑ ∫ i 3 i 0 4 5 ⋅π⋅ρ⋅ R 2 5 R= G 4 3 ⋅π⋅ρ⋅ R 3 12 / 2 3R 3 2 R== ;RR⋅ GG  5 5 M V(volume)= v⋅ ;v= the partial spesific volume N A 3 4⋅⋅π R 3⋅⋅ vM 3 V= ; R= 3 4⋅⋅π N A 713 / 12 /  3 3v  13 / R=⋅⋅ M   G  5 4⋅⋅π N A 1/3 R=const.⋅ M (3) G Rod: Rod: Centre of gravity The rod has a cross section with an area A. 3 L / 2 A⋅ρ⋅ L 2 2 r m=ρ⋅ r⋅ dr⋅ A= ∑ ∫ i i 24 i 0 L / 2 ⋅ρ⋅ A L m= A⋅ρ⋅ dr= ∑ ∫ i 2 i 0 3 A⋅ρ⋅ L 2 L 2 24 R== G A⋅ρ⋅ L 12 2 8 R=const.⋅ L G R=const.⋅ M (4) G Random coil Thermodynamic good conditions: 0.60 R ∝ M (”Mean-field” approximation) G 0.588 R ∝ M (”Renormalization group theory”) g 0.50 θ-Conditions: R ∝ M g The relation between R and the end-to-end G distance, r , in the molecule i 9 For linear flexible polymers the following relation between chain distance (r) and the radius of gyration is valid: 2 12// 12 r 2 2 2 R== ;rR 6 (5) G G ()() 6 1/2 (-) Root-mean-square (r.m.s.)-average. 12 / 2 22 12 /  nr +n r + •••n r 2 11 22 ii r=  () nn++••n  12 i  Models for random coils 1) Chain molecules with a kind of given, locked, rigid structure. 10 a) Totally extended chain: L = l·n K n = number of bonds; l = bond length; L = contour length K b) Chain with a zigzag structure θ rn =⋅l⋅sin( ) 2 This chain has a locked bond angle that assures that the chain may have only one conformation, and that it is completely rigid. 112) Chain without restrictions (bonds that may assume every possible direction with the same probability) This kind of chain is called freely joint chain, and it exhibits statistics that is called random walk, or drunk mans walk. n r (the resultant vector) = (l+ l+ l+•••l )= l ∑ 1 2 3 n i i=1 n n n n  2 rr =⋅r= l⋅ l=⋅ll ∑∑∑∑  i j ij   i== 11 j i=1j=1 2 r=() l+ l+ l⋅() l+ l+ l= 1 2 3 1 2 3 () l⋅l+ l⋅l+ l⋅l+ l⋅l+ 1 1 1 2 1 3 1 2 ll⋅+ll⋅ +l⋅l+ll⋅+l⋅l ()() 2 2 23 1 3 23 3 3 ( ) represents i=j 2 When i= j;l⋅l= l (l is the lenght of the vector) i i 12 If we have n monomers in the chain, we have (n-1)≈n vectors. We assume that all bonds is of the same length l and multiply out all i=j, we get the square- average of the end-to-end-distance n n 22 (6) 〈〉rn = ⋅l + 〈l ⋅l〉 ()i≠j ∑∑ ij i=1j=1 For l · l with i ≠ j, we get: i j 2 〈l⋅ l〉= l⋅〈cosθ〉; where θ is the angle between the vectors. i j For a random coil all values of θ are equally probable 22 〈〉 cosθ= 0; 〈rn〉=⋅l 12 / 〈〉rn = ⋅l (7) 13For a rod like particle the equivalent expression is: r = n · l (8) 3) Free rotation, fixed bond angle In this case one lets the bond angle be set at a fixed value. One allows free rotation around the bond. In this case, the last part of the equ. (6) is not zero due to the fixed bond angle. 1+ cosθ 22 〈〉rn = ⋅l ⋅ (9) 1− cosθ (This locking of the bond angle gives an increase of r ). 14Equ. (8) and (9) is only valid when the end-to-end distance exhibits a Gauss distribution. If we identify this bond angle with the tetraeder-angle o (θ = 109 ) we get equ. (9): 2 2 〈r〉= 2.00⋅ n⋅l (10) If we compare this result with the experimental result for polyethylene: 2 2 〈r〉= (6.7± 0.3)⋅ n⋅l we observe that this model gives too small values. 154) Hindered rotation We will now take into consideration a fact that is often the case for polymer chains, namely when the rotation around the single bonds is not free. For a complete description of the conformation of a model chain, we have to have information of both the bond angle (θ) and the rotation angle (φ) (torsion angle). 1+ cosθ 1+〈cosφ〉 22 〈〉rn = ⋅l ⋅⋅ (11) −θφ 1 cos 1−〈cos 〉 〈cosφ〉= 0 (free rotation) ;〈cosφ〉≠ 0 (hindered rotation) 16Equ. (11) takes into account trans- and gauche- conformations. If the gauche- and trans- conformations have the same energy: cosφ = 0 Real polymer chains (short-range interactions) The end-to-end distance, r, for a polymer chain with a fixed bound angle, θ, and the rotation angle, φ, may be written as: 1+ cosθ 1+〈cosφ〉 22 〈〉rn = ⋅l ⋅⋅ 1− cosθ 1−〈cosφ〉 Let us now replace the real bound length l with a fictive bound length, β, which is called the effective bound length: 22 〈〉rn = ⋅β 17β The ratio is a measure of the stiffness of the 1 polymer chain. 2 β C= : the characteristic ratio. 2 l Ex.: Polymer C Polyethylene 6.8 Polystyrene 9.9 Polyethylene oxide 4.1 Polybuthadiene 4.8 Definition of the Kuhn length l ku We may generally describe a statistic chain molecule with the aid of the concept equivalent statistic segment. In this case we imagine that instead of contemplating a chain that consists of real segments with a hindered rotation around the bonds and fixed bond angles, we make a hypothetical statistic chain 18with the same chain length and the same end-to-end distance as the real chain. (n=30, N =5) k Model: 22 We got the following equation: rC=nl By using the Kuhn-model, we get 22 rN=⋅l ku ku For a fully outstretched chain, we get the contour length L k L = n · l k For the hypothetical chain L = N · l k ku ku We got: n · l = N · l ku ku 1922 ln ⋅⋅l=N ⋅l =C⋅n⋅l ku ku ku lC =⋅l ku n N= ku C We see from these two equations that the stiffer the molecule, the longer is the Kuhn-segment, while there will be a smaller number of Kuhn-segments. Polymer chains and excluded volume effects (”long-range”-interactions) Non-perturbed Chain Excluded Volume Effect (Good conditions) (θ-conditions) 20

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