Chapter 3 Polymer Solutions

Transcription

1 Chapter 3 Polymer Solutions 3. Interactions in Polymer System 3. Criteria of Polymer Solubility 3.3 Thermodynamics of Polymer Solutions: Flory-Huggins Theory 3.4 Thermodynamics and Conformations of Polymers in Dilute Solution 3.5 Scaling Law() of Polymers 3.6 Conformations of Polymer in Semi-dilute Solution 3.7 Thermodynamics of Gels 3.8 Polyelectrolytes Solution 3.9 Hydrodynamics of Polymer Solutions 0

2 Chapt. 3 Polymer Solutions The solution process (linear Polymer) Swelled Sample Polymer Solution This process is usually slower compared with small molecules, and strongly dependent on the chemical structures and condensed states of the samples.

3 Crosslinked polymers: only can be swelled

4 Crystalline polymers Crystalline PE: dissolve at the temperature approached to its melting temperature. Crystalline Nylon 6,6: dissolved at room temperature by using the solvent with strong hydrogen bonds. 3

5 3. Interactions in Polymer System van der Waals interactions. electrostatic interaction Keesom force. Induction (polarization) interaction Debye force 3. Dispersion interaction London force 3 = between permanent charges between a permanent multipole on one molecule with an induced multipole on another between any pair of molecules, including nonpolar atoms, arising from the interactions of instantaneous multipoles. : Dipole moment : Polarizability = 3 I: Ionization energy 4

6 Long-range Coulomb interactions see 3.8 U r QQe k Tr 4 0 B Hydrogen bond interactions 5

7 The Lennard-Jones or Hard Core Potential The L-J (6-) Potential is often used as an approximate model for a total (repulsion plus attraction) van der Waals force as a function of distance. = = 0 > = 6

8 Mayer f-function and Excluded Volume The Probability P(r) of finding two monomers at distance r: Boltzmann factor exp Mayer f-function: difference of Boltzmann factor for two monomers at r and at = exp - Exclude Volume v: = = exp 7

9 3. Criteria () of Polymer Solubility Gibbs free energy of mixing Gmix Hmix TSmix S mix 0 Solubility occurs only when the G mix is negative. Entropy of mixing for ideal solution i S k N lnx N lnx 0 mix H mix??? 8

10 . Hildebrand enthalpy of mixing () : solvent; : polymer H E E EE vv mix v v E E / / H mix v v E v v / / E m V V P (E/v) cohesive energy density nv () P total pairs of [-] nv m nv V m Vm Vm nv V V m m E E V m v v / / =(E/v) / : solubility parameter () 9

11 . Huggin s Enthalpy of mixing Different pairs in solution: solvent-solvent molecule: [-], solute-solute segment: [-], solvent-solute: [-], Mixing process: [ ] [ ] [ ] N N with x segments V s V ps H P P total pairs of [-] mixing mixing P ( Z ) x cells surrounding a polymer volume fraction of solvent ~ Possibility of the cell occupied by solvent. N ( Z ) N Hmixing kt N RTn number of polymers N VN xn s and NxN V m H ( Z ) N = m Vs ( Z ) kt ( Z ) xn xnv N xn V V kt V N m / V s RT Flory-Huggins parameter: (interaction parameter) 0 m s

12 H H H rdr r r dr V Vs V r dr r r dr s Vs V m m Vs Vs V m Vs Hmixing H H H V m const. V s s V V V m V V m s s Vm V s const.

13 H H rdr r r dr V s V s H rdr r rdr V s Vs V V m V Vm V m V s s H H H H mixing Vm V s m H 0 0 Vm kt V s H s Vm V s V V VN s V m s xnv V m s H H V 0 0 s V s 0 V 0 p xn V s V V V 0 0 s p m N

14 3.3 Thermodynamics of Polymer Solutions + 3 h Sconf ( h, Ng ) kb C Nl g S conf 0?, > 0 3

15 3.3 Thermodynamics of Polymer Solutions () Entropy of mixing for ideal solution i S k N ln X N ln X mix () Entropy of mixing for polymer solutions The lattice model assumes that the volume is unchanged during mixing. Each repeating unit of the polymer (segment) occupies one position in the lattice and so does each solvent molecule. The mixing entropy is strongly influenced by the chain connectivity of the polymer component. 4

16 Flory-Huggins theory (Lattice Model ()) N + N x : N=N +xn N N xn N xn xn j,n-xj,j+ W j+???.j+ N-xj.j+ Z(N-xj-)/N 3.j+3 (Z-)(N-xj-)/N x.x (Z-) (N-xj-x+)/N. 5

17 Entropy of mixing from FH theory N xj N xj N xj 3 N xj x Wj N xj Z Z Z Z N N N N Z Z st nd 3 rd 4 th W N Z N! N x! N xn x! N! N! N N x! N x! N xn! N x Wj... j0 N ( x ) Z N! N! N N xn! Entropy of solution: j x Z N xj! N N xjx x th segment st st N st chain Z Ssolution k ln k N ( x )ln ln N! ln N! ln( N xn )! N! 6

18 Entropy of mixing from FH theory Using Stirling s approximation (lnx! xlnx x), we have: N N Z S solution kn ln N ln N ( x )ln N xn N xn e Entropy of the pure solvent and pure polymer: Therefore, Z Spolymer kn ln x( x) ln e S S ( S S ) mixing solution solvent polymer (N = 0) and S conf S solvent 0 0 where N n kn NV NnV s s NxN nxn V m Vm ln N N ln xn NxN NxN Smixing knlnnln Rn ln n ln xn xn xnv NxnV s NxN nxn V m Vm s V k V m s ln ln x 7

19 Free Energy of FH Theory Huggins Enthalpy: Gibbs Free Energy mixing V H kt N RT n kt mixing = m Vs xn xn xnv s NxnV s xn xn xn xn V s m V x m NxN xn xn V m Vm xn Gmixing Hmixing TSmixing G kt N ln N ln xn V xn RT n ln n ln xn V m kt ln ln V s x x G s m Fm ln ln Vm kt x x For Polymer Solutions x = x NV NxnV s

20 Chemical potentials (): G RT ln = m n x T, Pn, = G RT ln x x m n T, Pn, (for solvent) Vs Gm Fm Vm kt ln ln x (for polymer) = = ln ln In the case of <<, 3 RT w x =/,??? < /, good solvent = /, theta solvent > /, poor solvent 9

21 Osmotic pressure (): Polymer solution P T PT,, 0,, s s P P pure solvent V n xn V m p s s F m,,, G n n P T G PV G P n x n V V V p s m m m p s s, PT, Gn, n, PT, s p s ns s m G 0 m s n s T, Pn, p T, Pn, 0 m, P, T RT F F PV s s m s TPn,, p PV Gm F m RT Fm n s P T PT,, 0,, s s G kt m RT F V s m F m s p 0

22 Osmotic pressure (): RT F V s m F m RT V s x Vs Gm Fm ln ln V m kt x A Vs c M, c xv RT c s M A A c V second Virial coefficent <</x(0.5-)=/ >>/x, >> /x RT V s x???

23 Polymer Shapes in Dilute Solutions RT V x Expanded, unperturbed, and collapsed chains A > 0, < /; good solvent /RTc A = 0, = /; condition Coil-globule transition A < 0, > /; poor solvent concentration ~/kt!!! The coil-globule transition in a solution of polystyrene in cyclohexane. The radius of gyration R g and the hydrodynamic radius R h of the polymer show a dramatic change as temperature passes through the temperature. (Sun, S.T.; etc. J. Chem. Phys. 980, 73, 597.)

24 3.4 Chain Conformations in Dilute Solutions () Flory-Krigbaum s Theory V U F T S kt ln ln ln N kt N V or U V N U U / V U / V... U / V N 0 N N / N / U U V V N iu i0 V iu iu F V V V U R T V 3 ~ g ~?? c RT M NU c M A 3

25 3.4 Chain Conformations in Dilute Solutions () Flory 3/ 3 3h ideal chain W h, x exp 4 h real chain in solutions xl xl 0 E h Whx, W0hx, hexp kt 4

26 () (h) h m n h h 3 v c =l 3 v c /h 3 (-v c /h 3 ) x vc vc vc... i h h i0 h xv c 3 h x x vc vc exp ln i exp ln i 3 3 i0 h i0 h iu iu ln V V x vc vx c exp i exp 3 3 h i0 h 5

27 W(h,x) () E(h) H kt Eh kt V m V s 3 h v 3 3 h h v v c Vm c 3 h Vs vc const c xv c 3 h x xv c h 3 h v xv 3 c 6 c h, =, exp h x v c 3 3 xl, h exp W h x =/, h /,??? - 6

28 . Polymer chain in good solvents method I 3 xl h x v c 3, h exp W h x h W hx, h 0 W hx, 3h x v c 3 3h 3x v exp h xl h c hh 4 xl h * x v c *3 3h 3 0 xl 4 h W 0 (h,x)h 0* =(xl /3) / 5 3 * * h 9 6 v / c x * * h h h l / /5 * * x v c v 0 3 c 3 /5 h h x l l l v /5 x u v v c c, =/ 7

29 . Polymer chain in good solvents - method II G G~ ktln h x B 3h xvc ~ kt B 3... xl h G h 0 /5 h x 3/5 3h x v, exp c... 3 xl h G/ kt B xv c h conformation entropy + </ + second Virial coefficent two body interaction: excluded volume repulsion solvent-segment interaction Two body interactions are important in good solvents! 8

30 Temperature & Solvents h ~ T ( Z ) V s Solution kt RT Z V s Temperature k R T T 9

31 Excluded Volume T Z V s Temperature k R T / T c 3 u v l h x / x e * 5 /5, u 0, u 0, u 0 30

32 When does the freely jointed chain works () <h > 0 ~N 6/5 /5 h ~ N () - h ~ N 0 N e l e Coarse-grained () picture: R h ~ 0 e e chain end 3 Nl chain head

33 Crossover from Gaussian to Swollen Each polymer chain can be divided into many blobs. Polymers within blobs are ideal. u v v c R c Thermal Blob g l T / T Number of Segments in a Chain: x Number of Segments in a Blob: g T Numbers of Blob in a Chain: x/g T x / x 0 5 T T T gt gt R g l g xl /5 3/5 u x 3 l l g T T u 3 l u 3 l 4 l l u E u kbtgt 3 T B 3

34 . Polymer chain in poor solvents-method I: blob model Globule: g In a Globule: Blobs are densely packed T Blobs: u 3 l R Blob size: Interaction Energy in a blob: Excluded Volume: 3 3 glob T T N gt u 3 l T l 3 3 glob g / T u EkTg kt l l B T 3 B T u l 3 l 3 R l N Interaction Energy in a globule: R N l glob /3 /3 N F k T k TN int B B gt 33

35 . Polymer chain in poor solvents-method II: chemical potential 3 RT w x T ' " ' 0 ' " RT " " w" 0, x x " ~3 In Polymer rich Phase: xvc Concentration in Coil: " in 3 h /3 /3 /3 /3 c h~ v x / ~ x l w=/3 /3 34

36 . Polymer chain in poor solvents-method III: three body interactions >/ 3h xvc kt h<<h 0 W 3h x v, exp c... 3 xl h h x G~ B??? 3 xl h =xv c /h xvc xvv G/ kbt ~ w h w h 3 3 vc vc h h 3 3 second Virial coefficent two body interaction: excluded volume repulsion solvent-segment interaction + third Virial coefficent G=0 three body repulsion h 3 ~ x vc Three body interactions become important 35

37 Flory Formula in Dilute Solutions 3 R N N G k u v c BT u w 3 6 Nl R R N: Number of Segments in a chain In good solvents In poor solvents R h l: Length of the Kuhn Segment R N G kbt u 3 Nl R 3 N N G kbtu w 3 6 R R 36

38 Polymer Shapes in Dilute Solutions Coil-globule transition swelling chain unperturbed chain collapsed coil <h> athermal solvents =0 ~N 3/5 ~N / ~ N /3 =/ T >/ Good solvents theta()-solvents poor solvents Excluded Volume positive zero negative 37

39 From dilute to concentrated Semi-dilute T 38

40 Anomalous phenomenon in Semi-Dilute Polymer Solution RT V x c w M lim0 RTc w independent on M /c~c 5/4 c xv M Osmotic pressure measured for samples of poly(-methylstyrene) dissolved in toluene (5 C). Molecular weight vary between M = (uppermost curve) and M = (lowest curve). (Noda,I.; et al. Macromolecules 98, 4, 668.) << >> /x /c~c 0 M cxv RT V c c RT c V M M V /c~c /c~c 5/4??? 39

41 R N ~ RW 3/ 3 3h h, N exp Nl Nl R ~ N ~ N SAW 6/5??? Rg <R g >~N v RgN 40

42 - 4

43 3.5 Scaling Law()- () (a) h N / l N N / g l l g h R F( l, N) Fl g, N F( gl, N / g) a=/ a? F( l, N) const. gl N / g const. Nl 4

44 (b) h l N N N / g v-a=0, a=v h F, N g const N g a? const. N l lg /. lg / l lg Blob Model ~ gl Numbers of blobs: n=n/g Energy of each blob Thermal fluctuation energy k B T, F, N / h F l N g G k Tn k ~ B B T N g 43

45 () g k N Np k k R / g g, R F kln, F kr N g l l 44

46 () g( k) dr i g exp kr r ( k) F( kl, N) F( klg, N / g) F( kl, N) g g( k) g( k)/ g g( k) F( kl, N) NF( kln, ) NF( kr ) g k kr g >>g(kn a 0 g( ) const. N( kr ) k. ( ) g const N kln k a / g g F( kln,, ) N kr g k N / k R k k R g N k / v=/ ideal v=3/5 real 45

47 Form Factor of a Real Chain kr g >> p( k) k =3/5 / 46

48 (3) Blob Model - a. Athermal Chain Stretching Thermal Blobs / T gt l f ~ g T h x G h N g T h g ~ kt B N TT v=/ ideal v=3/5 athermal ~ / N T T / hx / f Number of Thermal blobs Number of Thermal Blobs in a large blob: g Number of large blobs: n=n T /g Energy of one blob thermal blob Thermal fluctuation energy k B T Total Free Energy G T ~ B h 3kT B N N kt g T f G N kt g T ~ B / h ~ kt B N TT h f ~ kt 0. 5 B N T T 3/ athermal v=3/5 thermal solution Energy of large blob thermal blob T / ~ N N / g k T B T 3/ T h N 3/ h kt N g T 5/ T ~ B 3/ 5/ T T 47

49 f f 0.6 h kt h 3/ ~ B N / gt T ~ f /3 3/ ~ B N / gt T f kt ~ B to kt h h 3/ 5/ N / g T T h ** 3/ 5/ ** f T k B / T N / g T T u 3 l 4 l * f kbt / l ** f T k B / T l T 48 l u

50 (b) : / h G~ kt B Nl G h R hn, ~ exp ~ exp kt h h RhN, ~ h exp.06 / h h Nl h x x x xh/ h R h = 3 exp 3 = 3 exp.5 49

51 D () Blob Model c. Biaxial Compression gl or D g T v=/ ideal v=3/5 athermal / / N Nl / R// ~ D~ D N l ideal g D / /3 N Nl l // ~ ~ ~ real / R D D Nl g D D N l R0 Gconf kt ideal B ktn B kt B g D D / R0 N l / / 5/3 N l N l Rr conf real B B B B G kt ktn kt kt g D D D 3/5 Rr N l 50

52 d. Uniaxial compression?? R G kt kt R G kt N / g D conf, blob B R conf B B Nl R0 Entropy of an ideal chain Entropy of an ideal blob chain G N k Tu R u l 3 int B 3 3/4 R N l G k TD N / g int, blob B G G R conf, blob int, blob R Answers real, d 3 d R N l Rideal N l 0 Interaction between blobs d: dimensions () 5

53 I. R ~ N RW, = 3 exp 3, = 6, l Path Integral() Edward s Minimum Model i interaction energy b N g 3/ 3hi exp 3 hi i l l 3 Ng/ N g 3 3 i i exp b R R n b 3 N g Rn const exp dn b 0 n h H Mean-field Free Energy 5

54 - R ~ N l Entropy of Chain Conformations 3 h Sconf ( h, Ng ) kb C Nl g In solution: R Gaussian Chain Model h 0 Nl, = 3 R 0 exp ll e 3 h, (h, N) Mechanic Properties Scattering Theory (R>/0) p f h kt Nl 3 B k k R / g h Scaling Concept & Blob Model N G~ kt B g 53

55 II R ~ N ~ N 3/5 SAW (N,l,k) blob model 54

56 Unique Features of Polymers Thermodynamics: () Large Spatial Extent () Connectivity a. Tacticity b. Polymer Topology c. Flexible vs. Rigid (3) Multiple Interactions (Enthalpy) Dynamics: (4) Entanglement d. Multiple Confirmations (Entropy) (5) Responsive Molecules a. Large-scale Relaxation Time Spectrum b &c. Temp, Rate and Time Dependent Behavior 55

57 Spatial Extent & Connectivity () (Monomer) (Segment) (Blob) (Chain) 56

58 Entropy Multiple Confirmations (Entropy) A Chain R Sconf kb Nl A Blob Chain S conf k B N R / g Flory-Huggins Entropy of Mixing for Multi Chains S k N ln N mixing B ln V m kb ln ln V s x x 57

59 Enthalpy Multiple Interactions (Enthalpy) Two-body interaction N H,int kbtu R 3 3 u vcvc vc l N d chain H,int kbtu R u s s s l In a 3d Chain c c within a Blob In a 3d Blob Chain d Blob Chain E H H g T ktu B 3 T kt N / g 3,int B 3 kt R N / g,int B c Vm Flory-Huggins Enthalpy of Mixing for Multi Chains Hmixing kt B Vs N Three-body 3d H3,int / kbt w R w R 6 R k T xn B 58

60 3.6 Semi-dilute Solutions of Polymers RT V x c w M lim0 RTc w independent on M /c~c 5/4 c xv M Osmotic pressure measured for samples of poly(-methylstyrene) dissolved in toluene (5 C). Molecular weight vary between M = (uppermost curve) and M = (lowest curve). (Noda,I.; et al. Macromolecules 98, 4, 668.) << >> /x /c~c 0 M cxv RT V c c RT c V M M V /c~c /c~c 5/4 59

61 () overlap concentration c* dilute c: < c* Overlap concentration: c = c* Semi-dilute regime: c > c* c <R g > c * Nl R g N 3 3 / 3 ~ N N N 3 3/5 3 3/5 ~ N 4/5 3/5 For good solvent, v = 3/5 N * * c N 33/ 3/5 / Apparent correlation length () R g > app > T >l T / T g /5 /5 l R N N l R g R 0 N / /5 60

62 () Osmotic pressure of semi-dilute solution 6 M xv c c RT M V c V Nv 3 x g i u i U R 3 g c c RT M M NR c u v * 3 g M c R ' * *... c c c RT B B M c c x c R c M T M V x i c c RT N u M M i x i x i * c c RTf M c 3 g c R NR M T c M RT c M Van t Hoff relation of osmotic pressure c c c RT M Nv x M c c RT N M M x u osmotic pressure from Flory-Huggins Theory * m c c RT M c f c

63 Scaling Law of semi-dilute solution Osmotic pressure: 5/4 c c kbtf N c* m c c kt B * N * 3 c ~ N c c c k m m(3v ) B Tc N c c In semi-dilute regime, is independent on N: N m( 3v) For good solvent, v = 3/5, therefore m = 5/4. c m 9/4 c k T B c c * N 0 < < * 6

64 (3) Apparent correlation length app : app m v /5 c Nl * c c* N 3v 3/5 app is independent on N: v Rg N l c c v m m(3) +3 m /5 N c N l c c * * v m( 3v) 0 app app c c Rg f * For good solvent, v = 3/5, therefore m = -3/4. 3/4 /4 app c l 63

65 (4) Polymer Shapes in Semi-dilute Solutions blob model: N/g R N / g g app -g() app ~c -3/4 --/4 l app ~ g l 6/5 /5 g app ~ g,n/g, app g ~ g 3/5 /5 l app app ~c -3/4 --/4 l 3/5 /5 ~ ~ c l 5/3 /3 5/4 3/4 app 3 3 c ~ gl / app N N R ~ ~ c l g c 6/4 g semi-dilute app 5/4 R ~ Nc l /4 /4 g semi-dilute Gaussian Chain 64

66 R g R 0 / app Rg app Rg ~ app u Rg R N R Nc R ~ ~ /4 g g dilute semi-dilute g concentrated ~ N The presence of monomers from the other chains begins to screen () the intramolecular excluded volume interactions. : :, : 65

67 (5) Regions of the Polymer-Solvent Phase Diagram I R ~ N g 3/5 /5 II R ~ N c g / /8 /8 I II I III III Rg ~ N I / Rg ~ N / IV V /3 /3 Rg ~ N c*: c* N 4/5 3/5 IV: BMc R ~( N / ) g /3 Rg ~ N / / AB EF N N / 66 66

68 Other Interesting Topics: Grafted Layer from Single Chain Adsorption: to Multichain l 67

69 (6) Concentrated Solutions. Polymer-Plasticizer. Spinning Solution 3. Gel 68

70 Fiber Spinning Wet Spinning Dry Spinning 69

71 app (6) Concentrated Polymer Solutions Concentrated regime c** Dilute c c* app Rg R N T gt 3 5 /5 3/5 u N 3 g T T l u 3 l u l 3 l l g / T l N gt T 3 5 l g / T g l T N l u v v E c vc l 3 u kbtgt 3 T c B Semi-dilute c c* 3/4 /4 app c l Concentrated c c** 3/4 /4 app c** l c** u 3 l gl 3 T 3 T T 70

72 (7) Polymer Melts A Polymer A + Solvent B c A = A /v c ca kt A c A vc NA Polymer A + Polymer B c A c A kt vc u v NA NB kt A v c NA Excluded Volume c 3 u v l c N B 0 v u N c B l N 3 B lg T / T E u kbtgt 3 T u gt B 3 3/ l gt kt kt B g T l u 6 N B lg ln / T T B 7

73 Polymer Chain in Melts B() R A N A T gt g T NB T ln B N / N ln ln A A N N B A B B Overlap parameter P 3D D / / 3 RA N l P 3 3 Nl Nl R Nl P Nl Nl Flory N 7

74 3.7 Flory-Huggins free energy of a gel V 0 swelling V 0 3 V <h > / 3 V 0 <h > / F F F mixing elastic F RT n ln n ln n F mixing elastic TS S kln ~ kln R, N elastic 3 NkT x y z h0 3 NkT x y z h0 x y z h 3 NkT 3 3 MN / N / V0 N VN 0 VN 0 M Mc 3 V Felastic RT M 0 0 /3 c 73

75 Basic Equation of Gel Swelling ln 0 gel x gel F gel n F n m gel F elastic n F m RT ln n x RT V 0 V nv V V V n n V 0 s 0 s s V0 nv 0 Q=V/V 0 =/ = 3 0 F elastic V V0 RT M /3 / RT 0 Mc M c V s Q 5/3 F elastic n () () 5/3 c Vs RT /3 c M 3/5 Q M c 74

76 Volume Phase Transition of Gels UCST T A 0 T Experiment LCST A 0 Theory, =- ~ -T /T Flory-Huggins Parameter A/ T B 75

77 Problem: semi-dilute solution??? R Nl 0 c 9/4? R N l /4 /4 fluct 76

78 3.8 Solutions of Polyelectrolytes + C C + 77

79 Charge Reversal & Layer by Layer Assembly Oppositely charged plane Polyion of charge Z 78

80 Li-ion Battery 79

81 Wrapping a chain on/in a sphere 80

82 Appendix: Introduction to the theory of electrolytes Long-range Coulomb interaction The force between two spherically symmetric charges in vacuum r Q Q f r U r / QQe 0 4 r The interaction energy (in unit of k B T) between two spherically symmetric charges in vacuum Compared to van der Waals Interaction kt B r 3 QQe 4 r / = 0 Summation of two-body Interactions within a Chain or a Blob E vc N / kt B 3 R 8

83 Linearized Poisson-Boltzmann Equation - Debye-Hückel Theory When the electric potential is low (<5mV), we can make the expansion and only keep the linear term. Thus the Debye- Hückel equation (linearlized Poisson-Boltzmann equation) is obtained. The Debye-Hückel treatment gives a simple (meanfield) description to the many-body interactions between ions. r r r r r 3 8en 0 8en0 e... kt B 3! kt B kt B D B r r where D D kt 8en 8ln 0 B 0 l B e k T B n 0 is the number density (per unit volume) of the ions. l B : Bjerrum In the limit of a strong electrolyte, the surface potential s is small enough so a linearization of the P B equation can be justified. 8

84 Debye-Hückel potential The effect of charge screening is dramatically different from the presence of a polarizable environment. As has been shown by Debye and Hückel 80 years ago, screening modifies the electrostatic interaction such that it falls off exponentially with distance. r l D =3Å (M NaCl) D =m (Water) B exp r/ r D 83

85 Charge screening Negatively charged object Bounded Counterions Salt ions of opposite charge are drawn to charged objects and form loosely bound counter-ion clouds and thus effectively reduce their charges. This process is called screening. 84

86 Polyelectrolyte in solution Semi-dilute concentrated 85

87 Theoretical Model of A Single Polyelectrolyte Chain F R F R F R kt kt kt e conf e electr e B B B efn or B Fconf Re R k T Nl e Guassian Chain l: Kuhn segment l B Bjerrum l B e / kt B F R R e e Felectr Re lb fn kt R 0 e or F R l fn R ln kt R N l electr e B e / e / u l l B B 86

88 () A Single Polyelectrolyte Chain in good solvents Good solvent, </: R Nlu f e /3 /3 B or /3 /3 R Nlu f ln en u f /3 /3 e B B Re N / 87

89 () A Single Polyelectrolyte in poor solvents Poor solvent, >/: N is small N is large 88

90 Blob Bead (globule) Beads+ Strings = necklace Rayleigh instability of a charged droplet 88 the break-up of a charged drop of liquid into droplets due to electrostatic repulsion Q Rayleigh drop 3/ F k T R Q R Rayleigh / B drop Rdrop F k Tl fn R electro B B / / T l glob F k TR / surf B glob T F / R 0 /3 /3 R l N glob f crit crit un B drop 89

91 Computer Simulations 90

92 Topics not Discussed: Hydration and Associating in Polymer Solutions 9

93 Hydration PNIPAM (N -) in Water and Mixed Solvents 9

94 Topics not Discussed: Associating Polymer Solutions 93

95 3.9 Hydrodynamics Properties of Polymer Solutions F A v + dv F/A v For Newtonian fluids D A dv dy v D 3D v v v T v r H F n nm m m v r dr' H rr' F r' Correlation function: H or nm H r r' 94

96 Diffusion of Suspensions in Solution v F Stokes formula F 6Rv v Stokes-Einstein relation kt B DkT B / 6R D D0 k... Dc 6R D 0 flux k M D b Fick s law c J D r c J t kt B 6R h c D r KMH a b 3 hydrodynamics radius: R h M a 95

97 Effective viscosity of suspensions For the solution of impenetrable spheres of radius R, Einstein derived the Effective viscosity of suspensions : volume fraction occupied by c c0 the suspensions in the solution. If each sphere consists of n particles (monomer units) of mass m, and their density is c, we have 4 Nc4 N R / V R 3 M 3 3 A 3 Instrinsic viscosity () R /3 Vh.5NA.5NA c M M 0 : viscosity of pure solvent N A : Avogadro Number nm=m V h hydrodynamics volume 96

98 [] dependence of MW: Flory-Fox equation 3/ h h M M 3/ 0 / 3/ h 0 M M mol Mark-Houwink Relation =/ / 0 h / h ~ N 0.5 M 3/ h 0 / 3 M M KM / a 0 is calculated by Rouse- Zimm Theory and confirmed by experiments For solution h 0 ~ M For flexible chain in good solvent 6/5 0.8 h ~ M For stiff chain h ~ M For flexible chain For stiff chain 0.5 ~ M ~ M ~ M a=0.5~0.8 a=0.8~. 97

99 Rouse-Zimm Model 3 RH Nb R 8 g D G kt B kt B kt B R 6 Nb 0 H N M M A Na Rg 3 3 0(RZ) 0 exp N A. ~

100 Chapter 4 Multi-component Polymer Systems 4. Thermodynamics of Polymer Mixtures 4. Properties of Polymer Interface 4.3 Thermodynamics of Block Copolymers 99

101 Mixing? Polymer A Polymer B Polymer Blends 00

102 4. Thermodynamics of Polymer Mixtures Why are two kinds of polymers not compatible? G kt N ln N ln xn mixing Entropy of Mixing N Polymer/N Polymer p xn xn S k N ln N ln xn, xn xnxn xnxn N Solvent/N Solvent xn Solvent/xN Solvent s N N S k N, ln Nln N N NN NN s xn xn S k xn ln xn ln xn, xn xn xn xn xn S p s S xn, xn N, N xn, xn S s 0

103 The importance of multi-component, multi-phase materials homogenous properties Polymer A contents Polymer B heterogeneous Polymer blends properties Polymer A contents Polymer B HIPS, ABS Polyolefins crystallinity 0

104 Typical Phase Diagram of One Component 03

105 Typical Phase Diagram of Mixture I: Liquid-Liquid UCST LCST LCST + UCST 04

106 Typical Phase Diagram of Mixture II: Liquid-Liquid & Liquid-Vapour There is no Vapour Phase in Polymer System 05

107 Typical Phase Diagram of Mixture III: Liquid-Solid & Solid-Solid & Liquid-Liquid There may exit Solid Phase in Polymer System 06

108 Relation between phase diagram and morphology 0.6 binodal curve spinodal curve ONE PHASE REGION quench T TWO PHASE REGION

109 Phase Diagrams of Polymer Blends ( Z ) kt UCST LCST A T 0 A A 0??? 08

110 Phase diagram of aqueous solutions of PEO 09

111 UCST/LCST Upper critical solution temperature (UCST) A 0 G TS A 0 A T Lower critical solution temperature (LCST) A B T >0 T, T, Endothermic symmetrical polymer mixture Exothermal symmetrical polymer mixture 0

112 Universal Phase Diagram or

113 Why UCST or LCST? A B Gmix RTV lna lnb A B N A N B is the key issue. Effective interaction parameter eff : Dispersion forces Free volume effects Specific interactions eff di sp f.v. s. i. A T B A>0 UCST A<0 LCST 0 disp T 0 f.v 0 T.s.i T Dispersion force: ~/T monotonic decreasing, disp 0 as T. Free volume effect: Monotonic increasing with T, small, but positive at low T. Specific interaction: Always < 0, decreasing magnitude with increasing T.

114 The Phase Behavior of Polymer Mixtures How to judge it s homogeneous state or inhomogeneous state? What is the mechanism of phase separation? G G~ A V g +(-V )g -G>0? G~ A V g +(-V )g -G<0?

115 The shape of the mixing free energy curve gm with one local minimum PPQQ P P Q Q Q P P Q Q P G kt g g mix P m P Q m Q Q P kt g g Q P Q P G R GR' m P m Q Phase Separation will not occur if free energy curve has one local minimum 4

116 Free energy curve with two local minima gm Phase Separation can take place if free energy curve has two local minima g m A g m B g m B m A B g A 5

117 Phase Diagram and Phase Equilibrium, p, p Phase Phase, p, p, p, p, p, p, p, p, p, p ln x x, p, p x, p ln, p, p x, p ln x x ln, p, x, p, p, p p, p x x x x x 6

118 I. Phase diagram of symmetric mixtures G mix RTV ln ln N N G mix 0 NG mix VkT ln ln RTV b N N N N RTV ln 0 N b b N ln A b Tb ln B T b N A ln B 7

119 II. Phase diagram of asymmetric mixtures G mix RTV ln ln N N G A T Phase Phase + V V * * dg dn dn G n n nxv n xvs, s Vm Vm V V G x x m m Vs Vs Vm x x x V s G Y=A+B, * G m * B B x x G*=A+B * V V s A x V m Vs

120 Finding the phase equilibrium conditions x G V m * Vs x V * m ** Vs ** Phase * Phase ** x x V m * Vs V m ** Vs * ** G V V Y x x x m * m * * Vs Vs V 0, Y x m * 0 Vs V, Y x m * V s * ** * ** * ** 9

121 Search the phase equilibrium point * ** * ** 0

122 Relations between Free EnergyChemical Potentials and Phase Diagram Phase * Phase ** * ** /RT, /RT * ** G 0 G/k B T

123 () Phase equilibrium curve binodal T () ~ [ *(T), **(T)] binodal curve

124 () Metastable/unstable limits - spinodal Phase Phase G V V + unstable metastable G 0 Spinodal curve G x x 0

125 (3) Critical point Spinodal binodal: Critical point 3 G 3 0 x x x /, c, / / c / / x x x x For symmetric blends x =x cn x=n For polymer solutions c x=n For symmetric di-blocks f=0.5 cn 0.5

126 Critical points dependence of N T For blends x /, c, / / c / / x x x x For solutions x = T solvent =0.5 x x x, 0, c T c, c,, c / c / x x c Poor solvent >0.5 5

127 6

128 Phase Separation Dynamics T T quenching() (),, spinodal decomposition () spinodal,, nucleation and growth 7

129 Phase diagram and phase separation mechanisms (),, spinodal decomposition () spinodal, Spinodal decomposition, nucleation and growth Nucleation and growth 8

130 Phase Separation Mechanisms. Nucleation and growth () mechanism In metastable region, separation can proceed only by overcoming the barrier with a large fluctuation in composition. 4 3 Nucleation barrier: G( r) r g 4r with g g( 0) g( '') 3 r: radius of the nuclear; : excess free energy per unit surface area. dropletdispersed phase Nucleation Growth * ** 9

131 Phase Separation Mechanisms. Spinodal decomposition () mechanism In unstable region, separation can occur spontaneously and continuously without any thermodynamic barrier. Early stage No sharp interface Late stage Well-established interfaces ** Coarsening process * Co/bi-continuous 30 phase

132 Examples of Phase Separation Dynamics Nucleation & Growth Spinodal Decomposition 3

133 Interfaces between weakly immiscible polymers A-rich B-rich interfacial tension is defined as the increase in Gibbs free energy of the whole system per unit increase in interfacial area for each i phase int G A A A B B G An n G A dg S dt V dp dn dn i i i i i A A B B G n n i i i A A B B for interfacial region int int int A A B B int A A A A int B B B n A int A B n A int B T, pn, 3

134 The width of the interface is determined by a balance of chain entropy, favoring a wider interface, and the unfavorable energy of interaction between the two different species, which favors a narrow interface. Suppose a loop of the A B-rich polymer with N loop units protrudes into the B side of the interface A-rich G N kt loop loop AB B Interfacial width: R AB A AB B / S g, loop kt N B loop AB bn / loop V k T S V S / kt b B AB 6 AB 6 b 6 b: Kuhn segment AB N Segment density: A 3 Nb (Helfand, E.; Tagami, Y. JPS, PL, 97, 9, 74) 33

135 4.3 Theromodynamics of Block Copolymers diblock triblock random multiblock four arm starblock graft copolymer Applications: Thermoplastics ()PU, SBS Compabilitzier () of Polymer Blends Nanotechnology, Biomaterials 34

136 Self-assembly of Diblock Copolymers Melts Microphase (mesophase, nanophase) separation () is driven by chemical incompatibilities between the different blocks that make up block copolymer molecules. Solutions Micellization () occurs when block copolymer chains associate into, often spherical, micelles () in dilute solution in a selective solvent (). In concentrated solution, micelles can order into gels (). Solids Crystallization of the crystalline block from melt often leads to a distinct (usually lamellar ()) structure, with a different periodicity from the melt. 35

137 Microphase Separation of Diblock Copolymers (BCPs) Phase diagram Homogeneous State order disorder f is the volume fraction of one component. f controls which ordered structures are accessed beneath the order-disorder transition. N expresses the enthalpicentropic balance. It is used to parameterize block copolymer phase behavior, along with the composition of the copolymer. Structurally Order State order order Structurally Order State 36

138 Microphase Separation of Triblock Copolymers 37

139 Thermodynamics of Microphase Separation Lamellar structure Minimize interfacial area and Maximize chain conformational entropy (MIN-MAX Principle) kt AB AB b 6 Z AB AB ( AA BB ) kt, AB L/ F: free energy per chain N: number of segments (=N A + N B ) b: Kuhn length v b b 3, b A b B L: domain periodicity : interfacial area per chain AB : interfacial energy per area AB : segment-segment interaction parameter or A AB B T 38

140 Thermodynamics of Microphase Separation 3 ( L / ) FLAM AB kt Nb 3 L Using Nb V enthalpic term entropic spring term 3 kt AB Nb 3 ( L/ ) we have FLAM kt L b 6 ( L/) Nb L F LAM 0 L L 0 opt L Free energy of lamellae: opt Thus, the optimum period of the lamellae and the lamellar free energy are: L opt Assume /3 /6 bn AB and FLAM ( Lopt ) Lopt L 3 V F m disorder AB A BkT NAB A BkT V s At the order-disorder transition: FLA M Fdisorde red /3 /3 For a 50/50 volume fraction, A B =/4, so:.ktn AB N ABkT 4 3/ BCPs ( N) c (4.8) ~ 0.5 Critical point Symmetric blends opt F LAM See Appendix /3 /3.kTN AB 39 cn

141 Appendix 3/ 3 3h Sel ( h) kln kln hn, hn, dhkln exp Nb Nb 3/ 3 3h kln kln exp Nb Nb 3h k Nb const. L/ Nb L/ L/ Fel ( h L /) kt kt kt Nb Nb Nb 40

142 4

143 : (segment) le h L / 0 (Thermal Blob) g T u 3 l T u 3 l l u EkTgT kt B 3 T (Single Chain) h Nl e () 4

144 () () (3) (4) (5) 43

145 H 0 G H TS lnph 0 G h 0 or G 0 3 h Sconf kb xl, H xv c 3 h ln P h xv c 3 h 3 h Sconf kb xl H h xv 3 c / ~ 3 vc h 3 xv v w h 3 S 0 conf +blob model +blob model 3 h Sconf kb xl 44

146 l B fn Helectr Re kt () B R or l fn R H R kt ln B e electr e B / Re N b e () 3 h S k B Nl Vm V m H kt V M Flory-Huggins Theory B s m HM kt B V V s SM kb ln V s x x S conf 0 Vm S kb ln V x 3 Nk s h 0 B h0 ln ln N 0 : 45 45

147 H M V m kt B V s GM HM TSM Phase Diagram H int erface V AB V m SM kb ln V s x x ln 3 ( L / ) Sconf kt Na H kt N kt 0 m AB A B AB A B s V F disoeder, i, i, i G M 0 3 F G M 0 3 Symmetric Blends: order F L S conf 0 S conf Phase equilibrium (binodal) Phase stability (spinodal) N c L opt Solutions: / c N Critical Point Critical Point: c