Polymers Physics. Michael Rubinstein University of North Carolina at Chapel Hill

Transcription

1 Polymers Physics Michael uinstein University of orth Carolina at Chapel Hill

2 Outline 1. eal Chains. Thermodynamics of Mitures. Polymer Solutions

3 Summary of Ideal Chains Ideal chains: no interactions etween monomers separated y many onds Mean square end-to-end distance of ideal linear polymer Mean square radius of gyration of ideal linear polymer g 6 / Proaility distriution function P d, ep Free energy of an ideal chain F kt kt Entropic Hooke s Law f 1 Pair correlation function g( r) r

4 eal Chains Include interactions etween all monomers Short-range (in space) interactions Proaility of a monomer to e in contact with another monomer for d-dimensional chain d * d 1/ If chains are ideal * 1 d / small for d> umer of contacts etween pairs of monomers that are far along the chain, ut close in space d / * small for d>4 For d<4 there are many contacts etween monomers in an ideal chain. Interactions etween monomers change conformations of real chains.

5 Life of a Polymer is a Balance of entropic and energetic parts of free energy Entropic part wants chains to have ideal-like conformations F ent F eng Energetic part typically wants something else (e.g. fewer monomermonomer contacts in a good solvent). Chain has to find a compromise etween these two desires and optimize its shape and size.

6 Flory Theory umer density of monomers in a chain is / Proaility of another monomer eing within ecluded volume v of a given monomer is v/ Ecluded volume interaction energy per monomer ktv/ Ecluded volume interaction energy per chain ktv / Entropic part of the free energy is of order kt /( ) Flory approimation of the total free energy of a real chain F kt v 1/5 /5 /5 Free energy is minimum at v Universal relation ~ - Flory scaling eponent More accurate estimate

7 g (nm) / M w ( g / m o l e ) adius of gyration of polystyrene chains in a q-solvent (cycloheane at 4.5 o C) and in a good solvent (enzene at 5 o C). Fetters et al J. Phys. Chem. ef. Data, 619 (1994)

8 Polymer Under Tension 0 f 1/ Unpertured size F /5 Tension los (Pincus los) of size contain g monomers Chains are almost unpertured on length scales up to 1/ /5 g g On larger length scales they are stretched arrays of Pincus los f Ideal chain g 0 f 0 f g Size of Pincus los eal chain 5/ / 5/ F / f 5/ F /

9 F f Polymer Under Tension 0 f Size of Pincus los 5/ F / f Free energy cost for stretching a chain is on order kt per lo kt Ideal chain g f 0 eal chain f f kt kt F kt kt kt g Tension force is on order kt divided y lo size kt kt kt f kt kt f / kt f f 0 0 5/ f 0 F F linear elasticity For = 1nm at room T kt/ = 4p kt/ f 0 1 / f 0 F ma F f F 5/ non-linear elasticity even at low forces Log log plot /

10 f Polymer Under Tension Ideal chain eal chain kt kt/ 5/ / 0 f linear elasticity f 1 / f 0 F ma f kt F f non-linear elasticity Challenge Prolem 1: Pulling a ing Consider a pair of forces applied to monomers 1 and 1+/ i. Show that the modulus of an ideal ring f is twice the modulus of a linear /-mer ii. How does modulus of a ring in a good solvent G r compare to twice the modulus of linear /-mer G /? a. G r = G / similar to ideal case. G r < G / ecause the entropy of sections of a ring is lower c. G r > G / ecause ring sections reinforce each other f f f f f

11 Ideal chain Biaial Compression D eal chain On length scales smaller than compression lo of size D chain is almost unpertured 5/ D g D g 1/ 1/ D g Occupied part of the tue D g D Free energy of confinement 5/ F conf kt kt F conf kt kt g D g D 5/ 0 F conf kt F kt F conf D D / D

12 Challenge Prolem : Biaial Confinement of a Semifleile Chain Consider a semifleile polymer e.g. doulestranded DA with Kuhn length =100nm and contour length L=16mm. Assume that ecluded volume diameter of doule heli is d=nm (larger than its actual diameter nm due to electrostatic repulsions). F Calculate the size F of this lamda phage DA in dilute solution and the length occupied y this DA in a cylindrical channel of diameter D (for d<d< F ). D

13 Uniaial Compression D Free energy of confinement in a slit is the same as in the cylindrical pore Longitudinal size of an ideal chain in a slit is the same as for an unpertured ideal chain. 1/ -dimensional Flory theory for real chain confined in a slit D is the ecluded area of a confinement lo with g monomers F kt D ( / g) ( / g) D Longitudinal size of a real chain in a slit / 4 1/ 4 / 4 D g D Fractal dimension of real chains in -d is D = 4/

14 Scaling Model of eal Chains Thermal lo - length scale at which ecluded volume interactions are of order kt kt v g T T Chain is ideal on length scales smaller than thermal lo umer of monomers in a thermal lo good solvent v>0 g 6 v T kt Size of a thermal lo poor solvent v<0 1/ T g T T 4 v T gl T /5 1/5 1/ v /5 T g 1/ T 1/ T g T v

15 Flory Theory of a Polymer in a Poor Solvent F F kt v In poor solvent v < 0 and 0 Cost of Confinement Compression lo of size g with g monomers Confinement free energy F conf kt kt g F kt v For v < 0 0 kt Three Body epulsion Size of a gloule v w 6 1/ w gl v

16 eal Chains in Different Solvents T good T g T /5 T athermal good /5 athermal /5 q-solvent /5 1/ poor 1/ nonsolvent 1/ non-solvent 1/ 1/ q T poor T g T 1/ 1 g T

17 Temperature Dependence of Chain Size Mayer f-function U r f r ep 1 kt 0 { -1 for r< where U(r)>>kT U r kt for r> where U(r) <kt Ecluded volume 4 q v 4 f kt T rr dr 4 r dr Urr dr 1 Interaction parameter z is related to numer of thermal los per chain z g T v 1/ 1/ 0 T q T g T v 6 Chain contraction in a poor solvent 1/ z 1/ Chain swelling in a good solvent 1 z 1/

18 Universal Temperature Dependence of Chain Size g /q g /q / (1 - q/t) / (1 - q/t) Monte-Carlo simulations Graessley et.al., Macromolecules, 510, 1999 & I. Withers Polystyrene in decalin Berry, J. Chem. Phys. 44, 4550, 1966

19 Summary for eal Chains Size of Linear Chains 1/ q v good athermal nearly ideal in q-solvents 1 v in athermal solvents Ecluded volume T q v T swollen in good solvents F poor v 1/ 1/ Stretching a real chain kt 1/(1 ) kt F F non-hookean elasticity collapsed into a gloule in poor solvents In good solvents.4 Confinement of a real chain F kt F D 1/ kt F D 1.7

20 Outline 1. eal Chains. Thermodynamics of Mitures. Polymer Solutions

21 Binary Mitures Homogeneous if components are intermied on a molecular scale Heterogeneous if there are distinct phases Assume, for simplicity, no volume change on miing Consider a miture with total numer n of monomers (sites) and total volume v 0 n where v 0 volume of a lattice site Assume that a monomer of each species occupies the same volume v 0 + = Volume fractions VA A V V A B B 1 V A V B V A +V B A n = n monomers of type A and B n B = (1- n monomers of type B

22 Entropy of Miing + = - volume fraction of A 1 - volume fraction of B nv 0 1nv 0 nv 0 v 0 volume of a lattice site umer of translational states of a molecule in a miture is the numer of sites n umer of states of molecule A in a pure A phase Entropy change upon miing of a molecule A AB 1 SA k ln AB k ln A k ln k ln k ln A n A / A and n B / B numer of A and B molecules S Total entropy change upon miing n A nb na nb S SA SB k ln ln1 A B A B AB V A A V v 0 V A v 0 Infinite slopes B n n 0 1

23 Energy of a Homogeneous Miture Average energy per A monomer??? A??? Mean Field Theory A monomer with proaility u ij interaction of i with j B monomer with proaility 1 z u z coordination numer A u AA 1 uab (numer of neighors) Average energy per B monomer u u 1 Average energy per monomer in homogeneous miture naua nbu U B u u 1 f 0 Average energy per monomer of pure components efore miing z ui uaa 1 u BB Energy of miing B u A B z n uaa 1 u AB 1 z u BB AB z u BB u u m BB u m u m 1 z u u AA f u i

24 Energy of Miing u u Um num n AB 1 Flory interaction parameter z u uab kt Proaility of AB contact 1 Energy change on miing per site F m U AA BB 1 z u n kt u m per site Fm 1 n kt A ln U m 0 1 kt( 1) Free Energy of Miing m TS m AA u B BB (1 ) ln 1 egular solution: A = B = 1 Polymer solution: A = >>1, B = 1 Polymer lend: A >>1, B >> 1

25 F mi /(kn) Flory-Huggins Free Energy of Miing F n mi kt A 1 ln ln At high T entropy of miing dominates, F mi is conve and homogeneous miture is stale at all compositions. B o K At lower T for >0 repulsive interactions are important and there is a composition range ' '' with thermodynamically stale phase separated state. This composition range, called misciility gap, is determined y the common tangent line A = 00 B = o K 50 o K o 5 K T

26 Phase Diagrams ktn F n mi 1 kt ln ln For cr miture is stale at all compositions For > cr A B there is a misciility gap for ' ' ' Critical composition For a symmetric lend A = B = cr =/ cr =1/ For polymer solutions A =, B = cr cr cr A A B B B 1 1 F mi cr sp1 sp Two Phases Single Phase cr A = B =.7

27 A B T Phase Diagram of Polymer Solutions Polymer solutions phase separate upon decreasing solvent quality elow q-temperature Upper critical solution temperature B>0 Solution phase separates elow the inodal in poor solvent regime into a dilute supernatant of isolated gloules at ` and concentrated sediment at ``. dilute supernatant concentrated sediment ` `` Single Phase ` inodal spinodal unstale (spinodal decomposition) metastale Two Phases (nucleation and growth) M=5.kg/mole Polyisoprene in dioane Takano et al., Polym J. 17, 11, 1985

28 Intermolecular Interactions 10 6 P/cT (moles/g) h solution memrane solvent 0 Osmotic pressure c P T A c... M n A second virial coefficient 5 Osmometer 0 M n = g/mole 15 Poly(a-methylstyrene) in toluene at 5 o C oda et al, Macromol. 16, 668, M n = g/mole 5 0 M n = g/mole c (g/ml)

29 Mitures at Low Compositions F n mi kt A 1 ln ln B 1 1 Epand ln(1-) in powers of composition F mi 1 1 kt ln... n A B B 6B Osmotic pressure Fmi F mi n kt 1 P... V n A B A B Virial epansion in powers c P n v kt c... of numer density c n = / n wcn A 1 6 Ecluded volume v -ody interaction w B B v T q A M In polymer solutions B = T Av

30 Polymer Melts Consider a lend with a small concentration of A chains in a melt of chemically identical B chains. o energetic contriution to miing = 0. 1 Ecluded volume v is very small for B >> 1 B B Flory Theorem 6 Thermal lo gt B T gt B v Chains smaller than thermal lo A < B are nearly ideal. In monodisperse A = B and weakly polydisperse melts chains are almost ideal. In strongly asymmetric lends A > B long chains are swollen A T g A T /5 B A B /5 1/ A A B 1/10 A A 1 1/ B /5 A

31 A /( A ) Challenge Prolem : Long A -mer in a -d Melt of B -mers B B B B A A A B 1/5 B B B A / B M. Lang Why doesn t Flory Theorem work?

32 Challenge Prolem 4: Miing of Polymers with Asymmetric Monomers A monomers per A chain B monomers per B chain v 0 volume of a lattice site = volume of a small B monomer mv 0 volume of an A monomer m times larger than B monomer Derive the free energy of miing F mi of A and B polymers. Calculate the size A of dilute A- chains in a -d of B-chains for m>>1 in the case of =0.

33 Summary of Thermodynamics of Mitures Free energy of miing consists of entropic and energetic parts. Entropic part per unit volume (translational entropy of miing S mi ) TS V mi Energetic part per unit volume U mi kt 1 V 1 kt ln ln v v v 0 Flory interaction parameter A A Many low molecular weight liquids are miscile Some polymer solvent pairs are miscile Very few polymer lends are miscile B T B 1 v A volume of A chain v B volume of B chain v 0 volume of a lattice site Chains are almost ideal in polymer melts as long as they are shorter than square of the average degree of polymeriztion A <

34 Outline 1. eal Chains. Thermodynamics of Mitures. Polymer Solutions

35 Quiz #1 Which of These Chains are Ideal? A. Polymer in a solution of its monomers B. Polymer dissolved in a melt of identical chains C. Chain in a sediment of a polymer solution in a poor solvent D. one of the aove sediment

36 T Polymer Solutions v Theta solvent good solvent T q v 1 0 T q * semidilute Chains are nearly ideal dilute q c qsolvent 0 1/ T at all concentrations c poor solvent Overlap concentration * 1 ` -phase `` q Chain is ideal if it is smaller than 4 thermal lo T v Boundaries of dilute q-regime * * * 1 T q v 1 T Temperatures at which chains egin 1 T q 1 to either swell or collapse

37 Poor Solvent 1 Critical composition cr Critical interaction parameter ` cr Solution phase separates elow the inodal into a dilute supernatant of isolated gloules at ` and concentrated sediment at ``. `` M = 4.6 kg/mole Sediment concentration `` is determined y the alance of second and third virial term (similar to the concentration inside gloules). '' v 1 gl M = 1,70 kg/mole Polystyrene in cycloheane ( ) and polyisoutylene in diisoutyl ketone ( ) Shultz and Flory, J. Am. Chem. Soc. 74, 4760, 195 gl 1/ v 1/ 1/ 1/ 1

38 gl Poor Solvent Dilute Supernatant of Gloules Gloules ehave as liquid droplets with size gl 1/ T v v T Surface tension is of order kt per thermal lo kt kt kt v 1 8 T 4/ ktgl kt v / Total surface energy of a gloule gl 4 T is alanced y its translational entropy ktln Concentration of a dilute supernatant 4/ gl v v ' ''ep ep 4 kt 1/ / is different from the mean field prediction. 4

39 concentrated Good Solvent Swollen chain size in dilute solutions v 1/5 /5 Overlap concentration /5 dilute good (swollen) T q T c * 4/5 dilute v poor (gloules) Semidilute solutions * 1 * dilute q semidilute good c ` -phase semidilute q `` v 0 Correlation length Distance from a monomer to nearest monomers on neighoring chains.

40 Correlation Length For r < monomers are surrounded y solvent and monomers from the same chain. The properties of this section of the chain of size are the same as in dilute solutions. 1/5 v /5 g g numer of monomers inside a correlation volume, called correlation lo. 1/ 4 Correlation los are at overlap g / 4 v For r > sections of neighoring chains overlap and screen each other. On length scales r > polymers are ideal chains - melt with /g effective segments of size. 1/ g v 1/8 1/

41 Semidilute Solutions T r T On length scales less than T chain is ideal ecause ecluded volume interactions are weaker than kt. r ~ n 1/ On length scales larger than T ut smaller than ecluded volume interactions are strong enough to swell the chain. r ~ n /5 1 1/ g T /5 n g 1/ On length scales larger than ecluded volume interactions are screened y surrounding chains. r ~ n 1/

42 Scaling Theory of Semidilute Solutions /5 * 4/5 At overlap concentration v 1/5 v /5 chain has its dilute solution size F In semidilute solutions is a power of that matches dilute size at * (1 ) /5 v (4) / 5 * F 4 1 Chains in semidilute solutions are random walks = -1/8 5 1/8 1/8 v 1/ F * y (1 y) /5 v (4 y) /5 y Similarly correlation length F * is independent of in semidilute solution + 4y = 0 y = -/4 / 4 1/ 4 / 4 F * v

43 Concentrated Solutions Correlation length decreases with concentration, while thermal lo size T is independent of concentration. At concentration ** the two length are equal T and intermediate swollen regime disappears. 1/ 4 / 4 ** v v This concentration is analogous to in poor solvent at which two- and three-ody interactions are alanced. In concentrated solutions chains are ideal at all length scales. 4 v On length scales less than chains are ideal ecause ecluded volume interactions are weaker than kt. On length scales larger than chains are ideal ecause ecluded volume interactions are screened y surrounding chains. T

44 concentrated dilute good (swollen) T q T c * dilute q Polymer Solutions semidilute good c ** v semidilute q 0 0 ** for * < < ** 1/8 dilute poor (gloules) ` In athermal solvent / 4 -phase * * v 1 `` T 1/ 1/8 F 0 T * -/4-1/8 ** -1 1

45 Osmotic Pressure In dilute solutions < * - van t Hoff Law kt P Osmotic pressure P in semidilute solutions > * is a stronger function of concentration P f { kt 1 for < * * f * (/ * ) z for > * Osmotic pressure in semidilute solutions P kt * z kt 1 z v is independent of chain length (4z/5-1=0). / 4 z /5 4z /51 z=5/4 kt v 9/ 4 kt P eighoring los repel each other with energy of order kt.

46 1. * * 1 P f kt Poly(a-methylstyrene) in toluene at 5 o C oda et al, Macromol. 16, 668, 1981 Concentration Dependence of Osmotic Pressure

47 concentrated dilute good (swollen) T q Semidilute Theta Solutions T c * semidilute good dilute q c ** semidilute q dilute ` -phase poor `` (gloules) g Correlation los are space-filling * 1 0 at 0 / Scaling assumption for > * Correlation length in semidilute solutions is independent of chain length (1+)/ = 0 v 0 Chains are almost ideal 1/ / g 1 * =-1 1 /

48 Quiz # What is the meaning of correlation length in semidilute theta solutions? How different are semidilute theta solutions from ideal solutions of ideal chains?

49 Osmotic Pressure in Semidilute Theta Solutions kt w Mean-field prediction P... 6 First term (van t Hoff law) is important in dilute solutions Three-ody term is larger than linear in semidilute solutions > 1/ 1/ P kt P Scaling Theory h * Osmotic pressure in semidilute qsolutions P kt kt { 1 for < * h * (/ * ) y for > * * y kt is independent of chain length (y/-1=0). y= 1 y y / 1 kt P kt

50 Osmotic Pressure P (Pa) kt kt P Correlation length in semidilute q-solvents is of the order of the distance etween -ody contacts. umer density of n-ody contacts ~ n / Distance etween n-ody contacts in -dimensional space r n Polyisoutylene in enzene at q4.5 o C (filled circles), in enzene at 50 o C (filled squares), in cycloheane at 0 o C (open circles) and at 8 o C (open squares) Flory and Daoust J. Polym. Sci. 5, 49, 1957 n /

51 concentrated dilute good (swollen) T q Summary of Polymer Solutions T c dilute poor (gloules) * dilute q ` semidilute good c semidilute q -phase ** `` v 0 1/ In poor solvent part of the diagram inodal separates -phase from single phase regions: Dilute gloules at low concentrations < Concentrated solutions with overlapping ideal chains at > ear qtemperature there are dilute and semidilute q-regimes with ideal chains. Dilute good solvent regime with swollen chains at < * and v > 0. Semidilute good solvent regime at * < < ** with chains swollen at intermediate length scales shorter than correlation length. Osmotic pressure in semidilute solutions is kt per correlation volume.

52 Challenge Prolem 5: Confinement of Polymers in Solutions & Melt D Calculate the pressure etween two solid plates fully immersed into a semidilute polymer solution or melt as a function of separation D etween plates for separations smaller than chain size. Assume no attraction etween plates and polymer. semidilute polymer solution or melt Separately consider cases with correlation length <D< and <D<

53 Single Chain Adsorption Ideal chain adsorption lo ads g ads eal chain Volume fraction in a chain section of size ads containing g monomers g ads ads 1/ adsorption lo g ads ads /5 ads g umer of monomers per adsorption lo in contact with the surface / ads ads Energy gain per monomer in contact with the surface is -ekt kte ads 4/ Energy gain per adsorption lo / kt kt ads e kt

54 Single Chain Adsorption Ideal chain ads eal chain e ads 1 e ads 1 Size ads and numer of monomers g in an adsorption lo / ads e 5/ g ads ads 5/ e / g ads e e Free energy of an adsored chain F kt g kte F kt g kte 5/

55 Flory Theory of Adsorption Fraction of monomers in direct contact with the surface is /D D umer of monomers in direct contact with the surface is /D Energy gain per monomer contact with the surface is -kte Energy gain from surface interactions per chain F int e kt Total free energy is a sum of interaction and confinement parts D Ideal chain F F conf F int eal chain F kt D kte D Optimal thickness F kt D D / e e D 5/ kte D

56 Multi-Chain Adsorption de Gennes self-similar carpet Single-chain adsorption z ads / e ads ads (z) z First layer is dense packing of adsorption los Polymer concentration decays from the high value in the first layer. The correlation length (z) corresponding to concentration (z) at distance z from the surface is of the order of this distance. z z dz g ads z / 4 5/ e ads z z 4/ z dz z 4/ ads Coverage is controlled y the first layer of los. 1/ e 1/ g ads ads

57 s Aleander de Gennes Brush Grafting density s numer of chains per unit area. Distance etween sections of chains s 1/ umer of monomers per lo H g ~ 1/ ~ s -1/() Thickness of the rush H ~ /g ~ s (1-)/() z Energy per chain E chain ~ kt /g ~ kts 1/() Energy per unit volume E V chain E chain s H kt g s H s kt kt P

58 Polymer Brush Height depends on Grafting Density Brush height H (nm) H ~ s 1/ Grafting density s (nm - ) If the grafting density is high, chains repel each other and stretch away from the surface, forming a polymer rush. Mushroom egime unpertured size H 0