Chemical Engineering 160/260 Polymer Science and Engineering. Lecture 15: Molecular Aspects of Polymer Rheology February 21, 2001

Transcription

1 Chemial Engineering 160/260 Plymer Siene and Engineering Leture 15: Mleular Aspets f Plymer Rhelgy February 21, 2001

2 Objetives! T intrdue the nept f saling analysis t aunt fr the nentratin and mleular weight dependene f mdulus, relaxatin time, and vissity.! T illustrate the nept f sreening that arises frm hain interatins in nentrated slutins and shw hw this allws simplifiatin f the analysis.! T intrdue the nept f reptatin fr diffusin in highly nentrated slutins and melts.

3 Outline! Mleular Thery f Semi-dilute Slutin " Overlap nentratin " Blbs " Sreening! Mleular Thery f Cnentrated Slutin and Melts " Reptatin

4 Summary f Dilute Slutin Results Cnentratin dependene Mleular weight dependene Mdulus G G 1 M Relaxatin time -independent λ M 18. Vissity η plymer η plymer M 08.

5 Semi-dilute and Cnentrated Slutins in Gd Slvents Dilute Semi-dilute Cnentrated < > At ν = ν r =, the hains begin t verlap and entangle. ν 1 3 R M = 3 R 1 [ η]

6 Saling Cnept due t DeGennes The gverning idea is that when >, the mdulus must be independent f the mleular weight, i.e., the mehanial interatins are sreened frm eah ther and the plymer des nt knw hw big it is. G = G f G = ν kt The measurement f the mdulus simply invlves lal jiggling f the hain, nt enter-f-mass mvement. f 1 as > 1

7 Mdulus in Semi-dilute Slutin Fr > 1 Prpse that G = G the mdulus must be independent f mleular weight. α and determine α. We knw that G 1 = M [ ] η M Thus, G = G

8 The Blb Size: a Simple View ξ ξ is the sreening length r rrelatin length In the saling analysis, the sreening length als annt be a funtin f mleular weight. Dimensinal analysis gives In a gd slvent R M 06. M. ξ = Rf ξ = R = R β = β

9 Relatin Between Mdulus and Blb Size The mdulus may be written as G kt = ξ 3 ξ = R 075. G = kt 3 R = kt R This is the same result as befre: = kt M This suggests that G = ν netwrk kt G = G = νkt By analgy t the ideal gas law, G is the istrpi pressure generated by the slutin (fr nentrated systems)

10 A Mre Realisti Piture f the Blb ξ A strand is any length f plymer between tw ntat pints; these are really randm walk ils (r blbs). If the plymer itself nsists f N statistial subunits, hw many suh units (all this number g) make up the blb f plymer that frms the strands? Beause R = 06. bn we prpse that ξ = bg 06. ξ = R 075. g = N = N 125.

11 Sreening The number f blbs per hain is btained frm: N N = blb g = 125. The exluded vlume interatin that swells the hain in gd slvents is nw sreened fr distanes larger than ξ and the hain bemes a randm walk with step length ξ. In nentrated slutin, the blb is nw the statistial unit, and the radius f gyratin then bemes: ξ blb = R R= N = R Thus, the hain shrinks with inreasing nentratin. (In the bulk slid, the hain will exhibit theta dimensins.)

12 Fritin Fatr f a Blb The lse prximity f the hains als sreens hydrdynami interatins between the hains. This means that the fritin fatr f a hain is simply given by ζ = N blb ζ blb A single blb an then be treated by the dilute slutin ase: ζ = ηξ = η R blb s s ζ = η sr Even thugh the fritin experiened by the hains nly mes frm the slvent, sreening auses an enhanement f fritin with inreasing nentratin.

13 Summary f Results fr Semi-dilute Slutins Diffusin effiient: Relaxatin time: R R ζ η λ = = = R s D kt kt Reall that D = kt = kt ζ η R s = λ M 1 = = M 3 R [ η] Thus, λ R [ η] M ( M ) = M Fr dilute slutin, λ M 18. Vissity: η = λ = λ G G

14 Highly Cnentrated Slutins and Melts In this regime we nsider the lng plymer hains t be fully entangled, whih leads t a fundamental hange in hw the system relaxes. (In the semi-dilute ase, the hains may interpenetrate, but are nt entangled.) entanglement ξ In the reptatin mdel f plymer dynamis, the hain an relax its nfrmatins and diffuse alng its ntur by slipping thrugh the entanglements.

15 New Length Sale fr Reptatin In the reptatin mdel a new length sale bemes imprtant beause fritin urs all alng the hain ntur. Cntur length: L = N blb ξ The time sale, λ, is nw alulated differently in that the ntur length, L, is used instead f the radius f gyratin, R, as the harateristi length L L ζ η ξ η λ = = = = snblb sr D kt kt kt Nte that this is a muh strnger nentratin dependene than fr the semi-dilute slutin. = λ

16 Saling Results fr Reptatin Relaxatin time: λ = λ Vissity: λ M ( M ) = M η = Gλ = G λ η M 3 Semi-dilute: 275. M 1 = = M 3 R [ η] λ M 18. Experiment yields an expnent f 3.4

17 Saling Results fr Reptatin Center f mass diffusin: Sine the plymer hain will eventually diffuse thrugh the tube, the relevant length sale is R, nt L. D M 2 R D R 2 D = = M 2 λ L N blb 2 Agrees with experiment The reptatin mdel is still the best mleular piture tday.