Lecture 24: 12.07.05 Flry-Huggins Thery Tday: LAST TIME...2 Lattice Mdels f Slutins...2 ENTROPY OF MIXING IN THE FLORY-HUGGINS MODEL...3 CONFIGURATIONS OF A SINGLE CHAIN...3 COUNTING CONFIGURATIONS FOR A COLLECTION OF CHAINS...4 THE ENERGY OF MIXING IN THE FLORY-HUGGINS MODEL...7 The Bragg-Williams apprximatin...9 PREDICTIONS FROM THE FLORY-HUGGINS MODEL...11 REFERENCES...14 Reading: Dill and Brmberg Ch. 31 Plymer Slutins, pp. 593-605. Supplementary Reading: Annuncements: Final quiz: Tuesday, Dec. 20 1:30pm-4:30pm DuPnt Same frmat and length as first 2 quizzes; yu will have 3 hrs but quiz shuld nly take 2 hrs Equatin sheet is allwed, as in prir quizzes Review sessins (tentatively): Therm/Stat Mech review: Friday, Dec. 16, 5-7pm (nrmal recitatin rm) Structure review: Sunday, Dec. 18, time TBD (nrmal recitatin rm) Therm cverage n Quiz 3: lectures 17-24 Lecture 25 Flry-Huggins thery cntinued 1 f 14
Last time Lattice Mdels f Slutins Lattice mdels are a frm f COARSE GRAINING, where nly the mst imprtant mlecular details f a system are retained in a statistical mechanical mdel. This is a pwerful apprach t create stat mech mdels where meaningful predictins can be made fr cmplex materials. We began a derivatin f the free energy f mixing fr a plymer slutin based n Flry-Huggins thery: n p = number plymer chains N = number f segments per chain ns = number f slvent mlecules M = ttal lattice sites = n s + Nn p Figure by MIT OCW.! Lecture 25 Flry-Huggins thery cntinued 2 f 14
Entrpy f mixing in the Flry-Huggins mdel CONFIGURATIONS OF A SINGLE CHAIN Hw d we calculate W, the number f uniques states in the plymer system? We start by lking at the number f ways t place a single plymer chain n the lattice, ν 1 : What are the number f cnfrmatins fr first bead? With the first bead placed n the lattice, what is the number f pssible lcatins fr the secnd segment f the chain? Mving n t placement f the third segment f the chain: Lecture 25 Flry-Huggins thery cntinued 3 f 14
We repeat this prcess t place all N segments f the chain n the lattice, and arrive at ν 1, the ttal number f cnfiguratins fr a single chain: COUNTING CONFIGURATIONS FOR A COLLECTION OF CHAINS We can fllw the same prcedure used fr a single chain t btain the number f cnfiguratins pssible fr an entire set f n p chains. We start by placing the FIRST SEGMENT OF ALL n p CHAINS. The number f cnfiguratins fr the first segment f all n p chains is ν first : " first = M(M #1)(M # 2)(M # 3) $$$ (M # (n p #1)) = M! (M # n p )! The number f cnfiguratins fr the (N - 1) remaining segments f all n p chains is ν subsequent : * $ " subsequent =, z #1 & + % ( ) M # n p M '-* $ )/, z #1 & (. + % ( ) M # n p #1 M '- * $ )/ 000, z #1 & (. + % ( ) M # 2n p M '- * $ )/ 000, z #1 (. & +, % ( ) M # N n p #1 ( ) # N #1 M ( ) '- / ) (./ $ " subsequent = z #1 ' & ) % M ( n p (N#1) ( M # n p )! (M # Nn p )! Putting these tw cnfiguratin cunts tgether, we have the ttal number f cnfiguratins fr the cllectin f n p chains f N segments each: W = " first" subsequent n p! The factr f n p! Crrects fr the ver-cunting since the plymer chains are indistinguishable, and we can t tell the difference between tw cnfiguratins with the same plymer distributins but different chain identities: Lecture 25 Flry-Huggins thery cntinued 4 f 14
Figure by MIT OCW. Pure Plymer Figure by MIT OCW. Figure by MIT OCW.
Applying Stirling s apprximatin: ln x!" x ln x # x, arriving at a final result: "#S mix = $k b [ n s ln% s + n p ln% p ] Lecture 25 Flry-Huggins thery cntinued 6 f 14
The energy f mixing in the Flry-Huggins mdel Mlecules frm bnds with ne anther, attract, and repel ne anther by electrstatic, plar, r Van der Waals interactins- changing the internal energy f the system. Figure by MIT OCW. Figure by MIT OCW. T accunt fr these interactins in a lattice mdel, we need t first calculate the enthalpy f mixing. At cnstant pressure, the enthalpy f mixing will be related t the internal energy f mixing in the fllwing way: We will assume in the lattice mdel that the change f vlume n mixing is zer- thus the secnd term drps ut and ΔH mix = ΔU mix. We can directly frmulate an expressin fr the energy f mixing in the mdel: The internal energy f the slutin is btained by cnsidering the cntacts between mlecules n the lattice: m ij =# i - j cntacts " ij = energy per i - j cntact # energy & U slutin = "(# cntacts) % ( $ cntact ' The ttal number f cntacts made by plymer segments r slvent mlecules n the lattice can be related t the crdinatin number and the cntact numbers m ij : Lecture 25 Flry-Huggins thery cntinued 7 f 14
ttal slvent cntacts: ttal plymer cntacts: Substituting int the expressin fr the internal energy f the slutin, we have: U slutin = m pp " pp + m ss " ss + m ps " ps $ U slutin = znn # m ' $ p ps & )" pp + zn # m ' s ps & )" BB + " ps m ps % 2 ( % 2 ( U slutin = znn p 2 " pp + zn s 2 " $ $ BB + " ps # " pp + " ss & & % % 2 '' )) m ps (( The internal energy f the umixed state has a simpler frm: U unmixed = (# s - s cntacts)" ss + (# p - p cntacts)" pp This leads us finally t an expressin fr the energy f mixing: "U mix = U slutin #U unmixed Lecture 25 Flry-Huggins thery cntinued 8 f 14
The Bragg-Williams apprximatin Our next step is t simplify the last equatin- hw can we determine m ps frm ur mdel? One way t achieve this is t make the apprximatin that the plymer and slvent mlecules mix randmly n the lattice. In that case, all we need t d t find m ps is calculate the prbability f P- S cntacts fr randm mixing. This is called the mean field apprximatin r the Bragg- Williams apprximatin. Nte that we have already implicitly invked this apprximatin in ur excluded vlume crrectin in the calculatin f the entrpy f mixing. The prbability that a lattice site cntains a plymer segment is: Fr a single slvent mlecule, the average number f cntacts with plymer segments is then: Thus, the ttal number f P-S cntacts is: Lecture 25 Flry-Huggins thery cntinued 9 f 14
This gives us a cmpletely knwn expressin fr the energy f mixing in terms f the interactin energies: Frm this expressin, the mlar enthalpy f mixing is: We define the Flry-Huggins interactin parameter as: χ is a unitless equivalent f Ω, the interactin parameter frm the regular slutin mdel. Nte that Ω is thus directly related t the mlecular energy f interactin between the tw cmpnents f a binary system! The change in internal energy nw reduces t: Lecture 25 Flry-Huggins thery cntinued 10 f 14
Predictins frm the Flry-Huggins mdel Nw, this expressin, derived directly frm ur mlecule-scale mdel, can be cmbined with the lattice mdel entrpy f mixing derived abve t prvide the free energy f mixing fr the regular slutin: free energy, enthalpy, entrpy f mixing 0.5 0.4 0.3 0.2 0.1 0-0.1-0.2 0 0.2 0.4 0.6 0.8 1! p "H mix "S mix "G mix "G mix 0.05 0 0 0.2 0.4 0.6 0.8 1-0.05-0.1-0.15-0.2-0.25! p Lecture 25 Flry-Huggins thery cntinued 11 f 14
The Flry-Huggins mdel predicts majr trends in the behavir f real plymer slutins: T ( 0 C) 30 Experimental phase diagrams f a plystyrene in cyclhexane. 25 b One phase Mlecular weights ( a ) 1.27 x 10 6, c ( b ) 2.5 x 10 5, ( c ) 8.9 x 10 4, 20 ( d ) 4.3 x 10 4. The dashed lines 15 d indicate the Flry-Huggins thery predictins fr the first and third 10 Tw phases curves frm the tp. 5 0.0 0.1 0.2 0.3 0.4 0.5 φ Figure by MIT OCW. 3.5 3.4 I/TC x 10 5 3.3 3.2 3.1 0 0 2 4 6 8 10 (1/x 1/2. 1/2x) x 10 2 A plt f the reciprcal f the critical temperature against the mlecular size functin fr plystyrene fractins in cyclhexane ( ) and fr plyisbutylene fractins in diisbutyl ketne ( ). Figure by MIT OCW. Lecture 25 Flry-Huggins thery cntinued 12 f 14
And this thery is used t predict new behavir in plymers in current research: Image remved fr cpyright reasns. Scanned image f article: Gnzalez-Len, Juan, Metin Acar, Ryu, Sang-Wg Ryu, and Ruzette, Anne-Valerie Ruzette, and Anne M. Mayes. "Lw-temperature prcessing f 'barplastics' by pressure-induced flw." Nature 426 (2003): 424-428. Lecture 25 Flry-Huggins thery cntinued 13 f 14
References 1. Dill, K., and S. Brmberg. 2003. Mlecular Driving Frces, New Yrk. 2. Flry, P. J. 1953. Principles f Plymer Chemistry. Crnell University Press, Ithaca. Lecture 25 Flry-Huggins thery cntinued 14 f 14