3.014 Materials Laboratory Dec. 9 th Dec. 14 st, 2005 Lab 4 Module α 2 Miscibility Gas OBJECTIVES Review miscibility gas in binary systems Introduce statistical thermodynamics of olymer solutions Learn cloud oint technique for exerimentally determining miscibility gas Study effect of chain length on miscibility of olymer-solvent mixtures SUMMARY OF TASKS 1) Preare solutions of olystyrene / methyl cyclohexane of varying concentration 2) Measure cloud oints of reared solutions by light scattering methods 3) Comare results to values ublished in archival literature READINGS See Dill, K. A., & S. Bromberg, Molecular Driving Forces. New York, NY: Garland Publishing, 2002. Chaters 15 "Solutions and Mixtures" & 31 "Polymer Solutions." 1
BACKGROUND In 3.012 we ve studied how ideal solutions of two comonents tend to mix together to increase the total entroy of the system, while regular solutions can have a miscibility ga in which the system hase searates into 2 distinct comositions of the same structure. This lab will exlore how this tendency is affected when one comonent is a macromolecule or olymer, and exerimentally determine hase diagrams for the olymer-solvent system olystyrene-methyl cyclohexane. Recall from 3.012 that the molar Gibb s free energy for an ideal mixture is given by: G C =! µ X, sol i i i= 1 where C is the number of comonents in the mixture and X i is the mole fraction of the ith comonent, whose chemical otential is given by: µ µ i = i,0 + RT ln X i For an A-B mixture: G = µ X + X RT ln X + µ X + X RT ln X sol A,0 A A A B,0 B B B In the heterogeneous (demixed) state, the free energy is given by the sum of the free energies of the unmixed comonents: G = X µ + X µ heter A A,0 B B,0 G The change in molar free energy on mixing is thus given by: G heter ( ln ln )! G = RT X X + X X mix A A B B G sol In an ideal solution, the mixing enthaly is zero so that:! G = " T! S mix mix X B 2
( ln ln )! S = " R X X + X X mix A A B B The ideal solution remains miscible at any temerature because the change in free energy on mixing is zero. The ideal solution can be contrasted with a regular solution. In a regular solution, interactions between comonents result in a mixing enthaly given by:! H = " X X mix A B The total change in free energy is then: ( ln ln )! G = RT X X + X X + " X X mix A A B B A B For Ω > 0, a regular solution can undergo hase searation as temerature decreases. The two-hase region of the T-comosition hase diagram is known as a misibility ga.!g mix T T Single Phase Two Phase X B X B 3
The boundary between the one- and two-hase regions which defines the miscibility ga is obtained by:!" G! X mix B ( B [ B] ) ( B ) = RT ln X # ln 1# X + $ 1# 2X = 0 The critical oint occurs at:! 1 1 " = RT ' + ( + % 2& = 0 # ) X B 1% X B * 2 # $ Gmix 2 X B X = 0.5 B An examle of a miscibility ga is found in the Ir-Pd hase diagram, shown below. In the solid state, mixtures of iridium and alladium exhibit an FCC structure. Above 1450 C, the system is single hase. Below this temerature, however, is a miscibility ga where the system slits into searate Pd-rich and Rh-rich comositions. Similar to the regular solution model above, the critical comosition is near 50 at% (M Ir =192.2 g/mol; M Pd =106.4 g/mol). Plotting the hase diagram in weight fraction introduces asymmetry to the miscibility ga. Figure removed for coyright reasons. Ir-Pd hase diagram. From Baker, H., ed. ASM Handbook Alloy Phase Diagrams. Vol. 3. 10th ed. Materials Park, OH: ASM International, 1992,. 265. ISBN: 0871703815. 4
How does the result change for olymer solutions? Polymers are long chain molecules of reeating chemical units called monomers. The molecular weight of a olymer increases with the number of monomers er chain. H C C H H Styrene monomermonomer Polystyrene For olymers, molecular weights tyically range from 10,000 g/mol to 1M g/mol while solvents have molecular weights tyically ~100 g/mol. For given mass ratio of olymer and solvent, m P /m S, the molar fraction of olymer is: X! m "! 1 " # $# $ ms NM mer = % &% &! 1 "! m "! 1 " # $ + # $# $ % M s & % ms &% NM mer & while that of the solvent is: X s! 1 " # $ M s = % & = 1' X! 1 "! m "! 1 " # $ + # $# $ % M s & % ms &% NM mer & where M s is the solvent molecular weight, M mer is the molecular weight of the monomer reeat unit and N is the number of reeat units. Table 1 gives values of X for given values of m P /m S and N. For equal mass fractions, the mole fraction is increasingly asymmetric, introducing similar asymmetry into the hase diagram. We can see this qualitatively by calculating for the ideal solution.!s mix 5
N m P /m S w = m P /(m S +m ) X /R (K -1 )!S mix (ideal soln) 100 0.1 0.0909 0.0010 0.008 100 0.2 0.1667 0.0012 0.009 100 0.5 0.3333 0.0050 0.031 100 1 0.5000 0.0099 0.056 100 2 0.6667 0.0196 0.096 100 5 0.8333 0.0476 0.191 100 10 0.9090 0.0909 0.305 100 50 0.9804 0.3333 0.636 100 100 0.9901 0.5000 0.693 100 1000 0.9990 0.9091 0.305 1000 1 0.5000 0.0010 0.008 1000 2 0.6667 0.0020 0.014 1000 5 0.8333 0.0050 0.031 1000 10 0.9090 0.0099 0.056 1000 100 0.9901 0.0909 0.305 1000 500 0.9980 0.3333 0.636 1000 1000 0.9990 0.5000 0.693 1000 10000 0.9999 0.9091 0.305 1 0.1 0.0909 0.0909 0.305 1 0.2 0.1667 0.1667 0.451 1 0.5 0.3333 0.3333 0.636 1 1 0.5000 0.5000 0.693 1 2 0.6667 0.6667 0.636 1 5 0.8333 0.8333 0.451 1 10 0.9090 0.9090 0.305 6
As the olymer chain length increases, there is less entroy gained by adding small amounts of olymer (e.g., by weight) into solvent comared to that gained by adding small amounts of solvent to olymer. This introduces strong asymmetry to the hase diagram. T Single Phase N Two Phase w Although qualitatively correct in showing that entroy of mixing diminishes with increasing chain length, the regular solution model cannot rigorously be alied to olymer mixtures and solutions. An imroved model for redicting hase behavior of olymer solutions was ut forth by P.J. Flory, 1 who won the Nobel Prize for his contributions to olymer science. Flory s model emloys statistical mechanics, a field that connects macroscoic behavior to the microscoic roerties of systems. His model builds off the statistical mechanics develoment of the regular solution model. Consider, for examle, a small molecule mixture of n A molecules of comonent A and nb molecules of comonent B. The number of distinguishable ways we could arrange these comonents on a lattice of n A +n B sites is:! = ( na + nb )! n! n! A B For examle, the total number of configurations for a system with na = 4 and n B =2 is: 6!! = = 15 4!2! 7
The entroy of the system is related to the number of configurations of the system Ω by: S = k ln! where k is Boltzmann s constant, k = R/N Av = 1.381 10-23 J/K. Using Stirling s formula: ln( x!) = x ln x! x we obtain the total entroy for the system:! na n " B Ssol = # k $ na ln + nb ln % & na + nb na + nb ' In the heterogeneous state, the number of distinguishable B configurations is: nb!! B = = 1 n!, B and similarly for A. The change in entroy on mixing is given by subtracting off the ure state entroy. For the small molecule mixture: [ ln ln ]! S = S " S = " k n X + n X mix sol heter A A B B Since our molecules A and B have equal volume, we can also write: [ ln! ln! ] " S = # k n + n mix A A B B Or, er lattice site we have: 8
[ ln ln ] " s = # k!! +!! mix A A B B where Vi nivo! i = = V n v + n v A o B o and ν ο is the volume of the lattice site. For olymer solutions, we can use a similar lattice model to obtain the entroy of mixing, according to the model develoed by Flory 1. Here we consider a olymer whose segmental volume is equal to the volume of a lattice site. Due to the connectivity of the segments, the number of configurations available to the system decreases. For a mixture of n c olymers and n s solvent molecules, the total number of lattice sites is now: No = nn + ns where N is the degree of olymerization of the olymer, i.e., the number of segments er chain. Incororating connectivity into the lacement of the olymer segments, the total number of ways of arranging the system was first shown by Flory to be 1 : 9
( N ) o! " z! 1# $ = % & nc! ns!' No ( n ( N! 1) where z is the number of neighbors in the lattice. Again alying Stirling s formula, we obtain the total entroy for the system:! n n " s! z # 1" Ssol = # k $ ns ln + n ln % + kn ( N # 1) ln ns + Nn ns + Nn $ & e % $ & %' ' Setting n s = 0 in the above exression gives the entroy associated with the various configurations of the olymer coil: " 1 # " z! S ln ( 1) ln 1 # ol =! k $ n + kn N! & N % ' $ & e % ' We are interested in the entroy gained by mixing solvent and olymer molecules together:! Smix = Ssol " S ol $ Smix = % k " & ns ln! s + n ln! # ' If we consider the entroy of mixing er site, 2 " n n s # $ smix = % k & ln! s + ln! ' ( N o N o ) 10
"! # = $ k %! s ln! s + ln! N & ' ( Here we see quantitatively the effect of chain length on molecular weight. As N increases, the amount of entroy gained by mixing the olymer into the solvent is reduced by 1/N. The enthaly of mixing is analogous to that obtained for the regular solution model for atomic mixtures. In the demixed or hase-searated state, the total interaction energy can be obtained by counting the number of air-wise interactions between monomermonomer and solvent-solvent airs: 3 " 1 # " 1 # H heter = nn $ % z! & + ns $ % z! ss & ' 2 ( ' 2 ( where ε ii is the attractive (negative) interaction energy of the j-j air. In the mixed state, the energy is calculated as: # 1 $ # 1 $ # 1 $ # 1 $ H sol = nn! %& z" ' + ns! s %& z" ss ' + nn! s %& z" s ' + ns! %& z" s ' ( 2 ) ( 2 ) ( 2 ) ( 2 ) where the volume fractions account for the reduced robability of the adjacent site being a olymer (φ ) or solvent (φ s ) secies. The change in enthaly on mixing er site is given as: H sol # H heter z $ h = = ( 2! #! #! )" " N 2 mix s ss s o The resulting exression for the free energy of mixing er site can be written as 2 : # g kt mix! =! ln! + ln! + "!! N s s s where χ is known as the Flory-Huggins interaction arameter, defined as: 11
z %! ss $! & " = '! s $ (# s# kt ) 2 * Phase diagrams can be constructed by taking the derivative of the free energy with resect to φ. The boundary of the miscibility ga, also called the binodal or coexistence curve, is defined by: "# g "! mix = 0 The limit of stability of the one hase mixture, called the sinodal, is obtained by equating the second derivative of the F-H free energy to zero: 2 # $ #" g mix 2 1 1 = + % 2! = 0 N" 1% " For values of χ > χ s (or T < T s ), the solution will sontaneously decomose into olymer-rich and solvent-rich hases. 2 It is debated whether cloud oint measurements are a measure the sinodal rather than the coexistence curve. Between the sinodal and coexistence lines, demixing occurs by nucleation and growth mechanisms. The critical oint is the extrema of the sinodal: 3 " # g "! mix = 0 3 χ Two Phase! = 1 1 + N, cr 1/ 2 One Phase! cr = 1/ 2 ( 1+ N ) 2 2N φ 12
/2 The critical concentration of olymer in a olymer-solvent mixture scales as N -1. Note that the critical temerature (defined by χ cr ) is also fixed, in accordance with the hase rule. The Flory-Huggins free energy of mixing exression can be used to calculate hase diagrams of olymer solutions using the using the Hildebrand solubility arameter formalism 4 : (! #! s ) " = v kt where v is an averaged volume of the olymer segment and solvent secies, and δj is the solubility arameter of secies j. For the system investigated here, the solubility arameter value is 17.52 (MPa) 1/2 for olystyrene and 16 (MPa) 1/2 for methyl cyclohexane. 5 Previous studies investigating hase diagrams of olystyrene-methyl cyclohexane are described in the archival literature. 6-8 Recently, this system has also been exloited for microencasulation. 9 2 References 1. P.J. Flory, Princiles of Polymer Chemistry, Cornell Univ. Press: Ithaca, NY, 1953. 2. P.G. degennes, Scaling Concets in Polymer Physics, Cornell Univ. Press: Ithaca, NY, 1979. 3. M. Rubenstein and R.H. Colby, Polymer Physics, Oxford Univ. Press: New York, 2003. 4. A.-V.G. Ruzette and A.M. Mayes, A simle free energy model for weakly interacting olymer blends, Macromolecules 34, 1894 (2001). 5. Polymer Handbook, 3 rd Edition, J. Brandu and E.H. Immergut, Eds., John Wiley & Sons: New York, 1989. 6. S. Saeki, N. Kuwahara, S. Konno, and M. Kaneko, Uer and lower critical solution temeratures in olystyrene solutions, Macromolecules 6, 246 (1973). 7. A. Imre and W.A. Van Hook, Demixing of olystyrene/methylcyclohexane solutions, J. Poly. Sci. Part B: Pol. Phys. 34, 751 (1996). 8. C.-S. Zhou, et al., Turbidity measurements and amlitude scaling of critical solutions of olystyrene in methylcyclohexane, J. Chem. Phys. 117, 4557 (2002). 13
9. T. Narita, et al., Gibbs free energy exression for the system olystyrene in methylcyclohexane and its alication to microencasulation, Langmuir 19, 5240 (2003). 14