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Polymer Solutions and Mixtures

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Polymer Solutions and Mixtures Johannes Gutenberg-University Mainz Institute for Physics Chapter 4: POLYMER SOLUTIONS AND MIXTURES 4.1 Excluded volume interaction & Flory theory • idea of excluded volume interactions • θ−temperature • Flory approach for the free energy and scaling behavior of RE with excluded volume interactions Up to here (see script Chap. 3) we looked at the properties of a single chain with increasing concentration polymer will interpenetrate each other. We can distinguish the following concentration regions: Range of concentrations single polymer dilute solution c << c* overlap concentration c ≈ c* semi-dilute solution c > c* (actual concentration low) melt Overlap concentration c∗ pervaded volume V~R3 Nv Nv Nv overlap of single polymer if c∗ =≈=mon mon mon (with ν = Flory exponent) VRNl33()ν 3 ⇒ cN∗−~~13ν N − 0.8 if ν = for good solvent conditions. 5 For high degrees of polymerization, the overlap concentration can be rather low! PD Dr. Silke Rathgeber Johannes Gutenberg-University Mainz phone:phone: +49 +49 (0) (0) 6131/ 6131/ 392-3323 392-3323 Institute for Physics – KOMET 331 Fax:Fax: +49 +49 (0) (0) 6131/ 6131/ 392-5441 392-5441 Staudingerweg 7 email:email: [email protected] [email protected] 55099 Mainz webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/ Germany Now we look at the thermodynamic properties of polymers in solutions at cc≥ ∗ and in polymer mixtures (dense systems). i.e. phase separations & osmotic pressure Flory-Huggins theory A mixture is miscible if the free enthalpy of mixing ΔGHTSmix=Δ mix − Δ mix <0 where ΔH mix and ΔSmix are the enthalpy and entropy of mixing, respectively. First we look at a mixture of low molecular weigth substances • distribution on lattice without change in volume • both components have lattice volume so in this theory; number fraction=volume fraction Entropy of mixing n1 = number of lattice sites for component 1 n2 = number of lattice sites for component 2 n=n1+n2 = total number of lattice sites Calculation of entropy change Δ=Skmix B ln Ω where Ω = number of possible distributions of component 1 & 2 on lattice sites n! Ω= the denominator takes permutations of component 1&2 among each nn12!! other into account with lnx !≈−xxx ln PD Dr. Silke Rathgeber Johannes Gutenberg-University Mainz phone: +49 (0) 6131/ 392-3323 Institute for Physics – KOMET 331 Fax: +49 (0) 6131/ 392-5441 Staudingerweg 7 email: [email protected] 55099 Mainz webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/ Germany ⎡⎤⎛⎞nn12 ⎛⎞ nn12 ⇒ Δ=−Sknmix B ⎢⎥12ln⎜⎟ + n ln ⎜⎟ with number fractions cc12= and = ⎣⎦⎝⎠nn ⎝⎠ nn since component 1&2 have same lattice volume: number fraction=volume fraction ⇒ Δ=−Snkccccmix B ⎣⎦⎡⎤1122ln( ) + ln ( ) ⇒ if n=N n per mole! A Δ=−SRccccmix ⎣⎦⎡⎤1122ln( ) + ln ( ) with gas-constant R=kBNA Calculation of enthalpy change Every lattice site is in the average surrounded by c1 z neighbors of component 1 by c2 z neighbors of component 2 (z = coordination number) Looking at averages implies that concentration fluctuations are small. The Flory-Huggins theory is a mean-field theory. Enthalpy of mixing z=6 1 ⇒ interaction enthalpy of component 1: Hncczwczw=+() 111112122 1 component 2: Hncczwczw=+() 221122222 with wij=pair-wise interaction energies The factor 0.5 arises from the fact, that all pairs are counted twice. 1 The expressions for the pure components are: Hnczwi0 ==,1,2 iiii2 PD Dr. Silke Rathgeber Johannes Gutenberg-University Mainz phone: +49 (0) 6131/ 392-3323 Institute for Physics – KOMET 331 Fax: +49 (0) 6131/ 392-5441 Staudingerweg 7 email: [email protected] 55099 Mainz webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/ Germany 1 Δ=+−+=HHHHHnzccwwwwithcc()()00 (2) −− += 1 mix 1 2 1 22 1 2 12 11 22 1 2 1 =Δnzc c w with Δ=−+ w w() w w 1 2 12 12 122 11 22 M ⇒ Δ=Hnzccwmix 12 Δ 12 per mole! Δ<w12 0 exothermal mixing process Δ>w12 0 endothermal mixing process Δ=w12 0 athermal mixing process 11 introduction of the interaction parameter: χ =Δw ∝ 12 12 RTT M ⇒ Δ=HRTccmix χ12 1 2 M ⇒ free enthalpy of mixing: Δ=GRTccccccmix [ 11221212ln + ln +χ ] per mole Polymer solutions The calculation of the enthalpy of mixing is the same for mixtures of low molecular weight and polymeric substances. It was based on pairwise interaction of segments/molecules. Connectivity of the segment plays no role! Calculation of entropy change Entropy of mixing PD Dr. Silke Rathgeber Johannes Gutenberg-University Mainz phone: +49 (0) 6131/ 392-3323 Institute for Physics – KOMET 331 Fax: +49 (0) 6131/ 392-5441 Staudingerweg 7 email: [email protected] 55099 Mainz webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/ Germany n1= number of solvent molecules n2= number of polymers N= lattice sites per polymer = degree of polymerization n= n1+n2N = total number of lattice sites Assume that i polymers are already distributed on the lattice ⇒ number of free sites is: n-N×i ⇒ for the first segment of the (i+1) polymer there are (n-N×i) possibilities to find a site ⇒ for the second segment there are z × (number of free sites) ⇒ for the third to xth segment there are (z-1) × (number of free sites) nNi− × number of free sites is replaced by the average number: n • n>>N otherwise there would be a significant reduction with each segment added Entropy of mixing X 2 1 X 1 X 3 4 X X X not considered: X X excluded volume interaction X 2 1 .. X 4 3 2 1 X X ⇒ number of possibilities to distribute the (i+1)th polymer on the lattice: NN−21if z>>1 − ()nNi−−NN−2 ⎡ () nNi⎤⎛⎞ z − 1 ν i+1 =−()nNiz ⋅⋅ ⋅− (1) z ≈ () nNi − ⎜⎟ 14243 nn⎣⎢⎥⎦⎝⎠ n 1st 142 43 14444244443 2nd all other (N-2) segments ⇒ number of possibilities to distribute all polymers on n2 lattice sites: ν ⋅⋅⋅νν... 12 n2 P2 = denominator takes into account indistinguishability of polymers! n2 ! nn(1)− ⎛⎞zn−1!2 for n>>N ⇒ P2 ≈ ⎜⎟ ⎝⎠nnnNn()!!− 22 PD Dr. Silke Rathgeber Johannes Gutenberg-University Mainz phone: +49 (0) 6131/ 392-3323 Institute for Physics – KOMET 331 Fax: +49 (0) 6131/ 392-5441 Staudingerweg 7 email: [email protected] 55099 Mainz webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/ Germany desorientation of polymers solution of polymers ni entropy of the solution: S=kB ln P2 with n=n1+n2N and ci = nnN12+ ⎧−⎫⎛⎞z 1 ⇒ Sk=−B ⎨⎬ n112ln cn − ln c 22 + nN ( − 1)ln ⎜⎟ ⎩⎭⎝⎠e The final stage has been reached by (1) disorientation of the polymer (creation of an amorphous state) (2) dissolution of the disoriented polymer substraction of the entropy change due to step (1) ⎛⎞z −1 entropy of a polymer melt sn(0)ln122== knBB N + kn (1)ln z − ⎜⎟ ⎝⎠e ⇒ change solely due to step (2): Δ=−==−SSSnmix(0)11122 kcccc B [ lnln + ] M ⎡⎤c2 ⇒ Δ=−SRcccmix 11ln + ln 2 polymer in solution ⎣⎦⎢⎥N M ⎡⎤cc12 Δ=−SRmix ⎢⎥ln c12 + ln c polymer mixture ⎣⎦NN12 M ⎧⎫cc12 ⇒ Δ=GRTcmix ⎨⎬ln121212 + ln c +χ cc ⎩⎭NN12 in general Δ∝w12χ 12 >0 (endothermal mixing) The entropy of mixing is for high molecular weights small. ⇒ Polymers are poorly miscible! PD Dr. Silke Rathgeber Johannes Gutenberg-University Mainz phone: +49 (0) 6131/ 392-3323 Institute for Physics – KOMET 331 Fax: +49 (0) 6131/ 392-5441 Staudingerweg 7 email: [email protected] 55099 Mainz webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/ Germany Osmotic pressure & membrane osmometry polymer solution pure solvent nn,,, pT npT,, sp s′ ′ semipermeable membrane: exchange of solvent until the chemical potential of the solvent is equal on both sides`. 0 chem. potential pure solvent = μμss(',)p TpT= (,) = chem. potential solvent in polymer solution 0 ∂ΔGmix Δ=μμss(,)pT − μ s (,) pT = enthalpy for dilution (*) ∂ns pTn,,p ⎧⎫cp ns np remember: Δ=GnkTccmix B⎨⎬ sln s + ln ccc p +χ s p with ccnnnNsp===+;; sp ⎩⎭N nn ∂ΔGmix ⎧⎫1 20 ⇒ Δ=μχμμsBsppss =nk T⎨⎬ln c +− (1 ) c + c = ( p , T ) − ( p , T ) ∂nNs ⎩⎭ M ⎧⎫1 2 ⇒ Δ=μχs RTc⎨⎬lnspp +− (1 ) c + c per mole (**) ⎩⎭N The osmotic pressure is the pressure difference pp− ' =Π In a linear approximation (in equilibrium): 0 00∂μs 0 μμs (,)p TpTpTpp==−−=−Πss (',) μ (,)( ') μ s (,) pTv s ∂p Tn, 14243s = vs = molar volume with (*) ⇒ Δ=−Πμs vs ⎧⎫1 2 with (**) ⇒ −ΠvRTcs =⎨⎬lnspp + (1 − ) c + χ c ⎩⎭N PD Dr. Silke Rathgeber Johannes Gutenberg-University Mainz phone: +49 (0) 6131/ 392-3323 Institute for Physics – KOMET 331 Fax: +49 (0) 6131/ 392-5441 Staudingerweg 7 email: [email protected] 55099 Mainz webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/ Germany 1 for low polymer concentrations ccccc<<=−≈−−+1: ln ln(1 )2 ... psppp2 ⎧⎫cp ⎛⎞1 2 ⇒ −ΠvRTsp =⎨⎬ +⎜⎟ −χ c +... ⎩⎭N ⎝⎠2 c with Π=vRTp van’t Hoff Law s N 1 and the other terms are derivations from ideal behavior, they disappear for χ = 2 remember: (1) χ ∝ T −1 (2) osmotic pressure is the pressure difference between polymer solution and pure solution ⇒ 1 1. for χ < (high temperatures): 2 osmotic pressure is higher compared to ideal gas condition; solvent is incorporated into the polymer and results in extension of the polymers; polymers repell each other. good solvent conditions 1 2. for χ > (low temperatures): 2 osmotic pressure is reduced compared to ideal gas condition; solvent is supressed from polymer and results in strinkage of the polymers; polymers attract each other. bad solvent conditions 1 3. for χ = : 2 osmotic pressure corresponds to ideal gas condition. θ-conditions Radius of PS in cyclohexane as a function of the solvent quality bad solvent good solvent PD Dr.
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