idea of excluded volume interactions θ temperature Flory approach for the free energy and scaling behavior of R E with excluded volume interactions

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1 Institute for Physics Chapter 4: POLYMER SOLUTIONS AND MIXTURES 4. Excluded volume interaction & Flory theory idea of excluded volume interactions θ temperature Flory approach for the free energy and scaling behavior of R E 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~R 3 Nvmon Nvmon Nvmon overlap of single polymer if c = = (with ν = Flory exponent) 3 ν 3 V R ( N l) c ~ N ν 3 ~ N if ν = for good solvent conditions. 5 For high degrees of polymerization, the overlap concentration can be rather low! phone: (0) (0) 63/ 63/ Fax: Fax: (0) (0) 63/ 63/

2 Now we look at the thermodynamic properties of polymers in solutions at c tures (dense systems). i.e. phase separations & osmotic pressure c and in polymer Flory-Huggins theory A ture is miscible if the free enthalpy of ing Δ G =ΔH TΔ S < 0 where Δ H and Δ are the enthalpy and entropy of ing, respectively. S First we look at a ture 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 ing n = number of lattice sites for component n = number of lattice sites for component n=n +n = total number of lattice sites Calculation of entropy change Δ S = kb ln Ω where Ω = number of possible distributions of component & on lattice sites n! Ω= the denominator takes permutations of component & among each n! n! other into account with ln x! xln x x phone: +49 (0) 63/ Fax: +49 (0) 63/

3 n n Δ S = k n + n n n with number fractions n n c = and c = n n B ln ln since component & have same lattice volume: number fraction=volume fraction Δ S = nk c ln ( c ) + c ln ( c ) B n Δ S = R cln c + cln c per mole! with gas-constant R=k B N A if n=n A ( ) ( ) Calculation of enthalpy change Every lattice site is in the average surrounded by c z neighbors of component by c z neighbors of component (z = coordination number) Looking at averages implies that concentration fluctuations are small. The Flory-Huggins theory is a mean-field theory. Enthalpy of ing z=6 interaction enthalpy of component : H = nc( czw+ czw) component : H = nc( czw + czw) with w ij =pair-wise interaction energies The factor 0.5 arises from the fact, that all pairs are counted twice. The expressions for the pure components are: H = nczw, i=, 0 i i ii phone: +49 (0) 63/ Fax: +49 (0) 63/

4 0 0 Δ H = ( H + H) ( H + H ) = nzcc ( w w w) with c + c = = nzcc Δw with Δ w = w ( w + w ) Δ H = nzcc Δ w per mole! M Δ w < 0 exothermal ing process Δ w > 0 endothermal ing process Δ w = 0 athermal ing process introduction of the interaction parameter: χ =Δw RT T Δ H M = RTχcc M free enthalpy of ing: G RT[ c ln c c ln c χ cc ] Δ = + + per mole Polymer solutions The calculation of the enthalpy of ing is the same for tures 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 ing phone: +49 (0) 63/ Fax: +49 (0) 63/

5 n = number of solvent molecules n = number of polymers N= lattice sites per polymer = degree of polymerization n= n +n N = 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+) 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-) (number of free sites) n N i 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 ing X X X X 3 4 X X X X X.. X 4 3 X X not considered: excluded volume interaction number of possibilities to distribute the (i+)th polymer on the lattice: ν i+ N if z>> N ( n Ni) N ( n Ni) N z ( n Ni) z ( z ) ( n Ni) 443 n n n st nd all other (N-) segments = number of possibilities to distribute all polymers on n lattice sites: ν ν... νn P = denominator takes into account indistinguishability of polymers! n! for n>>n n ( n ) z n! P n ( n n N)! n! phone: +49 (0) 63/ Fax: +49 (0) 63/

6 desorientation of polymers solution of polymers entropy of the solution: S=k B ln P with n=n +n N and z S = k n ln c n ln c + n ( N ) ln e B c i ni = n + n N The final stage has been reached by () disorientation of the polymer (creation of an amorphous state) () dissolution of the disoriented polymer substraction of the entropy change due to step () z entropy of a polymer melt sn ( = 0) = kn B ln N+ kn B ( z )ln e Δ S = S S( n = 0) = k c lnc + c lnc change solely due to step (): [ ] B c Δ S = R c c + c N M c c Δ S = R ln c+ ln c N N M ln ln polymer in solution polymer ture c M Δ G = RT ln c+ ln c + χcc N N in general Δw χ > 0 (endothermal ing) The entropy of ing is for high molecular weights small. c Polymers are poorly miscible! phone: +49 (0) 63/ Fax: +49 (0) 63/

7 Osmotic pressure & membrane osmometry polymer solution pure solvent nn s, p, pt, ns, p, T semipermeable membrane: exchange of solvent until the chemical potential of the solvent is equal on both sides`. chem. potential pure solvent = μ 0 s( p ', T) = μs( p, T) = chem. potential solvent in polymer solution ΔG 0 Δ μs = μs( pt, ) μs( pt, ) = enthalpy for dilution (*) ns p, T, np cp n n s p remember: Δ G = nkbt cs ln cs + ln cp + χcscp with cs = ; cp = ; n= ns + npn N n n 0 μ ΔG s nkbt ln cs ( ) cp cp s( p, T ) s( p, T ) ns N χ Δ = = + + = μ μ M Δ μs = RT ln cs + ( ) cp + χcp per mole (**) N The osmotic pressure is the pressure difference In a linear approximation (in equilibrium): p p' =Π μ μ μ μ μ p T, ns s 0 s ( p, T) = s( p', T) = s( p, T) ( p p') = s( p, T) Πvs with (*) Δ μs = Π vs with (**) Π vs = RT ln cs + ( ) cp + χcp N = vs = molar volume phone: +49 (0) 63/ Fax: +49 (0) 63/

8 for low polymer concentrations c c c c c cp Π vs = RT + χ cp +... N cp with Π vs = RT van t Hoff Law N and the other terms are derivations from ideal behavior, they disappear for p << : ln s = ln( p) p p +... χ = remember: () χ T () osmotic pressure is the pressure difference between polymer solution and pure solution. for χ < (high temperatures): 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. for χ > (low temperatures): 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 3. for χ = : 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 phone: +49 (0) 63/ Fax: +49 (0) 63/

9 Dependance of osmotic pressure on polymer concentration and solvent quality Πv s crt p A good solvent ( A ) χ 0 < > N θ solvent bad solvent ( A ) χ 0 = = ( A ) χ 0 > < c p Question: How can we determine N and χ? Answer: plot Πvs crt p versus c p! Πvs = χ cp... crt + + p N axis intercept = N slope = χ = nd virial coefficient A Question: How can we measure Π? Answer: Membrane osmometry The membrane is impermeable for the polymer. Solvent permeates through the membrane until the chemical potential of the solvent is the same in both chambers. The excess column is a measure of the hydrostatic pressure which is in equilibrium equal to the osmotic pressure. ρs gδ h=π = RT cp + χ cp +... Nvs with ρ s = solvent density; g = acceleration of gravity; Δ h = excess column height; v s = molar volume solvent phone: +49 (0) 63/ Fax: +49 (0) 63/

10 Δh μ S μ 0 S( p ' +Π, T) = μs( p', T) 0 μ S polymer solution pure solvent semipermeable membrane Question: What kind of molecular weight average is measured by membrane osmometry? [see Physical chemistry of polymer solutions, Kamide & Dobast] Π vs = RT ln cs + ( ) cp + χcp N for very (!) dilute solutions: Π vs = RTln cs = RTln( cp, i) with with (*) i c pi, n n i pi, i pi, = n + n n s i p, i s total concentration of solute RT n n i Π= ln RT v s n s vsns ( ) pi, i pi, ( ) v= v n + n v n s s i p, i s s i with n M n M n n c= = = 4443 i pi, i i pi, i i pi, i pi, Mn v n i pi, v vsns = M n n i () npi, << ns:ln i n s c Π= RT M n n pi, i pi, n s notice: concentration in weight volume Answer: Membrane osmometry determines the number averaged molecular weight! (in very dilute solutions) phone: +49 (0) 63/ Fax: +49 (0) 63/

11 Phase diagrams for polymer solutions and tures Phase diagrams (the basics) Φ Φ α Φ β Assume a ture with composition Φ separates into two phases with composition Φαand Φβ with volume fractions α and β, respectively. Φ = α Φ + βφ with α + β = The original phase is stable, if α β ΔG α ΔG ( Φ ) + ( α) ΔG ( Φ ) =Δ G αβ Gibbs stability criterion α β free enthalpy of ing as a function of volume fraction of one component phone: +49 (0) 63/ Fax: +49 (0) 63/

12 more complex situation: convex and concave curve regions stable, metastable and unstable regions ΔG at fixed temperature metastable unstable metastable stable stable spinodal binodal () The binodal defines the boundary between the one-and two-phase region and can be determined by tangent construction ΔG Φ T, p = 0 (minima) () The spinodal defines the boundary between the metastable and unstable region and can be determined from the points of inflection G Φ Δ T, p = 0 unstable: metastable: small fluctuations in concentration lead to spontaneous deing system is stable in respect to small fluctuations; only if a sufficient large nucleus with equilibrium composition is formed it growth in size. The nucleus has to be large enough because of the surface tension between the two phases. keyword: separation by nucleation (3) There is a temperature T c and a composition Φc where spinodal and binodal merge called critical point 3 ΔG = 0 3 (extremum of the spinodal curve) Φ T, p phone: +49 (0) 63/ Fax: +49 (0) 63/

13 Construction of a phase diagram T 5 free energy T c T 3 T T good solvent temperature T c bad solvent T concentration () Determination of the free enthalpy as a function of Φ at various temperatures () Determination of stable and unstable regions and projection into the ( T, Φ) space M Φ Φ Flory Δ G = RT ln Φ + ln Φ + χφφ < 0 N N χ T as condition for miscibility In Flory-theory always lower miscibility gap and upper critical point (miscibility at high temperatures) phone: +49 (0) 63/ Fax: +49 (0) 63/

14 Polymer solution N =N ; N = ; Φ =Φ ; Φ = ( Φ ) G M Δ () binodal: = ln Φ+ ln( Φ) + ( Φ ) χ = 0 Φ RT N N M ΔG () spinodal: = + χ = 0 Φ RT NΦ ( Φ) 3 M ΔG (3) critical point: = + = 0 3 Φ RT NΦ ( Φ ) c c Φ = c N N + for large N>> insertion of Φ c into (): ΔG ( Φ ) = 0 + χc = 0 ( ) M c Φ RT NΦc Φc χc = ( + ) = N + + N = + + N N N Tc Asymmetric miscibility gap which shifts to very small polymer contents for high molecular weight polymers. T c increases with molecular weight of polymer Binodal experimentally determined from precipitation temperature M w = binodal (Flory) binodal (experimental) spinodal (Flory) No good description of experimental data! The dependence of χ on Φ has to be taken into account; also bad description of T dependence! volume fraction polymer phone: +49 (0) 63/ Fax: +49 (0) 63/

15 Polymer tures Φ =Φ ; Φ = ( Φ ) M Φ Φ Δ G = RT ln Φ + ln Φ + χφφ < 0 N N as condition for miscibility () M ΔG ln( ) binodal: Φ = ln Φ+ + ( Φ ) χ = 0 Φ RT N N N () spinodal: M ΔG = + χ = 0 Φ RT NΦ N( Φ) (3) critical point: 3 M ΔG = + = 0 Φ 3 c = Φ RT NΦ c N( Φ c) N N + with () ( ) χc = N + N Tc T c increases with increasing N or N N << N or N << N Φc shifts towards very low concentrations of the longer component. Special case: Polymer with the same degree of polymerization: N = N = N Φ = and χ = N T ( χ ) = critical point c c c c N ( χ ) < one-phase region c N ( χ ) > two-phase region Phase diagram is symmetrical at Φ c for polymer tures with identical degree of Polymerization Mixture of symmetric polymers c N ΔG χn=. χ. (χn) c = temperatur two-phase. one-phase Φ Φ phone: +49 (0) 63/ Fax: +49 (0) 63/

16 In summary Connectivity of polymers degree of possible disorder is limited entropy of ing is reduced by prefactor monomer-monomer interaction determines ing behavior. N As higher the molecular weight as higher the critical temperature; weak repulsive interaction between the monomers leads to deing. Miscibility of polymers as significantly reduced compared to low molecular weight substances. Attractive interaction, such as dipole-dipole interaction / hydrogen bonds / donor acceptor interaction, is necessary. Flory-Huggins theory yields bad description of experimental data The simple lattice model does not describe the behavior of dilute polymer solutions very well () Approach is based on mean-field-theory, i.e. lattice site occupation is uniform and every molecule/ monomer sees the same averaged surrounding. It was assumed that the monomer/ molecule distribution on the lattice is purely statistically. only true if Δ w = 0 fluctuations in particular relevant in dilute solutions are neglected () Volume changes due to ing having an impact on the mobility/ flexibility ( entropy) are neglected. χ is concentration dependent! (3) Interaction is assumed to be between nearest neighbors only. Any specific interaction (longer ranged, inducing order or specific interaction involving only some specific groups) between component A and B like: dipole-dipole interaction, hydrogen bonding, donoracceptor interaction, solvent orientation in polar solvent close to the chain. Ansatz: χ parameter comprises on enthalpic χ H and entropic χ S contribution B χ = χs + χh = A + kt B dχ B d( χt) ΔS with χ H = T = and χs = = A = dt k B T dt k B from experiments major contribution from non-combinatorial change in entropy: χ S phone: +49 (0) 63/ Fax: +49 (0) 63/

17 UCST = upper critical solution temperature χ decreases with increasing temperature e.g. polybutadiene (88% vinyl, protonated; 78% vinyl, deuterated) ing leads to increase in volume; volume for local movements of the chains is larger increase in mobility/flexibility increase in entropy. effect more dominant with increasing temperature χ is negative and decreases with T, i.e. increases with /T S LCST = lower critical solution temperature χ increases with increasing temperature e.g. polyisobutylene (protonated); head-to-head polypropylene (deuterated) phone: +49 (0) 63/ Fax: +49 (0) 63/

18 Mechanism: () Mixing leads to reduction in volume; volume for local movements of the chains is reduced reduction in mobility/ flexibility reduction in entropy. Effect becomes more dominant with increasing temperature χ is positive and increases with temperature/ decreases with /T s () Competition between attractive interaction of special groups of both components and repulsive interaction between remaining groups (e.g. copolymers). Attractive interaction: dipole-dipole interaction; hydrogen bonds; donor acceptor interaction Increasing temperature weakens bonds and repulsive interaction dominates both LCST and UCST Phase diagram with lower and upper miscibility gap Flory-Huggins parameter as a function of temperature phone: +49 (0) 63/ Fax: +49 (0) 63/

19 4.4 Flory-Krigbaum Theory On the foundation of the excluded volume interactions an enthalpy parameter H and an entropy parameter Ψ for dilution are defined describing deviations from ideal behaviour. M d G M Flory-Huggins: G Δ μs RT Δ = =Δ = ln cs + cp + χc p n S N with c p << ln cs = ln( cp) cp cp +... as for osmotic pressure c d M p Δ G =Δ μs = RT[ + χ cp +...] N 443 deviations from ideal behavior Flory-Krigbaum: dilution enthalpy: dilution entropy: Δ H = RTHc d ni p d Δ Sni = RΨ cp ( p c due to pair contacts) free enthalpy of dilution: d Δ G = RT( H Ψ ) c ni comparison Flory-Huggins and Flory-Krigbaum theory: ( H Ψ ) = χ d d θ conditions: χ = H =Ψ ; i.e. Δ Hni = TΔ Sni H define: θ = T (only defined if H and Ψ have equal sign) Ψ p non-ideal behavior disappears for T = θ! replace θ χ =Ψ = nd virial coefficient A! T d M cp θ Δ G =Δ μs = RT +Ψ cp +... N T Determination of the θ temperature in polymer solutions (A) From the determination of the critical phase separation temperature T c as a function of the degree of polymerization θ χc =Ψ = N N = N + N + T Tc Ψθ θ θ can be determined from axis intersection N and N + N 0 Ψ from slope phone: +49 (0) 63/ Fax: +49 (0) 63/

20 θ (B) From A measurement (osmotic pressure) as a function of the temperature A =Ψ T intersection with T axis (A =0) T = θ phone: +49 (0) 63/ Fax: +49 (0) 63/

21 4.5 Phase separation mechanisms Separation process time resolution possible due to high viscosity of polymer or by quenching to temperatures below glas transition temperature Methods for the determination of the cloud point Depending on method of investigation different length scales can be resolved. Phase separation mechanisms cooling from one-phase region into: () binodal (metastable region) segregation by nucleation () spinodal (unstable region) spinodal segregation phone: +49 (0) 63/ Fax: +49 (0) 63/

22 Cooling from one-phase region into two-phase region Resulting structure depends strongly whether cooling down ends in metastable or unstable region. In unstable region any small concentration fluctuation can growth. In metastable region the concentration fluctuation has to be large enough in respect to concentration difference ΔΦ and size R () spinodal decomposition results in interpenetrating, continuous structure with specific length scale. () binodal decomposition by nucleation, spherical precipitates growthing in time. Spontaneous local concentration fluctuation δφ in volume 3 dr and Φ δ in adjacent volume 3 dr change of free enthalpy in terms of free enthalpy g per unit volume 3 3 δg = [ g ( Φ 0 + δφ ) + g ( Φ0 δφ )] d r g ( Φ 0 ) d r series expansion of g up to second order in Φ yields g 3 ΔG 3 ( 0) ( 0) δg = Φ δφ d r = Φ δφ d r Φ v Φ ΔG depending on the sign of the curvature a small fluctuation in concentration leads to Φ an increase or decrease of the free enthalpy (Gibbs stability criterion) phone: +49 (0) 63/ Fax: +49 (0) 63/

23 () () > 0 (convex) δg > 0 Φ ΔG < 0 (concave) δg < 0 Φ ΔG phase is stable against small pertubation; amplitude of small pertubation decays phase is unstable against small pertubation; amplitude of small pertubation growth Separation by nucleation a small concentration fluctuation decays. concentration difference of the nucleus must correspond to that of the equilibrium phases. nucleus must have critical size. free enthalpy for nucleation: Δ Gr = 4π rδ g+ πrσ Δ g= gφ gφ 3 ( ) 4 with ( 0) ( ) Term II Term I (*) Term I: volume term corresponds to reduction in enthalpy by generation of the new phases. Term II: surface term corresponds to increase in enthalpy due to surface generation with σ being the free enthalpy per unit surface. segregation by nucleation overcome energy barrier before nucleus can growth phone: +49 (0) 63/ Fax: +49 (0) 63/

24 ΔG b nucleation rate increases with Arrhenius law: ν nucleus exp kt B 3 6π σ with the barrier height Δ G b = which is the maximum of (*) Δ Gr () 3 ( Δg) energy barrier ΔGb increases with decreasing distance from the binodal (approaching from spinodal); at the binodal Δ g = 0 ΔGb ture needs to be supercooled (or overheated) to achieve relevant nucleation rates. In summary nucleus growth diffusion controlled; downhill-diffusion towards decreasing concentrations (chemical potential gradient supports feeding nucleus). decreasing temperature diffusion slower (nucleus growth slower); nucleation rate increases (for lower miscibility gap) number of nuclei increases droplet size, number and distance depend on time and temperature at later stages coalescence, coarsening and ripening until there are two large phases with composition Φ and Φ Spinodal Decomposition ΔG 0 at spinodal spontaneous deing; no activation energy Φ no sharp transition from phase separation by nucleation to spontaneous deing since ΔG Φ vanishes continuously Kinetics of spinodal decomposition in polymer tures local chemical potential difference caused by concentration fluctuations 0 ΔG Δ μ = μs μs = κ Φ Φ Term I Term II Term I: homogenous system Term II: gradient energy term associated with departure from uniformity κ = gradient energy coefficient (empirical constant) difference in the chemical potentials causes interdiffusion flux. Linear response between flux and gradient in chemical potential: phone: +49 (0) 63/ Fax: +49 (0) 63/

25 ΔG 3 j ( μ) =Ω Δ =Ω Φ κ Φ Φ with Ω =Onsager-type phenomenological coefficient substitution into equation of continuity which equates the net flux over a surface with the loss or gain of material within the surface Φ ΔG 4 = j = Ω κ Φ Φ t Φ Cahn-Hillard relation comparison to conventional diffusion equation Φ = D Φ t ΔG D Ω < Φ 0 (D=diffusion coefficient) in unstable region diffusion against concentration gradient at spinodal ( T T Sp ) D 0 critical slowing down general solution for Cahn-Hilliard relation: Φ Φ 0 = exp( Γ( β ) t) [ A( β)cos( βx) + B( β)sin( βx)] β π with λ = and β ΔG ( ) Γ β = Ωβ { κβ Φ > 0 stable region: unstable region: ΔG > 0 Γ ( β ) < 0 Φ ΔG < 0 Γ ( β ) > 0 Φ instabilities rapidly decay some wavelengths notably the long wavelengths (small β ) will grow phone: +49 (0) 63/ Fax: +49 (0) 63/

26 Spinodal segregation () early stages: amplitudes of fluctuations growth for preferred wavelength λ = (length scale) ; modes are independent. () intermediate stages: amplitudes growth, modes interact with each other; wavelength growth and structure coarsens; deviations from cosine wave form. (3) late stages: fluctuation reaches the upper and lower limit Φ and Φ of equilibrium coexistence curve; wave form becomes rectangular and domain structure appears. Domain boundaries have finite thickness but extension is small compared to domain size. Late stages of spinodal segregation Finally: system tends to minimize its interfacial free energy by minimizing the amount of interface area. Sphere contains the most volume and the least surface area. () interwoven structure coarsens by an interfacial-energy driven viscous flow mechanism () tendency to spherical precipitates phone: +49 (0) 63/ Fax: +49 (0) 63/

27 4.6 Summary Miscibility of tures if Δ G =ΔH TΔ S < 0 Flory-Huggins Theory (mean field theory) considers a random distribution of polymer monomers & solvent molecules on a lattice calculation of calculation of Δ H in terms of pair interaction energies; every molecule or monomer sees the same average surrounding. Δ S number possible distributions of both component on the lattice under preservations of the connectivity of the polymer(s). further assumptions: distribution on lattice without volume change; both components have same lattice volume: volume fractions = number fractions. number of lattice sites much larger than the degree of polymerization. large coordination number (nearest neighbors on lattice). M c c Flory-Huggins Theory Δ G = RT ln c+ ln c+ χcc N N zδw χ T Δ w = w w + w RT = with ( ) for polymers with high degree of polymerization the entropy gain due to ing very is small (reduced by factor /N) for most substances endothermal ing χ > 0 small enthalpic contribution leads to deing! in Flory-Huggins theory χ T ing at high temperatures & deing at low temperatures phone: +49 (0) 63/ Fax: +49 (0) 63/

28 Osmotic pressure is the pressure produced by a solution in a space that is enclosed by a semi-permeable membrane. Π= pressure polymer solution pressure pure solvent cp Π Vs = RT + ( χ) cp +... N χ < good solvent - high temperatures - polymer swells χ > bad solvent - low temperatures - polymer shrinks χ = θ solvent Membrane osmometry (measures excess column height) hydrostatic pressure = osmotic pressure dilute solutions: ΠVs = ( χ ) cp... RTc + + N P in very dilute solution: the number averaged molecular weight is measured Phase diagrams Gibbs stability criterion: ΔG αδg ( Φ ) ( ) G ( ) G αβ α + α Δ Φ β =Δ ΔG convex metastable concave unstable metastable convex at fixed temperature stable stable spinodal binodal phone: +49 (0) 63/ Fax: +49 (0) 63/

29 Construction of phase diagrams binodal (metastable region) free enthalpy T T T > > > > T T good solvent T ΔG Φ T,p = 0 spinodal (unstable region) Δ G Φ T, p critical point = 0 χ temperature bad solvent T 3 Δ G Φ 3 T, p = 0 concentration Φ binodal and spinodal merge; determined by (Tc,Φc) Flory-Huggins Theory : χ always lower miscibility gap & upper critical point T Polymer solution M w = / Φ C = N N + χc = + N + N T / C binodal (experimental) miscibility gap is highly asymmetric, shifts to very small polymer concentrations for high N binodal (Flory) Tc increases with N volume fraction polymer spinodal phone: +49 (0) 63/ Fax: +49 (0) 63/

30 Polymer tures temperature χn two-phase region Φ C = N + N χc = ( N + N ) T / / C for N <<N or N <<N : Φ C shifts towards very concentrations of the shorter component. one-phase region Φ T C increases with N or N miscibility gap is symmetric at critical point for N =N Φ C = χ C TC = N Limitation of the flory theory Mean field approach but distribution of the monomers + molecules on the lattice is not purely statistically due to non-equal interaction between identical and non-identical components: χ becomes concentration dependent. Concentration fluctuations are also neglected. Main lack of the Flory Huggins theory originates from an additional non-combinatorial entropical contribution to the enthalpy of ing resulting from volume changes due to ing. Ansatz: : χ=χ S + χ H =A(T)+B/(k B T) () ing leads to increase in volume - gain of entropy due to higher chain mobility - χ S is negative and decreases with temperature UCST- Lower miscibility gap () ing leads to decrease in volume - loss of entropy due to higher chain mobility - χ S is positive and increases with temperature LCST- Upper miscibility gap Combinations of enthalpic contribution and () to χ can lead to both UCST + LCST also in combination with: (3) competition of attractive bonds of certain groups on the polymer and repulsive interaction between remaining groups. Weakening of the attractive bonds with increasing temperature repulsive interaction dominates. phone: +49 (0) 63/ Fax: +49 (0) 63/

31 Phase diagram with lower and upper miscibility gap Flory-Huggins enthalpy of ing: M c c Δ G = RT ln c+ ln c+ χ cc < 0 N N (LCST) one-phase region two-phase region (UCST) Flory Krigbaum theory introduces an enthaply κ and an entropy ψ parameter for dilution describing deviations from ideal behavior due to excluded volume interaction, introduction of the θ temperature: θ=t κ/ψ M ΔG θ =Δ μ = RT( Ψ κ ) c = Ψ c n T S M S P P nd virial coefficient A ~ (/-χ)= ψ(-θ/t) θ temperature can be determined from A measurements θ temperature and entropy parameter ψ can be determined from molecular weight dependency on Tc phone: +49 (0) 63/ Fax: +49 (0) 63/

32 Phase separation mechanisms cooling from one-phase region into: () binodal (metastable region) segregation by nucleation results in spherical precipitates growthing with time. () spinodal (unstable region) spinodal segregation results in interpenetrating, continuous structure with specific length scale. Separation by nucleation a small concentration fluctuation decays. concentration difference of the nucleus must correspond to that of the equilibrium phases. and nucleus must have critical size. nucleus growth diffusion controlled; downhill-diffusion towards decreasing concentrations (chemical potential gradient supports feeding nucleus). decreasing temperature diffusion slower (nucleus growth slower); nucleation rate increases (for lower miscibility gap) number of nuclei increases droplet size, number and distance depend on time and temperature at later stages coalescence, coarsening and ripening until there are two large phases with composition Φ and Φ Spinodal segregation a small concentration fluctuation growth. diffusion against concentration gradient at spinodal ( T T Sp ) D 0 critical slowing down phone: +49 (0) 63/ Fax: +49 (0) 63/

33 Stages () early stages: amplitudes of fluctuations growth for preferred wavelength λ = (length scale) ; modes are independent. () intermediate stages: amplitudes growth, modes interact with each other; wavelength growth and structure coarsens; deviations from cosine wave form. (3) late stages: fluctuation reaches the upper and lower limit Φ and Φof equilibrium coexistence curve; wave form becomes rectangular and domain structure appears. Domain boundaries have finite thickness but extension is small compared to domain size. (4) Finally: system tends to minimize its interfacial free energy by minimizing the amount of interface area. Sphere contains the most volume and the least surface area. interwoven structure coarsens by an interfacial-energy driven viscous flow mechanism tendency to spherical precipitates phone: +49 (0) 63/ Fax: +49 (0) 63/

34 Literature. U.W. Gedde, Polymer Physics, Chapman & Hall, London, G. Strobl, The Physics of Polymers, Springer-Verlag, Berlin, M.D. Lechner, K. Gehrke, E.H. Nordmeier, Makromolekulare Chemie, Birkhäuser Verlag, Basel, J.M.G. Cowie, Polymers: Chemistry & Physics of Modern Materials, Blackie Academic & Professional, London, K. Kamide, T. Dobashi, Physical Chemistry of Polymer Solutions, Elsevier, Amsterdam, O. Olabisi, L.M. Robeson, M.T. Shaw, Polymer-Polymer Miscibility, Academic Press, San Diego, M. Rubinstein, R.H. Colby, Polymer Physics, Oxford University Press, New York, U, Eisele, Introduction to Polymer Physics, Springer Verlag, Berlin M. Doi, Introduction to Polymer Physics, Chapman & Hill, London, A. Baumgärtner, Polymer Mixtures, Lecture Manuscripts of the 33th IFF Winter School on Soft Matter Complex Materials on Mesoscopic Scales, 00.. T. Springer, Entmischungskinetik bei Polymeren, Lecture Manuscripts of the 8th IFF Winter School on Dynamik und Strukturbildung in kondensierter Materie, K. Kehr, Mittlere Feld-Theorie der Polymerlösungen, Schmelzen und Mischungen; Random Phase Approximation, Lecture Manuscripts of the th IFF Winter School on Physik der Polymere, H. Müller-Krumbhaar, Entmischungskinetik eines Polymer-Gemisches, Lecture Manuscripts of the th IFF Winter School on Physik der Polymere, T. Springer, Dichte Polymersysteme und Lösungen, Lecture Manuscripts of the 5th IFF Winter School on Komplexe Systeme zwischen Atom und Festkörper, H. Müller-Krumbhaar, Phasenübergange: Ordnung und Fluktuationen, Lecture Manuscripts of the 5th IFF Winter School on Komplexe Systeme zwischen Atom und Festkörper, 994. phone: +49 (0) 63/ Fax: +49 (0) 63/