Polymer Solutions and Mixtures

Home , Entropy of mixing, Miscibility gap, Osmotic pressure

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

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 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

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

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!

Osmotic pressure & membrane osmometry

polymer solution pure solvent

nnsp,,, pT npTs′,,′

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

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

Dependance of osmotic pressure on polymer concentration and solvent quality

Πv s A2 cRTp χ < 1 A > 0 122 () 2 good solvent

χ = 1 A = 0 θ solvent 122 () 2 1 χ ><1 A 0 N 122 () 2 bad solvent

c p

Question: How can we determine N and χ ?

Πvs Πvs ⎡ 11⎛⎞ ⎤ Answer: plot versus cp ! = ⎢ +−⎜⎟χ cp +...⎥ cRTp cRTp ⎣ N ⎝⎠2 ⎦ 1 ⇒ axis intercept = N 1 ⇒ slope = − χ = 2nd virial coefficient A2 2

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.

⎡⎤11⎛⎞ 1 2 ρχsppghΔ=Π= RT⎢⎥ c +⎜⎟ − c +... ⎣⎦Nvs ⎝⎠2

with ρs = solvent density; g = acceleration of gravity; Δh = excess column height; vs = molar volume solvent

Δh

μμ('p +Π ,)TpT = 0 (',) SS 0 μS μS

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]

⎧⎫1 2 −ΠvRTcs =⎨⎬lnspp + (1 − ) c + χ c ⎩⎭N for very (!) dilute solutions: Π=−vRTcRTcln =− ln(1 − ) s spi∑i , nn with c =≈∑∑iipi,, pi ∑i pi, nn+ n spis∑i ,

()× RT ⎛⎞nnpi,, pi with (*) ⇒ Π=−ln 1 −∑∑ii ≈RT ⎜⎟ vnss⎝⎠ vn ss ()× total concentration of solute vvn=+ n ≈ vn s spiss∑i ,

nM⎛⎞ nM⎛⎞ n n ∑∑∑∑iiiipi,,,, i pi i pi pi weight cM==⎜⎟⎜⎟ = notice: concentration in vnvvn⎜⎟⎜⎟n volume ⎝⎠∑i pi, ⎝⎠ s s 1442 443 =M n ⎛⎞nn ∑iipi,,∑ pi with ()×⇒nn << :ln1⎜⎟ − ≈− ∑i pi, s ⎜⎟ ⎝⎠nnss

c ⇒ Π=RT M n

⇒ Answer: Membrane osmometry determines the number averaged molecular weight! (in very dilute solutions)

Phase diagrams for polymer solutions and –mixtures

Phase diagrams (the basics)

Φ Φ α Φβ

Assume a mixture with composition Φ separates into two phases with composition Φαand Φβ with volume fractions α and β, respectively. Φ =Φ+Φα αββαβwith += 1

The original phase is stable, if

αβ Δ≤ΔΦ+−ΔΦ=ΔGGmixαα mix()(1)αβ G mix () G mix Gibbs stability criterion

free enthalpy of mixing as a function of volume fraction of one component

more complex situation: convex and concave curve regions stable, metastable and unstable regions

ΔGmix at fixed temperature metastable metastable

unstable

stable stable

spinodal

binodal

(1) The binodal defines the boundary between the one-and two-phase region and can be determined by tangent construction

⎛⎞∂ΔGmix ⎜⎟= 0 (minima) ⎝⎠∂Φ Tp,

(2) The spinodal defines the boundary between the metastable and unstable region and can be determined from the points of inflection

2 ⎛⎞∂ΔGmix ⎜⎟2 = 0 ⎝⎠∂Φ Tp, unstable: small fluctuations in concentration lead to spontaneous demixing metastable: 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 Tc and a composition Φc where spinodal and binodal merge called critical point 3 ⎛⎞∂ΔGmix ⎜⎟3 = 0 (extremum of the spinodal curve) ⎝⎠∂Φ Tp,

Construction of a phase diagram

T 5 T c T 3 T 2

free energy T1

Å good solvent

bad solvent

T temperature 1

concentration

(1) Determination of the free enthalpy as a function of Φ at various temperatures (2) Determination of stable and unstable regions and projection into the (,T Φ )space

M ⎧⎫ΦΦ12 Flory Δ=GRTmix ⎨⎬ln Φ+Φ+ΦΦ<121212 lnχ 0 as condition for miscibility ⎩⎭NN12 1 χ ∝ 12 T

⇒ In Flory-theory always lower miscibility gap and upper critical point (miscibility at high temperatures)

Polymer solution

N1=N ; N2=1 ; Φ=Φ12;(1) Φ= −Φ

∂ ΔGM 11 (1) binodal: mix =Φ+−−Φ−+−Φ=ln ln(1 ) 1 (1 2 )χ 0 ∂Φ RT N N ∂2 ΔGM 11 (2) spinodal: mix =+ −=20χ ∂Φ2 RT N Φ(1 − Φ ) 3 M ∂ ΔGmix 11 (3) critical point: 322=− + =0 ∂ΦRT N Φcc(1 − Φ )

1 1 − ⇒ Φ= ≈N 2 for large N>>1 c N +1

insertion of Φc into (2):

∂2 ΔGM 11 mix ()0Φ= ⇒ + − 2χ = 0 ∂Φ2 RTcc N Φ(1 − Φ ) cc 1 − 11211 11⎛⎞ 1 12 1− − ⇒=χc (1)21 +=⎜⎟ + +=++NNT ∝c 22NN⎝⎠N 22

⇒ Asymmetric miscibility gap which shifts to very small polymer contents for high molecular weight polymers.

⇒ Tc increases with molecular weight of polymer

Binodal experimentally determined from precipitation temperature

M = w • No good description of experimental data! binodal (Flory) • The dependence of χ on Φ binodal has to be taken into account; (experimental) also bad description of T dependence!

spinodal (Flory)

volume fraction polymer

Polymer mixtures

Φ=Φ12;(1) Φ= −Φ

M ⎧⎫ΦΦ12 Δ=GRTmix ⎨⎬ln Φ+Φ+ΦΦ<121212 lnχ 0 as condition for miscibility ⎩⎭NN12 ∂−Φ−ΔG M 11ln(1)1 (1) binodal: mix =Φ+−ln +−Φ= (1 2 )χ 0 ∂Φ RT N112 N N 2 M ∂ ΔGmix 11 (2) spinodal: 2 =+ −=20χ ∂ΦRT N12 Φ N (1 − Φ ) 3 M ∂ ΔGmix 11 1 (3) critical point: 322=− + =0 ⇒ Φc = ∂ΦRT N12 Φcc N (1 − Φ ) NN12+1 1 2 with (2) ⇒ χ =+∝11NNT−1 cc2 ()12 ⇒ Tc increases with increasing N1 or N2

⇒ N1 << N2 or N2 << N1 ⇒ Φc shifts towards very low concentrations of the longer component.

Special case: Polymer with the same degree of polymerization: N1 = N2 = N

−1 ⇒Φcc =1 2 and χc = 2 NT ∝ ⇒ ()2χc N = critical point

(χc N )< 2 one-phase region

(χc N )> 2 two-phase region

⇒ Phase diagram is symmetrical at Φc for polymer mixtures with identical degree of Polymerization

Mixture of symmetric polymers

ΔGmix χN=2. χ

two-phase

(χN)c=2 temperatur

1. one-phase

Φ Φ

In summary

• Connectivity of polymers ⇒ degree of possible disorder is limited ⇒ entropy of mixing is reduced by prefactor 1 ⇒ monomer-monomer interaction determines mixing behavior. N

• As higher the molecular weight as higher the critical temperature; weak repulsive interaction between the monomers leads to demixing.

• 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

(1) 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 Δw12 = 0 ⇒ fluctuations in particular relevant in dilute solutions are neglected

(2) Volume changes due to mixing 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, donor- acceptor interaction, solvent orientation in polar solvent close to the chain.

Ansatz: χ parameter comprises on enthalpic χH and entropic χS contribution B χχ=+SH χ =+A kTB ⎛⎞dBχ dT()χ Δ S with χH =−T ⎜⎟ = and χS ===−A ⎝⎠dT kB T dT kB

from experiments ⇒ major contribution from non-combinatorial change in entropy: χS

UCST = upper critical solution temperature χ decreases with increasing temperature e.g. polybutadiene (88% vinyl, protonated; 78% vinyl, deuterated)

mixing 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

⇒ χS is negative and decreases with T , i.e. increases with 1/T

LCST = lower critical solution temperature χ increases with increasing temperature e.g. polyisobutylene (protonated); head-to-head polypropylene (deuterated)

Mechanism:

(1) 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 ⇒

χs is positive and increases with temperature/ decreases with 1/T

(2) 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

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 dM∂ΔGmix ⎡ ⎛⎞1 2 ⎤ Flory-Huggins: Δ=GRTccc =Δ=μχSS⎢ln +−⎜⎟ 1 pp + ⎥ ∂nNS ⎣ ⎝⎠ ⎦

1 2 with cp << 1 ⇒=−≈−−+lncccc ln(1 ) ... as for osmotic pressure Sppp2

dM cp ⎛⎞1 2 ⇒ΔGRTc =Δ=−μχSp[ +⎜⎟ − + ...] N ⎝⎠2 142 43 deviations from ideal behavior

d 2 Flory-Krigbaum: dilution enthalpy: Δ=HRTHcni p d 2 2 dilution entropy: ΔSRcni=Ψ p (cp due to pair contacts)

d 2 ⇒ free enthalpy of dilution: Δ=GRTHcni() −Ψ p ⎛⎞1 ⇒ comparison Flory-Huggins and Flory-Krigbaum theory: −−Ψ=−()H ⎜⎟χ ⎝⎠2 1 θ conditions: χ =⇔=ΨH ; i.e. ΔHTSdd=Δ 2 ni ni H define: θ = T (only defined if H and Ψ have equal sign) Ψ non-ideal behavior disappears for T = θ !

1 ⎛⎞θ nd replace −=Ψ−χ ⎜⎟1 = 2 virial coefficient A2 ! 2 ⎝⎠T

dM ⎡ cp ⎛⎞θ 2 ⎤ ⇒ΔGRTc =ΔμSp =−⎢ +Ψ⎜⎟1 − + ...⎥ ⎣ NT⎝⎠ ⎦

Determination of the θ temperature in polymer solutions

(A) From the determination of the critical phase separation temperature Tc as a function of the degree of polymerization

111111⎛⎞θ −−12 1 ⎛ −− 12 1 ⎞ −=Ψ−=χc ⎜⎟1 NN − ⇒ = ⎜ NN + ⎟ + 22⎝⎠TTc Ψθ ⎝ 2 ⎠θ 1 ⇒ θ can be determined from axis intersection N →∞ and NN−−12+ 1 → 0 2 ⇒ Ψ from slope

⎛⎞θ (B) From A2 measurement (osmotic pressure) as a function of the temperature A2 =Ψ⎜⎟1 − ⎝⎠T intersection with T axis (A2=0) ⇒=T θ

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:

(1) binodal (metastable region)

segregation by nucleation

(2) spinodal (unstable region)

spinodal segregation

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

(1) spinodal decomposition results in interpenetrating, continuous structure with specific length scale. (2) binodal decomposition by nucleation, spherical precipitates growthing in time.

Spontaneous local concentration fluctuation

δΦ in volume dr3 and −Φδ in adjacent volume dr3

⇒ change of free enthalpy in terms of free enthalpy g per unit volume

1 δδδGg=Φ+Φ+Φ−Φ−Φ[( ) g ( )] drgdr33 ( ) 2 00 0 series expansion of g up to second order in Φ yields

111∂2 g ∂Δ2 G δδGdrdr=ΦΦ=()23mix () ΦΦ δ 23 22∂Φ2200v ∂Φ

∂Δ2 G ⇒ depending on the sign of the curvature mix a small fluctuation in concentration leads to ∂Φ2 an increase or decrease of the free enthalpy (Gibbs stability criterion)

2 ⎛⎞∂ΔGmix (1) ⎜⎟2 >>0 (convex) δG 0 phase is stable against small pertubation; ⎝⎠∂Φ amplitude of small pertubation decays 2 ⎛⎞∂ΔGmix (2) ⎜⎟2 <<0 (concave) δG 0 phase is unstable against small pertubation; ⎝⎠∂Φ 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=−Δ+Gr( )4π r32 g 4πσ r with Δ=Φ−Φ g g ( ) g ( ) (*) 3 123 02 14243 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

⎛⎞ΔGb ⇒ nucleation rate increases with Arrhenius law: ν nucleus ∝−exp⎜⎟ ⎝⎠kTB 16πσ3 with the barrier height Δ=G which is the maximum of (*) ΔGr() b 3(Δg )2

⇒ energy barrier ΔGb increases with decreasing distance from the binodal (approaching

from spinodal); at the binodal Δg = 0 ⇒ΔGb →∞

⇒ mixture 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 Φ1 and Φ2

Spinodal Decomposition

∂Δ2 G • at spinodal mix →⇒0 spontaneous demixing; no activation energy ∂Φ 2 • no sharp transition from phase separation by nucleation to spontaneous demixing since ∂Δ2 G mix vanishes continuously ∂Φ2

Kinetics of spinodal decomposition in polymer mixtures

• local chemical potential difference caused by concentration fluctuations ∂ΔG Δ=μμ − μ0 =mix −2 κ ∇Φ2 ss ∂Φ 14243 123 Term II Term I

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:

2 ⎡∂ΔGmix 3 ⎤ −=Ω∇Δ=Ωj ()μκ⎢ 2 ∇Φ−∇Φ 2 ⎥ ⎣ ∂Φ ⎦ with Ω =Onsager-type phenomenological coefficient

• substitution into equation of continuity which equates the net flux over a surface with the loss

2 ∂Φ ⎡∂ΔGmix 24⎤ or gain of material within the surface = −∇ ⋅j = Ω⎢ 2 ∇ Φ −2κ ∇ Φ⎥ ∂∂Φt ⎣ ⎦ Cahn-Hillard relation

• comparison to conventional diffusion equation

∂Φ =∇ΦD 2 (D=diffusion coefficient) ∂t ∂Δ2 G ⇒≈ΩD mix <0 in unstable region ∂Φ2

diffusion against concentration gradient at spinodal (TT→ Sp ) D → 0 “critical slowing down” general solution for Cahn-Hilliard relation:

Φ−Φ0 =∑exp( Γ (β ) ⋅tA ) ⋅ [ (ββ )cos( xB ) + ( ββ )sin( x )] β 2π ⎡∂Δ2 G ⎤ with λ = and Γ=−Ω()β βκβ22⎢ mix − 2 ⎥ β ∂Φ2 { ⎣⎢ >0 ⎦⎥

∂Δ2 G stable region: mix >⇒Γ<⇒0()0β instabilities rapidly decay ∂Φ 2

∂Δ2 G unstable region: mix <⇒Γ>⇒0()0β some wavelengths notably the long ∂Φ 2 wavelengths (small β ) will grow

Spinodal segregation

(1) early stages: amplitudes of fluctuations growth for preferred wavelength λ = (length scale)−1 ; modes are independent. (2) intermediate stages: amplitudes growth, modes interact with each other; wavelength growth and structure coarsens; deviations from cosine wave form.

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. ⇒

(1) interwoven structure coarsens by an interfacial-energy driven viscous flow mechanism (2) tendency to spherical precipitates

4.6 Summary

Miscibility of mixtures if Δ=Δ−Δ

Flory-Huggins Theory (mean field theory) considers a random distribution of polymer monomers & solvent molecules on a lattice

calculation of ΔH mix in terms of pair interaction energies; every molecule or monomer sees the same average surrounding.

calculation of ΔSmix 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 ⎡ cc12 ⎤ Flory-Huggins Theory Δ=GRTcmix ⎢ ln111212 + ln c +χ cc⎥ ⎣ NN12 ⎦ zwΔ 1 χ =∝12 T −1 with Δ=ww −() ww + 12 RT 12 122 11 22 for polymers with high degree of polymerization the entropy gain due to mixing very is small (reduced by factor 1/N)

for most substances endothermal mixing χ12 > 0

⇒ small enthalpic contribution leads to demixing!

−1 in Flory-Huggins theory χ12 ∝ T

⇒ mixing at high temperatures & demixing at low temperatures

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 1 2 Π=VRTsP +( −χ ) c + ... ⎣⎦⎢⎥N 2 1 χ < good solvent - high temperatures - polymer swells 2 1 χ > bad solvent - low temperatures - polymer shrinks 2 1 χ = θ solvent 2

Membrane osmometry (measures excess column height) hydrostatic pressure = osmotic pressure dilute solutions:

ΠV ⎡⎤11 s ( )c ... =+−+⎢⎥χ P RTcP ⎣⎦ N 2 in very dilute solution: the number averaged molecular weight is measured

Phase diagrams

αβ Gibbs stability criterion: Δ≤αΔΦ+−αΔΦ=ΔGG()(1)G()Gmix mixαβ mix mix

ΔGmix metastable metastable concave convex convex

unstable at fixed temperature

stable stable

spinodal

binodal

Construction of phase diagrams

binodal (metastable region) T > ⎛⎞∂ΔGmix T ⎜⎟= 0 > ⎝⎠∂Φ T,p T > T > spinodal (unstable region) free enthalpy

Å 2 ⎛⎞∂ΔGmix ⎜⎟2 = 0 ⎝⎠∂Φ Tp, good

Å solvent

T critical point

χ bad 3 solvent ⎛⎞∂ΔGmix ⎜⎟3 = 0 ⎝⎠∂Φ Tp, T1 temperature

concentration Φ binodal and spinodal merge; determined by (Tc,Φc)

1 Flory-Huggins Theory : χ ∝ always lower miscibility gap & upper critical point T

Polymer solution

Mw= 1 Φ= ≈N −1/2 C N +1

11 χ =+NNT−−−1/2 + 1 ∝ 1 CC22

binodal miscibility gap is highly (experimental) asymmetric, shifts to very small polymer concentrations for high N

Tc increases with N binodal (Flory)

volume fraction polymer spinodal

Polymer mixtures

1 Φ= C N χN 1 +1 N2

1 −1/2−− 1/22 1 χCC=+∝()NN12 T 2

two-phase region • for N1<

temperature of the shorter component.

• TC increases with N1 or N2 one-phase region • miscibility gap is symmetric at critical point for N1=N2 Φ Φ=1 C 2

χ =∝2 T −1 CCN

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 mixing resulting from volume changes due to mixing.

Ansatz: : χ=χS + χH =A(T)+B/(kBT)

(1) mixing leads to increase in volume - gain of entropy due to higher chain mobility - χS is negative and decreases with temperature ⇒ UCST- Lower miscibility gap (2) mixing leads to decrease in volume - loss of entropy due to higher chain mobility - χS is positive and increases with temperature ⇒ LCST- Upper miscibility gap

Combinations of enthalpic contribution and (2) 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.

Phase diagram with lower and upper miscibility gap

Flory-Huggins enthalpy of mixing:

M ⎡⎤cc12 ΔGRTcmix =++<⎢⎥ln111212 ln cχ cc 0 ⎣⎦NN12

two-phase region (LCST) one-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 ∂ΔGmix M22⎛⎞θ =ΔμκSPP =−RT() Ψ− c =−Ψ⎜⎟ 1 − c ∂nTS ⎝⎠

2nd virial coefficient A2 ~ (1/2-χ)= ψ(1-θ/T)

• θ temperature can be determined from A2 measurements • θ temperature and entropy parameter ψ can be determined from molecular weight dependency on Tc

Phase separation mechanisms

cooling from one-phase region into:

(1) binodal (metastable region)

segregation by nucleation

results in spherical precipitates growthing with time.

(2) 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 Φ1 and Φ2

Spinodal segregation

• a small concentration fluctuation growth.

• diffusion against concentration gradient at spinodal (TT→ Sp ) D → 0 “critical slowing down”

Stages

(3) late stages: fluctuation reaches the upper and lower limit Φ12and 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. (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

Literature

1. U.W. Gedde, “Polymer Physics”, Chapman & Hall, London, 1995. 2. G. Strobl, “The Physics of Polymers”, Springer-Verlag, Berlin, 1996. 3. M.D. Lechner, K. Gehrke, E.H. Nordmeier, “ Makromolekulare Chemie”, Birkhäuser Verlag, Basel, 1993. 4. J.M.G. Cowie, “Polymers: Chemistry & Physics of Modern Materials”, Blackie Academic & Professional, London, 1991. 5. K. Kamide, T. Dobashi, “Physical Chemistry of Polymer Solutions”, Elsevier, Amsterdam, 2000. 6. O. Olabisi, L.M. Robeson, M.T. Shaw, “Polymer-Polymer Miscibility”, Academic Press, San Diego, 1997. 7. M. Rubinstein, R.H. Colby, “Polymer Physics“, Oxford University Press, New York, 2003. 8. U, Eisele, “Introduction to Polymer Physics”, Springer Verlag, Berlin. 1990. 9. M. Doi, “Introduction to Polymer Physics”, Chapman & Hill, London, 1995. 10. A. Baumgärtner, “ Polymer Mixtures”, Lecture Manuscripts of the 33th IFF Winter School on “ Soft Matter – Complex Materials on Mesoscopic Scales”, 2002. 11. T. Springer, “ Entmischungskinetik bei Polymeren”, Lecture Manuscripts of the 28th IFF Winter School on “ Dynamik und Strukturbildung in kondensierter Materie”, 1997. 12. K. Kehr, “ Mittlere Feld-Theorie der Polymerlösungen, Schmelzen und Mischungen; Random Phase Approximation”, Lecture Manuscripts of the 22th IFF Winter School on “ Physik der Polymere”, 1991. 13. H. Müller-Krumbhaar, “ Entmischungskinetik eines Polymer-Gemisches”, Lecture Manuscripts of the 22th IFF Winter School on “ Physik der Polymere”, 1991. 14. T. Springer, “ Dichte Polymersysteme und Lösungen”, Lecture Manuscripts of the 25th IFF Winter School on “ Komplexe Systeme zwischen Atom und Festkörper”, 1994. 15. H. Müller-Krumbhaar, “ Phasenübergange: Ordnung und Fluktuationen ”, Lecture Manuscripts of the 25th IFF Winter School on “ Komplexe Systeme zwischen Atom und Festkörper”, 1994.