flory huggs

8/12/2019 Flory Huggs

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2.5 Thermodynamics of Polymer Solutions (1)

Notation: A = solvent; B = solute (polymer)

in case of copolymers or multi-component systems:1 = solvent; 2,3...polymer

Thermodynamic of low molecular weight solution

(revision):Gibbs free energy (Free Enthalpy): G = f(p,T,n)

dG =

G

TdT

G

pdp

G

ndn

p n T n i p T n

i

i i j

+

+

, , , ,

dG = - S dT + V dp + idni;

p = const; T = const:dG = idni

1.+ 2. law of thermodynamics (isothermal condition, dT = 0):dG = dH T dS + idni

partial molar entropy si: si= -(i/T)p,n

partial molar volume vi: vi= (i/p)T,n

Pressure dependence of chemical potential i:

iid

(p) = iid

(po) + RT ln (p/po); iid

(po) = i,o(standard pot.)

ire

(p) = iid

(po) + RT ln (f/fo) ; f = fugacity

Concentration dependence of chemical potential i:

iid

(p,T,xi) = i*(p,T,xi=0) + RT ln xi

ire

(p,T,xi) = i*(p,T,xi=0) + RT ln ai; ai(activity) = xifi

fiactivity coefficient

ire

(p,T,xi) = iid

(p,T,xi) + iexcess

(p,T,xi)

Entropy of mixing: )Sid = -R niln xi= -R nAln xA R nBln xB;


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Classification of solutions:

ex

sex

h

Ideal solutions

Athermic solutions

Regular solutions

Irregular solutions

= 0

0

0

0

= 0

0

= 0

0

= 0

= 0

0

0

Entropy of mixing: The Flory-Huggins theory (1)

Deviation of polymer solutions from ideal behavior is mainly due to

low mixing entropy. This is the consequence of the range of difference

in molecular dimensions between polymer and solvent.

Flory (1942) and Huggins (1942)

Calculation of Gm= G(A,B) - {G (A) + G (B)}H = 0Gm= -T Sm

Lattice model

volume of solvent molecule: VA;

each solvent molecule occupies

1 lattice cell

NA= number of solvent molecules

volume of macromolecule: VB

each macromolecule occupies

VB/VA= Llattice cells

NB= number of macromolecules

Number of lattice cells: K = NA + L NBCoordination number: z (two-dimensional: z = 4)

VAVB= L VA=10 VA


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Flory-Huggins theory (2)

transfer of the polymer chains from a pure, perfectly ordered stateto a state of disorder mixing process of the flexible chains with solvent molecules

Calculation of the number of possible ways a polymeric chain can be

added to a lattice:

1. Macromolecule

1st Segment K possibilities of arrangement on lattice

2nd

Segment z possibilities of arrangement on lattice

3rd Segment z 1 possibilities of arrangement on lattice

L segments of 1. macromolecule:

1= K z (z 1)L 2

i. Macromolecule

number of vacant cells: K (i - 1)L

probability to find a vacant cell: (K (i-1)L)/K(mean-field theory)

L segments of i. macromolecule:

i = (K (i-1)L z (K (i-1)L)/K {(z-1) (K (i-1)L)/K}L - 2

thermodynamic probability = (NB! 2NB

)-1

i

entropy (Boltzmann): S(NA,NB) = kBln (solvent: only 1 arrangement

Sm= S(NA,NB) - {S(NA) + S(NB)}

Sm= -R (nAln A+ nB ln B)

A= volume fraction solvent = NA/K = nAVA/( nAVA+ nBVB)

B= volume fraction polymer = L NB/K = nBVB/( nAVA+ nBVB)


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Flory-Huggins theory (3); chemical potential

A= RT ln aA = RT (ln A+ (1 - VA/VB) B)

B= RT ln aB = RT (ln B+ (1 VB/VA) A)

A= f (M) !

VA/VB= [BMA/AMO] 1/PVA/VB~ 1/P ~1/MB

Enthalpy change of mixingquasichemical process: (A-A) + (B-B)(A-B)

(A-B): solvent-polymer contact

interchange energy per contact: u = AB= AB (AA+ BB)

U = H if no volume change takes place on mixing

H = q AB; q = number of new contacts

calculation of number of contacts can be estimated from the lattice

model assuming that the probability of having a lattice cell occupied

by a solvent molecule is simply the volume fractionA, by a polymerB.

q = ABz K

H = ABz KAB

with: := z AB/kBT (definition of !)

and K = NLnA/A; R = NLkB

H = RT nAB

= Huggins interaction parameter


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Gibbs enthalpy based on Flory-Huggins theory:

Gm= RT (nAln A+ nB ln B + nAB)

(often in literature Gm/mol (monomer and solvent));

Gm/(nA+ PnB0) = RT (Aln A+ (B/P)ln B + AB)

A= RT (ln A+ (1 - VA/VB) B+ B2)Meaning of

combinatorial

comb

= entropy according F.-H.residualR= difference to the combinatorial solution, and

excessex

= difference to the ideal low-molecular weight solution

term of chemical potential

A= Acomb+ A

R

= AR/RTB

enthalpic and entropic parts of AR:

AR= TsA

R + hA = H+ S

with H= h/RTB; S= sAR/RB

Determination of H and S: H= -T(/T)p

S = d(T)/dT

S= 0 (combinatorial solution; F.-H. equ. valid)= a/T

experiments: = a + b/TS0

in most cases: S, H > 0; S> H;< 0 means: contacts between A and B are preferred (good solution)


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Theta-temperature and Phase separation (1)

phase stability conditions:

temperature (g/T)p < 0pressure (g/p)T < 0

concentration (g/x)p,T > 0binodal curve (local minima):

spinodal curve (reflection point):

Application on Flory-Huggins:

binodal curve

AB p T B

A

BBRT

V

V

=

+

+

,

1

11 2

(1)

spinodal curve( )

2

2 2

1

12

A

B p T B

RT

=

+

,

(2)

critical point: (1); (2) = 0

B cB A

cA

B

A

BV V

V

V

V

V,

/;=

+= + +

1

1

1

2 2

polydispers polymer:

( )( ) B c w z c z w zP PP P P

, / ; / /= + = + +

1

1

1

21 1 1

g

x

g

x

m

p T

m

p T

=

>

, ,

;0 02

2

2

2

3

30 0

g

x

g

x

m

p T

m

p T

=

, ,

;


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Theta-temperature and Phase separation (2)

( )

AR

Acomb

A A A

RT RT RT

h

RT

s

R= + = = = +B H s B

2 2

critical point: T = Tc; = c

(*)

cA

B

A

B

A

c

A

c

V

V

V

2 V

h

R T

M )=1

hR T M )

= + + = +

= +

1

2

2

12

2

2

B

s

c

B

s

(

(

s

1

2

(*)

1

2

1

22+ + = +

V

V

V

2 V

h

R T

A

B

A

B

A

c

B

h R T M )A c= B2 (

R T (M

R T

c B

c B

A

B

A

B

V

V

V

V

= + +

+)

2

2 2

1

T

1

T M

1

T Mc c c=

+

+

( ) ( )

V

V

V

V

A

B

A

B2

T M Tc ( ) =

Theta-temperature


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Second virial coefficient and

A= Aid+ A

ex

Osmosis: A= -VAreal solution, virial expression:

/cB= RT (1/M + A2cB+ A3cB+ )

A= f(cB)

expanding ln A= ln (1 - B) as far as the second term in a Taylorseries, B = cB/B

A2= (1/2 - )/(B2VA)

Aex

= - RTA2cBVA= - RT(1/2 - )BBVA/(B2VA)

Aex= - RT(1/2 - )B

Second virial coefficient and Thetatemperatur

A2= (1/2 - )/(B2VA);

= H+S= T/T + S= T/T + -

A2= ( 1 - T/T) /(B2VA)

T = T: A2= 0

pseudo-ideal solution


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FloryKrigbaum theory

to overcome the limitations of the lattice theory resulting from the

discontinuous nature of a dilute polymer solution

solution is composed of areas

containing polymer which were

separated by the solvent

Polymer areas: Polymer segments

with a Gaussian distribution about the center of mass

chain segments occupy a finite volume from which all other chainsegments are excluded (long range interaction)

see Excluded Volume Theory

Introduction of two parameters

enthalpy parameter

entropy parameter

to describe long range interaction effects:

A= Aid+ A

ex;

Aex

= - RT(1/2 - ) B = hex

-T sex

hex= RT B ; sex = R B

(1/2 - ) = (- )Theta condition A

ex= 0

= hex=T sex;

T:= T (/) ; Aex= - RT( 1 - T/T) B

Deviations from ideal (pseudoideal) behavior vanish

when T = T!


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2.5 Thermodynamic of Polymer Solution (2)

Solubility Parameter

The strength of the intermolecular forces between the polymermolecules is equal to the cohesive energy density (CED),

which is the molar energy of vaporization per unit volume.

Since intermolecular interactions of solvent and solute must be

overcome when a solute dissolves, CED values may be used to

predict solubility.

1926, Hildebrand showed a relationship between solubilityand the internal pressure of the solvent;

1931, Scatchard incorporated the CED concept intoHildebrands eq.

= HV2 (nonpolar solvent; )H= heat of vaporization)heat of mixing: Hm= VAB(A- B) = nAVA(A- B)

Acc. solubility parameter concept any nonpolar polymer will

dissolve in a liquid or a mixture of liquids having a solubility

parameter that does not differ by more than 1.8 (cal cm-3)0.5.Small:

F = Fi; Fi= molar attraction constant [in (Jcm)1/2mol-2]

-CH3 438 -CH2- 272 CH- 57

=C= -190 -O- 143 -CH(CH3)- 495

-HC=CH 454 -COO 634 -CO- 563

Like dissolves like is not a quantitative expression!

Problems: polymers with high crystallinity;polar polymers hydrogen-bonded solvents or polymers

additional terms

= = =E

VE F V V

Mcoh

B o

cohB o B o

o

B amorph,, ,

,

; / ;2


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The square root of cohesive energy density is called

solubility parameter. It is widely used for correlating

polymer solvent interactions. For the solubility of polymer Pin solvent S ( P- S) has to be small!


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Excluded-Volume-Effect

Dilute gas of random flight chains:

it is physically impossible to occupy the same volume elementin space at the same time

the conformations in which any pair of beads

(segments) overlap were avoided

when a pair of beads come close they exert a repulsion

force F on each other

Dilute solution of random flight chains:

The force that acts between a pair of beads becomes no longer

equal to F.interaction bead-solvent > interaction bead-bead => good solvent

interaction bead-solvent < interaction bead-bead => poor solvent

=> solvent-bead (segment) interactions: F

good solvent: F repulsive

bad solvent: F attractiveThe term excluded volume-effect is used to describe any effect

arising from intrachain or interchain segment-segment

interaction.

Excluded-volume of two hard spheres:

( ) = = =4

32R

4

3R Vsphere

3 38 8

excluded volume

Second virial coefficient A2and the

excluded volume:

A2M

R ~ M2 22

32

32=

Nhard sphereL

; : ~

A2~ M-1/2


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Excluded Volume Theory

volume of segments interaction between segments (repulsion forces)excluded volume depends on space-filling effects and interaction

forcesshort range, long range interactions

Problems:

Calculation of excluded volume in dependence on molecular

properties;

relation between interaction (A2) and excluded volume.

Excluded volume and lattice theory:

number of possibilities, that the molecule mass center is in the volume

V, excluded volume/molecule , proportionality constant k:

1.molecule: 1= k V2.molecule: 2= k (V - )i. molecule: i= k [V (i 1)]

A= - RT VAcB[1/MB+ ((NL)/(2MB))cB]

Qualitative Discussion:

excluded

volume

r

hard sphere < 0; A2< 0

= 0; A2= 0

> 0; A2> 0


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Scaling Law

o= nsls

eq. tell us, how o "scales" with ns

Global (universal) Propertiesproperties of polymer chains, which do not depend on

local properties (independent of the monomer structure, nature

of solvent, etc.)=> very large characteristic lengths

=> small frequenciesIt has been found that in the appropriate variables all

macroscopic polymer properties can be plotted on universal

curves (power laws, characteristic exponents).

The Blob-chain, () of a labeled chain

the labeled chain is made

of n/g blobs each of length (screening length) containing gsegments

a blob is an effective stepalong the contour of the chain

contains g segments

we assume:

- the segments inside the blob

obey the excluded volume

chain statistics, ~ g3/5

- the n/g blobs obey the random walk statistics such that = (n/g) is the distance up to which the native self-avoidance due to theexcluded volume interaction is completely correlated and beyond

which it is totally uncorrelated; since g ~ 5/3 ~ (n/g) ~ n 1/3 ~ n -1/4(see = f (c))


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Scaling Laws for polymer solutions(good solvent at nonzero concentrations)

We are in search of a dimensionless variable in order to applythe scaling method.

fundamental concentration to make the polymerconcentration dimensionless:

We introduce a reduced concentration (/*)with: = segment concentration (number of

segments/volume); N chains with nssegments* = overlap concentration~ ( N ns)/( N RF) ~ ns

(1-3)~ ns-4/5; (RF~ ns

3/5)

scaling laws: concentration dependence of (Radius of gyration)

= *: 1/2= RF

solid amorphous polymer, > *: ~ ns ~ RF (/*)

x

= *: ~ RF (/*)x~ RF

> *: ~ RF (/*)x~ ns

since: *~ ns-4/5; RF ~ns

3/5; x = - *)

1/2~ -1/4 ~ cB-1/4

concentration dependence of the screening length= *: = RF> *: ~ ns

0(no molar mass dependence)

~ RF(/*)y

> *: ~ RF(/*)y~ ns

0

since: *~ ns-4/5; RF ~ns

3/5 y = - 3/4

~ -3/4 ~ cB-3/4

*) ~ ns~ xns(6/5+4x/5); 6/5 + 4x/5 = 1 ; x = -1/4


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osmotic pressuredilute solution:

R T k T nB s

= =c

M

> *: /cBshow no molar mass dependence

k T nB s~

*

z

> *:

k T

n

B

s0~ ; *~ n

s

-4/5 z = 5/4

~ ~94 ns

0

experiments, poly--methyl-styrene in toluene, different molar mass

flory huggs

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