ORGANIC AND PHYSICAL CHEMISTRY OF POLYMERS phần 2 ppt

The same expression can be rewritten as a function of the volume fractions of thetwo components, with 1 being the volume fraction of the solvent and 2 beingthat of the polymer: 1= n1 n t

The same expression can be rewritten as a function of the volume fractions of the

two components, with 1 being the volume fraction of the solvent and 2 beingthat of the polymer:

1= n1

n t and 2= n2X

n t which results in a very simple expression for
Smix:

Smix= −k(n1ln 1 + n2ln 2 ) Substituting in this expression of
Smix the number of moles for the number of

molecules gives (with R = kNa, R being the gas constant)

appli-• When molecules interact strongly, the assumption of a random placement ofthe components in the solution is not realistic Strong interactions induce ashort-range order, which leads to a lower entropy of mixing;

• In dilute solutions, polymers are subjected to excluded volume Excluded

vol-ume prohibits the access of any other homolog into the vicinity of a chainsegment and thus causes a lower entropy of mixing;

• At high temperature, the density of a mixture can considerably decrease andthe contribution of the “free volume” to the entropy of mixing cannot beneglected

4.2.2 Enthalpy and Free Energy of Mixing

Macromolecular solutions deviate generally from ideality and are characterized

by a nonzero enthalpy of mixing In the Flory– Huggins theory the calculation of

Hmix is inspired by that of the enthalpy of mixing of regular solutions Threetypes of interactions between nearest neighbor pairs of molecules are considered:solvent–solvent, solvent–monomeric segment, and segment–segment interactionscharacterized byε11,ε12,ε22;εij corresponds to the potential energy of an i j pair

contact or the energy to dissociate it The proportion of the various interactionsdepends on the relative proportions of solvent and solute The enthalpy of mixing

of such a system can be calculated starting from the relation:

Hmix= H − (H1+ H2) where H , H1, and H2 are the energies of the interactions which develop within themixture and in the pure components (solvent and polymer), respectively

The energy required to break the n1/2 solvent–solvent interactions that occur in

a lattice exclusively constituted of n1 molecules of solvent is equal to

In the case of a binary mixture, each solvent molecule is surrounded by

zn1/(n1+ n2X ) molecules of solvent and zn2X /(n1+ n2X ) repetitive units The

energy corresponding to the interactions of the n1 solvent molecules involved insolvent–solvent and solvent–segment contacts is given by

and the enthalpy of mixing is established as follows:

ε12= ε12−ε11

2 +ε222

Strong interactions result in negative values of ε If interactions of 1–1 and 2–2

types are stronger than 1–2 type,
ε12 and
Hmix are positive and the mixture isthen endothermic The mixing will be exothermic in the opposite case

Hmixcan also be written as a function of the volume fraction of polymer (2):

zn12
ε12

This corresponds to
Hmix per unit of volume; to obtain the molar enthalpy of

mixing, it must be multiplied by V1, the molar volume Defining χ12= z
ε12V1

RT ,the expression for the enthalpy of mixing becomes

4.2.3 Miscibility Conditions and Phase Separation

In the case of an athermic solution, the replacement of a contact between similar

species by a “hetero-contact” (
ε12) between solvent and repetitive unit (segment)

does not cause any modification of
Gmix, since, in this case, χ12 is 0 Exceptfor some rare cases of athermic solutions, solvent–polymer mixtures are gener-ally endothermic, characterized by a positive enthalpy of mixing The interactionsdeveloped within such solutions are intermolecular repulsive forces of the van derWaals type.

In the case of nonpolar polymer–solvent mixtures, which are the only onesconsidered here, these interactions are indeed controlled by the polarizability ofthe components and are described by the relation

εij = −3/2[I i I jαiαj /(I i + I j ) ]r−6where Ii and Ij are the ionization potentials and αi, αj are the polarizabilities of

components i and j Hence, the interaction parameter that reflects the whole of

these interactions can be written as

χ12= A(α1α2)2where A is a constant and indices 1 and 2 correspond to solvent and polymer,

respectively This expression, which is established considering only London-typevan der Waals interactions, shows that interactions between dissimilar units arenecessarily repulsive (or zero); hence,χ12should be positive

Even in the case of toluene –polystyrene solutions,χ12is in the range 0.3–0.4.Thus a positive enthalpy of mixing tends to oppose the polymer dissolution in a

solvent (
G > 0) In the Flory– Huggins model, two components can mix with

each other only if the positive enthalpy term is compensated by the entropy term(−T
S ), which is always negative.

In the particular case of specific interactions of higher energy—such as hydrogenbonding—the interaction parameter can take negative values Solvent and polymerare then miscible in all proportions, but the Flory– Huggins theory does not accountfor this case, which implies a completely different calculation of the entropy ofmixing

Hence, the interaction parameter χ12 is a measure of the quality of a solvent,and its knowledge is essential to the prediction of the domains of concentrationcorresponding either to the miscibility or to the phase separation of the components

As a matter of fact, it is possible, using the Flory– Huggins theory, to delimit thesedomains as a function of the concentration of the species and of the interactionparameter

Figure 4.2 shows the variation of the free energy of mixing (
Gmix) as a

func-tion of the volume fracfunc-tion of component 2 for various values of the interacfunc-tionparameterχ12 A symmetrical variation of
Gmixis observed when the components

1 and 2 have the same size

The situation for a solution containing a polymer with a degree of polymerization

X is different The variation of
Gmixbecomes strongly asymmetrical in this case.The effect ofχ12can be seen in the form of the curves depicting the variation of

Gmixwith 2(Figure 4.2) When concave, these curves indicate a total miscibility

of the two components This occurs for χ values lower than 0.5 When χ is

Figure 4.2 Variation of the average free energy of mixing (in RT units) as the function of volume

fraction of the aqueous solution for various values of χ 12and the degree of polymerization X.

higher than 0.5, these curves show a maximum and two minima and therefore twoinflection points, characteristic of the presence of two phases in equilibrium

These inflection points, also called spinodal points, correspond to

∂2(
Gmix)/∂2 = 0They define the thermodynamic limits of metastability For concentrations corre-sponding to the spinodal points, the system is unstable and demixes spontaneouslyinto two distinct continuous phases which form an “interpenetrating system.” This

type of phase separation characteristic of spinodal regions, is also called spinodal

decomposition.

As for the minima, they are called binodal points and a common tangent line

passes through them The chemical potential of a component at these binodal points

is the same in each of the two phases in equilibrium (called prime and double

µ = µ − µ◦= µ− µ◦ =
µ

The chemical potential (µi) of a component i is by definition the variation of the free energy of mixing
Gmix resulting from the introduction of N moles of i :

Gmix thus varies linearly with 2 with a slope equal to (
µ2−
µ1), and the

chemical potentials are given by the intercepts of the function
Gmix= f (2) for

2→ 0 and 2→ 1; this function also corresponds to the tangent (P) to the curve shown in Figure 4.2 for a given composition 2.

When a polymer of degree of polymerization X is one of the two components,

The slope of the common tangent that passes through the two binodal points is

equal to (
µ1−
µ2/X) , and its intercept for 2 = 0 is equal to
µ1 Insofar asthe two binodal points possess a common tangent, the chemical potentials of the

two components are identical in both phases for p and pcompositions

As for the compositions located between the spinodal and binodal points, thefree energy of mixing of the corresponding systems, albeit negative, is higher thanthose of bimodal compositions These systems will thus demix into two phases withcompositions equal to those of the binodal points in order to minimize their freeenergy Indeed, even a negative energy of mixing is not necessarily synonymouswith miscibility: should a lower free energy be accessible, a system will tend to iteven if it requires that it demixes into two phases

To summarize, three areas can be distinguished at a given temperature:

• Between 2= 0 or 1 and the binodal points, a system forms homogeneoussolutions and only one stable phase

• Between the binodal points, two phases coexist whose composition is given bythe contact points of the tangent to the curve; the regions between binodal and

spinodal points form metastable solutions; in this case, the phase separation

is kinetically controlled by the nucleation and the growth of nuclei leading tothe dispersion of one phase into the other

• The regions between the spinodal points lead to unstable solutions that demixspontaneously into two phases

S 1

Figure 4.3 1: Phase diagram of a macromolecular solution whose phase separation occurs

through a decrease of temperature (UCST) 2: Variation of the average free energy of mixing

as a function of the volume fraction of the solute: (a) formation of a homogeneous solution at

T2; (b) demixing in two phases for compositions between pand pat T1 S, M, I indicate the regions of stability, metastability, and instability, respectively.

Curves of binodal and spinodal points can be drawn as a function of the

temper-ature up to the critical tempertemper-ature (Tc) (T2 in Figure 4.3) where these two curves

meet Beyond Tc, the system forms only one phase At this critical temperature, the partial first- and second-order derivatives of the chemical potential (
µ1) areequal to zero; the chemical potential is the derivative of the free energy of mixing

(
Gmix) relative to the number of moles (N1):

For values of the degree of polymerization tending to infinity,  2,crit thus tends to

0 andχ1,2,crit tends to 0.5

4.2.4 Determination of the Interaction Parameter (χ12 )

χ12 can be determined through osmometry measurements (see Section 6.1.2.1).The osmotic pressure, which is the pressure to apply to stop the flow of the sol-vent molecules through a semipermeable membrane (permeable to the solvent andimpermeable to the macromolecules), is related to the solution activity and to thechemical potential by the relation

The osmotic pressure can be easily deduced from equations (4.20) and (4.23)

after expressing them as functions of the concentration C2 Considering that

1= (1 − 2) and that ln (1− 2) can be developed into a series,

After observing that 2 can also be written as C2 V2or C2 V2 /M2, where C2 is the

concentration of polymer, V2 the partial specific volume of the polymer, and V2 its molar volume, and considering that X is also the ratio of the molar volume of polymer to that of the solvent (V2 /V1 ), one obtains

3V10C

3

2+ · · ·

3V10 (4.27)Knowing the osmotic pressure () and hence the chemical potential, one can

determine the interaction parameter (χ12).

4.2.5 Real Macromolecular Solutions

As already shown above, the Flory– Huggins theory appears particularly well-suited

to the case of regular macromolecular solutions whose components are nonpolar;indeed, their enthalpy of mixing is slightly positive and their entropy of mixing hasmainly a conformational origin Solutions of polyisobutene in benzene or naturalrubber in benzene belong to such category The phase separation in such sys-tems occurs upon decreasing the temperature (upper critical solution temperature(UCST)) (Figure 4.4), which means that their enthalpy of mixing is independent

of the temperature andχ12is inversely proportional to the temperature These aretwo basic assumptions of the Flory– Huggins theory

On the other hand, the same model does not predict the lower critical solutiontemperature case (LCST), which is observed in macromolecular solutions with polarcomponents Indeed, demixing upon an increase of the temperature (Figure 4.4) is

a well-known phenomenon for solutions of polar polymers that are characterized

by a high enthalpy of mixing Conscious of this shortcoming, Flory modified theinitial version of his model and reconsidered the assumptions of an enthalpy ofmixing and of an energy of contact independent of the temperature

Observing that in solutions containing polar components, segment–segmentinteractions can be favored and perturb a random distribution of macromolecu-lar chains, Flory proposed to take into account the existence of such interactions

1

1 2

2

U

U 2

Figure 4.4 Phase diagrams showing the temperatures of demixing (T) as a function of the

volume fraction of polymer, with one-phase region (1) and two-phase regions (2) U is used for

‘‘UCST’’; it means that phase separation occurs upon decreasing the temperature L is used for ‘‘LCST’’; in the latter case, phase separation occurs upon increasing the temperature.

in the calculation of the entropy of mixing In addition to the traditional tional component, he introduced a term reflecting such interactions or associations

conforma-in the expression of the entropy of mixconforma-ing In such an event, the expression of

the exchange (or contact) energy (
ε12) also necessitates a reformulation with anentropy term introduced in complement to the enthalpy term, the latter reflectingthe variation of enthalpy due to solvent–solute contact:

Only for contact energies really independent of the temperature andχ12varying in

a proportional manner to temperature —which are two assumptions of the theory

of Flory–Huggins —is the traditional expression (4.13) for
Hmix valid

One can also observe that the term 1/2 in the expression (4.25) for the

chemi-cal potential (
µ1) originates from the entropic term ln(1− 2) and results fromthe development into a power series of the latter Sinceχ12can also be written as(χ12,H+ χ12,S), this last term can be regrouped with 1/2 to define an entropic param- eter:  =1

2 − χ12,S Becauseχ12,Hrepresents the enthalpic contribution— and thus

has the dimension of an energy—the  term should also have the dimension of an

energy and must be divided by T A temperatureθ thus exists at which the variation

of the chemical potential resulting from solvent–solute interactions and from the

variation of the free energy of mixing are equal to 0 (
G =
H − θ
S = 0):

conditions, the excluded volume (u) is also equal to 0, which will be shown later

on The system is exactly at the boundary between the “good” and “bad” solventregimes; polymer segments do not exhibit a particular preference for the molecules

of solvent or for other segments of the chain under θ conditions, and they behavelike “phantoms.” A solvent is said to behave as a good solvent for a particularpolymer for values of χ12 in the range 0–0.3; it is a poor solvent if χ12 valueslie between 0.4 and 0.5 and a non-solvent for higher values From the expressions

(4.22) and (4.32), the critical temperature of demixing (T c) andθ temperature can

a lower critical solution temperature (LCST) Such behavior is observed wheneverpolymer (usually polar) and solvent exchange strong interactions

4.2.6 Kinetics of Phase Separation

Phase separation occurs in a macromolecular solution by one of the two followingmechanisms:

• Binodal transition (implying nucleation and growth) or spinodal

decomposi-tion Nucleation and growth are associated with the metastability zone betweenthe binodal and spinodal curves This process implies the existence of anenergy barrier and the occurrence of large composition fluctuations Uponraising or reducing the temperature, when the bimodal curve is crossed, spher-

ical domains of a minimum size, also called critical nuclei , are formed and

grow with time The growth of these domains is accomplished by diffusion

of the one of the two components (Figure 4.5a)

Figure 4.5 Schematic representation of the phase separation processes: (a) by nucleation and

growth and (b) by spinodal decomposition.

• The region enclosed by the spinodal curve is that of the spinodal

decom-position which refers to a phase separation of negligible energy barrier.

Overlapping worms are formed and grow through small fluctuations in position (Figure 4.5b), affording purer phases with time These fluctuations

com-in composition occur with a certacom-in periodicity, and their amplitude com-increaseswith time until reaching the gradient corresponding to phase separation Thedomains eventually formed are of about the same size as the original periods

of the fluctuations observed in the early stages of phase separation Cahn andHillard have described in detail the kinetics of such phase separation

4.3 DILUTE MACROMOLECULAR SOLUTIONS

The expression of the entropy of mixing proposed by the regular Flory–Hugginsmodel is not appropriate to the case of dilute macromolecular solutions Its assump-tion that the polymer chains are randomly distributed in the lattice is indeeduntenable in dilute solutions in which polymers instead exist as isolated rafts sur-rounded by a sea of solvent In such a case the local density in segments can

be high in the vicinity of long chains, whereas the overall concentration in ments in the solution can be very small As a matter of fact, the assumption of theFlory– Huggins theory of a local concentration in segments identical to the aver-age concentration applies only to concentrated or fairly concentrated media and iserroneous at large dilutions

seg-A tangible manner to perceive the shortcoming of the Flory–Huggins model is

to measure the osmotic pressure whose expression at low concentrations can bewritten as



RTC2 = 1

M2 + A2C2+ A3C22+ · · · (4.26a)

The second virial coefficient (A2), which reflects the quality of solvent and is

measured by taking the slope of the right-hand side of the equation /RTC2 = f (C2),

is assumed to be independent of the molar mass of the sample Experimentally,

one observes in contrast a decrease of A2 with the sample molar mass, which is

not accounted for by the Flory– Huggins model due to its oversimplification— inparticular, in the case of low concentration regions The concept of excluded vol-ume that applies to dilute macromolecular solutions was introduced to overcomethis shortcoming.

4.3.1 Concept of Excluded Volume: Case of Compact Molecules

In an ideal dilute medium, the chains are isolated and they are in contact onlywith solvent molecules In other words, chains and even segments of a samechain exclude each other from the volume they occupy, and the expulsion of alien

chains or segments of a certain volume is called “excluded volume”; this leads to

long-range interactions of a steric nature Steric exclusion affects only the entropyand not the enthalpy of the system, and therefore interactions of enthalpic naturecan be neglected in a first approximation The thermodynamics of such a systemcan thus be treated within the framework of ideal dilute solutions obeying theHenry law

As macromolecules are isolated, the lattice model used for concentrated solutions

is inappropriate to describe their behavior

The calculation of the entropy of mixing in such a case is based on the tion that the contribution of each macromolecule to the entropy of the systemdepends on the number of ways of placing it in the solution This number isproportional to the difference between the total volume of the solution and thevolume inaccessible to this macromolecule due to the presence of other macro-molecules

assump-Let V and R be the volume and the radius of a sphere (radius of the equivalent

sphere) containing a compact macromolecule of spherical form (Figure 4.6).

Because centers of gravity of two such spheres cannot approach each other

beyond a distance d = 2R, a certain volume of the sphere is excluded to the

other one Hence, the volume excluded by a compact spherical macromolecule

2d

R excluded volume

Figure 4.6 Representation of the excluded volume in the case of two spherical macromolecules

of identical volume.

corresponds to

u=4πd3

3 = 8

4

The excluded volume by a rigid rod of length L, circumference U and volume

V = LU for others is equal to

U r = 8V (1 + L/U sin γ)

The calculation of the entropy of mixing of a dilute macromolecular solution made

up of compact spheres may be developed as follows: the number of ways of placingthe first macromolecule is proportional to the volume V available (ν1= AV, where

A is a constant of proportionality), the number of ways of placing the j th chain

Expressed as a function of the numbers of moles of solvent (N1) and polymer (N2)

S m = S co, solute − (Sco)1− (Sco)2 (4.38)

(Sco)2 can be deduced from the expression (4.36b) with N1 = 0; as for (Sco)1, it is

equal to 0, and
S m can be written as

S m = −kN a N2ln XN2

N1+ XN2

+k2

u

V10

N2X

Under these conditions, the free energy of mixing (
Gmix) is simply equal to

−T
Smix, a dilute solution being considered as athermic Hence, the variation of

the chemical potential (
µ1) can be written as

Substituting the concentration of polymer for its volume fraction gives

2 =C2V10X

M2Thus,
µ1 becomes

µ1 = −C2RTV01

1

4.3.2 Flory– Krigbaum Theory: Case of Flexible Polymer Coils

When applying the concept of excluded volume to the case of real polymer coils,

it appears that they certainly do not have the same degree of compactness as that

of the spheres described in the model of the preceding paragraph In the case ofcompact particles, an element of volume is considered excluded or not whether it

is occupied or not For particles that are flexible, such as polymer coils, the degree

of exclusion can take any value between 0 and 1 The excluded volume, in thiscase, is the integral of the degree of exclusion over the entire volume occupied bymacromolecular coils However, the spatial aspect is only one part of the problem.Insofar as segment–segment interactions are present, they contribute to lower theenergy of the system (χ12is positive in the Flory– Huggins theory) and they favorthe coil interpenetration; steric exclusion is then counterbalanced by an “interseg-mental” attraction so that flexible coils can even interpenetrate freely (Figure 4.7).Flory and Krigbaum treated this case and established the expressions of theexcluded volume and the second virial coefficient for flexible chains They con-sidered the interactions existing between pairs of macromolecules whose segmentdistribution follows a function of radial distribution ρ(R), which starts from the

center of gravity of each one of them

The segment distribution within the envelope of all possible conformations lows a radial Gaussian function of the type

Figure 4.7 Curve describing the variation of the probability of placement  as the function of

the distance separating the macromolecular barycenters of two coils.

ρ(R) can also be written as a function of the radius of gyration:

ρ(r) = X

3

The first step consists in writing the expression for the free energy of mixing

within dV i and dV jvolumes using a calculation similar to that of the Flory–Hugginsmodel:

d(
G m )/dn1 = kT [ln(1 − 2)+ χ122] (4.42)

where n1 is the number of molecules contained in an elementary volume dV and

2 is the volume fraction of polymer Because the densities in segments [ρ(r)]

are small inside the volume occupied by the chain, only the first two terms of theseries resulting from the development of ln (1− 2) can be retained so that thepreceding expression now becomes

To express d (
G m) as a function ofρ(R) of the two entities, one assumes them

constant inside the two elementary volumes The volume fractions of polymer in

dV i and dV j areρ(Ri )v1 andρ(Rj )v1 , respectively (v1 is the volume occupied by asolvent molecule and considered identical to that of a repetitive unit) The numbers

of solvent molecules contained in dV i and dV j are given by:

dV i

v1

[1− ρ(R i ) ]v1 and dV j

v1[1− ρ(R j ) ]v1

From these elements the total free energies of mixing in dV i and dV j can be ten as

closer, and whose centers of gravity are at a distant a1, can be expressed as

d(
G m ) a∞= d(
G m ) a − [d(
G m ) i + d(
G m ) j]

= 2kT

1

From this relation the expression for the excluded volume can be derived; but prior

to that, it is necessary to determine the probability of placement  (a) of the two

macromolecular coils considered in a dilute solution This probability obviouslytends to decrease as the distance separating the two coils decreases, owing to anunfavorable free energy of overlapping This decrease can be accounted for by

multiplying the probability  ( ∞)—close to 1—by (
G m ) a∞/ kT:

(a) = (∞)e −(
G m ) a

In the volume 4π a2da surrounding the center of gravity (i ), the volume really

available for another macromolecule corresponds to

4πa2(a)da = 4πa2( ∞)e −(
G m ) a



1− e −(
G m ) a

∞/ kT

By substituting in the expression for (
G m ) a

∞the distribution functions established

by Flory and Krigbaum forρiandρj, one obtains the following expression for the

excluded volume (u):

u= 2

1

2 − χ12



The function F (Y ) is a complex integral which has no analytical solution, but

which can be evaluated graphically (Figure 4.9)

For low values of Y , F (Y ) can be developed into a series of the form

2 − χ12



χ2v1

3

Figure 4.9 Variation of F(Y) versus Y obtained by graphic integration (according to Flory).

into the preceding expressions delivers

u = (2/N a )

1

4πs2

3/2

(4.51a)

where V1 is the molar volume of the solvent in the three last expressions Hence,

the second virial coefficient, A2= u

2M2N a, can be written as follows:

A2 =

1

2 − χ12



V22

The Flory–Krigbaum model leads to an expression for the second virial

coef-ficient (A2) that differs by the factor F (Y ) from relation (4.27), given by the

Flory– Huggins model:

A2=

1

2− χ12



Because F (Y ) is always less than unity, the value of the second virial coefficient

predicted by the Flory–Krigbaum theory is necessarily lower than that predicted

by the theory of concentrated solutions

The relation (4.32) relatingχ12toθ shows that the excluded volume and Y are equal to 0 and F (Y ) is equal to 1 at the temperature θ even in a dilute medium.Thus the chains interpenetrate freely under θ conditions At temperatures higherthanθ, the excluded volume takes positive values; but at temperatures lower than

θ, segment–segment attractions prevail and the excluded volume is negative

4.3.3 Excluded Volume and Expansion Coefficient

In their “unperturbed” dimensions, chains —or rather their repetitive units —develop only short-range interactions The mean square average end-to-end dis-

tance separating the ends of a polymer chain made up of X segments and with a length L is given by

r20= CXL2

where C is a parameter that reflects the existence of short-range interactions The

introduction of a good solvent generates long-range interactions that affect units

in non-immediate vicinity and belonging to a same chain These interactions aredue to the fact that each of these units tends to maximize its solvation, which isaccounted for by the concept of excluded volume This results in the expansion ofthe chain, which will occupy a larger volume

Flory and Fox described the expansion of a macromolecular coil due to thepresence of a good solvent by using an approach identical to that of the swelling

of a three-dimensional network They considered all the repetitive units as subjected

to the same force field—the mean field— whose effect is to impose an energeticpenalty on any placement that would correspond to the the random walk Gaussianstatistics This force field is comprised of two components of opposite nature, one

of repulsion—and hence the expansion—and the other one of retraction which

is of entropic origin The expansion of a chain and its conformation are in factaffected by the balance between these two forces; in its perturbed dimensions, thechain adopts a gyration radius s21/2that is related to the nonperturbed dimension

by the expression

whereα, the expansion coefficient, is an empirical parameter.

The chain expansion results from forces of osmotic origin exerted by the vent, which imbibes the macromolecule The stretching causes the reduction of thenumber of possible chain conformations, but it is opposed by forces of entropicorigin The free energy of the system becomes

sol-
G =
Gosm+
GelThe volume of a spherical shape macromolecule is

V d =4

3πs3 and hence V d = 4

3πα3s03the increment of volume (dV d) associated with an increase in the radius of gyration

ds0 is proportional to

where÷ is a sign of proportionality

Since the extent of swelling depends on the degree of polymerization, the fraction

of chain segments present in this element of volume dV d is

2÷ X/dV d and thus 2÷ X/α3s02ds0

The variation of free energy corresponding to this swelling and hence to the

intro-duction of dn1 molecules of solvent is thus

d
G = (µ1− µ◦1)dN1 with dN1÷ dV d (1− 2)/v1 (4.56)

Since the product of a force and a distance (dα being assimilated to the variation

in size of the sample) is equivalent to an energy, one can write

Knowing that (µ1− µ1 ) can also be written as



(4.58)and that ln(1− 2) reduces to

ln(1 − 2) ∼ = −2−1

222with 1/X → 0 for X→∞, one obtains the following for (µ1− µ1 ):

As swelling occurs in the three dimensions of space, the expression of S will be

slightly different from that derived in the case of rubber elasticity, which considersonly one direction of stretching:

S÷1

2[αx + αy + αz − 3 − ln(α xαyαz )] (4.60)whereαx,αy, andαz denote the expansion in directions x , y , z , thus leading to

(or X ) Consequently, the excluded volume can be expressed as a function of αfrom relations (4.49) and (4.62):

u ÷ (α5− α3)



4πs203

3/2

with

The expressions (4.62) and (4.53) show that the radius of gyration of a chain

subjected to the phenomenon of excluded volume is proportional to X 3/5, whereas

that of an unperturbed chain is proportional to X 1/2 For example, the average

dimension of a chain with a degree of polymerization of X n= 1000 is expected

to grow by a factor of 2, and its volume under unperturbed conditions is expected

to increase by a factor of 23= 8

4.3.4 Perturbations Theory: Other Expressions of the Expansion Coefficient

In the Flory– Krigbaum theory, the chains are regarded and treated as isolatedspecies whose volume would be eight times larger than their own volume Inthe interior of the sphere, the segments are assumed to adopt a Gaussian dis-tribution, a statement that has been questioned by various authors According tothem, the excluded volume phenomenon occurs intramolecularly at the level ofeach monomer unit of the macromolecule and not only intermolecularly betweenmacromolecules In the improvements introduced in the Flory– Krigbaum theory,the elementary excluded volumeβ corresponds to the level of each polymer segment

and is related to the exclusion volume u by the expression

Depending on whether these models consider the contacts between segments asinvolving just pairs of them or more than two of these segments, the expressionsrelatingα to z —the parameter of excluded volume—differ:

z=

1

(4.68)Using the Flory formalism, one obtains

ments introduced in the Flory– Krigbaum theory lie within the traditional work of “mean field” theories

frame-4.4 SEMI-DILUTE MACROMOLECULAR SOLUTIONS

If the Flory theory is indisputably a reference for the thermodynamics of polymersolutions, it suffers from a lack of accuracy in its description of dilute polymersolutions as previously mentioned Well suited to the case of concentrated solutions,this theory depicts the behavior of dilute solutions and describes the forces due toexcluded volume as the result of a perturbation to “random walk statistics”; forexample, it does not account for the significant variations experienced by the density

of segments in dilute media Indeed, the replacement of the radial variation of thisfunction (which describes the density of interaction in the medium) by an averagevalue is not satisfactory

In particular, the transition from a dilute regime (s ≈ X 3/5) to a concentrated

one (s ≈ X 0.5) that is accompanied by a progressive contraction of the chain asthe concentration increases is ill-explained by a mean field theory such as that ofFlory The mean field approach predicts a result close to reality for the relationbetween the dimensions of a chain in a dilute medium and its degree of polymer-

ization (s ∼ X 3/5) This result is, however, fortuitous according to de Gennes It

results from the cancellations of the errors introduced into the calculation of
S m and
H m

De Gennes tackled the problem of polymer solutions with another point of viewand treated it as a second-order transition, a process characterized by a continu-ous variation of thermodynamic potentials and by the divergence of some of their

second derivatives According to de Gennes, the concentration C * at which the

chains overlap—defined as the start of the semi-dilute concentration region—is asecond-order transition that can be described by the renormalization of certain vari-ables, using tools developed by Wilson In the language of modern thermodynamics,critical points designate points that are subject to a second-order transition; in thevicinity of such critical points, the physical behavior of a system can be described

in the form of scaling laws that contain critical exponents (see Appendix)

Even if nothing peculiar occurs at this critical concentration (C *) with respect

to the solution properties, the medium is the subject of fluctuations of concentrationwhile passing from the dilute regime to the semi-dilute one (Figure 4.10).According to de Gennes, the fluctuations in this order parameter (the concen-tration) are reflected in the correlation length (ξ)—characterizing their amplitude

in space —and in the correlation function g(a) between pairs of repetitive units.

The existence of a critical point is observed when the correlation length ated with the correlation function of the order parameter diverges By definition, amean field theory ignores the fluctuations of the order parameter and affords satis-factory results only far away from the critical point; in addition to the calculation

associ-of the entropy and enthalpy associ-of mixing, de Gennes criticized this main point in the

classical theories and observed that the variation of g(a) with the concentration and the distance (a) considered cannot be overlooked or neglected:

g(a)= 1

2[C(0)C(a) − C2

C >> C *

C > C*

C < C * C = C *

(b) (a)

(d) (c)

Figure 4.10 Chain overlapping; (a) dilute solution; (b) overlap concentration; (c) semi-dilute

solution; (d) concentrated solution.

where C (a) denotes the local concentrations at the reference point (0) and at the point (a) under consideration, whereas C is the average concentration in the

medium

As for the correlation length (ξ), it represents the average distance that rates the segments of various chains which come into contact above the critical

sepa-concentration of overlapping C * C *( ≡ X/r3) is attained when the concentration

of the solution is roughly equal to the average concentration in segments inside

a macromolecular coil Hence, the macromolecular solutions can be classified inthree categories depending upon their concentration domains:

C < C∗ dilute solution

C ≥ C∗ semi-dilute solution

C ∗ concentrated solution

In the dilute regime, polymer chains behave like isolated spheres called “blobs”;

the radius of gyration or the end-to-end distance follows a variation in r ÷

Xγwithγ = 0.586 ≡ 0.6 and 2( ≤ ∗≡ XL3/r3), the volume fraction of polymer,

follows a variation in 2 ÷ X/(X 3/5)3, which gives X −4/5 This is in agreementwith the predictions of the mean field theories The correlation lengthξ is close to

r and g(a)= 0

As the average concentration increases, polymeric chains overlap and establishinteractions They can be depicted as a collection of “blobs” of a diameterξ whoseperiphery witnesses intersegmental interactions (Figure 4.11) The latter contributes

to create a sort of network whose mesh size corresponds precisely to the correlation

length Above C * the end-to-end distance is not relevant as characteristic length controlling the solution properties r has to be replaced byξ, which then decreasesfrom ξ = r for C = C * to ξ = L as the concentration increases Inside the “blob” (a <ξ) the subchain is subjected to the excluded volume effect As for the excludedvolume forces generated by segments located outside the “blob,” they are screened

by the presence and the interactions of immediatly adjacent chains according to amodel proposed by Edwards: beyondξ, chains adopt random walk Gaussian con-formations whereas the correlation length decreases exponentially with the volume

fraction—or the concentration—above C *:

ξ ÷ r[∗

In such a representation of the medium above C *, ξ is independent of X In other

words, when macromolecules overlap, the characteristic lengthξ is not determined

by the size of the chains but by their concentration y , the exponent in expression

(4.74), must take a value that yields an independent ξ with respect to the molarmass of the chain:

Thus, for concentrations higher than C * beyondξ, a macromolecule can be viewed

as Gaussian chain made up of X e () segments of size ξ and therefore r() can be

expressed as

Figure 4.11 Representation of a chain in a semi-dilute solution.

As forξ, it is equal to

where g(a), the correlation function between pair of repetitive units, also represents

their number inside the “blob” (good solvent and effect of excluded volume).Because we have

and its variation is represented in Figure 4.12

As the concentration increases, g tends to 1 ( ξ = L) due to the “screening” of the

excluded volume phenomenon: in the high concentration regimes the de Gennesapproach delivers the same traditional well-known expression for the unperturbeddimensions of a chain as the Flory model, but with its renormalization meth-ods it better accounts for the shrinkage undergone by the chain as the polymerconcentration increases

a

g(a)

0 c

4.5 POLYMER –POLYMER BLENDS

When two polymers of different chemical nature are mixed, a macroscopic phaseseparation is generally observed, due to the fact that they tend to minimize theirsurface of contact Their macromolecular nature is responsible for a low entropy ofmixing, which, associated with an endothermic enthalpy of mixing, explains theirimmiscibility For two polymers, the higher their molar mass the more difficulttheir mixing

Until the mid-1970s, polymer chemists used to think that attempts at mixingtwo polymers would necessarily wind up in immiscible blends so that very fewexamples of homogeneous blends were known The polystyrene/poly(2,6-dimethyl-1,4-oxyphenylene) blend (sold under the trademark Noryl) was an exception This

field witnessed a real development only after it was realized that most neous blends phase separate upon raising the temperature (LCST) (in general theresearchers tend to increase the temperature in order to improve the miscibility ofpolymers)

homoge-From this observation, studies were carried out to identify the window of cibility by varying the temperature or by introducing specific interactions betweenpolymers Thus, miscible blends could be obtained through the use of statisti-cal copolymers; the most well known example is the blend of poly(vinyl chloride)(PVC) with poly[ethylene-co-(vinyl acetate)], which is homogeneous although PVC

mis-is not mmis-iscible with either polyethylene or poly(vinyl acetate) It could be shownthat specific attractive interactions develop in such miscible blends between thePVC repeating units and the copolymer These interactions counterbalance the

repulsive ones opposing the formation of a homogeneous blend
G is, of course, negative in the case of homogeneous blend formation Expression (4.14) for
G

using the Flory– Huggins model is

Gmix= RT(N1ln 1+ N2ln 2+ χmixN12) (4.82)whereχmixis now equal to

χmix= 1χ13+ 2χ23− χ1212 (4.83)

where 1 and 2 are the volume fractions of the two comonomers in thecopolymer

4.5.1 Theories Based on Free Volume and on the Equation of State

Although it is very useful in the description of phenomena such as those tioned above in blends involving statistical copolymers, the Flory– Huggins theory

men-is unable to account for the variation of the properties of polymer–polymer blends

as functions of the temperature Indeed, the coexistence of both a lower criticalsolution temperature and an upper critical solution temperature of mixing, which

is a rather common phenomenon in polymer blends, could not be described by theFlory– Huggins theory

The assumptions that the volume of the solution remains constant upon mixing

(
Vmix= 0) and thus it is incompressible are the reasons generally put forward toexplain the deficiencies of this theory These assumptions indeed imply no variation

of a variable such as the pressure and more generally no variation of the equation

of state For the calculation of the enthalpy and the entropy of mixing, a theorythat would include relations between the temperature, the pressure, and the volume

of the system considered is thus a necessity

From the observation that free volumes —due to the difference between specific

volumes at a given temperature T and at 0◦K—can be very different, dependingupon the nature of the components, theories based on the concept of free volume andincluding the equations of state were thus proposed Polymer connectivity impliesthe existence of free volume and a lower coefficient of thermal expansion thanthat of a solvent, and under these conditions an increase of temperature necessarilyinduces differentiation in the densities of the components, which cannot adjustone with another to make a homogeneous solution As a result of an increase intemperature, the solution demixes and this phase separation can occur at an evenlower temperature if the polymer molar mass is higher

To overcome the limitations due to the hypothesis of volume incompressibility,Lacombe and Sanchez proposed to introduce vacant cells in the lattice model as a

means to accommodate free volume (V0) A system of m constituents would thus

be mixed with n0 vacant cells and V0 /V would be the free volume fraction Under

these conditions the entropy of mixing can be shown to be

Smix

k = −noln



V0V

i ) is the incompressible density of component i

The total incompressible volume is written as

The Lacombe–Sanchez model then defines reduced values ˜v and ˜ρ (reduced

volume and reduced density) as being equal to

˜v = V /V∗≥ 1 and ˜ρ = 1

The free volume fraction is easily deduced as

V /V = (V − V∗)/V = 1 − ˜ρ

The volume fraction of the component i considered in the incompressible state

denotes the characteristic pressure or the cohesive energy density

Consequently, the expression for the free energy density can be written as

respec-From the expression for the free energy, the expression for the equation of

state for a mixture of m constituents can be established from the condition that at equilibrium the system occupies a given volume (∂G/∂ ˜v = 0).

measured by taking the slope of the right-hand side of the equation /RTC2 = f (C2) ,

is assumed to be independent of the molar mass of. .. values of χ 12< /small> in the range 0–0.3; it is a poor solvent if χ 12< /small> valueslie between 0.4 and 0.5 and a non-solvent for higher values From the expressions

(4 .22 ) and. .. solutions of polar polymers that are characterized

by a high enthalpy of mixing Conscious of this shortcoming, Flory modified theinitial version of his model and reconsidered the assumptions of