CHAPTER THREE

THERMODYNAMICS OF POLYMER SOLUTIONS AND THEORY OF POLYMER GELS

CHAPTER - III

ll£RMODYNAMICS OF POLYMER SOLUTIONS AND

T.-£ORY OF POLYMER GELS

III.A THERMODYNAMICS OF POLYMER SOLUTIONS

29

The behavior of polymers towards a given solvent is characteristic and different

from that of other solvents. This is quite expected as the differeilces in molecular

weight of a polymer and that of a low molecular weight substance (solvent) is quite

large.· The size and conformations of dissolved polymer molecules require special

theoretical treatment to explain their solution properties. Conversely, it is

possible to obtain information about the size and shape of polymer molecules from

studies of their solution properties.

III.A.. Ideal Solutions

The ideal solution is defined as one which obey the RaouLt's law. This law states

that the partial vapor pressure of each component in the mixture is proportional to

its mole-fraction, i.e.,

(III. I)

where p~ is the vapor pressure of the solvent, p, is the vapor pressure of the same

solvent in the solution and N, is the mole-fraction.

For a binary solution, the Eq.(III.I) can be written as

and

30

From the condition of equilibrium between two phases, the free energy of dilution

of a solution is given by, the Claussius-Clayperon equation

(111.2)

where llGI is the free-energy of dilution resulting from the transfer of one mole of o

liquid one from a pure liquid state with vapor pressure PI to a large amount of

solution with vapor pressure Pl.

From Eq.(III.l) and (111.2) it is clear that

(111.3)

So, the total free-energy of mixing for a binary mixture is,

(111.4)

The condition for ideal mixing demands that the heat of mixing 6H = 0, i.e., the

components mix without change in enthalpy. Since llG = llH - TllS, the ideal entropy

of mixing is given by

(111.5)

The expression for ideal entropy of mixing as given in Eq. (II 1.5) is true for a

binary solution, consisting of two types of molecules virtually identical in size,

spatial configuration and external field so that if one type of molecule in the

solution is replaced by the other types, it does not affect the immediate neighbor

in the solution.

III.A.2 Non-Ideal Solutions

But in practice, few mixtures are ideal. They can deviate from the ideality in one

of these three ways.

31

1. Athermal solutions, in which f1H - 0 but f1S is no longer given by Eq.(IIl.S)

2. Regular solutions, in which f1S has the ideal value but f1H ~ o. 3. Irregular solutions, in which both f1S and f1H deviates from ideal values.

It is usually found in most of the systems having similar size molecules, f1S is

nearly ideal when f1H = 0; therefore athermal solutions are nearly ideal. However,

many mixtures are found with f1H ~ o. Such cases arises when the inter-molecular

force fields around the two types of molecules are different [Billmeyer, F.W, 1984)

III.A.3 The Entropy of Mixing According to Liquid Lattice Theory

The molecules in the pure liquids as well as in their solution are arranged with

enough regularity to justify the assumption of a lattice. In a liquid the first

neighbor of a molecule is relatively well defined and the subsequent neighbors are

less accurately defined. But since we are only interested in the first neighbor,

the lattice model representation is a valid assumption. The third assumption that

the same lattice to be used for both pure compounds and solution is a serious one

for real solutions. It demands the geometrical shape of the two molecules to be

identical.

Let us consider the solution has n l identical solvent molecules and nz identical

solute molecules and the lattice consists of no = n l + nz cells. According to the

Boltzman relation the entropy of mixing is given as [Tagger, A., 1978)

f1Smlx = (Entropy of solution) - (Entropy of solvent) - (Entropy of solutes)

Using stirling's approximation, i.e., In n! = n In n - n, one can get

f1Smlx = K8 [(n l + nz) In (n l + nzl - n l In n l - nz In nz)

-K.( n, In n l nz

) = + nz In n l + nz n l + nz

32

(111.6)

which is same as Eq.(III.S). In spite of such drastic approximations, the ideal

entropy of mixing also looks valid for solutions in which the components differ

considerably in molecular shape as much as two fold in size.

III.A.4 The Entropy of Mixing of Polymer Solutions

The above treatment rests on the assumption of interchangeability of molecules of

solute and solvent which is not the case where the solute is a polymer, since the

polymer molecule is at least a 1000 times larger in size than the solvent molecule.

In this case one can think of a polymer having 'x' chain segments, each having the

size of a solvent molecule. 'x' is roughly defined as molar volume of solute

divided by the molar volume of solvent. This is equivalent to a solution having an

equal proportion of monomers to replace a polymer except in the case of a polymer

'x' continuous cell in the lattice are to be filled up by a polymer. This is shown

in Fig.(III.D. In this case a solvent molecule can interchange its position with a

polymer segment.

J(

-- • K J( . I( 1C " " " " " " " • )f l( )t )( )( · )( " " " K " " 'k . .. · I(' )( 't( J( " '""'\ " " i) " • w )( · I(' K · • ~ " " ~ " l) " )r 'It' Jt 'tf )( 1ft I(' )t )( )( v ~ " V " " • Jt )( J( )f . 1ft J( )( "

J )( )( " • "II . · J( 1ft )( • )( )( 1 )( )( )( )( " 'tf )( 't( )( J( 11( • )( )( " " )( " )( )( )(

-Col (6)

I'ig.m.l Two-dimensional representation of (a) nonpolymer liquids and (b) a polymer molerule in the liquid lattice.

Let us calculate the configurational entropy of the polymer solution arising from

variety of ways of arranging the polymer and solvent molecules .. The solution has n t

solvent molecules and nz solute molecules, each having 'x' chain segments. So the

total number of sites in the lattice is (n t + xnz) = no. Let us consider the

filling of these lattice sites. Let the solute molecule be introduced first. At any

point of time nl solute molecules is already introduced. nl can vary from 0 to nz·

Let fl be the fraction of lattice sites occupied, i.e.,

fl = ------(n1 + xn2)

33

(111.7)

Then the next step is to find out the number of ways to put (nl+Ost solute

molecule in the lattice. The 1st segment can be put in anyone of the free sites of

the lattice, i.e., in (1 - f l ).(n1 + xn2) ways. The second segment can only be put

in anyone of the nearest neighbor of the 1st segment. Let 'z' be the nearest

neighbor coordination number. It can vary from 6 (for cubic lattice) to 12 (for

hexagonal lattice), so the 2nd segment of the polymer can be filled by (l - fl).z

ways. The number of ways the third segment can be filled is (z - 1).(1 - f l). But

it also depends on the flexibility of the polymer. For a less flexible molecule it

is less than (z 0.0 - f l) and is '1' for completely rigid polymer. The number

of ways of placing 4th segments and onwards is equal to the number of ways 3rd

can

be placed, i.e., (z - 0.0 - f l ).

So, the number of ways of introducing (nl+Ost polymer molecule (vI+1) [Tanford, C,

1961] is

1 x x-2 vI+1 = - 0 - f l) .(n1 + xn2).z.(z - 1)

2 (111.8)

The reason for 112 is that the two sides of the polymer is' indistinguishable. The

total number of ways of putting n2 polymer in the lattice is the product of VI+1

with nl ranging from '0' to n2-1. Then n1 solvent molecules has to be introduced.

Since all the solvent molecules are indistinguishable, there is only one

distinguishable way of arranging them. So the total number of distinguishable ways

of arranging the total mixture is

1 n2-1

Q = -- IT vI+1 n2! 1=0

(III.9)

It is divided by nz!, since nz polymers are indistinguishable. Taking the logarithm

of Eq.(III.9) and putting the value of vI+" one gets,

34

(111.10)

'=0

Since a real solution contains a large number of molecules the summation in

Eq.(III.lO) can be replaced by an integration

n2-,

[[ I-to x.n 1 .. - f,) = .dn n, + xn2 '=0

n1 n1 = - --In - n2

(111.11) x n, + xn2

Applying stirling's formula In n! = n In n-n and putting Eq.OIl.ll) in Eq.(IlI.lOl

we get,

In Q = n2 In (111.12)

The entropy of mixing is given by

for: n2 = 0 and for:

(111.13)

where VI = nV(nl +xn2 ) and v2 = xnz/(nl +xn2 ) are the volume fraction of the solvent

and solute respectively.

This is called as configurational entropy. This is called so, because only the

external arrangements of molecules and their segments are considered without

bothering about the contribution to the entropy term because of the interaction

between the segments of the polymers.

35

, Comparison of Eq.(III.5) and (III. 13) gives mole-fraction in the expression of

ideal entropy of mixing is replaced by volume-fraction. In the case of ideal mixing

the volume of solvent and solute molecules are same. In this case the mole-fraction

is same as the volume fraction and hence Eq.(III.S) and Eq.OIl.13) are equivalent.

So one can consider Eq.(1II.5) as a special case of Eq.OIl.13).

If the solution contains more than one polymer and only one solvent then

Eq.(1II.13) can be written in a more general way as (Flory, P.J., 1953],

-K.[ n, In T In v, 1 flSmlx = VI + L nl (111.14) I

where I L vI = v2 = 1 - VI

Both Eq.(I1I.13) and (111.14) appears to be a very simple equation Qut this model

has some limitations because of this assumptions and approximations involved in the

derivation of tJlis model either implicitly or explicitly. The foremost limitation

arises because of the assumption that this solution configuration is random. But in

a. r.eal polymer solution this is not true because of some preferential attraction or

repulsion exists between the molecules. Ore [1944] and Guggenhei.m (1944] derived an

expression for this entropy of mbcing taking into account the preferential

attraction. They found that the modified entropy of mixing caused by non-randomness

is very important in comparison to the effects due to other approximations (Flory,

P.J., 1953].

IILA.S The Heat and Free-Energy of Mixing

Inter-molecular interaction plays a great role in the liquid state due to close

proximity of the molecules. But since the forces between the molecules decrease

rapidly with the increase in distance between them, we are only interested in the

energy arising due to nearest neighbor interactions. We are only interested in

finding the difference of interaction energy (flHmlx ) between the solution and that

for pure liquid components. The heat of mixing originates due to replacement of

some of the contacts between the like molecules with an unlike molecule. With the

lattice model only three type of nearest neighbor contacts are possible, i.e.,

(1,1], (2,2] and [1,2].

".

36

If c ll ' c22 and C 12 are energy associated with formation of bond between like

solvent particles, solute particles and unlike particles then the change of energy

for the formation of an unlike pair at the expense of one like pair is given by

[Tanford, C., 1961),

(111.15)

The total energy content due to weak inter-molecular forces of n1 solvent molecules

and xn2 solute molecules can be derived by the following process. Surrounding each

solvent molecule, there will be z.nJ(n1+xn2) other solvent molecules and

z.x.nz!(n1+xn2) solute segments. For all n1 solvent molecules, this will give rise

to an energy content of -1/2 n~.z.cIJ(nl+xn2) - z.nln2x.clz!(nt+xn2). The factor

of 1/2 in the 1st term is introduced to avoid double counting in this case of

solvent-solvent interaction. Now consider the energy content due to solute

molecules (segments). Each solute segment will be surrounded by z.nJ(nt+xnz)

solvent molecules and z.x.nz!(nl+xn2) solute segments. But the energy due to

solute-solvent interaction is already taken care of. So the energy content due to

xn2 solute segments is -112 z.n~x2czz!(nl+xn2). So the complete expression for

energy becomes

z 011.16)

Using Eq.(III.l6), the heat of mixing of n1 molecules of solvent with n2 molecules

of solute becomes

-z z f1Hmlx = -----

2

(111.11)

which can be written as [Flory, P.J., 1953)

011.18)

31

The above expression is the well known Van-Laar expression for heat of mixing of

any two component systems. From the above expression it is clear that the polymeric

character of the solute does not change this form of heat of mixing expression. If

this solvent molecule has Xl segments instead of one, then the above expression

(Eq.(III.18» can be written as,

The above Eq.(III.18) or (111.19) can be written as,

where XI

"I = z.llc1Z• -­KaT

011.19)

011.20)

(111.21)

is a dimensionless quantity which characterizes the interaction energy per solvent

molecule divided by KaT. "lKa T represents the difference in energy of a solvent

molecule surrounded by solute molecules only (i.e., VZ ... 1) , compared to one

surrounded by molecules of its own kind only (i.e., v1"'1). "1 is sometimes

expressed as,

BVI

"1 = KBT

(m.22)

where Ac1Z

B = z (111.23) v.

VI is the molar volume of the solvent and v. is the molecular volume of the

segment. 'B' represents the energy density of the solvent-solute pair.

If for a moment we assume the configuration entropy of mixing as the total energy

of mixing than the total energy of mixing can be written as,

AF mix = AHmix - TAS ... (III.24)

(111.25)

38

The above Eq.(III.2S) represents the total free-energy for the formation of the

solution from pure disoriented polymer and pure solvent.

Taking configurational entropy of mixing as the total entropy of mixing means

neglecting the possible contributions which may arise due to specific interaction

between neighboring components of the solution. We have only considered these

interactions to contribute to the heat of mixing. But in general the mixing

processes represented by Eq.(III.1S) should represent a standard state of entropy

change as well as a change in energy or heat content. The entropy change due to 1st

neighbor interactions must be proportional to the number of pair contacts formed as .

in the case of heat change. So to rectify the mistake done by omitting the change

in entropy one has to consider Eq.(III.lS) which consists of two parts, one

representing the change in heat content and the other the product of absolute

temperature (T) with the change of entropy for conversion of reactants to products.

Then we can write

(111.26)

where IlWh due to heat change and IlWs due to entropy change.

Then the parameter '" described in Eq. (111.21) will contribute as entropy

contribution divided by (KB) in addition to heat of mixing terms divided by KBT.

and KBT",n,v2 should be considered a standard state of free-energy change rather

than heat of mixing only. Since the total free-energy change IlF mix consists of this

standard state of free-energy change plus the configurational free-energy change

-TIlSm due to change in entropy, the term of the free energy as expressed by

Eq.(III.2S) will remain unaltered.

Expression for entropy and heat of mixing which are in line with the above revision

can be obtained through the standard thermodynamic relations

From Eq.(III.2S). we have

8(IlFm/T) ]

8T p

39

(1II.27)

and

(111.28)

If ~c12 is independent of T, ". = (z.~cI2 "l)/KsT Is Inversely proportional to 'T' • and the 3rd term is Eq.(III.27) is equal to zero and t.Sm - ~Sm. Also -T(8"1/8T) =

"I and t.Hmlx reduces to Eq.(III.20). So If ~C12 is not independent of 'T', i.e., it

does depend on nearest neighbor interaction, then one should use Eq.(III.27) for

entropy of mixing instead of Eq.(II1.l3l.

The expression for free-energy of mixing (Eq.(III.25» behaves satisfactorily

within the limits of assumptions. Due to the discontinues nature of very dilute

polymer solution, the thermodynamic relations derived thus far are only applicable

to the solution which are concentrated enough such that the randomly coiled

molecules overlap one another extensively. The total entropy of mixing found

experimentally may contain a term (8(".Tl/BT) in it, which is difficult to

evaluate separately. So the theoretical configuration entropy of mixing can not be

compared with the experimentally found entropy of mixing t.Smlx.

III.A.6 Partial Molar Quantities

The chemical potential J.l. (which is free energy per mole) of the solvent in the

solution relative to the chemical potential Il~ in the pure liquid state (solvent)

can be obtained by differentiating Eq.(III.25) with respect to n.. Multiplying this

with the Avogadro's number one will get the chemical potential per mole or relative

partial molar free-energy (~Fl) [Flory, P.J., 1953].

(III.29)

For a so'lution having heterogeneous polymer Eq.(III.29) can be written as,

40

III - Il~ = RT ( In (1 - V 2 ) + (1 - 1/ Xn) V 2 + "I v~ ) (111.30)

where xn is the number average degree of polymerization. Equation (111.29) can be

written as,

(111.31)

where

65: = -R ( In (I - v 2) + ( I - I/X ),v2 ) (111.32)

is the relative partial confi.gurational entropy of the solvent in the solution. If

"I is inversely proportional to 'T', i.e., the interaction contributes to energy

only then the lilt two terms in Eq.(1II.29) represent the relative partial molar

entropy and the last term represents the relative molar heat content

2 = RT "I v2 (111.33)

But if "I contains an entropy term along with the energy term, then the form of the

chemical potential will remain unchanged but its separation into entropy and heat

contents has to be carried out by the operation like those given for the free­

energy of mixing. From pure thermodynamic relation we know that,

(111.34 )

where VI is the molar volume of the solvent and 'n' is the osmotic pressure. Using

Eq.(1lI.34) in Eq.(lII.29), we have

011.35)

However, the osmotic pressure method is most useful for dilute solution which is

inv.alid for above theory. Expanding the logarithmic term and keeping terms only

3 upto V 2 ' we will have

n = (111.36)

41

It is more convenient to use c (concentration) in gm/ml. Now v2 = cv where v is the

partial specific volume of the polymer and 'x' is the ratio of the molar volume of

the polymer and solvent, then,

CLI c --=--=- (111.37)

M

So Eq.(1II.36) becomes

TI RT =-- (JII.38)

c M

The 1st term in the above equation is called as Van't Hoff's term and it approaches

this limit at infinite dilution. The higher order terms represent the deviation in

the theory. These deviation are the errors as limitations of this theory in the

dilute polymer solutions, basically originating. from finite inter-molecular

interactions.

The second term can be represented by (Azc).(RT) where Az is the second virial

coefficient which is equal to

(111.39)

It is clear from Eq.(Ill.39) that "1 is a dimensionless quantity equal to the ratio

of energy of interaction of the polymer with the solvent molecules to the kinetic

energy Ks T; which should not depend on the concentration of a solution.

For an ideal solution the TI/c versus c dependence obeys Van't Hoff's law and its

graph is a straight line parallel to the abscissa. Hence the second virial

coefficient Az = O. The solvents of such solutions are called ideal solvents. For

good solvent Az > 0 and for poor solvents Az < O.

From Eq.(Ill.38) it is clear that

For ideal solvent Az = 0

For good solvent Az > 0

For poor solvent Az < 0

"1 = 112

"1 < 1/2

"1 > 112

42

"1 measures the thermodynamic affinity of a solvent for the polymer. Thus it

defines the quality of the solvent. Smaller the "1 better is the solvent quality.

For a good solvent it can be a negative quantity and for a poor solvent it can be

greater than unity. We shall see the implications of these in gelation experiments

to be discussed later.

III.B THEORY OF POLYMER GELS

A linear polymer formed by condensation of bi-functional units differ only in one

parameter, the chain length, apart from the possible differences in end groups. But

to describe a polymer having some units of higher functionality (greater than 2) is

quite complex. In this case apart from size of the molecule and the total number of

units it contains, one need to know about parameter like: degree of branching, the

number of units of higher functionality in the molecule. etc.

The presence of polyfunctional unit along with the bi-functional unit. under

appropriate condition helps in formation of infinite network (geL). It is called so

because the macroscopic dimension of the network is equal to the dimension of the

sample it self. The sudden onset of gelation marks the division of the mixture into

two parts: the gel which is insoluble in nondegrading solvent and the sol which

remain soluble and can be extracted from the solution. As the polymerization

proceeds beyond the gel point, the amount of gel increases at the expense of sol.

In this proce~ses the mixture transforms from a highly viscous liquid to an elastic

material of infinite viscosity.

III.B.1 Critical Condition for the Formation of Gel (or Infinite Network)

In order to calculate the point in the reaction at which gelation takes place, a

branching coefficient 'cx' is defined as the probability that a given functional

group on a branch unit (Le., a unit of functionality greater than 2) is connected

to another branch unit.

43

Let us consider the branching unit to be tri-functional. Then each chain which

terminates in a branch unit is succeeded by two more chains. If both these chains

terminate in branch units, four chains are reproduced and so on. If « < 1/2, then

there is less than even chance, that each chain will lead to a branch unit and

there will be greater than even chance that it will end at an unreacted functional

group. In this case the termination of chains will out weigh the continuation of

the network through branching. So for « < 112 the molecular structure must be

limited, i.e., finite in size.

On the other hand if « > 112 then, there is a fair chance that each chain will

reproduce two new chains. So 'n' chains are expected to head to 2n new chains,

which is greater than 'n' when « > 1/2. In this case, branching of successive

chains may continue the structure indefinitely and the formation of an infinite

network is possible. « = 112 tepresents the critical condition for formation of an

infinite network structure in the case of a tri-functionally branched system.

However, it should be noted that beyond « "" 1/2, all the materials are not the part

of the infinite network. There will be still some finite structures coexist with

the infinite structure as long as 1/2 < « < 1. As '«' approaches unity more and

more materials will belong to the infinite structure.

In general the mixture may have branched unit of higher- functionality than three,

or may have more than one type of branched unit. A general statement for the

critical condition for the formation infinite structure is the following: "Infinite network formation is possible {f expected number of chains which wUl succeed 'n'

chaw through branching of some of them, exceeds n". That is if 'f' is the

functionality of the branching unit, gelation will occur when «(f-O > 1. The

critical value of '«' is, therefore [Flory, P.l., 1953)

1 «c = -(-f---l-) - (111.40)

If more than one type of branching unit is present. (f-I) must be replaced by the

appropriate average.

III.B.2 The Relation Between 'a' and Extent of Reaction

Consider the polymerization a bi-functional unit A-A, a tri-functional unit

44

A A-<.. A

and a bi-functional unit of opposite character B-B. where condensation occurs only

between A and B.

Assumptions

1. All the functional groups of each kind A and B are equally reactive. The

reactivity is independent of size and structure of molecule to which it is

attached.

2. The reaction between A and B groups on the same molecule are forbidden. i.e .•

no intra-molecular reaction.

A The condensation of a bi-functional unit. A-A. a tri-functional unit A-< and a bi­

A functional unit of different kind B-B can be represented by the equation

A A A-A + A-< + B-B ~ >A[B-BA-A) B-BA-<

A 1 A (111.41)

where 'i' can have value from 0 to CIII. Under the assumption (j) the probability that

the> 1st A group of the chain shown in the right hand side has reacted is given, by

PA and the probability that the B on the right of the first B-B unit has reacted is

given by PB' Let 'p' be the ratio the ratio of A's belonging to the branch unit to

the total number of A's in the mixture then the probability that a 'B' group is

reacted to a branch unit is PPB and to a bi-functional unit A-A is (l-P)PB' Then

the above Eq.(II1.41l can be written a probability equation as

(III. 42)

The probability ·a·. that the chain ends in a branch unit regardless of number 'i'

of pairs of bi-functional units, is given by summation of such expressions having i

= O. 1. 2 ...... etc .• respectively, that is

CIII 1 a = L [PAPe(l - p») PAPeP (I1I.43)

1=0

PAPeP ~ a = ------------------ (III. 44)

[l - PAPB(l - p))

If A and 8 group initially present at ratio 'r', then

substituting this in Eq.(1II.44) we have

or

« = --------------

2 PaP

« = ----------------2

[r - PB(l - p»)

'«' is a calculable from experimentally observed and controlled quantities.

Special Cases

1. When there are no A-A units, then p = I, so from Eq.0I1.46) and 011.47)

2 2 « = rPA = PB

45

(III. 45)

011.46)

(III. 47)

011.48)

2. When A and 8 groups are present in equal proportion, i.e., r = I, PA = PB = p.

3. If r = 1 and p = I, then

2 pp « = -------------

2 [I - p (I-p))

2 « = p

011.49)

011.50)

4. In a system consists of a bi-functional unit and a f-functional unit R-Af ,

where A can condense with itself

pp « = -------------- (III.51)

[I - pO - p))

If the branch unit is other than tri-functional, i.e., tetra-functional the same

equation holds for '«'.

46

In Several cases investigated by FLory and other workers, the observed value of (Xc

is slightly higher than the theoretical value. This discrepancy is because of the

reaction of some functional groups to form inter-molecular links, do not contribute

to network structures. The reactions must therefore be carried slightly to further

to reach the critical point.