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J Solution Chem (2012) 41:318–334

DOI 10.1007/s10953-012-9795-6

Modification of the Two-Point Scaling Theory

for the Description of the Phase Transition in Solution.

Analysis of Sodium Octanoate Aqueous Solutions

Henryk Piekarski ·Michał Wasiak ·Leszek Wojtczak

Received: 8 November 2010 / Accepted: 13 May 2011 / Published online: 2 February 2012

© The Author(s) 2012. This article is published with open access at Springerlink.com

Abstract On the basis of conventional scaling theory, the two-point scaling theory was

modified in order to describe the influence of composition on the partial molar heat capacity

and volume during the micellization process. To verify the theory, isobaric heat capacities

and densities of aqueous sodium octanoate solutions were measured over wide composition

and temperature ranges and the modified approach was used to analyze the calculated partial

molar heat capacities and volumes of the surfactant in water. The results obtained indicate

that the micellization process is subject to the scaling laws. The results were compared with

those for other systems. Peculiar behavior of the critical indices was observed and correlated

with the structure of the micelles.

Keywords Scaling theory · Phase transition · Colloids · Heat capacity · Volume

1 Introduction

Research on colloidal systems is carried out using many different experimental techniques.

Interpreting the experimental data requires a model which combines a description of the

thermodynamics together with a description of the structure. One of such models is scaling

theory [1], particularly the two-point scaling theory introduced by Wojtczak et al. [2, 3] for

the description of paramagnetic–ferromagnetic phase transitions, which extends the theory

to the case of noncontinuous phase transitions. This idea came from the fact that some

analogies can be perceived between the phase transitions that occur in the two systems

Electronic supplementary material The online version of this article

(doi:10.1007/s10953-012-9795-6) contains supplementary material, which is available to authorized

users.

H. Piekarski (?) · M. Wasiak

Department of Physical Chemistry, University of Łód´ z, Pomorska 165, 90-236, Łód´ z, Poland

e-mail: kchfpiek@uni.lodz.pl

L. Wojtczak

Department of Solid State Physics, University of Łód´ z, Pomorska 149/153, 90-236, Łód´ z, Poland

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J Solution Chem (2012) 41:318–334 319

(i.e. paramagnetic–ferromagnetic and regular solution–micellar systems): a dominant role

is played by surface effects and changes in degree of order of surface particles. Moreover,

the similarity between the shape of the curve describing the partial molar heat capacities at

constant pressure of surfactant on molality, and the shape of the temperature dependence of

the specific heat capacities under the constant external field in the case of magnetic systems,

gave us additional encouragement to consider whether phase transitions in these solutions

also follow scaling laws.

Our research showed indeed that the thermodynamic relations following from the scaling

theory can be used for the description of colloidal [4] and microheterogeneous systems

[5, 6]. The application of the theory, however, was completely “intuitive” and, in order to

interpret the results, a reformulation of the theory with respect to the original basis was

necessary.

2 Theoretical Study

In physics, critical phenomena is a term related with critical points, which are conditions

at which a phase boundary ceases to exist. For chemists, commonly known critical points

are the critical temperature (or pressure) at which vapor pressure curve terminates, or in

liquid systems the critical temperature of mixing. Another kind of critical phenomena are

the paramagnetic–ferromagnetic phase transitions, which are described by the conventional

scaling theory [1].

Scaling theory assumes that thermodynamic potentials are homogeneous functions with

respect to the external field and the reduced temperature ε. According to Stanley [1] the

static scaling hypothesis states that they are of more general form, namely: f(λax,λby) =

λf(x,y) rather then f(λx,λy) = λpf(x,y). The reduced temperature is defined as:

ε =|T −Tc|

where Tcis the critical temperature.

The external field term can have many meanings. In the conventional theory it was the

magnetic field [1] but it can be the pressure p when solutions are considered.

Application of scaling theory to the description of solutions requires the introduction of

one more variable, namely the reduced amount of solute μ defined as:

μ =|n2−nc|

where n2is the amount of the solute and ncis the critical amount of the solute; by analogy to

the conventional theory it is the amount of solute at which the critical phenomenon occurs.

As a result the thermodynamic potential, in our case the Gibbs energy, can be expressed

as a function of three variables:

Tc

(1)

nc

(2)

G = G(ε,p,μ)

(3)

When the amount of solvent n1is constant and its mass is equal to 1000 g, the amount of

solute n2is equal to the molality of solution by means of:

n1=1000

M1

which allows use of the molality instead of the amount of solute, and equates the critical

amount of the solute with the critical micellar concentration c.m.c.

⇒

n2= m

(4)

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320J Solution Chem (2012) 41:318–334

The conventional theory describes the behavior of the molar properties of pure sub-

stances. In the case of solutions, partial molar quantities should be considered instead. The

singularities in the behavior of thermodynamic functions related to the critical phenomena

in solutions occur with increasing molality, not temperature, as in the case of conventional

theory.

On the basis of the fundamental assumption of scaling theory about the homogeneity of

thermodynamic potentials, we can write:

λG(ε,p,μ) = G?λaε,λbp,λcμ?

for any λ and arbitrary a, b, c (non-vanishing simultaneously).

Differentiating both sides of Eq. 5 we obtain:

(5)

λ∂2G(ε,p,μ)

∂p∂n

=∂2G(λaε,λbp,λcμ)

∂p∂n

= ±1

ncλbλc∂2G(λaε,λbp,λcμ)

∂(λbp)∂(λcμ)

(6)

Taking into account the thermodynamic fundamentals (∂2G

to the homogeneous function:

?∂2G(λaε,λbp,λcμ)

we obtain:

±1

and:

∂p∂n2)T,n1= V2and applying it

∂(λbp)∂(λcμ)

?

T,n1

= V2

?λaε,λbp,λcμ?

(7)

ncλV2(ε,p,μ) = ±1

ncλb+cV2

?λaε,λbp,λcμ?

(8)

V2= λb+c−1V2

?λaε,λbp,λcμ?

λ = μ−1

(9)

Since Eq. 9 is valid for all values of λ it must hold also for the particular choice:

c

(10)

which is equivalent to:

λcμ = 1(11)

If we consider a constant (or zero, in conventional theory) field (the pressure in our

case), constant temperature, and the above (Eqs. 10 and 11) value for λ, then the expression

V2(λaε,λbp,λcμ) becomes constant and we denote it as Vo

2. This leads to:

V2= μ

1−b−c

c

Vo

2

(12)

Taking into account the above results, the equation for the partial molar volume, as a

function of solution molality, at constant pressure and temperature, can be derived in the

form of a power law as follows:

V2= Vo

2μβ?

(13)

with the critical index β?:

β?=1−b −c

c

(14)

A similar relation can be derived for the partial molar heat capacity:

?∂3G(λaε,λbp,λcμ)

∂(λaε)2∂(λcμ)

?

p,n1

= −1

TCp,2

?λaε,λbp,λcμ?

(15)

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J Solution Chem (2012) 41:318–334 321

with an expression Cp,2(λaε,λbp,λcμ) which becomes constant at constant temperature

and pressure:

Cp,2= μ−2a+c−1

Cp,2= C◦

c

C◦

p,2

(16)

p,2μ−α?

(17)

and the critical index α?:

α?=2a +c −1

c

(18)

Relations for other thermodynamic functions can also be derived.

For the partial molar isothermal compressibility we obtain:

κT,2= κ◦

γ?=2b +c −1

T,2μγ?

(19)

c

(20)

Analogously, the relation for the pressure dependence of the partial molar volumes in

terms of the scaling law has the form:

V2= V◦

δ?=

2p

1

δ?

(21)

b

1−b −c

(22)

Similar to the original theory [1] where the relation ξ ∝ μ−νcan be proven, we suspect

the relation for the “partial molar correlation length”, a property which characterizes only

the solute, is in the form:

ξ2= ξ◦

2μ−ν?

(23)

with the critical index ν?.

In the conventional theory some relations between the critical indices, e.g. Rushbrooke

relation α +2β +γ = 2 or Widom relation β(δ −1) = γ, can be proved [1]. The modified

theory causes some of these relations to have different forms:

α?+2β?+γ?=2a +c −1+2−2b −2c +2b +c −1

but they can be transformed into the original form for the case when the temperature, rather

than molality, is a variable and consequently the constant a is the denominator of the fraction

of Eq. 24:

c

=2a

c

?= 2(24)

c → a

⇒

2a

c

= 2(25)

Some other relations remain unchanged:

β??δ?−1?=1−b −c

c

?

b

1−b −c−1

?

=2b +c −1

c

= γ?

(26)

In the conventional approach the so called geometrical relation can be assumed to be in

the form:

G

kTV∝ ξ−d

(27)

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322J Solution Chem (2012) 41:318–334

resulting from the fact that thermodynamic potential divided by kTV has the same dimen-

sion as the inverse of volume, which can be expressed, using the correlation length, as ξ−d.

Taking into account that heat capacity is the second derivative of the thermodynamic po-

tential, the behavior of the Gibbs energy can be described using the heat capacity critical

index α in the form of G ∝ ε2−α. Correlation length, in turn, in terms of scaling can be

written as ξ ∝ ε−ν. Comparison of the indices of both functions results in the relation which

links the critical index α (describing the behavior of heat capacity) with the dimensional-

ity of the system d, by means of the critical index ν which describes the behavior of the

correlation length [7], namely:

2−α = νd

(28)

The modification of the theory makes that we cannot be sure if this relation is true in the

case of a solution. However, assuming the relation:

2−α?= ν?d?

(29)

by analogy to Eq. 28, we can see that d?denotes the dimensionality with respect to solutions,

by analogy with the conventional approach, but a precise definition at this stage is difficult,

due the lack of appropriate experimental data.

Agreement of the relations between the critical indices indicates the possibility that our

treatment of the scaling theory could be applied to the description of solutions and phase

transitions induced by changes of solution composition.

The conventional scaling theory can be used for the description of the continuous phase

transitions(secondorder),inthecasewheretheorderparameterchangescontinuously.Itcan

be extended, however, to the case of noncontinuous phase transitions (first order). Such an

extension is described by the two-point scaling theory [2, 3], which is considered when the

system can appear in n phases confined by the stability points of each phase. The two-point

scaling is also based on the assumption that the thermodynamic potentials are generalized

homogeneous functions, although their singularities are related to the stability points of each

phase instead of the phase transition points which now are situated in the intervals between

stability points. This approach results in a different temperature scale for each phase. The

values of mspdenote the stability points for lower limits of the concentration of phase s

while the values msf represent the stability points for upper limits of the concentration of

the phase s. In our case it refers to the regular solution of surfactant monomers (s = 1) and

the phase of micellar structures (s = 2), respectively. In the case of a regular solution of

surfactant monomers we consider only the upper stability point, and therefore we denote

it simply as mf. Similarly for the micellar phase we consider only the lower limit denoted

as mp.

The crossing points of the curves for each phase indicate the critical molality mc, which

can be considered as the boundary between the concentration range in which the solution

predominantly exhibits properties characteristic for a regular solution (m < mc) and the

range in which the solution properties become typical for micellar systems (m > mc). This

point corresponds to the transition point in the conventional approach to scaling.

Considering the two-point version of scaling theory, the relations for the heat capacities

for each phase with respect to the molality variable m can be written as follows:

Cp,2(m ≤ mf) = C◦1

p,2

?

?m

1−

m

mf

?−α?

?−α?

1

(30a)

Cp,2(m ≥ mp) = C◦2

p,2

mp

−1

2

(30b)