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Applied Surface Science 253 (2006) 1469–1472

Kinetics of slow collapse process: Thermodynamic

description of rate constants

M. Weis *

Faculty of Electrical Engineering and Information Technology SUT,

Ilkovicova 3, 812 19 Bratislava 1, Slovakia

Received 6 February 2006; accepted 14 February 2006

Available online 11 April 2006

Abstract

Insoluble monolayer formed at the air/water interface compressed at a surface pressure above the equilibrium spreading pressure is unstable. In

presented paper the classical theory for homogeneous nucleation is modified adding the activation energy term. It allows quantitative

thermodynamic interpretation of the slow collapse phase transformation. The collapse of stearic acid Langmuir films has been carried out by

systematic measurements of the area loss–time isobaric dependencies at various temperatures and isothermal dependencies at various pressures.

Activation energies (activation enthalpies) and activation entropies have been evaluated for the nucleation (Ea = 1.32 eV) and the growth

(Eb = 1.55 eV) processes. The experimental data for various pressures are discussed on the basis of the Gibbs energy analysis.

# 2006 Elsevier B.V. All rights reserved.

PACS: 68.18.Fg; 64.70.Nd; 68.35.Rh

Keywords: Langmuir–Blodgett films on liquids—Structure: measurements and simulations; Structural transitions in nanoscale materials; Phase transitions and

critical phenomena

1. Introduction

Monolayers spontaneously formed at the air/water interface

represents a two-dimensional system with various potential

applications [1,2]. More recently, with increasing interest in the

deposition technique on the solid substrate (Langmuir–Blodgett

technique), the applications have appeared in the field of physics

[3,4], chemistry, or biology. However, the monolayer instability

[5] is one of the main problems, which obstructs its practical

application. Over the past decades the collapse phenomena in

Langmuir films have been a subject of numerous experimental

studies in physics or chemistry (e.g. [6]).

In general, there are two classes of structural changes in the

Langmuir film, transitions between different two-dimensional

phases [7], and transitions in which dimensionality changes

from the degree of two to three [8,9]. We will now deal only

with the latter class of transitions, which is usually termed as a

monolayer collapse. Different basic types of monolayer

* Tel.: +421 2 602 91 846; fax: +421 2 654 27 427.

E-mail address: Martin.Weis@stuba.sk.

0169-4332/$ – see front matter # 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.apsusc.2006.02.029

collapse can be distinguished according to their different

mechanisms. At higher surface pressures the monolayer

structure is abolished by fracture collapse, sometimes also

called catastrophic collapse. There are several mechanisms,

which can likely contribute to the process, but the final effect is

easy to observe as a sudden pressure fall in the surface

pressure–area isotherm. At lower pressures, the degradation of

the monolayer is caused by so-called ‘‘slow collapse’’. Also

here the final effect can be realized through various microscopic

processes—buckling instability, folding of the monolayer, or

the formation and growth of cracks [10–12]. Therefore, various

models were proposed for the description of the experiment

[13–15]. These theories are in excellent agreement with many

direct and indirect experimental results, the choice of the theory

usually varies with the used surfactant and the dominated

degradation process.

2. Theory

Gaines [16] proposed that there might be a monolayer

stability limit, which he defined as the pressure of equili-

brium between the film and the freshly collapsed material.

M. Weis / Applied Surface Science 253 (2006) 1469–14721470

Fig. 1. Area loss–time dependencies at surface pressure of 30 mN/m for various

temperatures.

Fig. 2. The Arrhenius plot for experimentally obtained parameters a and b.

The three-dimensional phase in this case would have a degree

of stability lying between that of the monolayer and the

equilibrium bulk phase. Degradation of the monolayer in the

pressure range between the equilibrium spreading pressure and

the fracture pressure was a center of interest in several

experimental works (e.g. [17]).

Smith and Berg [18] were the first who described the slow

collapse mechanism by homogenous nucleation and growth. It

is possible to express the change of molecular area (relative

area loss) as a dependence on time:

A

A0

¼ expð�at � bt2Þ (1)

where A0 is the initial area and parameters a and b are

characteristics of the nucleation and the growth pro-

cesses, respectively. A more complex view on the nucleation

phenomena was presented by Vollhardt and Retter [19],

who considered shape effects of the critical nuclei. Later

works showed that on the basis of these nucleation-growth

theories important parameters of the classical nucleation

theory such as, critical size of nuclei and free energy for

the formation of the critical nuclei can be determined

[13–15].

In general, parameters a and b are considered only as

experimental constants. However, if we describe the slow

collapse of the monolayer as a phase transition, it is possible to

assign the activation energy Ea to this process:

a ¼ aT ;0 exp

�� Ea

kBT

�¼ kBT

hexp

�DSa

R

�exp

�� DHa

RT

�

(2)

or to express the activation enthalpy DHa and the activation

entropy DSa by the Eyring equation, where kB is the Boltzmann

constant and h is the Planck constant. A formally similar

relation to Eq. (2) can be applied for parameter b.

Rate parameters a and b are proportional to the rate of

formation of critical nuclei J what can be expressed [18] for the

Langmuir film in the form:

J ¼ k1p exp

�� k2

ln2ðp=pEÞ

�(3)

where k1 and k2 are constants and pE is the equilibrium

spreading pressure. Therefore, the pressure dependence of

the rate parameter can be written:

a ¼ ap;0p exp

�� k

ln2ðp=pEÞ

�(4)

Using Eqs. (2) and (4) the behaviour of the Gibbs energy can

be described by the relationship:

DGa ¼ RT

�ln

ap;0hp

kBT� k

ln2ðp=pEÞ

�(5)

and in analogy a formally similar relation to Eqs. (4) and (5) can

be applied for rate parameter b. Constants k and pE are the same

for the both rate parameters.

3. Experimental

Surface pressure–area isotherms were measured in a

Langmuir trough (Type 611, NIMA Technology Ltd., UK).

The surface pressure measurements were carried out with a

filter-paper Wilhelmy plate. The Langmuir monolayer of

stearic acid (purchased from Fluka, Switzerland) was formed

by careful casting a 1 mg/ml chloroform solution to the water

surface (bidistilled deionized water, 15 MV/cm). The solvent

was allowed to evaporate for at least 15 min prior to

compressing the surface. Before a collapse measurement the

monolayer situated on the surface of water was firstly two times

slowly compressed up to the pressure of 10 mN/m and

subsequently released for monolayer homogenization. All

the chemicals used were of spectroscopic grade purity.

M. Weis / Applied Surface Science 253 (2006) 1469–1472 1471

Fig. 3. Area loss–time dependencies at temperature 24 8C for various surface

pressures.

Fig. 4. Experimentally obtained rate parameters a and b vs. surface pressure.

Solid line represents theoretical results of Eq. (4) for k = 40, ap,0 = 70 s�1 and

bp,0 = 8 s�2. The equilibrium spreading pressure is pE = 5 mN/m [9].

4. Results and discussion

The stability of stearic acid monolayers was studied by

monitoring their relative area loss (A/A0) with respect to time at

a constant surface pressure of 30 mN/m. The temperatures were

varied in the range from 10 to 26 8C. The obtained experimental

results are shown in Fig. 1.

As shown in [18] for the stearic acid monolayer collapse, the

description by Eq. (1) is sufficient. The parabolic part of ln(A/

A0) was fitted by a polynomial of the second degree. The

obtained values of parameters a and b from Fig. 1 for each

temperature are shown in the Arrhenius plot in Fig. 2.

The activation energies (activation enthalpies) of nucleation

and growth are Ea = 1.32 eV (DHa = 127.4 kJ/mol) and

Eb = 1.55 eV (DHb = 150 kJ/mol), respectively. The difference

between the activation energy and enthalpy (DH = E � RT) is

lesser than the error of experimental data. The activation

entropies are positive, DSa = 1.83 eV/K (176.5 J/K mol) and

DSb = 2.43 eV/K (234.4 J/K mol), which agrees with theoretical

predictions [14]. According to this fact a new three-dimensional

Fig. 5. The Gibbs energy vs. surface pressure. Solid line represents theoretical

results of Eq. (5) for k = 40, ap,0 = 70 s�1 and bp,0 = 8 s�2. The equilibrium

spreading pressure is pE = 5 mN/m [9].

M. Weis / Applied Surface Science 253 (2006) 1469–14721472

phase is generated, providing a more disordered system in

comparison with monolayer.

The surface pressure dependencies of slow collapse

mechanism of stearic acid monolayers was studied by their

relative area loss (A/A0) versus time functions at a constant

temperature of 24 8C. The surface pressures were varied from

25 to 30 mN/m. The recorded experimental results are shown in

Fig. 3. The values of rate parameters a and b from Fig. 3 for

each surface pressure were compared with theoretical results

of Eq. (4) for k = 40, ap,0 = 70 s�1 and bp,0 = 8 s�2 in Fig. 4.

The value of the equilibrium spreading pressure was taken

pE = 5 mN/m [9]. More general view provides development of

the Gibbs energy with respect to surface pressure (Fig. 5),

where the values of the Gibbs energy were obtained from

Eq. (2).

5. Conclusions

The stability of stearic acid monolayers at the air/water

interface was evaluated by their area loss on time at a constant

surface pressure for various temperatures and at a constant

temperature for various surface pressures. The experimental

data can be interpreted by the theory of homogenous nucleation

and growth [18] as the phase transition between two and three-

dimensional molecular systems.

After initial regular collapse of the monolayer as described

by Eq. (1), the dependence ln(A/A0) versus time became more

linear. This effect is more pronounced at higher temperatures.

This is probably caused by a mutual contact of growing three-

dimensional aggregates, which results in stopping its growth.

Our course of approach in the quantitative interpretation is

appropriate to evaluate the activation energies and the Gibbs

energies of nucleation and growth, which (according to [18,20])

control the slow collapse process.

Acknowledgement

The work was supported by the Slovak grant agency VEGA,

project no. 1/0277/03.

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