Table Of ContentECAFERP
In the small world of micrometer to nanometer scale many natural and industrial processes
include attachment of colloid particles (solid spheres, liquid droplets, gas bubbles or protein
macromolecules) to fluid interfaces and their confinement in liquid films. This may lead to the
appearance of lateral interactions between particles at interfaces, or between inclusions in
phospholipid membranes, followed eventually by the formation of two-dimensional ordered
arrays. The present book is devoted to the description of such processes, their consecutive
stages, and to the investigation of the underlying physico-chemical mechanisms.
For each specific theme the physical background is first given, that is the available
experimental facts and their interpretation in terms of relatively simple theoretical models are
presented. Further, the interested reader may find a more detailed theoretical description and
review of the related literature.
The first six chapters give a concise but informative introduction to the basic knowledge in
surface and colloid science, which includes both traditional concepts and some recent results.
Chapters 1 and 2 are devoted to the basic theory of capillarity, kinetics of surfactant adsorption,
shapes of axisymmetric fluid interfaces, contact angles and line tension.
Chapters 3 and 4 present a generalization of the theory of capillarity to the case, in which the
variation of the interfacial (membrane) curvature contributes to the total energy of the system.
Phenomenological and molecular approaches to the description of the interfacial bending
moment, the curvature elastic moduli and the spontaneous curvature are presented. The
generalized Laplace equation, which accounts for the latter effects, is derived and applied to
determine the configurations of free and adherent biological cells; a convenient computational
procedure is proposed.
Chapters 5 and 6 are focused on the role of thin liquid films and hydrodynamic factors in the
attachment of solid and fluid particles to an interface. The particles stick or rebound depending
on whether repulsive or attractive surface forces prevail in the liquid film. Surface forces of
various physical nature are presented and their relative importance is discussed. In addition, we
consider the hydrodynamic interactions of a colloidal particle with an interface (or another
particle), which are due to flows in the surrounding viscous liquid. Factors and mechanisms for
detachment of oil drops from a solid surface are discussed in relation to washing.
Chapters 7 to 10 are devoted to the theoretical foundation of various kinds of capillary forces.
When two particles are attached to the same interface (membrane), capillary interactions,
mediated by the interface or membrane, may appear between them. Two major kinds of
capillary interactions are described: (i) capillary immersion force related to the surface
wettability and the particle confinement into a liquid film (Chapter 7), (ii) capillary flotation
force originating from interfacial deformations due to particle weight (Chapter 8). Special
attention is paid to the theory of capillary immersion forces between particles entrapped in
spherical liquid films (Chapter 9). A generalization of the theory of immersion forces allows
vi
one to describe membrane-mediated interactions between protein inclusions into a lipid bilayer
(Chapter 10).
Chapter l l is devoted to the theory of the capillary bridges and the capillary-bridge forces,
whose importance has been recognized in phenomena like consolidation of granules and soils,
wetting of powders, capillary condensation, long-range hydrophobic attraction, bridging in the
atomic-force-microscope measurements, etc. The treatment is similar for liquid-in-gas and gas-
in-liquid bridges. The nucleation of capillary bridges, which occurs when the distance between
two surfaces is smaller than a certain limiting value, is also considered.
Chapter 21 considers solid particles, which have an irregular wetting perimeter upon
attachment to a fluid interface. The undulated contact line induces interfacial deformations,
which are theoretically found to engender a special lateral capillary force between the particles.
Expressions for the dilatational and shear elastic moduli of such particulate adsorption
monolayers are derived.
Chapter 31 describes how lateral capillary forces, facilitated by convective flows and some
specific and non-specific interactions, can lead to the aggregation and ordering of various
particles at fluid interfaces or in thin liquid films. Recent results on fabricating two-
dimensional (2D) arrays from micrometer and sub-micrometer latex particles, as well as 2D
crystals from proteins and protein complexes are reviewed. Special attention is paid to the
methods for producing ordered 2D arrays in relation to their physical mechanisms and driving
forces. A review and discussion is given about the various applications of particulate 2D arrays
in optics, optoelectronics, nano-lithography, microcontact printing, catalytic films and solar
cells, as well as the use of protein 2D crystals for immunosensors and isoporous ultrafiltration
membranes, etc.
Chapter 41 presents applied aspects of the particle-surface interaction in antifoaming and
deJoaming. Three different mechanisms of antifoaming action are described: spreading
mechanism, bridging-dewetting and bridging-stretching mechanism. All of them involve as a
necessary step the entering of an antifoam particle at the air-water interface, which is
equivalent to rupture of the asymmetric particle-water-air film. Consequently, the stability of
the latter liquid film is a key factor for control of ~baminess.
The audience of the book is determined by the circle of readers who are interested in
systems, processes and phenomena related to attachment, interactions and ordering of particles
at interfaces and lipid membranes. Examples for such systems, processes and phenomena are:
formation of 2D ordered arrays of particulates and proteins with various applications: from
optics and microelectronics to molecular biology and cell morphology; antifoaming and
defoaming action of solid particles and/or oil drops in house-hold and personal-care
detergency, as well as in separation processes; stabilization of emulsions by solid particles with
application in food and petroleum industries; interactions between particulates in paint films;
micro-manipulation of biological cells in liquid films, etc.
Consequently, the book could be a useful reading for university and industrial scientists,
lecturers, graduate and post-graduate students in chemical physics, surface and colloid science,
biophysics, protein engineering and cell biology.
vii
.yrotsiherP An essential portion of this book, Chapters 7-10 and ,31 summarizes results
and research developments stemming from the Nagayama Protein Array Project (October 1990
- September 1995), which was a part of the program "Exploratory Research for Advanced
Technology" (ERATO) of the Japanese Research and Development Corporation (presently
Japan Science and Technology Corporation). The major goal of this project was formulated as
follows: Based on the molecular assembly of proteins, to fabricate macroscopic structures (2D
protein arrays), which could be useful in human practice. The Laboratory of Thermodynamics
and Physicochemical Hydrodynamics (presently lab. of Chemical Physics and Engineering)
from the University of Sofia, Bulgaria, was involved in this project with the task to investigate
the mechanism of 2D structuring in comparative experiments with colloid particles and protein
macromolecules. These joint studies revealed the role of the capillary immersion forces and
convective fluxes of evaporating solvent in the 2D ordering. In the course of this project it
became clear that the knowledge of surface and colloid science was a useful background for the
studies on 2D crystallization of proteins. For that reason, in 1992 one of the authors of this
book (K. Nagayama) invited the other author (P. Kralchevsky) to come to Tsukuba and to
deliver a course of lectures for the project team-members entitled: "Interfacial Phenomena and
Dispersions: toward Understanding of Protein and Colloid Arrays". In fact, this course gave a
preliminary selection and systematization of the material included in the introductory chapters
of this book (Chapters 1 to 6). Later, after the end of the project, the authors came to the idea to
prepare a book, which is to summarize and present the accumulated results, together with the
underlying physicochemical background. In the course of work, the scope of the book was
broadened to a wider audience, and the material was updated with more recent results. The
major part of the book was written during an 8-month stay of P. Kralchevsky in the laboratory
of K. Nagayama in the National Institute for Physiological Sciences in Okazaki, Japan
(September 1998 - April 1999). The present book resulted from a further upgrade, polishing
and updating of the text.
Acknowledgments. The authors are indebted to the Editor of this series, Dr. Habil.
Reinhard Miller, and to Prof. Ivan B. Ivanov for their moral support and encouragement of the
work on the book, as well as to Profs. Krassimir Danov and Nikolai Denkov for their expert
reading and discussion of Chapters 1 and ,41 respectively. We are also much indebted to our
associates, Dr. Radostin Danev and Ms. Mariana Paraskova, for their invaluable help in
preparing the numerous figures. Last but not least, we would like to acknowledge the important
scientific contributions of our colleagues, team-members of the Nagayama Protein Array
project, whose co-authored studies have served as a basis for a considerable part of this book.
Their names are as follows. From Japan: Drs. Hideyuki Yoshimura, Shigeru Endo, Junichi
Higo, Tetsuya Miwa, Eiki Adachi and Mariko Yamaki; From Bulgaria: Drs. Nikolai Denkov,
Orlin Velev, Ceco Dushkin, Anthony Dimitrov, Theodor Gurkov and Vesselin Paunov.
October 2000 Peter A. Kralchevsky and Kuniaki Nagayama
CHAPTER 1
PLANAR FLUID INTERFACES
An interface or membrane is one of the main "actors" in the process of particle-interface and
particle-particle interaction at a fluid phase boundary. The latter process is influenced by
mechanical properties, such as the interfacial (membrane) tension and the surface (Gibbs)
elasticity. For interfaces and membranes of low tension and high curvature the interracial
bending moment and the curvature elastic moduli can also become important. As a rule, there
are surfactant adsorption layers at fluid interfaces and very frequently the interfaces bear some
electric charge. For these reasons in the present chapter we pay a special attention to surfactant
adsorption and to electrically charged interfaces.
Our purpose is to introduce the basic quantities and relationships in mechanics,
thermodynamics and kinetics of fluid interfaces and surfactant adsorption, which will be
further currently used throughout the book. Definitions of surface tension, interfacial bending
moment, adsorptions of the species, surface of tension and equimolecular dividing surface,
surface elasticity and adsorption relaxation time are given. The most important equations
relating these quantities are derived, their physical meaning is interpreted, and appropriate
references are provided. In addition to known facts and concepts, the chapter presents also
some recent results on thermodynamics and kinetics of adsorption of ionic surfactants. Four
tables summarize theoretical expressions, which are related to various adsorption isotherms and
types of electrolyte in the solution. We hope this introductory chapter will be useful for both
researchers and students, who approach for a first time the field of interracial science, as well
as for experts and lecturers who could find here a somewhat different viewpoint and new
information about the factors and processes in this field and their interconnection.
2 Chapter 1
1.1. MECHANICAL PROPERTIES OF PLANAR FLUID INTERFACES
1.1.1. THE BAKKER EQUATION FOR SURFACE TENSION
The balance of the linear momentum in fluid dynamics relates the local acceleration in the fluid
to the divergence of the pressure tensor, P, see e.g. Ref. 1 :
dv
,o~=-V.P (1.1)
dt
Here o, is the mass density of the fluid, v is velocity and t is time; in fact the pressure tensor P
equals the stress tensor T with the opposite sign: P = -T. In a fluid at rest v - 0 and Eq. (1.1)
reduces to
V.P=O (1.2)
which expresses a necessary condition for hydrostatic equilibrium. In the bulk of a liquid the
pressure tensor is isotropic,
P=PsU (1.3)
as stated by the known Pascal law (U is the spatial unit tensor; P~ is a scalar pressure). Indeed,
all directions in the bulk of a uniform liquid phase are equivalent. The latter is not valid in a
vicinity of the surface of the fluid phase, where the normal to the interface determines a special
direction. In other words, in a vicinity of the interface the force acting across unit area is not the
same in all directions. Correspondingly, in this region the pressure tensor can be expressed in
the form 2,3:
P = Pr (exex +eyey)+PNeze: (1.4)
Here ,xe ye and ~e are the unit vectors along the Cartesian coordinate axes, with ze being
oriented normally to the interface; NP and Pr are, respectively, the normal and the tangential
components of the pressure tensor. Due to the symmetry of the system NP and Pr can depend
on ,z but they should be independent of x and .v Thus a substitution of Eq. (1.4) into Eq. (1.3)
yields one non-trivial equation:
NP9 =0 (1.5)
9z
ranalP diulF secafretTh 3
In other words, the condition for hydrostatic equilibrium, Eq. (1.3), implies that NP must be
constant along the normal to the interface; therefore, NP is to be equal to the bulk isotropic
pressure, NP = P8 = const.
Let us take a vertical strip of unit width, which is oriented normally to the interface, see Fig.
1.1. The ends of the stripe, at z = a and z = b, are supposed to be located in the bulk of phases
1 and 2, respectively. The real force exerted to the strip is
b
)laer(TF - I er (z)dz (1.6)
a
On the other hand, following Gibbs [4] one can construct an idealized system consisting of
two uniform phases, which preserve their bulk properties up to a mathematical dividing surface
modeling the transition zone between the two phases (Fig. 1.1). The pressure everywhere in the
idealized system is equal to the bulk isotropic z=b pressure, 8P .NP= In addition, a surface tension yc
laeR[ ,metsyS ] dezilaedI[ [metsyS
y e ~ l ~ z e
PT...PN
Phase 2
Phase 2
~."0 Z=Z 0
transition zone \
ecafrus !
[,,~ividing
.TP NP
Phase 1 Phase 1
z=a
Fig. 1.1. Sketch of a vertical strip, which si normal to the boundary between phases 1 and .2
4 Chapter I
is ascribed to the dividing surface in the idealized system. Thus the force exerted to the strip in
the idealized system (Fig. 1.1) is
b
FT idealized) = f PN de - rC (1.7)
a
The role of yc is to make up for the differences between the real and the idealized system.
Setting )dezilaedi(7/r- ---- ~er(v"7- from Eqs. (1.6) and (1.7) one obtains the Bakker [5] equation for the
surface tension"
+oo
r - I (PN - rP &) (1.8)
--oo
Since the boundaries of integration z = a and z = b are located in the bulk of phases 1 and 2,
where the pressure is isotropic (PT- PN), we have set the boundaries in Eq. (1.8) equal to +oo.
Equation (1.8) means that the real system with a planar interface can be considered as if it were
composed of two homogeneous phases separated by a planar membrane of zero thickness with
26
24
22
20
0
~ 18
)..e
Liquid, phase I Gas, phase II
;a~ 14
r"
M 10
% 8
4
2
0 ....... - -.
-2
-18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10
z-zv , Angstroms
Fig. 1.2. Anisotropy of the pressure tensor, AP, plotted vs. the distance to the equimoleqular dividing
surface, ,v.Z-:. for interface liquid argon-gas ta 84.3 ;K Curves 1 and 2 era calculated by the
theories ni Refs. [8] and [10].
Planar Fluid Interfaces 5
tension 6 given by Eq. (1.8). The latter equation gives a hydrostatic definition of surface
tension. Note that this definition does not depend on the exact location of the dividing surface.
The quantity
AP - NP - rP (1.9)
expresses the anisotropy of the pressure tensor. The function AaD(z) can be obtained
theoretically by means of the methods of the statistical mechanics [6-9]. As an illustration in
Figure 1.2 we present data for AP vs. Z-Zv for the interface liquid argon-gas at temperature T
= 84.3 K; vZ is the position of the so called "equimolecular" dividing surface (see Section 1.2.2
below for definition). The empty and full points in Fig. 1.2 are calculated by means of the
theories from Refs. [8] and [10], respectively. As seen in Fig. 1.2, the width of the transition
450 .................. ~ ............... , ......
...............................4.00. ............................................ ............... .).1.(...................................... ........................i ...
~ 300 .... i ! i
00~ i -i
.~ ..... ............ i ............ i ........... ........................ i ..... ~' .... .......................................i .............
0 I ~ 115000 i ..... ......i ..... ..........i ... ........i .... ........i .... ........! .... .j.i,../,.1,. .. . ! ? " ik...... ....." . ... ..........{ ... .......} ..... .....{ ...... ......i ..... i
~o ~ ~ i:: I .............. i
"~ ..................... ~ ........... ........................
~.~ o~- iiiiiiiiiiii. iiiiiiii
v -100
150
--.o-(1)
OO2
....... (2)
-250
-14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 01 1 ..... (3)
Angstroms
z-z v ,
Fig. 1.3. Anisotropy of the pressure tensor, ,PXz plotted vs. the distance to the equimoleqular dividing
surface, =-Zv, calculated by the theory ni Ref. [10] for the phase boundaries n-decane-gas
(curve ,)1 gas-water (curve 2) and n-decane-water (curve 3).
6 Chapter 1
zone between the liquid and gas phases (in which AP :/: 0) is of the order of 10 A. On the other
hand, the maximum value of the anisotropy AP(z) is about 2x 801 dyn/cm, i.e. about 200
atmospheres, which is an impressive value. The area below the full line in Fig. 1.2 gives the
surface tension of argon at that temperature, a = 13.45 mN/m, in accordance with Eq. (1.8).
Curves ,1 2 and 3 in Fig. 1.3 present AP(z) calculated in Ref. 10 for the interfaces n-
decane/gas, gas/water and n-decane/water, respectively. One see that kP(z) typically exhibits a
single maximum for a liquid-gas interface, whereas AP(z) exhibits a loop (maximum and
minimum) for a liquid-liquid interface. For all curves in Fig. 3.1 the width of the interfacial
transition zone is of the order of 10 A.
1.1.2 INTERFACIAL GNIDNEB MOMENTAND SURFACE FO NOISNET
To make the idealized system in Fig. 1.1 hydrostatically equivalent to the real system we have
to impose also a requirement for equivalence with respect to the acting force moments (in
addition to the analogous requirement for the acting forces, see above). The moment exerted
on the strip in the real system (Fig. 1.1) is
b
)laer(
M j PT (Z) zdz (1. lO)
O
Likewise, the moment exerted on the stripe in the idealized system is 11:
b
--)dez~aedi(M -f Pu zdz - OZo + 7 l B 0 1( 11)
a
Here z = 0z is the position of the dividing surface and 0B is an interfacial bending moment
(couple of forces), which is to be attributed to the dividing surface in order to make the
idealized system equivalent to the real one with respect to the force moments. Setting
M=~de~il'edi~M 'l''~'t fiom Eqs. (1.8), (1.10) and (1.11) one obtains an expression for the
interfacial bending moment:
P al n a r Flu id In te ifa ce s 7
~c,o
Bo -2 ~ (PN - iT ) (~.o - z)dz (1.12)
-co
As in Eq. (1.8) we have extended the boundaries of integration to +~,. From the viewpoint of
mechanics positive Bo represents a force moment (a couple of forces), which tends to bend the
dividing surface around the phase, for which xe is an outer normal (in Fig. 1.1 this is phase .)1
The comparison of eqs (1.8) and (1.12) shows that unlike o, the interfacial bending moment Bo
depends on the choice of position of the dividing surface .0z The latter can be defined by
imposing some additional physical condition; in such a way the "equimolecular" dividing
surface is defined (see Section 1.2.2 below). If once the position of the dividing surface is
determined, then the interfacial bending moment 0B becomes a physically well defined
quantity. For example, the values of the bending moment, corresponding to the equimolecular
dividing surface, for curves No. ,1 2 and 3 in Fig. 3.1 are, respectively 10: 0B = 2.2, 2.3 and
5.2x 01 -11N.
One possible way to define the position, ,0z of the dividing surface is to set the bending
moment to be identically zero:
I =0 (1.13)
Bo zo:z,
Combining eqs (1.8), (1.12) and (1.13) one obtains 2
+co
s.~ --- -IT )41.1~
7(
-co
Equation (1.14) defines the so called surface of tension . It has been first introduced by Gibbs
4, and it is currently used in the conventional theory of capillarity (see Chapter 2 below). At
the surface of tension the interface is characterized by a single dynamic parameter, the
interfacial tension ;yc this considerably simplifies the mathematical treatment of capillary
problems. However, the physical situation becomes more complicated when the interfacial
tension is low such is the case of some emulsion and microemulsion systems, lipid bilayers
and biomembranes. In the latter case, the surface of tension can be located far from the actual
transition region between the two phases and its usage becomes physically meaningless.
