Seetharaman S. Fundamentals of metallurgy
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5.6 References
Darken, L.S. (1949) Trans. A.I.M.E., vol. 180, 430.
Fuller, E.N., Schettler, P.D. and Giddings, J.C. (1966) Ind. Eng. Chem. Vol. 58 no. 5, 18.
Kucera, J. and Stransky, K. (1982) Mater. Sc. Eng., Vol. 52, 1.
Morita, Zen-ichiro (1996) ISIJ Int., Vol. 36, Supplement, S6.
Reid, R.C., Prausnitz, J.M. and Sherwood, T.K. (1977) Properties of Gases and Liquids,
3rd edn. McGraw-Hill, New York.
236 Fundamentals of metallurgy
6.1 Introduction
We have obtained great technological and academic benefits from describing the
nature of metals processing and also by controlling such processes with the aid
of thermodynamics and kinetics of bulk phases such as gas, slag and metal. In
order to achieve further progress in this area, it is necessary to understand the
surface and interface, and to clarify the participation of interfacial phenomena in
metallurgical processes. Surface and interfacial tensions of high temperature
melts such as slag and steel in iron and steelmaking processes are about 5 to 20
times as large as that of water. In addition, the high temperature melts have
remarkably strong surface active agents such as oxygen, sulfur, etc., which are
inevitably included in the melts, as shown in Section 6.3.
The subject of interfacial phenomena in relation to iron and steelmaking
processes has recently been one of the most attractive fields in metallurgy. There
are a number of reasons for this. Further understanding of the kinetics in
heterogeneous reactions in metallurgical systems requires more detailed
information on interfacial phenomena that occur during the progress of these
reactions. These phenomena can reveal more of the microscopic stages of
kinetics than did the conven tional treatments used in previous studies. Smelting
and refining processes developed or improved recently often include various
dispersion actions such as the injection of powder agents into liquid steel,
degassing of liquid steel with RH, bubble injection through porous plug
refractories and slag foaming in bath smelting or during pre-treatment of pig iron
in torpedo cars. Control of non-metallic inclusion during the steel refining
process is important for the production of much higher quality steels. Detailed
information on interfacial phenomena is essential to advance practical
operations in these processes.
It is very important to understand the fundamental treatments of the surface
and the interface by using interfacial physical chemistry in order to clarify the
interfacial phenomena and to apply the results to the improvement and develop-
ment of metallurgical processing. Interfacial physical chemistry used here
6
Interfacial phenomena, metals processing
and properties
K M U K A I , Kyushu Institute of Technology, Japan
means an academic field which treats interfacial phenomena by using surface
chemistry, thermodynamics and kinetics. This is the subject matter for Section
6.2.
In Section 6.3 typical interfacial properties obtained by previous investigators
are concisely introduced for the metallurgical systems containing liquid metal
and slag.
Section 6.4 focus es briefly on several interfacial phenomena which have been
recently shown to partic ipate or possibly participate in the processes of metal
smelting and refining. Many technological problems have been linked with the
interfacial phenomena.
6.2 Fundamentals of the interface
6.2.1 Thermodynamics of interface
Gibbs' dividing surface
Gibbs supposed a dividing surface S (see Fig. 6.1), in order to describe
macroscopic surface properties. The dividing surface is always taken parallel to
the surface of tension defined later in this section and the thickness of the
surface is taken as zero. Figure 6.1 shows concentration distribution of
component i, c
i
(mol/m
3
) in the real surface region and bulk phases of and .
n
i
is the total mole number of the component i, in the volume of V
V
. If we
suppose that c
i
(concentration in ( or phase)) keeps constant from bulk
6.1 Gibbs' dividing surface.
238 Fundamentals of metallurgy
phase to the dividing surface S, mole number of the component i in this area is
equal to c
i
V
. Then, n
s
i
given with equation 6.1 means excess mole number of
component i at surface and corresponds to the shaded portion of Fig. 6.1:
n
s
i
n
i
ÿ c
i
V
c
i
V
6 :1
Excess quantity per unit surface area, that is, n
s
i
=A is called a surface excess
quantity of component i, ÿ
i
(mol/m
2
), and expressed as equation 6.2:
n
s
i
Aÿ
i
6:2
where A is the surface area.
Similarly the other surface excess quantities are defined in equations 6.3 to
6.6:
U
s
U ÿ u
V
u
V
6:3
where U is internal energy and u
( ; ) is internal energy per unit volume
of the phase .
S
s
S ÿ s
V
s
V
6:4
where S is entropy and s
is entropy per unit volume of phase .
F
s
F ÿ f
V
f
V
6:5
where F is Helmholtz energy and f
is Helmholtz energy per unit volume of
phase .
The surface excess quantities mentioned above can be understood to be the
excess quantities which are altogether brought to the dividing surface with
infinitesimal thickness. Therefore those excess quantities, for example, n
s
i
, vary
with the position of the dividing surface within the real surface region as shown
in Fig. 6.1. The surface excess quantitiy is not the absolute value, besides that
depends on the position of the dividing surface. In order to solve the above-
mentioned inconveniences, Guggenheim introduced two dividing surfaces.
However, his treatment is complicated and nowadays is not so popular.
Surface tension
Thermodynamic interpretation
When the surface area A of pure liquid increases by dA, dU
s
, the change in
internal energy at the surface, is given by equatio n 6.6:
dU
s
dQ
s
dA 6:6
where dQ
s
is the heat absorbed by the surface, dA means the work which acts
on the surface at constant temperature by the change in the surface area dA, and
is the surface tension which is described later.
Interfacial phenomena, metals processing and properties 239
For a reversible process,
dQ
s
T dS
s
6:7
therefore
dU
s
T dS
s
dA 6:8
Surface excess Helmholtz energy F
s
, and its total differential, dF
s
, are expressed
by equations 6.9 and 6.10, respectively:
F
s
U
s
ÿ TS
s
6:9
dF
s
dU
s
ÿ TdS
s
ÿ S
s
dT 6:10
Equation 6.11 is derived from equations 6.8 and 6.10:
dF
s
dA ÿ S
s
dT 6:11
Therefore,
f
s
@F
s
@A
T
6:12
Equation 6.12 means that surface tension is the surface excess Helmholtz
energy per unit surface area. Therefore,
f
s
u
s
ÿ Ts
s
6:13
Equation 6.13 indicates that surface tension contains the entropy term, Ts
s
, as
well as the internal energy term, u
s
U
s
=A, that is known as surface energy.
The relation between f
s
and for an r -component liquid±gas system is given
by equation 6.14:
f
s
X
r
i1
ÿ
i
i
6:14
where
i
@G
@n
i
T; P; n
j
j1...r; j6i
is the chemical potential of the component i.
If we choose the dividing surface so as to make
X
r
i1
ÿ
i
i
0
f
s
is equal to from equation 6.14, even for an r-component liquid±gas system.
Position of dividing surface
It can be shown on thermodynamic grounds that the surface tension of a plane
surface at thermodynamic equilibrium is not dependent upon the position of the
240 Fundamentals of metallurgy
dividing surface. The surface tension of a curved surface, however, varies with
the position of the dividing surface as shown in equation 6.15 and Fig. 6.2.
s
r
2
s
3r
2
2r
3r
s
6:15
where r is the radius of the dividing surface. As show n in Fig. 6.2, the surface
tension has a minimum
s
at r r
s
. The dividing surface at r r
s
is called the
surface of tension. The surface of tension is within the surface layer (Fig. 6.2).
Therefore when r
s
is much larger than , should be regarded as almost constant
within the surface layer from equation 6.16:
s
1
2
O
3
6:16
where =r
s
.
Radius of curvature
The surface tension
s
at the surface of tension varies with the change in the
radius of curvature r
s
for the incompressible liquid as expressed with Tolman's
equation 6.17:
s
1
1=2
0
=r
s
1 6:17
6.2 Relation between surface tension and radius of the dividing surface.
Interfacial phenomena, metals processing and properties 241
where
1
is the surface tension at r
s
! 1 and
0
is the distance between
surface of tension and the dividing surface at which
P
r
i1
ÿ
i
i
0. If we take
the value
0
1:6 10
ÿ10
m for water,
s
=
1
is calculated as follows from
equation 6.17:
s
=
1
0:997 at r
s
100 nm,
s
=
1
0:97 at r
s
10 nm.
These results indicate that we should take account of the dependency of surface
tension on the radius of curvature for materials with the radii less than 100 nm
(10
ÿ5
cm), such as a droplet, nucleus at nucleation, etc. We should also notice
that the thermodynamic model cannot be expected to hold for a system
containing so few molecules, such as water droplets of radii smaller than 1 nm,
for example.
Temperature
For pure liquid, we can obtain equation 6.18 from equation 6.13:
d
dT
df
s
dT
ÿs
s
6:18
For pure liquid, d /dT has generally negative values, which indicate s
s
> 0.
s
s
> 0 means that entropy at the surface is larger than for the bulk liquid. Several
empirical relations between surface tension and temperature are propos ed, for
example, as expressed by equations 6.19 and 6.20:
k=v
1 2=3
T
c
ÿ T 6:19
where v
1
is molar volume, T
c
is critica l temperature, k is constant and is about
2.1 10
ÿ7
J/deg for most liquids. Equations 6.19 was proposed by EoÈtvoÈs in
1886. Equation 6.19 does not hold accurately in the vicinity of T
c
. Katayama and
Guggenheim proposed equation 6.20 in order to improve equation 6.19:
o
1 ÿ T=T
c
11=9
6:20
where is constant and depends on the kind of liquid.
Surface stress
In the case of a solid, we have to distinguish between surface tension and surface
stress. Surface tension is defined in terms of work, dA (see equation 6.6)
retaining the same surface condition, such as keeping the number of atoms at the
unit surface area constant. Another way to expand the surface by dA is to stretch
the distance between atoms at the surface. In this case the number of atoms at
the surface is kept constant and the work is expressed by gdA. g is called surface
stress, and is related to surface tension by equation 6.21 (Shuttleworth's
equation):
g @=@!
N
6:21
242 Fundamentals of metallurgy
where ! is the elastic surface strain and N is the total number of atoms at the
surface. Since a liquid surface deforms in a completely plastic manner, g is equal
to for the liquid surface.
6.2.2 Mechanical aspects of surface tension
Mechanical definition
As pointed out by Young, a system composed of two fluids such as liquid and
gas which contact each other, behaves, from a mechanical standpoint, as if it
consisted of two homogeneous fluids separated by a unifomly stretched
membrane of infinitesimal thickne ss. Surface tension has been defined from a
macroscopic standpoint as the tractional force, , acting across any unit length of
line on this fictitious membrane. Surface tension has the dimensions of force per
unit length and is usually expressed in Nm
ÿ1
.
The real surface in the above mentioned system has a finite thickness, that is,
the surface layer (or region) as shown in Fig. 6.3. For this system, the surface
tension is defined from a mechanical standpoint by equation 6.22 (the Bakker
equation):
Z
1
ÿ1
p
N
ÿ p
T
zdz 6:22
where p
N
is normal pressure to the surface and equal to p, the hydrostatic
pressure in the system, p
T
z is tangential pressure. p
T
z exerts normally across
a plane parallel to the z axis and varies with the value of z as shown in Fig. 6.3,
6.3 Variation of tangential pressure P
T
(z) with z.
Interfacial phenomena, metals processing and properties 243
while p is required, from the condition of hydrostatic equilibrium, to be the same
value even in the surface layer. If we can describe p
T
z from the molecular
distribution functions of the fluid by using `molecular dynamics', for example,
the surface tension can be rigorously expressed with the aid of equation 6.22.
Surface tension, , as a force of traction, can be available even for the system
which is not in a thermodynamic equilibrium state.
Laplace's equation
When a spherical surface of radius with curvature r maintains mechanical
equilibrium between two fluids and phases at different pressures p
and p
and the interface is assumed to be of zero thickne ss, the condition for
mechanical equilibrium provides a simple relation between p
and p
:
p
ÿ p
2=r 6:23
Equation 6.23 is known as the Kelvin relation.
If, instead of a spherical surface, we consider any surface, the condition of
mechanical equilibrium at each point in the surface is given by Laplace's
equation (6.24):
p
ÿ p
1
r
1
1
r
2
6:24
where r
1
and r
2
are the two principal radii of curvature of the surface as shown
in Fig. 6.4.
b
p
a
p
r
1
r
2
6.4 Pressure difference between phase and in mechanical equilibrium.
244 Fundamentals of metallurgy
Laplace's equation (6.24) provides a fundamental base for measuring surface
tension in static state such as the capillary rise method, the sessile drop method,
the pendant drop method, the maximum bubble pressure method, etc.
Marangoni effect
Surface tension as a tractional force is available for describing the behaviour of
the syst em which is not in a thermodynamical equilibrium state. Th e surface (or
interfacial) tension difference, or gradient, on the surface (or interface) of liquid ,
for example, in the direction x, can change the motion of liquid due to the
surface shear stress
s
, written as
s
d
dx
@
@T
dT
dx
@
@c
dc
dx
@
@'
d'
dx
6:25
Equation 6.25 indicates that the surface (or interfacial) tension gradient is caused
by the gradients of temperature T, concentration c of the surface active
component in the liquid and electric potential ' at the interface between two
liquids.
In hydrodynamics, the surface (or interfacial) tension difference or gradient
participating in the above dynamics is called the Marangoni effect.
1
The motion
of liquid induced by the Marangoni effect is called Marangoni flow or
Marangoni convection. The non-dimensional number defined by equations 6.26
and 6.27 is called the Marangoni number, M
a
, which characterizes the
Marangoni convection:
M
a
@=@T T L=a 6:26
M
a
@=@c c L=D 6:27
where T is temperature difference, L is the characteristic length of the system,
a is thermal diffusivity, is viscosity, c is the concentration difference of the
surface active component and D is the diffusion coefficient of the surface active
component.
Since liquid metals and slags generally have high surface or interfacial
tension and also have strong surface active components such as oxygen and
sulfur in liquid iron (see Section 6.3.1), both factors are favorable to the
occurrence of Marangoni convection in systems where these are present.
Even in the field of gravity on the Earth, occurrences of the Marangoni effect in
metallurgical systems have been observed in the following: (a) Marangoni
convection of molten silicon
2
and salts
3
due to temperature gradient, (b) spreading
and shrinking of slag droplets on the metal due to changes in applied potential,
4
and (c) Marangoni flow of slag film
5
and metal surface
6,7
due to concentration
gradient, which is described in further detail in Sections 6.4.1 and 6.4.2.
Since the Marangoni convection is most intensive at and around liquid±gas
and liquid±liquid interfaces, it effectively promotes mass transfer in the region
Interfacial phenomena, metals processing and properties 245