Table Of ContentThis is Volume 19 in
PHYSICAL CHEMISTRY
A series of monographs
Edited by ERNEST M. LOEBL, Polytechnic Institute of Brooklyn
A complete list of the books in this series appears at the end of the volume.
Phase Equilibria
Basic Principles,
Applications,
Experimental Techniques
Arnold Reisman
IBM Watson Research Center
Yorktown Heights, New York
Academic Press
New York and London 1970
COPYRIGHT © 1970, BY ACADEMIC PRESS, INC
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PREFACE
Until the mid to late 1950's, Phase Equilibria was a topic of concern
primarily to the classical metallurgist. The emphasis, naturally, was
then aimed at understanding the behavior of metal alloy phenomena and,
indeed, was heavily oriented to the steel industry. With the advent of
the electronics, nuclear, and space ages, however, we have witnessed
an increasing involvement in this field by the physicist, the physical
chemist, the inorganic and organic chemist, the device oriented electrical
engineer, and the interdisciplinary product of these ages, the materials
scientist. Accordingly, the contributions to the literature by this
grouping have increased enormously, with the emphasis shifted to
nonmetallic-compound and solid-solution systems. In addition, where
previously the domain of interest of the classical metallurgist was
confined mainly to interactions among condensed phases, the recent
entrants in the field, including the modern metallurgist, have been
confronted by a need to better understand condensed-vapor phase
behavior. This need exists because many systems of interest to them are
volatile, or employ the vapor as a vehicle for the transport, synthesis,
purification, and single crystal growth of materials.
It is because of this change in emphasis and need, as well as the fact
that existing treatments are basically descriptive in nature that the
present work was undertaken. Analytical treatments, for example, have
in the past been provided in texts on chemical thermodynamics. These
have in general focused on the phenomenological approaches involving
activities and fugacities, and therefore, on the largely defensive concepts
of ideal and nonideal behavior. In addition, except for advanced thermo-
dynamic treatments, heterogeneous equilibria has been treated generally
in a cursory fashion. Books devoted to phase equilibria exclusively have,
for the most part, minimized analytical concepts. Instead, the subject
has been developed graphically. Even a topic as widely employed by
the experimentalist as the lever arm principle is almost always presented
without rationalization.
Xlll
XIV
PREFACE
The present treatment has been formulated with these and related
considerations in mind. It has been organized in a fashion that affords
the theoretical elements sufficient emphasis for the casual participant
to understand at least the principles involved, and the serious worker
may be sufficiently motivated to delve further. The method of species
model systems is employed broadly following introduction of pertinent
thermodynamic concepts. In a number of chapters, computer analysis of
derived relationships is presented to provide graphical visualization
of the consequences of species changes within coexistent phases. The
chapter arrangements have been ordered so that in a course of formal
instruction the presentation may include or omit the more mathematically
oriented material depending upon the needs of the audience. This
feature has been utilized successfully in formal presentations by the
author within the IBM Research Professional Education Program.
The subject content includes the substance of classical treatments of
the subjects and also more modern aspects such as concern with con
densed phase-vapor phase equilibria and vapor transport reactions,
zone refining techniques, nonstoichiometry, among others. Further,
a fairly extensive treatment of experimental techniques is provided in
the text and appendix. In addition, the appendix presents additional
commentary on specialized topics,* as well as selected general and
specific references to major areas treated within the body of the text.
It was the intent to provide a reference work for the involved experi
mentalist, a step by step introductory graduate text for departments of
chemistry, metallurgy, and materials science. While it would be presump
tuous to assume that all these needs have been met, it is hoped that
together with existing treatises, obvious voids have been reasonably
filled.
* For example, a fairly detailed discussion of the effects of vacancy concentration on the
dissociation pressure of binary compounds (defect structures) is given in the appendix on
p. 524 et seq. Other topics, e.g., phase transformations, Vegard's law, crystal growth,
vapor pressure, evaporation rate, and accommodation coefficient, are also discussed in
some detail with appropriate references.
ACKNOWLEDGMENTS
During the several years that this manuscript was being formulated,
a number of the author's associates at IBM provided invaluable assistance
without which the complicated undertaking would probably have never
seen fruition. During the early stages, the late Dr. G. Mandel acted as
a sounding board for a number of the proposed approaches and it was
through these interactions that the current format began to take shape.
Mr. G. Cheroff devoted many hours to a critical evaluation of the
mathematical approaches, and in addition offered numerous suggestions
which were subsequently incorporated. Mrs. S. A. Papazian provided
much needed help in validating derivations and her contributions as
a co-worker in the evaluation of the theoretical approach employed in
Chapter 34 for the description of multicomponent systems were key
ones. The application of the model equations developed in Chapters
16-20 and 27-31 to the generation of hypothetical phase diagrams using
computer techniques is due in large measure to Mrs. J. E. Landstein.
Her patience in the face of innumerable obstacles, both those of a
mathematical nature and those due to the author's inadequacies, has
been commendable. Appreciation is due also to the late Dr. H. R.
Leonhardt for many stimulating discussions and to Dr. T. O. Sedgwick
for many excellent suggestions about improvement of the presentations
in the introductory chapters. The author is particularly indebted to
Mrs. M. H. Lindquist for her tireless efforts in bringing the manuscript
through its several drafts, then in putting it into final form ; proofreading,
correcting, typing, and what have you. Finally, the author is grateful to
the IBM Corporation for providing the extensive time and necessary
resources.
XV
I
Some Preliminary Remarks and Observations
The starting point for all that ensues is the body of accumulated
knowledge concerned with interrelationships among work, heat, and
energy for macroscopic systems existing in a state of chemical and
physical equilibrium. From this body of knowledge, known as thermo
dynamics, a mathematical generalization called the "phase rule" was
first deduced by Willard Gibbs. This equation functions usefully as a
constraint on the final state of almost any conceivable macroscopic
chemical or physical process. In terms of the parameters, temperature,
pressure, and the number of independent chemical variables, none of
which is dependent on the total quantity of material under scrutiny, one
is able to predict without exception the total number of variables that
must be specified in order to define uniquely the state of the system.
This total number of variables is termed the degrees of freedom possessed
by the system. Specifically, for a system possessing a specified number
of independently variable chemical quantities (system components)
present in one or more distinctly identifiable phases, all at a defined
temperature and pressure, one is able to predict precisely the total
number of variables that may be varied independently of the others,
namely, how many may be varied simultaneously without changing the
physical state of the system. Once this number of required variables is
1
2 1. SOME PRELIMINARY REMARKS AND OBSERVATIONS
specified, all other intensive properties of the system, i.e., density,
thermal conductivity, etc., are uniquely, if not explicitly, defined. The
equation known as the phase rule then, is not an equation of state that,
for example in the case of an ideal gas, relates what change in pressure
results from a specified change in temperature at constant composition.
It functions in more general terms, and defines whether the temperature
may or may not be varied in the first place without causing a change in the
number of phases present. It tells us that by varying only the pressure, or
only the temperature, or only the composition, or any two of these, or all
three of them (if such is an allowable condition), the remaining
parameters will dependency acquire unique values.
Based on the implications of universality inherent in the phase rule,
an extensive field of study has developed whose impact is felt but
frequently not realized in many outwardly unrelated areas. The designa
tion given an equation of thermodynamic origin has thus become
synonymous with a broad field of intellectual and experimental endeavor
having considerable practical significance. Since the practical implications
are of greatest import in situations where the properties of nonhomo-
geneous (heterogeneous) agglomerations of matter are under scrutiny,
alternative names such as phase equilibria and heterogeneous equilibria
have been invoked frequently to characterize the subject. Furthermore,
as the culmination of an effort in the field is in the quantitative description
of the variations in temperature, pressure, and composition of a given
system, such description being amenable to graphical capsulization in the
form of a phase diagram or diagram of state, the name phase diagrams
has also been employed as a description of the field.
In any event, all of these perfunctory designations are somewhat
misleading since they imply a concern with a restricted and self contained
area of activity which presumably is divorced from other areas of scientific
endeavor. Coincidentally, then, one finds that phase equilibria is treated
as a separate, unrelated topic in existing treatises on thermodynamics and
physical chemistry. The role of the phase rule in treatments of analytical,
organic, and inorganic chemistry is almost never mentioned, and workers
in these fields frequently entertain only vague notions of its existence or
implications concerning their particular activity. This state of affairs is
certainly surprising when it is realized that there does not exist a single
area of materials studies, that comes to mind, that does not depend on
a sound knowledge of the phase rule and its ramifications. For example,
it is generally the case that the practical and theoretical bases for analytical
gravimetry, inorganic and organic synthesis, physical measurements of
systems in equilibrium, and, for that matter, in transit toward
equilibrium, are bounded by the constraints of the phase rule.
1. SOME PRELIMINARY REMARKS AND OBSERVATIONS 3
To be more specific, consider, for example, the precipitation of an
insoluble sulfate in an analytical procedure, the melting of snow by
deposition of calcium chloride, the fractional distillation or crystallization
of a mixture of organic or inorganic materials, the zone refining and the
single crystal growth of an ingot of germanium, silicon, or other semi
conductor from the melt, the synthesis of a compound of defined
stoichiometry from two or more starting materials, the manufacture of
synthetic gems, the dehydration of a hydrate, the measurement of the
vapor pressure of a compound formed from one or more elements, or
the fabrication of a transistor via an alloying process. The above and
hundreds of other seemingly unrelated phenomena, e.g., the making of
potable water from sea water, are all describable using only a very few
generalized principles concerned with phase equilibria. The limiting
states of such processes are graphically depictable in a phase diagram
and in principal at least, these diagrams can be described analytically
by equations whose parameters include many of the useful variables of
physics and chemistry, e.g., temperature, pressure, composition, energy,
electrical conductivity, thermal conductivity, compressibility, solubility,
etc.
While heretofore, the classical metallurgist, almost uniquely, has been
exposed to an extensive formalism in the tenets of phase equilibria,
and has then alone contributed to the available literature, such is no
longer the case. In recent years the hybrid known as the materials
scientist has contributed increasingly to the literature on phase equilibria.
Unfortunately, because of his generally deficient formalism in the subject,
too often one encounters implausible conclusions based on imperfectly
defined experimental procedures.
The enhancement of activity in this field with the advent of the
electronics and space ages has indeed provided the activation energy
necessary for the writing of this monograph. Hopefully, it will serve the
function of filling the void that exists between general texts concerned
primarily with the thermodynamics of homogeneous systems and those
concerned with purely descriptive introductions to the field of phase
equilibria. Hopefully also, it will provide, together with the extant
formal treatments of the subject, a sufficiently useful tool to allow for
wider treatment of this technologically significant field in departments of
chemistry, physics, ceramics, materials science, etc.
2
Thermodynamics and the Phase Rule
The science of thermodynamics represents an attempt to summarize
analytically the sum total of man's experiences with respect to processes
involving energy changes attending the interplay of heat and work.
Classical thermodynamics is concerned solely with the initial states
from which such processes originated and the final state in which
they terminate. It is not concerned at all with any transient aspects of
the processes involved. Based on the underlying principle that an energy
change in a given process confined to a given system is the same,
irrespective of the pathway followed in achieving the final state, a network
of relationships has evolved which enables one to predict whether
processes can occur spontaneously and what energy changes are
associated with such processes. Thermodynamics does not have the time
element as one of its parameters. Consequently, the knowledge that a
process can take place does not imply that the process will take place in a
practical time interval. The transient aspects of processes occurring
within the boundary or equilibrium states considered in thermodynamics
lie within the realm of the science of kinetics, and one may bridge the
gap between thermodynamics and kinetics by invoking certain assump
tions between rate and equilibrium parameters as the transient
occurrences approach the limiting boundaries. These assumptions
