Structure of Liquids / Struktur der Flüssigkeiten

325 Pages·1960·13.13 MB·English

by  H. S. Green, Syu Ono, Sohei Kondo, Frank P. Buff (auth.)

ENCYCLOPEDIA OF PHYSICS
EDITED BY
S. FLOGGE
VOLUME X
STRUCTURE OF LIQUIDS
WITH 41 FIGURES
SPRINGER-VERLAG
BERLIN· GOTTINGEN . HEIDELBERG
1960
HANDBUCH DER PHYSIK
HERAUSGEGEBEN VON
S. FLOCCE
BAND X
STRUKTUR DER FLOSSICKEITEN
MIT 41 FI GUREN
SPRINGER-VERLAG
BERLIN· GOTTINGEN . HEIDELBERG
1960
ISBN-13: 978-3-642-45949-8 e-ISBN-13: 978-3-642-45947-4
DOl: 10.1007/978-3-642-45947-4
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Contents.
Page
The Structure of Liquids. By Dr. HERBERT S. GREEN, Professor of Mathematical Phy-
sics, University of Adelaide (South Australia). (With 7 Figures)
I. General nature of liquid structure.
a) Molecular constitution of liquids
b) Liquid models 7
c) Actual structure of liquids. 15
II. The quantitative description of liquid structure 25
a) Distribution functions and their properties. 25
b) Molecular mechanics and liquid structure 33
III. Structure of uniform liquids 47
a) The radial distribution function 47
b) Structure-dependent properties. 61
c) Structure in the surface zone. 77
IV. Structure of non-uniform liquids Hi
a) Effect of irreversible processes Hi
b) Deformation of structure by viscosity. H9
c) Theory of non-uniform liquids 96
d) KIRKWOOD'S theory of dissipative processes 112
V. Structure of quantum liquids. 119
a) Liquid helium 119
b) Quantum theory of .structure 126
Molecular Theory of Surface Tension in Liquids. By Dr. SYU ONO, Associate Pro
fessor of Physics, University of Tokyo and Dr. SOHEl KONDO, Chief on Radiation
Laboratory, National Institute of Genetics, Misima, Sizuoka-ken. (Japan). (With
32 Figures) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
A. Thermodynamics and quasithermodynamics 136
I. Thermodynamics . . . . . . . . . . 136
a) Thermodynamic quantities of interface layer. 136
b) Plane interface. . 138
c) Spherical interface 145
II. Hydrostatic approach 157
III. Quasithermodynamics. 163
IV. Application of thermodynamics of irreversible processes 16H
V. Empirical equations for temperature dependence of surface tension 172
B. Statistical mechanics 177
I. Statistical thermodynamic method 177
a) Canonical ensemble. . . 177
b) Grand canonical ensemble 190
VI Contents.
Page
II. Mechanical definition of surface tension 208
III. Numerical calculations. 216
a) Pure liquid . . . . 216
b) Electrolyte solution. 223
c) Gas adsorption on a solid surface. 231
IV. Quantum statistical mechanics 237
C. Lattice theory approaches 240
I. Pure liquids . . . . 240
a) Free volume theory. 240
b) Hole theory . . . . 249
II. Solutions of non-electrolytes 262
a) Regular solutions. 262
b) Polymer solutions 272
References. . . . . . . . 277
The Theory of Capillarity. By Dr. FRANK P. BUFF. Associate Professor of Chemistry,
University of Rochester, New York (USA). (With 2 Figures). 281
Introduction . . . . . . . . . . . . . .. 281
A. General theory 281
B. Applications . 295
General references. 304
Sachverzeichnis (Deutsch-Englisch) . 305
SUbject Index (English-German) . . 313
The Structure of Liquids. *
By
H. S. GREEN.
With 8 Figures.
1. General nature of liquid structure.
a) Molecular constitution of liquids.
1. Introduction. A large part of modern physics is concerned with the relations
between the physical behaviour of a composite system and that of its constituent
parts. It is desired, for example, to explain the properties of an atomic nucleus
in terms of the elementary neutrons and protons of which it is believed to consist;
to explain the properties of an atom in terms of the nucleus and electrons from
which it is constructed; to explain the properties of a molecule in terms of its
constituent atoms; and finally to explain the properties of macroscopic matter
in terms of its molecular structure. The kinetic theory of matter, and of liquids
in particular, is designed to provide the last link in this chain.
The object of relating the physical behaviour of a liquid with the properties
of the molecules which it contains is two-fold. In the first instance, it may happen
that the structure of the individual molecules of the liquid and the nature of their
mutual interactions, can be determined from purely theoretical considerations;
this is so in the case of hydrogen, helium, neon, argon, etc. Then, by study of
molecular structure, one can hope to predict macroscopic properties of the
corresponding liquid, such as its equation of state, its coefficients of viscosity
and thermal conduction, etc. In the second instance, the molecular constitution
may be too complicated to be determined by theoretical considerations alone.
Then the unknown features can often be inferred from the observed macroscopic
behaviour of the substance, with the help of the general molecular theory which
has been developed. In addition, one may hope to reach a more profound under
standing of the fundamental mechanism governing the behaviour of liquids.
The molecular theory of the liquid state is very much more difficult than that
of the gaseous or solid state. This could be expected from a purely superficial
comparison of the physical behaviour of the phases. There is no parallel for
liquids of the gas laws, or DEBYE'S law concerning the specific heat of solids.
Instead one is confronted with anomalies such as the negative coefficient of
expansion of water near its melting point. Some liquids obey ANDRADE'S law
of viscosity precisely, while others deviate from it very widely. There is no
simple way of representing the coefficient of thermal conduction of any wide
class of liquids. Obviously there is little hope of describing in the same way the
behaviour of the glasses at low temperatures, and other liquids near their critical
point. It is the thesis of this and the following chapters that such complexities
of physical behaviour correspond very closely to complexities of molecular
* The original of this contribution was communicated in April, 1956. Important work
published after this date has been mentioned in additions to Sects. 21 ce, 23 ce and 23, made
in proof.
Handbuch der Physik, Ed. X.
2 H. S. GREEN: The Structure of Liquids. Sect. 2.
structure. Regularities of behaviour will be found only within groups of liquids
with similar molecular constitution.
The group of liquids which offer the most favourable conditions for theoretical
study is that consisting of the noble gases in their condensed state, particularly
liquid neon, liquid argon and liquid krypton. The spherical symmetry and
lack of reactivity of the atoms of these substances reduces the intricacies of
molecular structure to a minimum, but even so the quantitative treatment of
this structure remains one of the most difficult problems of statistical mechanics.
Next in complexity come substances like liquid nitrogen and methane, with
molecules which are chemically inert, though they do not possess perfect spherical
symmetry. Some of the properties of liquids-the specific heat and thermal
conductivity are the main exceptions-do not depend sensitively on the structure
of the individual molecules, which can therefore be disregarded when such proper
ties are considered. This is unfortunately no longer true if the departure from
spherical symmetry is great, as in the long chain-like molecules of the higher
homologues of the paraffin series. Nor is it true of molecules which have a dipole
or quadrupole moment, still less of molecules with localized centres of attraction
for neighbouring molecules. The molecular theory of liquids of these types must
for some time be either rather formal and qualitative, or else based on models
so crude that their permanent value is very doubtful.
A quite tractable generalization of the theory of simple liquids with inert,
spherically symmetrical molecules can be applied to mixtures of liquids of the
same type. Here one is interested in the dependence of the physical properties
of the mixtures on the concentrations of its components, and also in the possi
bility of additional types of irreversible processes: diffusion and thermal diffusion,
the last of which is called the Soret effect in liquids. Just as in simple fluids,
such macroscopic phenomena must be related to the underlying molecular
structure.
The molecular structure of a liquid obviously depends to some extent on
externally imposed conditions, such as temperature and pressure. Equally
obviously, it must depend on the nature of the molecules themselves: not simply
on their structure, but on their mutual interactions. In this work, it is necessary
to consider not so much how the intermolecular forces arise as what their effect
is on the disposition of the molecules relative to one another. However, these
questions are not quite separate from one another, and it will be found useful
to pay some attention to the first before proceeding to a detailed study of liquid
structure.
2. Intermolecular forces. On a sub-microscopic scale, a liquid may be regarded
as a giant assembly of atomic nuclei and electrons. These do not form isolated
molecular groups, as in gases; on account of the greater density, the chances
of finding such a group which is not in continual interaction with other neighbour
ing groups is negligible. From a certain point of view, therefore, the liquid con
stitutes one giant molecule. However, the existence of certain atomic groups,
which remain essentially unchanged, in spite of the kinetic motion of the individual
particles, allows one to regard the liquid as truly composite. The molecules are
those irreducible collections of atoms which are very rarely, if ever, mutually
separated, through a wide range of temperature. One does not, accordingly,
regard as a single molecule a structure which, though sufficiently permanent at
low temperatures, is broken up by the kinetic motion when the temperature is
somewhat raised.
Sect. 2. Intermolecular forces. 3
However, the concept of separate molecular groups of nuclei and electrons
is sometimes of restricted value when one wishes to consider the interaction of
two or more molecules, since the electrons, at least those which do not form closed
shells, have to be regarded as common property of the interacting molecules.
The interaction energy has to be computed on a quantum mechanical basis,
which takes into account not only the classical Coulomb forces between charged
particles, but also the" exchange" forces required by quantum statistics. The
difficult problems which consequently arise are considered elsewhere1; the follow
ing, somewhat simplified picture emerges.
The interaction between inert molecules, which are not ionized or polarized
and do not possess unsaturated chemical bonds, is of the simplest type: it is,
on the average, attractive at sufficiently great intermolecular distances, owing
to the mutual polarization of different molecules. The attraction increases as
the molecules approach one another, the negative potential energy between two
molecules increasing approximately as the inverse sixth power of their distance
apart. At shorter distances, however, a repulsive force arises, which finally
becomes the dominant feature of the intermolecular field. This is due to the
repulsion between the outer closed electron shells of the molecules concerned,
and is very great at short distances, but decreases exponentially with their
distance apart. Probably the best simple approximation to the interaction
cp (r) between two spherically symmetrical molecules whose mass-centres are at
+
distance r apart is cp (r) = _ fl r-6 N e-(r/Q) , (2.1)
with constants fl, Nand (} depending on the nature of the molecules. From a
theoretical calculation for a pair of helium molecules, SLATER and KIRKWOOD 2
deduced the values fl = 0.68 (}6 X 10-10 erg, N = 7.7 X 10-10 erg, and (} = 0.527 X
10-8 c. These values can be verified by calculating physical quantities like the
second virial coefficient and comparing with the experimentally determined quan
tities; the agreement is generally excellent. Conversely, where the quantum
mechanical calculations are too difficult to allow the direct calculation of the
interaction energy, the constants fl' Nand (} can be inferred from experimental
data, such as the second virial coefficient and its temperature dependence.
Even with the simple radial dependence of the interaction energy indicated
by (2.1), calculations for liquids are apt to be unmanageable, and many authors
+
have preferred to assume cp (r) = _ fl r-6 ')I r-12• (2.2)
If the values of fl and are suitably chosen, there is not much difference between
')I
(2.1) and (2.2), except for very small values of r, which are not realized physic
ally, and where (2.1) does not apply anyway. It is important to keep in mind,
however, that some properties of liquids are very sensitive to the exact interaction
energy, especially the repulsive term which is represented only approximately
by either (2.1) or (2.2). For this reason it is desirable in theoretical calculations
to keep the form of interaction potential as general as possible. Values of the
constants fl and in (2.2) have been determined for many types of molecules,
')I
from data on the second virial coefficient, following the method of LENNARD
]ONES3; but, though these are very useful, more accurate determinations of the
intermolecular forces will soon be required.
1 e.g., see Chap. 8 to 9 of J. C. SLATER: Quantum Theory of Matter. New York, Toronto
and London 1951. - Also J. C. SLATER in Vol. XIX of this Encyclopedia.
2 J. C. SLATER and J. G. KIRKWOOD: Phys. Rev. 37, 682 (1931).
3 J.E.LENNARD-JONES: Proc. Roy. Soc. Land., Ser.A 106, 463 (1924). - Proc. Phys.
Soc. Land. 43, 461 (1931); also R. A. BUCKINGHAM: Proc. Roy. Soc. Land., Ser. A 168,
264 (1938)
1*
4 H. S. GREEN: The Structure of Liquids. Sect. 2.
Another assumption which is commonly made, and will be adopted for most
practical purposes in what follows, is that the force experienced by anyone of
a group of molecules is the resultant of the forces exerted by each of the others
separately. On this assumption, the interaction energy of a group of molecules
can be expressed in terms of the mutual potential energy of a pair of molecules;
specifically for a group of q molecules whose mass centres are at the points x(I),
X(2) , ••• x(q), the interaction energy will be
q
if>q = L<I'(lx(il- X(i) I) , (2·3)
i>i
where 1x (i) - xU) 1 represents the distance between xU) and xU). Quantum me
chanical considerations indicate that this assumption is not rigorously justified,
but that it should involve no serious error.
The restriction to molecules which possess spherical symmetry is one which
often has to be made in the interests of mathematical simplicity, but one which
is found only occasionally in nature. The configuration of a molecule must
usually be described by reference to a set of internal coordinates {}1' {}2 etc., as
well as the position x of its mass centre, and the interaction energy of a pair of
molecules is then a function <I' (I x - x' I, ~, ~') of both sets of internal coordinates
as well as the distance x - x' between the mass-centres. It will still often be
1 1
possible to express the interaction energy of a group of q molecules approximately
in the form
q
L
if>q = <I'(ij) , (2.4)
i>i
where
<I'(ij) = <I' (J x(i) - xU) I, ~(i), ~U») , (2.5)
i.e., as a sum of the mutual potential energies of pairs of molecules. A possible
exception arises where chemical bonds are formed between different molecules.
This does not necessarily exclude liquids whose molecules have a complex structure,
or polar liquids.
Water should most likely be regarded as an example of a polar liquid. The
H 0 molecule has a pair of electrons in the outer shell which are available for
2
bond formation. Probably the bonds are not of the usual chemical type; accord
ing to LENNARD-JONES and POPLEl, who describe them as "lone-pair" bonds,
they can be understood in terms of classical electrostatics. As a first approxima
tion, the water molecule has a tetrahedral structure, with lone-pair electrons
at two vertices, and the hydrogen bonds at the other two vertices. According
to this picture, the edge of the tetrahedron connecting the lone-pair electrons is
a region of predominantly negative charge, and the edge connecting the imper
fectly screened protons is a region of predominantly positive charge. Water has,
therefore, the essential feature of a polar liquid, that its molecules should possess
a dipole or quadrupole moment which is largely independent of the presence of
neighbouring molecules. In any liquid the structure of a molecule is deformed
to some extent by its neighbours, and its electric moments are therefore subject
to fluctuations; but, in a polar liquid, these should be secondary to the moments
of the unperturbed molecule.
Some polar liquids are referred to also as associated liquids, especially those,
like water, where the molecule has localized regions with an excess of positive
or negative charge. They have properties very similar to those of a general type
of associated liquids, whose molecules possess unsaturated chemical bonds.
1 ].LENNARD-]ONES and ].A.POPLE: Proc. Roy. Soc. Land., Ser.A 205,155 (1951).
Sect. 3. Structure of liquids, solids and gases. 5
Such bonds arise through the presence of unpaired electrons on the molecules;
association is caused by the pairing of electrons on different molecules. When
the pairing is effected, the bond is saturated: that is to say, no further molecule
can be held. Consequently, if there is just one unpaired electron per molecule,
the molecules will associate in pairs. If, on the other hand, there are two unpaired
electrons per molecule, the molecules can associate in chains. If there are more,
complicated branching associations can be formed. The interaction energy of
such an association of molecules cannot be exactly represented by a formula
like (2.4); but in view of the close resemblance between chemical and certain
electrostatic bonds, probably to assume the correctness of (2.4) would not lead
to serious error in a wide range of circumstances.
It is an essential feature of association that somewhere in the liquid state
the bonds formed between molecules can be broken sufficiently frequently by
the thermal motion; otherwise one would regard the liquid as made up of complex
molecules. The glasses1 illustrate what happens when branching associations
are formed between the molecules. As the temperature is reduced, the difficulty
in breaking the intermolecular bonds gradually increases and it is hard to say
at what point the liquid first becomes an amorphous solid. Rather than make
a purely arbitrary distinction, a glass will here be regarded as a liquid so long
as it does not crystallize.
3. Structure of liquids, solids and gases. The essential features of the liquid
state can be described best by comparison with those of the solid and gaseous
states. A liquid is usually conceived as a substance with no rigidity but possess
ing a free surface. However, the example just considered of the glasses, which
are perfectly rigid at low temperatures, but cannot without arbitrariness be
distinguished from other liquids, shows that rigidity is not a universal criterion.
Similarly the state of a highly compressed substance just above the critical tem
perature has no free surface but is indistinguishable from corresponding states
below the critical temperature which mayor may not possess a free surface;
so that the second criterion is also inadequate. Indeed, there seems to be no
satisfactory way of defining a liquid except by reference to its molecular structure.
If attention is restricted to substances with a well defined molecular consti
tution, a solid will be taken to mean a crystalline solid. The molecular structure
of a crystal is easy to visualize: it consists of an orderly arrangement of similarly
spaced, and often similarly oriented molecules extending over a macroscopic
region. Apart from occasional irregularities, the atoms are distributed so that
just one of a given kind will be found in the neighbourhood of a lattice site. There
is a high degree of correlation between the positions of even widely separated
atoms in the crystal. The solid state is, accordingly, an ordered state. The
liquid state, by contrast, is a disordered state, and the phenomenon of melting
consists of an order-disorder transition.
This need not imply that the liquid state is completely disordered; even the
molecules of a gas possess a degree of short-range order. The implication is simply
that in a liquid there is no system of occupied lattice positions extending for
more than a few molecular diameters. There is little or no correlation between
the positions of widely separated atoms or molecules; in melting, long-range
order is replaced by a special type of short-range order. Together with the
transition, there is a discontinuous change in the internal energy, specific heat
and density of the substance. The discontinuities are much smaller than those
1 Cf. the contributions on glasses by J. STEVELS in Vols. XIII and XX of this Encyclopedia.