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Melting and Crystal Structure Ч Some Current Problems.

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Alkylidenephosphoranes are also formed on thermal
decomposition of triphenyl-cr-methoxycarbonylalkylphosphonium salts (35) [29].
with mercury. The solid residue is then filtered off using rl
G-3 frit. The filtration can be dispensed with if the subsequent
reactions are not affected by excess sodamide and undissolved
sodium halide.
Preparation of' Ylide Solutions by the
DimethylsulJinate Method
The ylides formed can be isolated if they are stable. They
can also be intercepted by various reagents, or undergo
intramolecular reactions.
Procedures :
Salt-free Alkylidenephosphorane Solutions
All operations are carried out under nitrogen or argon.
Ammonia is condensed in a trap cooled with liquid air and
containing sodium. The resulting dry ammonia is distilled
into a second tube (75- 100 ml), and the required quantity of
excess) and a few grains
finely chopped sodium (up to 25
of ferric nitrate are added. When the blue sodium solution
has turned grey, the absolutely dry and finely powdered
phosphonium salt is added, stirred for a short time with a
glass rod, and the ammonia is then evaporated off through
a mercury valve. The evaporation is accelerated by heating
the tube with hot air. To the residue is added 100 ml of an
inert, anhydrous solvent, e.g. benzene, toluene, ether, or
tetrahydrofuran, and the solution is boiled for about 10 min
to remove residual gas, the reflux condenser being sealed
_ - - ~ .. .
I291 H . J . Bestmonn, H. Hortimg, and I. Pils, unpublished work.
The preparation must be carried out under nitrogen and i n
the strict absence of moisture. The stoichiometric quantity of
sodium hydride (as a 50 % suspension in mineral oil) is freed
from mineral oil by repeated washing on a sintered glass
filter with anhydrous petroleum ether, and placed in the reaction vessel. Dimethyl sulfoxide (dried over calcium hydride) is
added (50 ml of dimethyl sulfoxide per 100 mmole of NaH),
a reflux condenser with a mercury valve is fitted to the vessel,
and the mixture is slowly heated to 70-80 ' C , with stirring
(magnetic stirrer). The evolution of hydrogen stops after
about 45 min. The resulting solution of the dimethyl sulfoxide
anion is cooled in ice and the solution or suspension of the
phosphonium salt in dimethyl sulfoxide (50 mmole of salt
in about 100 ml of dimethyl sulfoxide) is added. The mixture
is stirred for a further 10-20 min at room temperature before
being used for further reactions.
Chloromethylidenetr@henylphosphorane Solution [24b]
A solution of 40 mmole of butyl-lithium in about 25 ml of
anhydrous ether is added dropwise, over a period of 40 min,
to a well-stirred solution of 35 mmole of triphenylphosphine
in 45 ml of absolute methylene chloride at -60 ' C . The resulting solution of the chloromethylidenetriphenylphosphorane
is then used for further reactions.
Received: November Znd, 1964 [A 443a/231 I€]
German version: Angew. Chem. 77, 609 (1965)
Translated by Express Translation Service, London
Melting and Crystal Structure - Some Current Problems
BY PROF. DR. A. R. UBBELOHDE
DEPARTMENT OF CHEMICAL ENGINEERING AND CHEMICAL TECHNOLOGY,
IMPERIAL COLLEGE, LONDON (ENGLAND)
The "classical" definition of liquid and solid phases acording to mechanical criteria is
unsatisfactory. It is now known that ''premeltinp'' efects occur in crystals, and in many
liquids the formation of clusters is observed near the melting point, with structures which
can correspond to the short-range order of the crystals from which the melt is derived. -In particular, the implications of the theories of Lennard-Jones and Devonshire and of
Mizushima and Ookawa are discussed.
As is now well known, many melts have structures which
resemble the crystals from which they are formed more
or less closely. This is shown, for example, by the similarity of density. X-ray diffraction shows, too, that the
interatomic or intermolecular distance in many melts is
quite close to that in the crystals. However, the conventional radial distribution function of melts gives data
which are far too condensed to be really informative.
Methods are required which give a three-dimensional
representation of regions or domains in the liquid, as a
Angew. Chem. internnt. Edit. 1 Vol. 4(1965)
No. 7
counterpart to the elaborate three-dimensional information usually available about the crystals.
This gap i n o u r direct structural knowledge about the molecular arrangements in melts is a serious obstacle to systematic
development of theories of the liquid state. It has tended
to direct mathematical theories to the discussion of
liquids whose units of structure are extremely simple, such
as atoms or diatomic molecules [I]. Actually, such melts form
only a minute fraction of the range of known liquids; their
[ I ] K . Fiirukawo, Rep. Progr. Physics 25, 396 (1950).
587
behaviour is not necessarily representative of melting in
general. Alternatively, statistical theories of liquids may be
developed on a quasi-gaseous analogy [2]. Whilst this
approach is rather more broadly based, it is applicable most
generally near the critical temperature of the liquids, and
does not yield particularly refined structural information
about the solid-liquid transformation at the melting point.
This consideration makes it convenient t o refer to liquids at
temperatures around their freezing point as metts, to distinguish them in broad terms from quasi-gaseous fluids near
the critical temperature.
The primary distinction between solid and liquid,
regarded as condensed states of matter, is mechanical.
But at the present time, for these as for other states of
matter, the most general approach to theories of transitions between them has to be based on thermodynamic
and statistical considerations. With reference to the role
of structural features in melting, some of the concepts
from which new lines of experimentation may originate
will be briefly reviewed here.
Two principal thermodynamic parameters of general
interest in discussing structural aspects of melting are the
entropy of fusion S, and the fractional change in molar
volume AV,/V,, where V, is the molar volume of the
crystal at the melting point Tf. These are related by the
Clausius-Clapeyron equation
dP/dT = SfiAV,.
Subsidiary thermodynamic properties, of great interest
for particular structures, are derived parameters such as
the compressibility, the thermal expansion coefficient
and the heat capacities. It can be very informative to
known what is the change in these quantities at the
melting point, but very often the data are not available.
Also of great importance are any premonitory changes
in these subsidiary parameters, in premelting or prefreezing effects. The entropy of fusion is particularly useful in
its correlation with the melting of different kinds of
crystal structures by virtue of the Boltzmann relationship
Sf/R = In (WLIWS)
where WL and Ws are the number of independent ways
of contriving the liquid :and solid states respectively.
For all known cases of melting, Sf is positive, i.e. the
fluid condensed state has a higher entropy than any state
with rigidity. In general the higher entropy of the melt
can be attributed to its greater disorder. More specifically, various types of crystal structure can be disordered
in different ways. Each distinguishable mode of disordering constitutes a mechanism of melting. Some
crystal structures exhibit a single mechanism of melting,
whereas others combine more than one mechanism of
disordering on passing from the crystal to the melt.
A representative but not exhaustive discussion of
mechanisms of disordering may be conveniently given,
with emphasis on directions in which research is currently particularly active, or where problems seem ripe
for attack.
[2] Cf. R. Fowler and E. A . Guggenheim: Statistical Thermodynamics. University Press, Cambridge 1939.
588
1. Positional Melting
As is apparent for example from X-ray diffraction
studies, all crystals undergo loss of long-range order on
melting. What is not quite as well established is the
extent of the disordering of local packing. Current
diffraction methods [ l ] establish a smooth average, but
for some theories of the structures of melts, particularly
in the temperature range around Tf, it is important to
know to what extent cooperative fluctuations about the
smoothed average lead to clusters of any kind. Some
types of cluster can act as crystal nuclei; other types of
assembly may have an enthalpy lower than the average,
but lack 3-dimensional lattice ordering and thus are noncrystallizable. This kind of information about domains
or cooperative fluctuations in the melt is only beginning
to receive attention by the use of diffraction methods [3].
Other aspects will be referred to later in connection with
prefreezing phenomena.
Numerical values of the entropy of positional disordering may be given a kind of norm from data for
crystals of the rare gases. Where quantum effects, i.e.
contributions from zero-point energy, can be neglected,
the entropy of positional disordering on melting of these
crystals is about 3.3 e.u. [4]. This value can be built into
a particularly straightforward statistical thermodynamic
model of positional melting, first discussed in mathematical details by Lennard-Jones and Devonshire in
1939 [5]. In their model, positional disordering is
restricted to the insertion of atoms in interstitial sites, in
the so-called Frenkel positional defects. According to
this theory, equilibrium can be established at Tf between a slightly defective crystal (the solid) and a highly
defective crystai (the liquid). Changes of vibrational
entropy are neglected, and the cooperative influence of
crystal defects upon neighboring crystal sites is allowed
for in terms of the expansion on melting by a BraggWilliams type of volume-dependent interaction. Despite
its obvious oversimplifications, this well-known model
helps to clarify the main features of positional melting.
It is one of the simplest ways of describing the structure
of a quasi-crystalline liquid.
Amongst current developments of this simple version of
positional melting, the following may be noted :
1. Dislocation Melting
Any crystal defect whose energy of creation is lowered
by virtue of defects on neighboring lattice sites may be
termed cooperative to distinguish it from an isolated or
point defect. As an alternative or an addition to Frenkel
defects the intervention of cooperative defects in mechanisms of melting has been proposed on qualitative
_ ~ _ [3] V. K . Prokhorenko and J. 2. Fisher, Zh. Fiz. Khim. 33, 1852
(1959).
141 A . R . Ubbelohde, Quart. Rev. (Chem. SOC.,London) 4, 356
(1950).
[ 5 ] J. E. Lennard-Jones and A. F. Devonshire, Proc. Roy. SOC.
(London) A 169, 317 (1939); ibid. 170, 464 (1939).
Angew. Chem. internat. Edit. 1 VoI. 4 (1965) 1 No. 7
grounds [6], but a quantitative theory has proved
difficult to elaborate. Recently, an attempt has been
followed through to describe the positional melting of
metals on a dislocation theory [7,8].
metals as seen from the high value of T,,it./Tf or Tb/Tf = 3,
compared with Tb/Tf of about 1.03 for rare gases and
Tb/Tf = 1.6 for the structurally related halides with rare-gas
configurations [lo] (Tb is the poiling point at 1 atm).
In its present form, this theory still involves some harsh
simplifications. However, this development of a cooperative defect model appears to be significant for at least
two aspects of the behavior of melts.
a) It points to the kinds of positional disordering in a
quasi-crystalline liquid on a more realistic basis than the
interstitial defect model of Lennard-Jones and Devonshire. A dislocation type of approach can be of importance for describing thermodynamic behaviour, and may
also be more readily applicable when dealing with any
transport processes that show some chain-mechanism
character in the melt. Thus the Lennard-Jones model
harmonises most readily with the theory that transport
processes in liquids all occur by the activated hopping
discussed by Eyring [9], according to which momentum
transfer is the outcome of mass transfer. Dislocation
models suggest that in addition to any hopping of units,
momentum transfer may occur by dislocation movement,
in a kind of zip fastener action.
b) In calculating the total entropy of fusion, a particular
dislocation model gives [ 7 ] :
sf= A%.
(0.69e.u.)
+
ASpos. (1.08 e.u.)
This contribution of a change of vibrational entropy
Asvib,to the total entropy of fusion S, marks a difference
from the Lennard-Jones theory in which AS,,,. is
assumed zero.
2. Melting of Metals
Types of molecules are known which have quasispherical
(globular) repulsion envelopes, when close packed in the
crystals. For these too, Sf 3 e.u., showing that a
simple model for positional melting has quite wide
generality. However, a large class of crystals is known,
mostly metallic in properties, in which the units of
structure are atoms, but for which Sf < 3 e.u. The
reason why many metals have a lower Sf and thus a
lower overall disorder in the melt than rare gases has not
yet been fully elucidated.
-
The repulsion force fields may be lower, since AVfjv, for
metals is markedly lower than for inert gases. Reasons for
lower repulsion force fields are however not at all apparent:
even alkali metals have Sf and AVf/V, much lower than the
rare gases whose electronic core is nevertheless similar.
Another possible explanation of the low values of melting
parameters of metals is that factors contribute t o Sf that do
not greatly increase the enthalpy of fusion Hf. resulting in a
lower Tf according t o the expression Tf = HfiSf, where Hf is
the molar enthalpy of fusion. lndications of this possibility
may be found from the abnormally long liquid range of
[61 J . W. H. Oldham and A . R. Ubbelohde, Proc. Roy. SOC.(London) A 176, 50 (1940).
[7] S. Mizushima, J. physic. SOC.Japan 15, 70 (1960).
181 A. Ookawa, J. physic. SOC.Japan 15, 2119 (1960).
[9]S. Glasstone, K . J. Laidler, and H. Eyring: Theory of Rate
Processes. McGraw-Hill, New York 1941.
Angew. Chem. internat. Edit.
Vol. 4(1965)
/ No. 7
3. Melting of Ionic Crystals of Simple Structure
For halides of inert-gas structure with low polarizability,
for example salts of lithium, Sf and A Vf/V, are close to
values for the inert gases, suggesting that positional disordering is similar in both types of crystals. However,
for the more polarizable ions of higher atomic number in
similar salts, both these parameters fall below the inert
gas values. Again, a review of all the evidence suggests
that the liquid state is abnormally favored relative to the
crystalline state for the more polarizable ions. This
polarizability might favor cluster formation in ways to
be discussed later.
4. Premelting Effects in Crystals
The Lennard-Jones and Devonshire model for positional
melting indicates considerable premelting in the crystals.
It does not allow for any quasi-crystalline or microcrystalline domains in the melt, since the Bragg-Williams
treatment of cooperative interactions between defects
excludes such considerations. For solid argon, the
fraction of atoms in defect sites at the melting point T,,
where there is a discontinuous jump to the melt, is about
0.05 in the model originally proposed by these authors.
In this theory, the extent t o which positional thermal defects
are present in the cIystals at Tf is unfortunately too sensitive t o the precise assumptions made about intermolecular
force fields to permit significant correlation with expcriment.
It is claimed that precise experiments o n Ar, CO, NzD, HCI,
and HBr d o not indicate any appreciable excess specific heat
[ t l ] , such as premelting defect formatio.? might require i n
crystals to which the model is expected t n apply.
5 . Orientational Disorder of Molecular
Axes in Crystals
For molecules whose repulsion envelope is not wholly
spherical, a model can be readily contrived which may
be combined with positional disordering in a LennardJones type of statistical theory of melting. This analysis
confirms the experimental finding that in some crystals,
two mechanisms of fusion, for which Sf = AS,,,, +
ASorient., occur simultaneously at T,. For others,
orientational randomization occurs at a transition
temperature T, in the solid well below Tf [12,13]. For
some molecules, orientational randomization may not
be complete even above Tf; this may be the case when
molecular repulsion envelopes do not permit free
_
~
-.
_
[lo] W. A. Weyl and E. C.Marboe, J. SOC.GlassTechnol. 43,417
(1959).
[I 11 K. Clusius and L. Staveley, Z . physik. Chem. 49 B, 1 (1941).
[ 121 A. R . Ubbelohde, Quart. Rev. (Chem. SOC.,London) 4, 365
(1950).
[I31 J . A. Pople and K. E. Karasz, J . physic. Chem. Solids 18, 28
(1961); ibid. 20,294 (1961).
589
rotation about the three principal axes in the space
available in the melt [14-161.
As a guide to mechanisms of melting, a great merit of
any reasonably workable theory of positional disorder is
that it permits more precise discussions of various
departures from the norm. Many of those referred to
above can still be discussed within the framework of a
quasi-crystalline model for the melt. A quasi-crystalline
model signifies an assembly which can be arrived at by
positional (and orientational) disordering of units of
structure definable with respect to the crystal lattice. The
molecular arrangement of a quasi-crystalline melt
defined in this way is arrived at by increasing the entropy
of position (including vibrations) and orientation, i.e. :
St = ASvib.
+ ASpos. + Asorient.
11. Melts that a r e Not Quasi-Crystalline
Considerable attention is being paid at present to melts
whose structure is not quasi-crystalline in the sense
described, but is more complex.
One important group of crystals whose melts cannot be
described on any quasi-crystalline model is formed by
flexible molecules, such as n-alkane hydrocarbons and
their derivatives, or rneta-polyphenyls, or in general
whenever the molecules can assume a variety of different
configurations in free space. In the case of n-decane, for
example, some of these may be represented by (1)-(3).
When the different configurations in free space have free
energies differing by not more than one or two units of
kT, a mixture of configurations will be formed in the
vapour, and somewhat less freely in the melt. On the
other hand, the crystal normally selects only one configuration (usually the fully stretched one) for economical packing into a crystal lattice. For such crystals, the
overall entropy of fusion includes an additional contribution due to configurational disorder i. e.:
Sf = ASvib. t ASpos.
+ Asorient. + Asconfig
It has long been known that this additional mechanism leads
to convergence melting points in homologous series of
molecules whose skeleton is flexible. Currently, increasing
attention is being paid to the fact that these melts cannot
be represented by a quasi-crystalline model, for example in
[14] A. A. K . A1 Mahdi and A . R . Ubbelohde,;Proc. Roy. SOC.
(London) A 220, 143 (1953).
[15] J . N . Andrews and A. R . Ubbelohde, Proc. Roy. SOC.(London) A 228,435 (1955).
[I61 J . H . Magill and A. R . Uhbelohde, Trans. Faraday SOC.54,
1811 (1958).
590
theories of transport processes, since they constitute a
mixture of different molecular configurations. Similar remarks apply to molecular tautomerism in which the free
energies of the tautomers are comparable and rates of
change are sufficiently high. Anomalies have been reported
in some of the transport properties, e . g . in o-terphenyl, as
Tf is approached [17]. These may indicate the onset of
prefreezing phenomena, which are best discussed in general
terms of cluster formation on approaching the melting
temperature.
111. Cluster Formation and Prefreezing
Phenomena in Melts
Fluctuation theories of phase transformations, such as
the theory of Frenkel[18], indicate a kind of symmetry
in the concentration of micro-regions or domains of any
phase, in the bulk of the other phase, near a transition
temperature Tc or melting point Tp In discussing possible structure anomalies of a melt as its freezing point Tf
is approached or traversed, conventional forms of cooperative fluctuation theory as discussed by Frenkel [18],
Lukasik [19], and Turnbull and Fisher [ZO] indicate the
presence of minute concentrations of micro-domains of
crystalline structure, swimming in the more disordered
melt. Below a critical nucleation temperature T,,
crystalline micro-domains can act as crystal nuclei
inducing spontaneous crystallization of the melt. For a
variety of crystal types, it has been established that
normally TN = 0.8 Tf [21]. However, certain melts fail
to nucleate spontaneously in this way [22,23]. Brief
comment may be made about the origin of this difference in behaviour.
One view attempts to focus any anomaly in the prefreezing region in terms of the “free volume” in a melt.
This parameter, which has analogies with the BraggWilliams treatment of cooperative effects in solids, represents the difference between the occupation volume
of a melt (the specific volume) and the volume of repulsion envelopes of all the molecules. If the “free
volume” falls below critical limits before T, is reached,
spontaneousnucleation may be inhibited, and a melt may
pass smoothly and progressively into a glass as its
temperature falls. This approach does not predict any
particular anomaly of properties as Tf is approached,
and traversed, on cooling the melt. However, it appears
to be grossly oversimplified. An alternative view draws
attention to the fact that many molecules can be closepacked or clustered together with their repulsion
envelopes in contact, in domains with various configurations. Whe<&e number of units in a cluster is small,
space can usually be filled more economically by this
1171 J. K . Horrocks and E. McLaughlin, Proc. Roy. SOC.(London) A 273, 259 (1963).
1181 J . Frenkel, J. chem. Physlcs 7, 538 (1939).
1191 S.J. Lukosik, J. chem. Physics 27, 523 (1957).
I201 D. TurnbrtN and J. Fisher, J. chem. Physics 17, 71 (1949).
[21 J E. R. Buckle and A . R. Ubbelohde, Proc. Roy. SOC.(London)
A 259, 325 (1960); A 261, 197 (1961).
[22] E. R. Buckle and C. N . Hooker, Trans. Faraday SOC.58,
1939 (1962).
I231 D.G.Thomas and L. A. K . Stoveley, J. chern. SOC.(London)
(1952)4569.
Angew. Chem. internat. Edit. I Vol. 4 (1965)
1 No. 7
denser packing than is given by the mean density of the
melt or vapour. A reduction in enthalpy of the order
Hf/N per unit may be expected to accompany clustering,
where N is Avogadro's number. A melt near Tf may
contain such clusters, regarded as cooperative fluctuations from more open packing, in appreciable concentrations. However, as more and more units attempt to
aggregate onto any particular cluster, an important discrimination must be made. If all the molecules in a
cluster are arranged on a crystalline lattice so that they
form regular repeat sequences in space, the cluster can
act as a crystal nucleus. It can grow indefinitely by
aggregation, without change of packing. This leads to
spontaneous crystallisation below Tf. By contrast, if a
cluster is close-packed in a non-crystallizable way,
attempts to aggregate more and more units inevitably
lead to the formation of fronds or inclusions of voids
and clefts, with consequent lowering of its mean density.
Quite soon, a non-crystallizable cluster ceases to grow
by aggregation because the reduction in enthalpy resulting from closer packing ceases to operate if the
number of units it comprises is too large. This second
model emphasises local, more condensed regions. If it
fails to crystallize on a regular crystal nucleus, such a
melt eventually congeals, through contact between noncrystalline clusters.
Cooperative fluctuations of packing in melts, as they are
progressively cooled, may be expected to involve more
or less marked anomalies with respect to quasi-crystalline behaviour regarded as the norm. The extent to which
departures from a quasi-crystalline model are found
will depend on intermolecular forces. For example,
different non-crystallizable types of aggregation have
been proposed in hydrogen-bonded melts, such as
water 1241. Clustering is also suspected in various molten
metals, as well as in certain ionic melts [25].
Methods of studying the micro-structure of melts experimentally are still too scanty to investigate theories
of cooperative fluctuations directly. Indirect methods
include the studies of spontaneous crystallization or
failure to crystallize spontaneously already mentioned.
Another line of approach is to measure various transport
processes in the melt as this is progressively cooled to
and below T,. When cooperative fluctuations lead to
significant aggregation into clusters, the relaxation time
for their dispersal may be compared with the normal
relaxation time characteristic for the property under
examination. For example, in viscous flow, the characteristic normal relaxation time in a quasi-crystalline melt
seems to be of the order indicated by the Eyring theory
[24] G. Nemetfiy and H . A. Scheraga. J. chem. Physics 36, 3382
(1962).
[251 A. R. Ubbelohde: Melting and Crystal Structure. University
Press, Oxford 1965.
Angew. Chem. internnt. Edit.
Vol. 4(1965)
No. 7
[9] of hopping frequency of viscosity and of mass
transport. If clusters are formed whose duration is long
compared with this time, these can only be moved as a
whole in the melt, i. e. they behave like colloidal particles
suspended in it. On this basis, an increase of viscosity
should be observed of the order
where <D is the volume fraction occupied at any instant
by the clusters regarded as spherical. A decrease of mass
diffusion constant should likewise be observed, due to
the volume blocking of clusters [26].
In order to avoid complications from possible modifications in attractive forces due to changes of local configurations in hydrogen bonding or intermetallic bonding, a series of studies on prefreezing anomalies have
been carried out on melts of polyphenyls [27]. In these,
opportunities for molecular close-packing in the melt
are determined predominantly by the shapes of the repulsion envelopes. Determinations of room to rotate in
the melt show that for the majority of such molecules,
nearest neighbors must in any case be roughly aligned,
since as already stated, the free space is much too small
to permit random orientation of molecular axes of
neighbors. Transport properties do not necessarily show
marked prefreezing anomalies; in many cases an Arrhenius type of equation is obeyed over considerable ranges
of temperature, with fair accuracy, down to Tf. However, when the repulsion envelopes have marked reentrant cavities, as for o-terphenyl or for 1,3,5-tri-ornaphthylbenzene, marked excess viscosities are found as
T, is approached and traversed. It is quite difficult to
cause such melts to crystallize. Normally, they pass
smoothly into glasses on sufficient cooling [16,271.
As a working hypothesis, excess viscosities defined b y
the Arrhenius equation as norm (with the melt at high
temperatures regarded as a free fluid and treated as
reference), and other excess properties above those of a
normal fluid whose behaviour can be studied, may be
interpreted on the hypothesis that transient cluster
formation blocks easy configurational rearrangements
that involve hopping in the melts.
What is urgently needed is a greater diversity of physical
means for examining the structures and local density
fluctuations of melts. When the latter are not quasicrystalline, this can have a close bearing on their freezing
and melting behaviour.
Received: July 27th. 1964
[A 4251230 IE]
German version: Angew Chem. 77. 614 (1965)
[ 2 6 ] A . R. Ubbelohde, unpublished work.
[27] E. McLaughlin and A . R . Ubbelchde, Trans. Faraday SOC.
53, 628 (1957).
59 I
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