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Owing to their unusual properties, carbon nanostructures such as nanotubes and fullerenes have caused many new nanomechanical devices to be proposed. One such application is that of nanoscale oscillators which operate in the gigahertz range, the so-called gigahertz oscillators. Such devices have potential applications as ultrafast optical filters and nano-antennae. While there are difficulties in producing micromechanical oscillators which operate in the gigahertz range, molecular dynamical simulations indicate that nanoscale devices constructed of multi-walled carbon nanotubes or single-walled carbon nanotubes and C 60 fullerenes could feasibly operate at these high frequencies. This paper investigates the suction force experienced by either an atom or a C 60 fullerene molecule located in the vicinity of an open end of a single-walled carbon nanotube. The atom is modelled as a point mass, the fullerene as an averaged atomic mass distributed over the surface of a sphere. In both cases, the carbon nanotube is modelled as an averaged atomic mass distributed over the surface of an open semi-infinite cylinder. In both cases, the particle being introduced is assumed to be located on the axis of the cylinder. Using the Lennard-Jones potential, the van der Waals interaction force between the atom or C 60 fullerene and the carbon nanotube can be obtained analytically. Furthermore, by integrating the force, an explicit analytic expression for the work done by van der Waals forces is determined and used to derive an acceptance condition, that is whether the particle will be completely sucked into the carbon nanotube by virtue of van der Waals interactions alone, and a suction energy which is imparted to the introduced particle in the form of an increased velocity. The results of the acceptance condition and the calculated suction energy are shown to be in good agreement with the published molecular dynamical simulations. In part II of the paper, a new model is proposed to describe the oscillatory motion adopted by atoms and fullerenes that are sucked into carbon nanotubes.

The work discussed in this paper was begun in 1924 in London, and was later on resumed in Durham. It forms part of an investigation which began eight years ago, of which the purpose was to gain information from a new quarter about intermolecular action; and the first part was published in 1923 by Masson and Dolley. In that part of the work, which dealt with the compressibilities of argon, of oxygen, of ethylene, and of a number of mixtures of each pair of these gases, a systematic study was made of deviations from Dalton’s law of partial pressures. It was possible to exhibit the cohesion of the molecules of one gas with those of another gas; and to prove, without using any special theory, that this type of cohesion is neither caused nor influenced by the possibility of chemical action between the atoms composing two colliding molecules. Thus this kind of cohesion differs from that which gives rise to unimolecular films, where the cohesion is chiefly traceable to the specific action of some particular atom within one molecule upon a particular atom in another, or in a surface. The bearing of this distinction has been discussed elsewhere, and thermodynamic aspects of the first paper have been recently dealt with by other authors. Meanwhile, it becomes clear from the results of that paper that any theory, proposed to explain the van der Waal’s cohesion between molecules of two different chemical kinds, need not invoke any types of force beyond those that could explain cohesion in a single pure gas. The present paper deals experimentally and theoretically with such forces. The first stage of the present experiments (1924) with helium and hydrogen indicated a new property in gases; for the data showed a certain range of mixtures of these two as being more incompressible than either constituent; a maximum being suggested at about 30 per cent, of helium in the mixture. Incidentally, this displaced hydrogen from its position as the least compressible gas. The effects were, however, slight; and, chiefly in order to test their genuineness, we resumed the work in 1926, and improved the methods in various ways. One result has been to confirm the new property, though the effect is very small; and meanwhile, Lennard Jones has independently noticed a similar effect in the experimental data of Holborn for a mixture of neon and helium; and in his recent theoretical work on gaseous mixtures he has been able to show it as a special case of the general properties of gases. The same general treatment proves to be applicable to the work we have done, as will be shown.

The quantum theory has provided a means of calculating the interaction energies of two atoms by a perturbation method. It appears that, the short range interaction forces are due mainly to electron exchange phenomena between the two atoms, while the van der Waals forces arise from mutual polarization effects. The theory gives the first of these forces in the first approximation, while the van der Waals forces appear only in the second approximation, At large distances, where the interaction is small, it is somewhat surprising that the first approximation is not sufficient, and one is led to doubt the accuracy of the method when applied at distances at which the first and second approximations give comparable results. At these distances the mutual potential energy is comparable with the mean kinetic energy of a gas atom at ordinary temperatures, and it is therefore clear that a study of gas-kinetic collision phenomena should provide a satisfactory test of the validity of the perturbation method in this region. It is the object of this paper to carry out a number of calculations with this aim in view. In a previous paper the quantum theory of collisions was applied to gas-kinetic collisions, and it was shown that, although the classical theory can be used with accuracy to determine the law of force from viscosity and diffusion phenomena associated with heavy gases, it cannot he applied with safety to hydrogen and helium. The method to he used in such cases was given, and it was also shown that the existence of a definite total collision area—a feature of the quantum theory of scattering by a centre of force, the potential of which falls of more rapidly than r -2 at large distances—provides a further means of determining the law of force. As this collision area can now be directly measured with accuracy by molecular ray experiments, the range of applicability of tins method is considerably greater than that of methods based on transport phenomena.

1. In a recent note I have called attention to the degree of correspondence between W. L. Bragg’s estimates (based on X-ray crystal measurements) of the dimensions of certain atoms, and those deduced by means of the kinetic theory of gases from determinations of viscosity. I was not then fully aware of the modern work of Chapman‡ in relation to the latter subject, and made calculations by means of a formula which has been definitely superseded. The result was that the comparison gave only qualitative agreement. A closer examination of the data, in the light of Chapman’s results, together with a new aspect of the matter which forms the main subject of the present paper, reveals certain interesting facts regarding the probable arrangement of atoms in molecules, and, in addition, leads to substantial quantitative agreement with Bragg’s values. 2. From the very close agreement between the values of the atomic radius calculated by means of Chapman’s formula, and from van der Waal’s law, there appears now to be no room for doubt that, in the case of monatomic gases, at any rate, the atoms, when in thermal agitation, behave like hard spheres which exert mutual attraction on one another. It is also evident that the absolute diameters of these spheres are now known with considerable accuracy, sufficient to justify quoting to three significant figures. They are given for four monatomic gases in column 3 of Table I for comparison with Bragg’s values, which appear in column 2. They differ to a slight extent from those calculated by Chapman himself, owing to the adoption in the present calculation of Millikan’s|| recent value 2.705 x 10 19 for the number of molecules per cubic centimetre at N. T. P. This differs by about 2 per cent, from that used by Chapman, and the substitution results in an increase of about 1 per cent, in the estimated atomic diameter.

The object of the first paper on "Critical Phenomena in Gases" (referred to in this paper as Paper I) was to develop a simple method of dealing with dense gases and to calculate critical temperatures in terms of atomic fields of force. Each atom in a dense gas was pictured as caged for most of its time by a cluster of neighbours, equal in number to those which surround it in the solid (and presumably also in the liquid) phase. The model was intended to provide a general average of the potential field in which any one atom moved by replacing its varying environment by an arrangement of neighbours which could be regarded as typical. This arrangement was taken to be the one in which the neighbours were situated at their mean positions as determined by the density of the gas. The potential energy of any one atom could thus be expressed as a function of the volume of the gas-a step which is probably the crucial one in a theory of critical phenomena. This point of view brings the process of condensation within the category of those described by Fowler (1936) as co-operative phenomena. In passing we may observe that the derivation of van der Waals' equation provides a particular example of this method, for in it the potential energy of each atom is assumed to be inversely proportional to the volume. The present theory goes beyond this simple model, for the potential energy of an atom is considered to be not only a function of volume but also a function of the position of the atom relative to its neighbours. The probability of finding an atom in any assigned position can be calculated by statistical means and its average potential energy and its available free volume easily deduced. The equation of state can then be deduced by thermodynamic methods, as has been pointed out elsewhere (Lennard-Jones 1937). The success of this method in calculating critical temperatures has encouraged an attempt to push the theory a stage further so as to give the boiling points of gases. At the boiling-point the conditions in the gas are very different from those at the critical point. The density is much less and

It is the purpose of this and the following paper to consider the type of broadening of a spectral line, produced by approaches of other atoms to the radiating atom, which we shall call throughout “transits". The aspect with which we are particularly concerned is the transits of single atoms producing the wing of the broadened line. The intensity distribution in spectra of this type, which are intermediate between ordinary pressure broadening and continuous molecular spectra, can be interpreted in a simple way, yielding results on the van der Waals forces. The first paper gives an account of intensity measurements in the single transit region of the mercury line 2537 A. broadened by the admixture of argon.

J. E. Mayer (1937) has published, together with some collaborators, several papers under the same title as the present one. We consider these papers as a most important contribution to statistical mechanics, and this opinion was shared by the International Conference held in Amsterdam, 26 November 1937, in commemoration of Van der Waals’ birth. One of the present authors gave to this meeting a report on Mayer’s work (published in Physica, 1937) which was followed by a vigorous dis­cussion on the question as to whether Mayer’s explanation of the pheno­mena of condensation is correct. Doubts about this point were raised by the referee, because it is difficult to comprehend how a method of approxi­mation such as that of Mayer, starting from the gaseous state, can lead to the discontinuity of the density on an isothermal curve which corresponds to condensation. The usual methods for treating the equilibrium of two phases introduce the equation of state of both phases and derive the con­dition for their co-existence. Mayer’s theory does nothing of this kind, but treats all possible molecular arrangements with their proper weight, as if there were only one phase. How can the gas molecules “know” when they have to coagulate to form a liquid or solid? Mayer’s mathematical method is too involved to make this point quite clear.

Andrews and Thomson worked in Belfast on the liquefaction of gases during the 1860s and early 1870s but much of their work was, for several reasons, not published until many years after it was done, and then only in part. Their surviving notebooks and letters show that their results anticipate some of the better-known and more systematic work of the Dutch school of van der Waals and Kamerlingh Onnes in the 1890s. The dating of the experiments and ideas of Andrews and Thomson on the continuity of the two fluid states and on the unusual behaviour of gas mixtures on liquefaction is the subject of this paper.

In this paper, a multiscale moving contact line (MMCL) theory is presented and employed to simulate liquid droplet spreading and capillary motion. The proposed MMCL theory combines a coarse-grained adhesive contact model with a fluid interface membrane theory, so that it can couple molecular scale adhesive interaction and surface tension with hydrodynamics of microscale flow. By doing so, the intermolecular force, the van der Waals or double layer force, separates and levitates the liquid droplet from the supporting solid substrate, which avoids the shear stress singularity caused by the no-slip condition in conventional hydrodynamics theory of moving contact line. Thus, the MMCL allows the difference of the surface energies and surface stresses to drive droplet spreading naturally. To validate the proposed MMCL theory, we have employed it to simulate droplet spreading over various elastic substrates. The numerical simulation results obtained by using MMCL are in good agreement with the molecular dynamics results reported in the literature.

In this paper, the snap-through buckling of an initially curved microbeam subject to an electrostatic force, accounting for fringing field effect, is investigated. The general governing equations of the curved microbeam are developed using Euler–Bernoulli beam theory and used to develop a new criterion for the snap-through buckling of that beam. The size effect of the microbeam is accounted for using the modified couple stress theory, and intermolecular effects, such as van der Waals and Casimir forces, are also included in our snap-through formulations. The snap-through governing equations are solved using Galerkin decomposition of the deflection. The results of our work enable us to carefully characterize the snap-through behaviour of the initially curved microbeam. They further reveal the significant effect of the beam size, and to a much lesser extent, the effect of fringing field and intermolecular forces, upon the snap-through criterion for the curved beam.

From the Lifshitz theory, Hamaker coefficients have been computed for all the monoglyceride-alkane thin films in polar liquids described in the previous two parts of this paper. The calculations have been based on three different models of the films. The first model represents the film simply as an isotropic layer of liquid hydrocarbon, in which it is assumed that the lipid and solvent chains are intimately mixed. The second is similar to the first except that it includes two 0.4 nm layers of glycerol to represent the glyceride polar groups of the lipid. The theoretical Hamaker coefficients for these two models are identical to withinca. 10% and, for twelve out of the twenty-four systems examined, they agree to withinca. 20 % with the corresponding experimental coefficients. In the remaining systems, which mainly involve very thin films, the theoretical and experimental results differ by as much as a factor of three. In seeking an explanation for these discrepancies, calculations were carried out for the third model in which, instead of assuming complete mixing of the lipid and solvent chains, it is assumed that they are completely segregated. The hydrocarbon layer is thus split into three layers, i. e. lipid chains–alkane solvent–lipid chains. As with the first and second models, for about half the systems, theory and experiment agree to within 20 % but, for the remaining films, the third model is a considerable improvement as it yields no discrepancies with the experimental data greater than a factor of two. Such discrepancies as do still exist may be accounted for by recognizing thatsomesolvent must in­evitably be present in the lipid chain regions and that, in fact, the state of the interior of a film is intermediate to those assumed in the second and third models. The results exemplify well how successful the Lifshitz theory can be in liquid systems, but they also emphasize the very large effects which small differences in dielectric properties can have on the van der Waals forces in thin films.

A simple model of the low temperature phases of the diatomic molecular solids is examined. The model consists of molecules, interacting via a Lennard-Jones atom-atom potential and quadrupole-quadrupole interactions. The internal energy of any crystallographic structure (excluding thermal effects) can then be given in terms of two dimensionless parameters, which describe the deviation of the molecular shape from a sphere and the relative importance of the quadrupole energy. The minimum energies and optimum molecular configurations in several structures are computed, for values of these dimensionless variables which span the values appropriate to the actual homonuclear diatomic molecular solids, H2, N2, O2, F2, Cl2, Br2and I2. Despite its great simplicity, the model is able to explain several features of these structures. These are (i)o-H2and N2have the optimum quadrupole structure, Pa3; (ii) β-O2is one of the optimum van der Waals’ structures, R3¯m; (iii) the monoclinic α-F2structure is the most stable structure for parameter values very close to those appropriate to F2; (iv) the ortho-rhombic Cmca structure (observed for Cl2, Br2and I2) is the most stable structure for a large range of quadrupole moments which may be appropriate to these molecules. The model, is, of course, unable to take into account intermolecular bonding or spin-dependent interatomic forces. The former is important for the halogens and the latter for the (magnetic) oxygen molecule. The case of α-O2is treated in the following paper.

An examination has been made of the hypothesis that the rate of the para-hydrogen conversion on tungsten is determined by the rate of condensation of hydrogen molecules into an almost full monolayer of chemisorbed atoms. Using the Peierls equation, and employing parameters consistent with the adsorption isotherm and heat data (in particular an interatomic repulsion energy of NV=4800 cal/mole), it is shown that the calculated values of absolute rate, activation energy, and pressure dependency are in disagreement with those observed experimentally. This paper is concerned with one of the postulated mechanisms by which parahydrogen conversion and surface deuterium exchange may occur on aclean tungsten surface. The arguments may also have relevance for other transition metals. At temperatures below 0? C and in the presence of low pressures of hydrogen, e.g. up to 1 mmHg, the surface of tungsten is largely covered by chemisorbed hydrogen (Roberts 1935). The conversion mechanisms that have been suggested are: (1) p-H + HW - WH +o-H2. This involves an exchange of atoms between a molecule in the van der Waals layer or on a gap in the chemisorbed layer, with a chemisorbed hydrogen atom, and was supported by the result of low pressure deuterium exchange experiments (Eley & Rideal 1940; Eley 1941).

The following work was undertaken with a view to obtaining data for the tentative application of Van der Waals’ equation to concentrated solutions. It is evidently probable that if the ordinary gas equation be applicable to dilute solutions, then that of Van der Waals’, or one of an analogous form, should apply to concentrated solutions—that is, to solutions having large osmotic pressures. Saturated solutions were taken for investigation because they presumably have the greatest osmotic pressures, and also because there is reason to believe that, in concentrated solutions at a given temperature, the greater the concentration the less the relative dissociation. For the purpose in view, measurements of volume, pressure and temperature are required.

This paper aims at a simple treatment of van der Waals dispersion forces by applying classical electrostatics of two basic quantum-mechanical equations. Expressions are derived for the forces between macroscopic bodies, and the effects of adsorbed surface layers is considered both quantitatively and qualitatively. Approximate expressions are given for non-retarded dispersion forces in terms of the refractive indices and absorption frequencies of the materials. The results should prove particularly useful in calculating the van der Waals forces in colloidal and biological systems.

In a recent communication to the Royal Society by Mr. Hartley and myself, the results are given of some measurements of the osmotic pressures of strong aqueous solutions of cane sugar, dextrose, galactose, and mannite. It may be of interest to inquire how far these results can be represented by an equation on the plan of that introduced by Van der Waals to express the relation between the pressures, volumes, and temperatures of gases and vapours.

In a communication* to the Royal Society by one of us it was stated that an attempt would be made to determine directly the osmotic pressures of strong solutions. The purpose for which these determinations were required was to obtain data for the tentative application of VAN DER WAALS’ equation of state to solutions, but we propose, in this communication, to restrict ourselves to the actual results obtained, reserving for a future occasion the theory of the subject. Some time ago, in the proceedings of the Royal Society,! we gave an account of some preliminary experiments. We there described the method and apparatus we used, and at the same time stated that we hoped to get more accurate results by modifying the apparatus. This hope has been fulfilled, and we trust that a somewhat full description of the methods used both in the actual determinations and in the preparation of the membranes will, therefore, not be out of place.

von Stackelberg & Muller (1951) and Muller & von Stackelberg (1952) have shown that two structural types of water lattice exist in which various species are intercalated to give solid hydrates. In the present paper a statistical thermodynamic interpretation is given of the properties of these clathrate phases of water. This method is a development of a procedure applied by van der Waals (1956) to clathrate compounds of quinol, and shows that these hydrates should be non-stoicheiometric. It is possible to give a satisfactory explanation of the stabilizing action of ‘hilfsgase’ (von Stackelberg & Meinhold 1954); to interpret various thermochemical quantities and indicate limitations in some of these quantities as previously determined; and to clarify the conditions under which the clathrate phases form. Calculations of heats of intercalation for several inert gases give values close to those derived from thermochemical data. Calculations of equilibrium constants for intercalation of these inert gases lead to reasonable estimates of dissociation pressures of the corresponding clathrates. On the basis of the dissociation pressures the fractionation of mixtures to be expected by formation of clathrate phases may be estimated.

The apparent stability and freedom from settling shown in practice by many colloidal suspensions has in recent years prompted renewed study of the distribution of the particles of colloidal systems under the influence of gravity, and results greatly at variance with earlier work have been obtained. The object of the present communication is (1) to show that the later conclusions are not justified by the experiments on which they are based, and (2) to describe experiments which favour a considerable extension of the applicability to colloidal solutions of a law similar to that which defines the behaviour of ordinary dilute solutions. Perrin, in 1909, showed that there was, within a depth of 0·01 cm. from the surface, a logarithmic decrease of concentration with height for suspensions of gamboge or mastic in gravitational equilibrium. In 1914 Costantin, working at lower depths than Perrin, and with a higher volume concentration of gamboge and correspondingly higher numerical concentration, obtained evidence of a repelling force between the particles at concentrations greater than 8 X 10 10 particles per cubic centimetre. Perrin showed how his equation could be modified to take this into account by assuming a law of the van der Waals’ type, which was found to agree with the observations up to the highest concentration investigated (6 X 10 11 particles per cubic centimetre). In the following year Westgren published the results of a series of measurements on selenium and gold sols with coarse particles, which were in complete harmony with Perrin’s law over a distance of 0·05 cm. and at concentrations up to 5 X 10 10 per cubic centimetre. These investigators were precluded from studying the distribution over greater ranges of depth solely by the restrictions of their experimental method, which demanded that for accurate counting the number of particles in the field of view should not be too large. For this reason also they were limited to moderately dilute sols and to rather large particles, that is, to specially prepared sols.

Broadly speaking, three types of crystal structure may be recognised: ( a ) interpenetrating arrangements of atomic and electric lattices, the electrons being symmetrically disposed with reference to the atoms; ( b ) neutral lattices, of which sulphur and phosphorus among the elements are examples; ( c ) ionic lattices, to which class most salts belong, in which the electrons are unsymmetrically arranged so as to be in close relationship with particular atoms. Sir J. J. Thomson pointed out that by a decrease of the closeness of this association between atoms and the linkage electrons, this type merges into the first. Franck similarly distinguishes between three kinds of linkage: (i) where the electrons remain in unchanged quantum orbits, i. e . homopolar linkage due to van der Waal’s forces owing to mutual polarisation of the atoms; (ii) where the linkage electrons occupy new quantum orbits which may be binuclear, i. e ., homopolar linkage such as occurs in molecules of many gases; and (iii) true heteropolar linking where electrons have been bodily transferred from one atom to the other, and occupy new quantum orbits around a single nucleus. Linkages of type (i) clearly produce crystal lattices of class ( a ), usually possessing electric conductivity, while those of class (ii) are concerned in the formation of insulators having neutral lattices of type ( b ). True homopolar linkage should give no active frequencies of vibration, or, if any exist, they should be, according to Schaefer, of extremely weak resonant amplitude, and should lie in the region where the wave-length of the radiation is comparable to the atomic spacing. Nernst suggests that all crystals are built up by homopolar linkages, but this would appear to be contrary to experience, crystals generally showing sharply defined absorption maxima in the infra-red. Kossel is quoted by Bragg as supposing the CO 3 group in carbonates to be held together by heteropolar attraction between the ionised atoms of the group. De, however, considers the binding to be homopolar, while Bragg and Chapman, Chapman, Topping and Morral, Chapman and Ludlam, Kornfeld, and Lennard-Jones all agree in supposing the group to consist of ions having effective charges modified by deformation of the electronic shells. Fajans and Joos, on the other hand, state that, contrary to Kossel’s assumption, there seems to be little reason for supposing the internal binding of the CO 3 ion to be by heteropolar linkages, they themselves being inclined to think that the setting up of an induced electric moment in the oxygen atoms, by reason of the deformability, provides a sufficient explanation of the stability of the group.

A work entitled “Le Point Critique des Corps Purs” has recently been published by E. Mathias, whose opinion on matters relating to the critical state must always carry weight. In this he discusses at length the various theories which have been put forward to explain certain irregularities observed in the behaviour of substances, which were supposed to be pure, at their critical temperatures. These irregularities are not accounted for by the simpler theories of Andrews and Van der Waals. He calls attention to the experiments of certain investigators, which appear to suggest that the currently-accepted values of the critical constants of many common substances may be vitiated, either owing to the time allowed for the establishment of equilibrium between the coexisting phases near the critical point being insufficient, or the temperature at which the dividing surface vanishes not being independent of the relative masses of the two phases at the temperature at which this takes place. According to I. Traube, substances contain different kinds of aggregates, which he calls “gasogenic” and “liquidogenic” molecules. It follows that if equilibrium demands that there shall be a certain concentration of these molecules in the vapour and liquid phases respectively, then unless dissociation and association take place instantaneously, there must elapse a time, following any change of condition, before equilibrium can be established between the two phases.

We have used the quantum theory of radiation, within the time-dependent Hartree approximation, to study exciton states of a van der Waals molecular crystal. The radiation variables are eliminated to give a semi-classical picture of molecular dipoles interacting through a retarded potential, and the solutions of the Hartree equations are closely connected with the quasi-particle excited states in Agranovich’s theory. In the Lorentz-Lorenz approximation the crystal has excited states which correspond to both longitudinal and transverse exciton weaves, and the refractive index behaves classically. The paper concludes with a brief discussion of metallic reflexion by dye crystals.