Frederick Charles Frank

One-dimensional dislocations. I. Static theory

The theory of a one-dimensional dislocation model is developed. Besides acting as a pointer to developments of general dislocation theory, it has a variety of direct physical applications, particularly to monolayers on a crystalline substrate and to conditions in the edge row of a terrace of molecules in a growing crystal. Allowance is made in the theory for a difference in natural lattice-spacing between the surface layer or row and the substrate. The form and energy of single dislocations and of regular sequences of dislocations are calculated. Critical conditions for spontaneous generation (or escape) of dislocations are determined, and likewise the activation energies for such processes below the critical limits. Various physical applications of the model are discussed, and the physical parameters are evaluated with the aid of the Lennard-Jones force law for the above-mentioned principal applications.

Supercooling of Liquids

I shall concentrate upon reviewing the important recent change in our appreciation of the facts of supercooling which has been brought about particularly by the work of Turnbull at the General Electric Research Laboratory in Schenectady. I suppose that most of us, talking about supercooling a couple of years ago, would have divided substances into two classes, one with simple crystal structures like gold, and all the other ‘good’ metals on the one hand, and those with complex crystal structures, such as glycerol and the silicates on the other; saying that whereas the latter class can be very much supercooled, and will form glasses, the former class can only be supercooled a very few degrees. Then we would have added that there are some ‘ bad ’ metals, with moderately complex crystal structures, such as antimony or bismuth, which can be supercooled some tens of degrees, forming an intermediate class. I think we would then have added that this is quite comprehensible. In particular, that the X-ray diffraction patterns of the monatomic liquids show us that most of the atoms have the right numbers of nearest neighbours in a first co-ordination shell, all ready in place to start the growth of a crystal; which readily explains why these substances cannot be supercooled very much—a nice simple experimental fact, with a straightforward theoretical interpretation—and both are wrong.

One-dimensional dislocations. II. Misfitting monolayers and oriented overgrowth

The equations derived in part I of this series for a one-dimensional dislocation model are applied in this paper to the case of a monolayer on the surface of a crystalline substrate, particularly when the natural lattice spacing of the monolayer differs from that of the substrate. Justification is given for this extension of the equations to the two-dimensional case. It is shown that the theory predicts a certain critical amount of misfit (9% difference in lattice spacing in a simple case) below which the monolayer in its lowest energy state is deformed into exact fit with the substrate, and above which it is only slightly deformed in the mean, having many dislocations between it and the substrate. The energy of adsorption as a function of misfit is also calculated, becoming almost constant above the critical limit. Up to a larger critical misfit (about 14% in the same simple case) the monolayer can be deposited metastably in exact fit on the substrate, at sufficiently low temperature. Since the dislocated layer is mobile on the surface, completely oriented overgrowth of one crystal on another can only be expected if the first monolayer can be formed over the complete surface under subcritical conditions. This is in general agreement with observation.

On the theory of Hertzian fracture

The fracture of a brittle solid under a spherical indenter is the best studied case of fracture in a strongly inhomogeneous, well defined, stress field. Two principal topics are discussed, the path of a crack in a field of non-uniformly directed stress, and the stability of cracks of various length when the prior stress on the crack path is non-uniform. For the first, it is shown that the crack growth should, to a first approximation, be orthogonal to the most tensile principal stress, and thus correspond, in a torsion-free stress field, to a surface delineated by the trajectories of the other two principal stresses: while, to a second approximation, the crack should deviate from this path by having a larger radius of curvature at every bend, thus exhibiting a pseudonertia even in slow growth. This is in accordance with the known experimental facts about the Hertzian crack, particularly the fact that the crack at the surface forms systematically outside the edge of the circle of contact, at which the maximum tensile stress occurs. On the second question, it is found that there are four crack lengths, c0, c1, c2, c3, corresponding to stationary values of energy. c0 and c2 represent unstable equilibria, and diminish with increasing load; cx and c3 represent stable equilibria and increase with increasing load. With small indenters, c0 soon becomes less than the size of pre-present surface flaws, and an unobserved shallow ring crack of depth c1 is produced: the critical condition for observed fracture is then the merging of c1 with c2, allowing unstable growth to the cone crack of depth c3. This explains Auerbach’s law, that the critical load for production of a cone crack is proportional to the radius, r, of the indenter sphere. With larger indenters, of several centimetres radius for a typical case, c1, and c2 merge and disappear before c0 exceeds the size of pre-present flaws. The critical load for cone fracture then becomes nearly propor¬tional to r2, as observed. The previous calculations of Roesler (1956 a, b) relate to the second stable crack dimension, c3, though his energy scaling principle is also applicable to the critical condition at which c1 and c2 merge. The Hertzian fracture test, within the validity range of Auerbach’s law, affords a means of measuring surface energy at the fracture surface independent of knowledge about the pre-present flaws.

Radially symmetric phase growth controlled by diffusion

Particular solutions of the diffusion equation, with radial symmetry, in three and in two dimensions, found originally by Rieck, represent the diffusion field around a spherical or cylindrical new phase, growing from a negligible initial radius in an initially uniform medium, maintaining equilibrium conditions at the growing surface. The resulting diffusion field is most simply described by saying that the radial gradient is the same as that of the corresponding potential problem with fixed boundaries, multiplied by a factor exp (-r2/4Dt), where r is radius, D diffusivity, t time. The result is applied to phase growth controlled by the diffusion of heat, solute, or both together. It differs appreciably from the static approximation unless the supercooling, or the supersaturation and the solubility, are small.

The strength and stiffness of polymers

Fully alined long chain polymers would have a Young modulus in the alinement direction similar to that of steel. Practically available polymeric materials have moduli less than one-tenth, and usually less than one-fiftieth of this, even with a high degree of molecular alinement. The paradox is of course resolved in principle by recognizing the predominance of chain folding in polymer crystallization, which allows strong alinement to occur with little or no extended chain continuity in the alinement direction. Questions which arise from this are: (1) what are the actual mechanisms of compliance in the material of relatively low modulus? and (2) by what means may it be possible to achieve full extension and alinement of a high proportion of the chains? Some tentative answers can be given in the light of current researches. Fully extended alinement for a small proportion is obtainable. Extension of a substantial proportion, rather than all, is the desideratum, since the folded chains contribute toughness. It can be useful to think of polymers as intrinsically composite materials, even when chemically homogeneous.

Polymer chain extension produced by impinging jets and its effect on polyethylene solution

Abstract Longitudinal velocity gradients in a flowing system have a powerful, molecular weight dependent orienting influence on macromolecules. A system of impinging jets has been constructed for the in situ observation of this phenomenon in two simple, analysable flow patterns, corresponding to uniaxial extension and compression. The system studied was polyethylene in a solution of xylene. Up to a limiting temperature (108–112°C dependent on the molecular weight) fibrous crystallization was observed in a predictable manner. Above this temperature strong transient birefringence was seen in the regions of extensional flow which disappeared when the flow ceased, the phenomenon persisting up to 200°C the highest temperature examined. There is a sharp lower concentration limit for this effect indicative of a cooperation between molecules. Implications of these observations for chain extension, crystallization, technical flow processes and for molecular entanglements in solution are indicated.

One-Dimensional Dislocations. III. Influence of the Second Harmonic Term in the Potential Representation, on the Properties of the Model

In the previous one-dimensional dislocation model, a single sinusoidal term was taken to represent the potential energy of the deposit as a function of its position on the substrate. In this model a more general representation of the potential, containing a second harmonic term as well, is used, and it is shown that the solution in this case is also expressible in terms of elliptic integrals. The displacements corresponding to a sequence of dislocations (or a single one) are calculated. The work done in generating a single dislocation by a force on a free end is derived and the stability conditions for such a chain determined. It turns out that the properties of single dislocations, especially as concerns their application to misfitting monolayers and oriented overgrowth, remain almost uninfluenced, unless the amplitude of the second harmonic term is so large as to produce a new minimum and provided the overall amplitude of the potential energy is taken to be constant. When the amplitude of the second harmonic term is large, so that the potential curve has a second minimum, a complete dislocation splits up into two halves which are the one-dimensional analogues of Shockley’s ‘half-dislocations’ in close-packed lattices. The equilibrium separation of the two halves, as well as the stability conditions for the existence of a single half, are determined.

Water damage in polyester resins

The nature of cracks, produced in three cured polyester resins during exposure to water has been studied by the combined techniques of optical microscopy, scanning electron microscopy and electron probe X-ray microanalysis. There are many cracks totally enclosed within the resin and it is shown that these must be attributed to pockets of high pressure produced at impurity inclusions by interaction with water : in most cases, osmotic pressure from water soluble inorganic inclusions. The distribution and orientation of cracks gives evidence of relatively complex laminar stress distributions generated by water exposure in a polyester resin plate. There can be a laterally compressive stress in the interior of a plate, with laterally tensile stress at or nearer to the surface. It is pointed out that the interactions of water with a cured polyester resin have the necessary complexity to generate such stress systems since the first effects, water uptake and hydrolysis, causing swelling, also enhance molecular mobility and can promote further double-bond polymerization, accompanied by shrinkage.

Flow-induced crystallization of polyethylene melts

In situ optical observations have been made of the way polyethylene melt can crystallize whilst subject to certain longitudinal velocity gradients. In general crystallization is seen to occur as the generation of fibres 5 to 50 μm in diameter. Hydrodynamic considerations lead to the conclusion that the externally applied velocity field is responsible for the nucleation of the fibrous crystals, subsequent growth is then influenced by both the local streaming of polymer melt around the growing tip of the fibre and the external velocity field. The effect enhanced pressure has on flow induced crystallization is also examined.