M. A. Fortes

The Cellular Structure of Cork from Quercus Suber L.

The main characteristics of the cellular structure of cork from Quercus suber L. are reviewed and complt;mented with new observations of virgin and reproduction cork by scanning electron microscopy. Particular emphasis is given to cell geometry and topology and to the corrugations that are observed in the cell walls. The effect of the growth season in these features is described. Large variations in cell size, wall thickness and corrugations are reported.

Grain size distribution: The lognormal and the gamma distribution functions

Summary The distribution of grain sizes (volume, intersected area or intercept length) in a polycrystal is usually fitted to a lognormal distribution. An alternative distribution function that can be used is the gamma function. The experimental data available in the literature seem to indicate that the latter distribution may be more adequate, since it gives log plots which are not symmetrical. The two distributions (lognormal and gamma) are in fact fairly similar, as we have shown, and the experimental results may not be precise enough to favour one or the other. There is, however, a good argument in favour of the gamma distribution. This is the fact that the cell model leads to a distribution of cell sizes which is very well described by gamma functions, with α values (Table 2) that depend on the definition of size used. The values of α in Table 2 for the 3D partition indicate that the distribution of cell volume is narrower than the distributions of intercept length and area. The adequacy of gamma functions in relation to the cell model is not entirely surprising, considering that the statistics of the cell model is essentially the Poisson statistics. Of course, the cell model may not be an adequate model for a polycrystal. Other models, which admit, for example, a sequential, rather than simultaneous, nucleation have been considered (13), including the Johnson-Mehl model (7,9), but the data available is not sufficient for an analysis of the type undertaken here for the cell model. There is indication, however, that the grain size distributions become broader (smaller values of α, if gamma functions are still appropriate) as the time delay in the nucleation of successive grains increases (e.g.the Johnson-Mehl distribution is broader than the cell distribution). Grain growth is also expected to broaden the distribution of grain sizes (e.g.ref.4).

Axisymmetric liquid bridges between parallel plates

Abstract Axisymmetric liquid bridges between two identical parallel plates are studied under two different configurations, namely, r bridges and θ bridges, according to whether the fluid interface touches the circumference of the plates or ends within the plates. Gravity is neglected. It is shown that, for a given volume of liquid, there may be more than one type of equilibrium configuration for the same separation of the plates. The stability of the various shapes is discussed using simple arguments and taking into account the possibility of transitions between r and θ bridges. Attention is also given to situtations of complete wetting of the plates by one of the fluids. The force of coehsion due to the bridge, identified with the rate of change of the Helmholtz energy with plate separation, 2d, is calculated as a function of d. When the bridge radius is very large compared to its width 2d, the force increases with d and r bridges and is proportional to d−2 in θ bridges.

Deformation of solid surfaces due to capillary forces

Abstract The system of forces acting on a solid partly contacted by a liquid region is described in terms of surface tension forces acting on the line of contact and pressure forces on the surface of the solid with a discontinuity at the line of contact. The forces parallel to the surface are in equilibrium at each point of the line of contact. The normal components produce deformation of the solid. The elastic deformation of thin plates with flexural rigidity and of membranes contacted by drops and liquid bridges with axial symmetry is calculated and compared with that produced in semi-infinite solids. Youngs equation for the contact angle in membranes is thermodinamically derived and it is shown that it can be interpreted in terms of equilibrium of forces at the line of contact, provided the tension in the membrane is included among those forces.

Stress relaxation and creep of cork

The stress relaxation and creep behaviour of cork under compression were characterized in tests done with the compression axis parallel to each of the three principal directions in the tree (radial, tangential and axial). All stress relaxations lead to a linear variation of stress with the logarithm of time, the slopes being nearly independent of stress and direction of compression. Creep stresses in the range 0.36 to 1.72 MPa were used. The strain rate continuously decreases during creep, from initial values around 10−4sec−1 to ∼ 10-7 sec−1 after 8 h, but its dependence on the creep stress and direction of compression is not simple, mainly because different deformation regimes may operate during a single creep test. Compression loading-relaxation-unloading cycles were imposed on specimens, with compression either in the radial or in the tangential direction, with the purpose of simulating the performance of a cork stopper. A “work softening” is observed, i.e. the resistance decreases in successive compressions, particularly between the first two. This is explained in terms of an increased undulation of the cell walls produced in the first compression.

Experiments on defect spreading in hexagonal foams

Defects were introduced in a hexagonal liquid foam prepared by a novel technique and their coarsening was observed. Both isolated defect clusters, with and without dislocation character, and grain boundaries were produced. As the defects coarsen, a gradient of cell size develops along the direction of growth. Differences in the growth characteristics of isolated defects were found, depending on whether the average size of the cells in the defect clusters is larger (type I) or smaller (type II) than the size of the surrounding honeycomb cells. Clusters of type I show an approximately linear increase of diameter with time, while clusters of type II have an initial contraction. The evolution to the scaling regime of a honeycomb containing a distribution of type I clusters is discussed.

Friction properties of cork

The friction coefficient, μ, of cork sliding on another material (glass and steel in most experiments and also cork) was measured for various compressive stresses and sliding velocities. There is a strong effect of stress and a negligible effect of velocity on the friction coefficient. Values of μ are in the range 0.4 to 1.2. The effect of moisture content of cork was also evaluated. For dry cork (6% moisture content) there is anisotropy of the friction coefficient related to the orientation of the sliding plane of cork, with larger values for sliding in the tangential plane (compression in the radial direction) as compared to sliding in planes perpendicular to this. At larger moisture contents, the anisotropy of μ decreases. No in-plane of sliding anisotropy was detected. The friction coefficients for sliding on glass and on steel are comparable, but an effect of roughness was detected. The friction coefficients for sliding on glass and on steel are comparable, but an effect of roughness was detected. The friction against cork is large, with μ close to unity. The interplay between the friction coefficient and the compression properties of cork is discussed.

Characterization of cells in cork

Various topological and metric properties of the cells in the phelogen of the cork oak have been measured in tangential sections of cork by image analysis methods. These include the fractions of cells with i sides (i-cells), the fractions of adjacencies between i- and k-cells and various distributions of cell areas in relation to topology.

Distribution of cell volumes in a Voronoi partition

Abstract The distribution of cell volumes in a Voronoi partition of three-dimensional Euclidean space was obtained by a computer simulation method in which the volume of a cell is derived from the number of lattice points of a reference cubic lattice which are closer to the centre of that cell than to the centre of any other cell. The cell centres have a Poisson distribution. It has been found that the distribution of cell volumes is fairly well fitted by a gamma distribution function with parameter α ≃ 5·56, which is the value of α that gives the exact second moment of the volume distribution, as calculated by Gilbert in 1962. This is an extension to three-dimensions of the applicability of the gamma function to describe the area distribution in a two-dimensional partition (with α ≃ 3·6) and the length distribution in a one-dimensional partition (with the exact value α = 2).

Average topological properties of successive neighbours of cells in random networks

Abstract Random planar (trivalent) networks are studied with the purpose of finding correlations between the number i of sides of a cell (i cell) and average properties of its successive neighbours. The properties investigated are the average number n i k of neighbours of order k of i cells and the average number m i k of sides of these neighbours. The neighbours of a given order k are classified into various types according to their ‘proximity’ to k+1 neighbours. Approximate relations are anticipated, such as a linear variation in n i k with both i and k, and a generalization of the Aboav-Weaire relation to more distant neighbours, that is a linear relation between n i k m i k and i. These relations were ‘experimentally’ assessed by analysing two types of random trivalent network (Voronoi and Poisson) using image analysis methods.