Two Parallel Hard Plates

Before considering the interaction between two ion-penetrable membranes, we here treat the interaction between two similar ion-impenetrable hard plates 1 and 2 carrying surface charge density cr at separation h in a salt-free medium containing counterions only (Fig. 18.1) [2]. We take an x-axis perpendicular to the plates with its origin on the surface of plate 1. As a result of the symmetry of the system, we need consider only the region 0 x h 2. Let the average number density and the valence of counterions be o and z, respectively. Then we have from electroneutrality condition that 

Note that Mq, which is a function of h, is proportional to h. We set the equilibrium electric potential j/(x) to zero at points where the volume charge density Pe x) resulting from counterions equals its average value —zen). 

Biophysical Chemistry of Biointerfaces By Hiroyuki Ohshima Copyright 2010 by John Wiley Sons, Inc. 

FIGURE 18.1 Schematic representation of the electrostatic interaction between two parallel identical hard plates separated by a distance h between their surfaces. 

Here we have assumed that the relative permittivity is assumed to take the same value inside and outside the membrane. We also assume that the distribution of counterions n x) obeys a Boltzmann distribution, namely, 

---

Equations (15.49) and (15.50), respectively, agrees with the expression for the electrostatic interaction energy between two parallel hard plates at constant surface charge density and that for two hard spheres at constant surface charge density [4] (Eqs. (10.54) and (10.55)). 

We follow the work of Mao et al. [47]. The same results as presented here were obtained earlier by Walz and Sharma [48] using a somewhat different method. The starting point for our treatment is a result by Henderson [49] that the force per unit area between two parallel hard plates immersed in a suspension of hard spheres is given by... 

Fig. 2.1 Schematic picture of two parallel flat plates in the presence of penetrable hard spheres dashed circles)...

Fig. 2.3 The overlap volume (hatched area) of depletion layers due to penetrable hard spheres between two parallel flat plates equals A a — h)...

Figure 15. Density profiles of cations, anions, and of total charge near an uncharged wall. Two parallel uncharged plates at at = 0 and x = 6nm confine all particle centers. The dielectric constant is the same across an interface. The Bjerrum length is Aj =0.72nm, the ion sizes are cr = 0.8nm, oo, while in the numerical model we used e = 1.

When bounding walls exist, the particles confined within them not only collide with each other, but also collide with the walls. With the decrease of wall spacing, the frequency of particle-particle collisions will decrease, while the particle-wall collision frequency will increase. This can be demonstrated by calculation of collisions of particles in two parallel plates with the DSMC method. In Fig. 5 the result of such a simulation is shown. In the simulation [18], 2,000 representative nitrogen gas molecules with 50 cells were employed. Other parameters used here were viscosity /r= 1.656 X 10 Pa-s, molecular mass m =4.65 X 10 kg, and the ambient temperature 7 ref=273 K. Instead of the hard-sphere (HS) model, the variable hard-sphere (VHS) model was adopted in the simulation, which gives a better prediction of the viscosity-temperature dependence than the HS model. For the VHS model, the mean free path becomes ... 

In this chapter, we discuss two models for the electrostatic interaction between two parallel dissimilar hard plates, that is, the constant surface charge density model and the surface potential model. We start with the low potential case and then we treat with the case of arbitrary potential. 

In Chapter 11, we derived the double-layer interaction energy between two parallel plates with arbitrary surface potentials at large separations compared with the Debye length 1/k with the help of the linear superposition approximation. These results, which do not depend on the type of the double-layer interaction, can be applied both to the constant surface potential and to the constant surface charge density cases as well as their mixed case. In addition, the results obtained on the basis of the linear superposition approximation can be applied not only to hard particles but also to soft particles. We now apply Derjaguin s approximation to these results to obtain the sphere-sphere interaction energy, as shown below. 

FIGURE 14.1 Interaction between two parallel dissimilar hard plates 1 and 2 at separation h. 

Large deformation methods usually are based on the determination of the amount of applied force required to induce a change in a sample. Parameters determined include hardness, spreadability, cutting force, or yield force (4). One such method involves the compression of a sample between two parallel plates to determine relative hardness values via the measurement of yield force. 

Ohwada, T., Sone, Y., and Aoki, K., (1989) Numerical Analysis of the Poiseuille and Thermal Transpiration Flows Between Two Parallel Plates on the Basis of the Boltzmann Equation for Hard-Sphere Molecules, Physics of Fluids A, Vol. 1(12), pp.2042-2049. 

T. Ohwada, Y. Sone, and K. Aoki. Nnmerical analysis of the Poiseuille and thermal transpiration flows between two parallel plates on the basis of the Boltzmann eqnation for hard sphere molecnles. Phys. Fluids A, 1(12) 2042-2049(1989). 

The A-A extruders are equipped with a flat perforated plate or screen, combined with two flat extrusion heads. The wet mass is pressed axially through the screen parallel to the feed screws. The extrusion forces are the highest, compared to other low pressure extruders, resulting in a hard and dense extrudate. It is commonly used in the extrusion of food, thermoplastics, and other industries, where a large pellet diameter and thermosetting properties are desired. 

We have already discussed confinement effects in the channel flow of colloidal glasses. Such effects are also seen in hard-sphere colloidal crystals sheared between parallel plates. Cohen et al. [103] found that when the plate separation was smaller than 11 particle diameters, commensurability effects became dominant, with the emergence of new crystalline orderings. In particular, the colloids organise into z-buckled" layers which show up in xy slices as one, two or three particle strips separated by fluid bands see Fig. 15. By comparing osmotic pressure and viscous stresses in the different particle configurations, tlie cross-over from buckled to non-buckled states could be accurately predicted. 

Stress Shielding. Beyond the traditional biocompatibility issues, hard tissue biomaterials must also be designed to minimize a phenomenon known as stress shielding. Due to the response of bone remodeling to the loading environment, as described by Wolffs law, it is important to maintain the stress levels in bone as close to the preimplant state as possible. When an implant is in parallel with bone, such as in a bone plate or a hip stem, the engineered material takes a portion of the load— which then reduces the load, and as a result, the stress, in the remaining bone. When the implant and bone are sufficiently well bonded, it can be assumed that the materials deform to the same extent and therefore experience the same strain. In this isostrain condition, the stress in one of the components of a two-phase composite can be calculated from the equation ... 

The slice compression test has been applied to polymer-matrix composites even though it was developed to probe the interface in ceramic matrix composites [16]. A thin slice sample of unidirectional composite is produced with the cut surface perpendicular to the fiber axis. The surfaces are cut and polished to be parallel to each other and perpendicular to the fibers. The thin slice is loaded in compression in the fiber axis direction with two plates. One of the plates is made of a very hard material such as silicon nitride and the other of a soft material, e.g. pure aluminum which can deform as the fibers are compressed into it. The thickness of the slice must be controlled to allow the fibers to debond without failing in compression as well as allowing them to slide inside through the matrix. The depth of the fiber indentation into the plate can be related to the interfacial shear strength [17]. 

---