Circular contact area

Figure 7.2 Liquid drop with circular contact area on a planar solid surface.

Here, a is the radius of curvature of the three-phase contact line. For a drop with circular contact area it is the contact radius. 

It can be readily seen that ft is less for a circular contact area ik. = 1) than for "line" contact between cylinders with the same effective values of Q, 5 and . ... 

The above result reduces to the case of a circular contact area e = 0 on a half-space where 4 kbRs = 32/(3tt2). 

Martin et al. [61] used a numerical technique to determine the effect of a uniform contact conductance h on the spreading resistance of square and circular contact areas. The dimensionless spreading resistance values were correlated with an accuracy of 0.1 percent by the following expression ... 

General Expression for Circular Contact Area With Arbitrary Flux on Circular Flux Tube. 

Effect of Flux Distribution on Circular Contact Area on Half-Space... 

Simple Correlation Equations of Spreading Resistance for Circular Contact Area... 

Circular Contact Area on Square Flux Tube. Sadhal [155] reported the general solution for an isoflux or equivalent isothermal elliptical contact area on a rectangular flux tube. His general solution gives the dimensionless spreading resistance for an isoflux square contact area on a circular flux tube that has the form... 

Circular Contact Area on Multiple Layers on Circular Flux Tube... 

The effect of single and multiple isotropic layers or coatings on the end of a circular flux tube has been determined by Antonetti [2] and Sridhar et al. [107]. The heat enters the end of the circular flux tube of radius b and thermal conductivity k3 through a coaxial, circular contact area that is in perfect thermal contact with an isotropic layer of thermal conductivity k, and thickness This layer is in perfect contact with a second layer of thermal conductivity k2 and thickness t2 that is in perfect contact with the flux tube having thermal conductivity k3 (Fig. 3.22). The lateral boundary of the flux tube is adiabatic and the contact plane outside the contact area is also adiabatic. The boundary condition over the contact area may be (1) isoflux or (2) isothermal. The dimensionless constriction resistance p2 layers = 4k3aRc is defined with respect to the thermal conductivity of the flux... 

Introduction. Transient spreading resistance occurs during startup and is important in certain micro-electronic systems. The spreading resistance can be defined with respect to the area-average temperature as a single point temperature such as the centroid. Solutions have been reported for isoflux contact areas on half-spaces, circular contact areas on circular flux tubes, and strips on channels. 

Spreading Resistance of Isoflux Circular Contact Area on Half-Space. Beck [6] reported the following integral solution for a circular area of radius a that is subjected to a uniform and constant flux q for t > 0 ... 

Isoflux Circular Contact Area on Circular Flux Tube. Turyk and Yovanovich [118] reported the analytical solutions for transient spreading resistance within semi-infinite circular... 

The operational part of the device is a pair of cylindrical lenses (called disks in Fig. 1). The lenses are crossed, which provides for a theoretically circular contact area. One of these lenses is fixed to the upper part of the SFA and is usually stationary (unless driven by the piezoelectric). The other lens is mounted on the end of the cantilever beam. The sample of interest is mounted on the surface of the lenses. Approximately 50 nm of silver is applied to the back of the sample before mounting. This amount of silver provides for sufficient optical transparency such that enough light can make it through both samples, both lenses, any reflecting optics and into the spectrometer for analysis. 

Kingery assumed a model consisting of two spherical particles of the same radius a, for which the geometrical parameters are similar to those outhned in Fig. 8.10. If each sphere is dissolved away along the center-to-centCT hne by a distance h to give a circular contact area of radius X, then... 

For this rough estimate a circular domain shape was assumed. The longest induction period was assigned to the idealized situation that the circular contact area (with radius r ) is centered on this domain. Based on JKR theory the original contact area was calculated (K = 4.7 x 10 Pa W12 (t = 0) = 50 mN/m). A reaction rate was defined as ratio of ro and reaction time tR. Using this rate the original domain size can be calculated based on the observed induction time. 

A is a constant valid for parallel circular contact areas and a is the distance between the contacting surfaces. A can be calculated using the Lifshitz-van der Waals constant has = 10 ° — 10 J) or the Hamaker constant H = 10 — 10 J). 

Ft is the force with which the particles are pressed together, t the compression time, p and E the viscosity and elasticity, respectively x the diameter of the circular contact area and d the particle diameter. Combining Eq. 7.5 and 7.6 yields a relationship that enables calculation of the van der Waals forces between two viscoelastically deformed spheres ... 

G - particle radius, m b — radius of circular contact area. 

Sliding experiments were carried out at room temperature in a UHV condition of less than 1x10 Pa throughout the experiment A normal load of 250 mN was applied on the Si wafer by a spherical diamond slider (3 mm curvature radius), and the sliding speed was 0.1 mm/sec controlled by a pulse-drive motor. The sUder was driven at 100 pulses per second. In this contact condition, approximately 350 MPa of Hertzian pressure was applied at maximum on a circular contact area 32 pm in diameter, as determined by our calculations. In the observed tracks, the width was less than 40 pm, which was close to the calculated width after 1000 cycles of reciprocal sliding, so that the lateral fluctuation of the contact point was stable enough to assume that the iterated sliding was in contact approximately on the same track. 

Generally, our software is capable of modeling contacts of surfaces that can be locally approximated as ellipsoids or cylinders.) The velocities of both surfaces were 7 m/s. The applied normal load was 150 N. The material of both surfaces was steel with Young s modulus of 210 GPa. In the Hertzian approximation (smooth surfaces with no lubricant), the maximum pressure in the center of the contact would be ph =1.5 GPa with a circular contact area of radius a... 

Using a simple Hertzian model for a sphere (the tip) pressing a flat surface (the Au film), the radius of the circular contact area can be obtained from (39) ... 

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