An Exact Solution Technique for Effective Linewidth Measurement

This technique solves the two-dimensional Poisson equations in geometries involving cylindrical objects. The method uses three fundamental solutions, corresponding to a line force, a line couple and a pressure gradient, on each cylinder. Superposition of the fundamental solutions due to all the cylinders involved, while approximately satisfying the no-slip condition on each cylinder, yields a mobility matrix relating the various forces and motions of all the cylinders. Any specific problem can be solved by prescribing the motions of the cylinders and solving the matrix. For problems involving few cylinders or with a sufficient degree of symmetry this can be done analytically.Once constructed, the general method is applied analytically to a series of specific problems. The permeability of an eccentric annulus is derived. The result is numerically indistinguishable from the exact solution to the problem, but unlike the exact solution the present one is obtained in closed form. The drag on two parallel rods moving past one another is also derived and compared to the exact solution. In this case the result is accurate for rod separations down to about 0.2 times the rod diameter. Finally the drag on a rod moving in a triangular array of identical rods is derived. Here it is shown that due to screening it is sufficient to include the six nearest neighbours, regardless of the rod separation. Although the present examples are all worked out analytically, the matrix can also be solved numerically, in which case any two-dimensional arrangement of cylindrical objects can be studied.

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Exact Solution of the Start-Up Electroosmotic Flow of Generalized Maxwell Fluids in Triangular Microducts

Journal of Fluids Engineering ◽

10.1115/1.4050940 ◽

2021 ◽

Author(s):

F.Talay Akyildiz ◽

Dennis A. Siginer

Keyword(s):

Exact Solution ◽

Electroosmotic Flow ◽

Fractional Derivatives ◽

Maxwell Fluid ◽

Solution Technique ◽

Maxwell Fluids ◽

Start Up ◽

Triangular Region ◽

Overshoot Phenomenon ◽

Poisson Boltzmann

Abstract Unsteady electroosmotic flow of generalized Maxwell fluids in triangular microducts is investigated. The governing equation is formulated with Caputo-Fabrizio time-fractional derivatives whose orders are distributed in the interval [0, 1). The linear momentum and the Poisson-Boltzmann equations are solved analytically in tandem in the triangular region with the help of the Helmholtz eigenvalue problem and Laplace transforms. The analytical solution developed is exact. The solution technique we use is new, leads to exact solutions, is completely different from those available in the literature and is applicable to other similar problems. The new expression for the velocity field displays experimentally observed 'velocity overshoot' as opposed to existing analytical studies none of which can predict the overshoot phenomenon. We show that when Caputo-Fabrizio time-fractional derivatives approach unity the exact solution for the classical upper convected Maxwell fluid is obtained. The presence of elasticity in the constitutive structure alters the Newtonian velocity profiles drastically. The influence of pertinent parameters on the flow field is explored.

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A strong-interaction theory for the motion of arbitrary three-dimensional clusters of spherical particles at low Reynolds number

Journal of Fluid Mechanics ◽

10.1017/s0022112088003155 ◽

1988 ◽

Vol 197 ◽

pp. 1-37 ◽

Cited By ~ 44

Author(s):

Qaizar Hassonjee ◽

Peter Ganatos ◽

Robert Pfeffer

Keyword(s):

Exact Solution ◽

Reynolds Number ◽

Low Reynolds Number ◽

Fluid Velocity ◽

Three Dimensional ◽

Particle Diameter ◽

Simple Expression ◽

Spherical Particles ◽

Solution Technique ◽

Low Reynolds

This paper contains an ‘exact’ solution for the hydrodynamic interaction of a three-dimensional finite cluster at arbitrarily sized spherical particles at low Reynolds number. The theory developed is the most general solution to the problem of an assemblage of spheres in a three-dimensional unbounded media. The boundary-collocation truncated-series solution technique of Ganatos, Pfeffer & Weinbaum (1978) for treating planar symmetric Stokes flow problems has been extensively modified to treat the non-symmetric multibody problem. The orthogonality properties of the eigenfunctions in the azimuthal direction are used to satisfy the no-slip boundary conditions exactly on entire rings on the surface of each particle rather than just at discrete points.Detailed comparisons with the exact bipolar solutions for two spheres show the present theory to be accurate to five significant figures in predicting the translational and angular velocity components of the particles at all orientations for interparticle gap widths as close as 0.1 particle diameter. Convergence of the results to the exact solution is rapid and systematic even for unequal-sized spheres (a1/a2 = 2). Solutions are presented for several interesting and intriguing configurations involving three or more spherical particles settling freely under gravity in an unbounded fluid or in the presence of other rigidly held particles. Advantage of symmetry about the origin is taken for symmetric configurations to reduce the collocation matrix size by a factor of 64. Solutions for the force and torque on three-dimensional clusters of up to 64 particles have been obtained, demonstrating the multiparticle interaction effects that arise which would not be present if only pair interactions of the particles were considered. The method has the advantage of yielding a rather simple expression for the fluid velocity field which is of significance in the treatment of convective heat and mass transport problems in multiparticle systems.

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Linewidth measurement using the low voltage SEM

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100144681 ◽

1986 ◽

Vol 44 ◽

pp. 652-653

Author(s):

M.G. Rosenfield

Keyword(s):

Electron Microscope ◽

Scanning Electron Microscope ◽

Integrated Circuits ◽

Secondary Electron ◽

Low Voltage ◽

Fabrication Process ◽

Critical Dimensions ◽

Linewidth Measurement ◽

Threshold Technique ◽

Scanning Electron

Minimum feature sizes in experimental integrated circuits are approaching 0.5 μm and below. During the fabrication process it is usually necessary to be able to non-destructively measure the critical dimensions in resist and after the various process steps. This can be accomplished using the low voltage SEM. Submicron linewidth measurement is typically done by manually measuring the SEM micrographs. Since it is desirable to make as many measurements as possible in the shortest period of time, it is important that this technique be automated.Linewidth measurement using the scanning electron microscope is not well understood. The basic intent is to measure the size of a structure from the secondary electron signal generated by that structure. Thus, it is important to understand how the actual dimension of the line being measured relates to the secondary electron signal. Since different features generate different signals, the same method of relating linewidth to signal cannot be used. For example, the peak to peak method may be used to accurately measure the linewidth of an isolated resist line; but, a threshold technique may be required for an isolated space in resist.

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