Capillary forces between spheres during agglomeration and liquid phase sintering
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I.
INTRODUCTION
IN liquid
phase sintering and powder agglomeration, the system consists of solid particles, pores, and a liquid. In the initial stage of liquid phase sintering the solid particles often preserve their initial size and shape while undergoing rearrangement without significant dissolution of the solid in the liquid.I Likewise, in agglomeration processes such as spray drying a wetting liquid is used to pull particles into a dense cluster. An important parameter necessary to mathematically treat these processes is the capillary force existing between the solid particles and the liquid. The first step in calculating the capillary force is to consider a system consisting of two solid spherical particles connected by a liquid bridge. Even though such a system appears to be simple, the task of calculating the force may become quite formidable. The reason is that the capillary force depends on the shape of the liquid bridge which is a function of the contact angle, particle size, and amount of liquid. The true configuration of the bridge is called a nodoid and is described by a differential equation given below.
II.
BACKGROUND
The capillary force F between two solid bodies connected by a liquid bridge depends on the shape of the liquid; an accurate description of the liquid profile is therefore desirable. In Figure 1, two spheres with radius A are shown separated by the distance D. The angle a is the semi-angle subtended by the perimeter where the solid, liquid, and vapor phases meet. The angle between the liquid-vapor surface tension vector Yl-v and the x-axis is ~b. The contact angle between the solid and the liquid is 0. The radius of curvature of the liquid surface in the x-y plane is S and its radius in the orthogonal plane is T. The equations describing these two radii are given by 2 S = [1 + (y,)Z]3n/y,,
[1]
K.S. I/WANG is Materials Engineering Manager, Power Semiconductor Division, General Instrument Corporation, 600 West John Street, Hicksville, NY 11802. R. M. GERMAN and E V. LENEL are Professors, Materials Engineering Department, Renssetaer Polytechnic Institute, Troy, NY 12180-3590. Manuscript submitted July 7, 1986.
METALLURGICALTRANSACTIONSA
(2
o
Fig. 1--The geometry and definition of variables ofa nodoid liquid profile which connects two identical spheres.
and T = y/sin fl = y[1 + (y,)2]~n
[2]
where y describes the liquid profile as a function of x, y' designates dy/dx, y" designates d2y/dx 2, and fl is given in Figure 1. Due to the curvature of the liquid-vapor interface, there will be a pressure differential P between the liquid meniscus and the vapor phase. According to the Laplace equation, the pressure deficiency of the liquid (neglecting the gravity effect) is given by
,(1 1 ) : ,
i3,
Substituting Eqs. [1] and [2] into Eq. [3] gives P/Y,-v = y"/[1 + (y,)213/2 _ l/[y(1 + (y,)2)]1/2. [41 Since the pressure is equal everywhere in the liquid, the pressure deficiency P in Eq. [4] is a constant. Equation [4] is thus a differential equation describing the profile of the liquid-
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