Washburn equation

In this method, mercury (which is a non-wetting liquid) is forced into the pores of a dry sample. For each applied pressure, the volume of mercury entering the sample porous structure is determined very accurately (e.g. by measuring the variation of capacity induced by the reduction in height of the Hg column connected to the measuring cell). The relationship between pressure P and pore radius rp is given by the modified Laplace equation (Washburn equation)... 

Contact angles on finely divided solids are more difficult to measure, but are often more desired and more important than those on large solid surfaces. One method of obtaining such contact angles is to pack the powder into a glass tube and measure the rate of penetration of the liquid into it (Brail, 1974). The distance of penetration l in time t of a liquid of surface tension yM and viscosity T is given by the modified Washburn equation (Washburn, 1921) ... 

Calculation of Total Osmotic Pressure from Freezing Point Data — From the lowering of fieezing point, At the total osmotic pressure may be readily and accurately calculated by means of the equation (Washburn, loc cit)—... 

The reconstitution behavior of porous particles (their so-called instant properties ) can be explained by the fact that the solvent has to penetrate first into the pores of the particles in order to loosen contact forces by, for example, dissolving contact bridges formed by crystals or amorphous material between primary particles (Pfalzer et al., 1973 Schubert, 1975) or by reducing inter-particulate van der Waals forces. The penetration time of the liquid into a porous spray-dried particle with diameter d (wetting time) is given by the well-known Washburn equation (Washburn, 1921 Schubert, 1990) ... 

The combination of Equations 9.2 and 9.3 gives the Washburn equation (Washburn 1921),... 

The Washburn equation has most recently been confirmed for water and cyclohexane in glass capillaries ranging from 0.3 to 400 fim in radii [46]. The contact angle formed by a moving meniscus may differ, however, from the static one [46, 47]. Good and Lin [48] found a difference in penetration rate between an outgassed capillary and one with a vapor adsorbed film, and they propose that the driving force be modified by a film pressure term. 

The Washburn model is consistent with recent studies by Rye and co-workers of liquid flow in V-shaped grooves [49] however, the experiments are unable to distinguish between this and more sophisticated models. Equation XIII-8 is also used in studies of wicking. Wicking is the measurement of the rate of capillary rise in a porous medium to determine the average pore radius [50], surface area [51] or contact angle [52]. 

Section 3.7, the gas adsorption method breaks down for practical reasons. Since the angle of contact of mercury with solids is 140° (see later), and therefore more than 90°, an excess pressure Ap is required to force liquid mercury into the pores of a soh d. The idea of using mercury intrusion to measure pore size appears to have been first suggested by Washburn who put forward the basic equation... 

Fig. 2. Liquid flow-through capiUary (Washburn equation). Time rate of penetration = dl/dt = l/4[7/ 7] x [r/l] x cos0, where 7 = surface tension and 77 = viscosity. A, contact angle 9 between Hquid and capiUary waU B, penetrating Hquid C, partiaUy fiUed capiUary, r = radius, and I = length already filled.

If the driving pressure is taken to be the capillary pressure, 2yivCOS0/r, Eq. 23 may be integrated, assuming 9 and r] are constant to give the Washburn equation [43] which shows the penetration jCt is proportional to the square root of time t... 

With the advent of mercury intrusion porosimeters, it is advantageous to perform a pore size distribution of investigational batches of a drug [43]. The Washburn equation [44] states that the pressure, P, necessary to intrude a pore is given by... 

The second approach considers the ability of the fluid to penetrate the powder bed and involves the measurement of the extent and rate of penetration into a column of powder, better known as the Washburn Test. By considering the powder to consist of capillaries of radius R, as illustrated in Fig. 19, the equilibrium height of rise, he, is determined by equating capillary and head pressures, or... 

Mercury porosimetry is based on the fact that mercury behaves as a nonwetting liquid toward most substances and will not penetrate the solid unless pressure is applied. To measure the porosity, the sample is sealed in a sample holder that is tapered to a calibrated stem. The sample holder and stem are then filled with mercury and subjected to increasing pressures to force the mercury into the pores of the material. The amount of mercury in the calibrated stem decreases during this step, and the change in volume is recorded. A curve of volume versus pressure represents the volume penetrated into the sample at a given pressure. The intrusion pressure is then related to the pore size using the Washburn equation... 

For a circular pore opening, the Washburn equation [37] relates the applied pressure and the radius of the pores intruded with a nonwetting liquid ... 

Inert polymer matrices, studied for use in possible controlled release applications, have used porosimetry to investigate a number of properties [54-56]. The kinetics of liquid capillary penetration into these matrices was explored using a modified Washburn equation [54]. It was shown that water... 

Relation 9.77 is usually called the Washburn equation [55,237], One should consider it as a special case of the fundamental Young-Laplace equation [3,9-11], Washburn was the first to propose the use of mercury for measurements of porosity. Now, it is a common method [3,8,53-55] of psd measurements for a range of sizes from several hundreds of microns to 3 to 6 nm. The lower limit is determined by the maximum pressure, which is applied in a mercury porosimeter the limiting size of rWl = 3 nm is achieved under PHg = 4000 bar. The measurements are carried out after vacuum treatment of a sample and filling the gaps between pieces of solid with mercury. Further, the hydraulic system of a device performs the gradual increase of PHg, and the appropriate intmsion of mercury in pores of the decreasing size occurs. 

A commonly used simple method for determining if there are any cracks or pinholes in microporous membranes is the so-caUed bubble point test. It has been used by many organic membrane manufacturers and users alike and is also being adopted by some inorganic membrane manufacturers. The method utilizes the Washburn equation... 

The ability of a system to draw water can be summarized by the Washburn equation (1). 

The principle of measurement is based on the fact that mercury does not wet most substances and thus, it will not penetrate pores by capillary action. Surface tension opposes the entrance of any liquid into pores, provided that the hquid exhibits a contact angle greater than 90° [115,116]. Therefore, external pressure is required to force the liquid (mercury in this case) into the pores of the material. The pressure that has to be applied to force a liquid into a given pore size is given by the Washburn equation. 

Equation (10.23) was first derived by Washburn and is the operating equation in mercury porosimetry. For wetting angles less than 90°, cos 6 is positive and AP is negative, indicating that pressure greater than ambient... 

According to the Washburn equation (10.23) a capillary of sufficiently small radius will require more than one atmosphere of pressure differential in order for a nonwetting liquid to enter the capillary. In fact, a capillary with a radius of 18 A (18 x 10 ° m) would require nearly 60 000 pounds per square inch of pressure before mercury would enter-so great is the capillary depression. The method of mercury porosimetry requires evacuation of the sample and subsequent pressurization to force mercury into the pores. Since the pressure difference across the mercury interface is then the applied pressure, equation (10.23) reduces to... 

An alternate derivation of the Washburn equation can be pursued as follows. For a pore of circular cross-section with radius r the surface tension acts to force a nonwetting liquid out of the pore. The force developed due to interfacial tensions is the product of the surface tension y of the liquid and the circumference (2nr) of the pore, that is. 

Another approach to the Washburn equation involves the work required... 

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