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Dick (1988) used a quasi-crystalline approximation (Lax 1952) for the radiative transfer in disordered media to obtain the range of validity of the Beer law. He calculated the volume concentration, nv [%], above which the Beer law fails, for spherical homogeneous monodisperse particles. That concentration depends on a dimensionless parameter x |n - 1|, where x = 2π a / λ, a is the particle radius, λ is the wavelength of light in the medium surrounding the particle, and n is the complex refractive index of the particle. Dick found that the maximum volume concentration can range from tens of percent to a fraction of 1 percent. Concentration nv varies most for 1 < x |n - 1| < 1.5. For x |n - 1| > 2, nv ≈ 0.01% for the transmission measurement error, δ = 0.02τ, where τ is the optical thickness of the dispersion.
Liang and Shinohara (2001) examined experimentally the applicability of the Beer law to dense (concentrations up to ~3% by volume) aqueous suspensions of monodisperse silica microspheres (0.1 to 2.5 µm) and polydisperse (halfwidth on the order of several micrometers) polyethylene microscpheres (median volume-weighed diameters ranging from 10 to 30 µm). They used a spectrophotometer (Shimadzu UV2400PC, unspecified acceptance angle) to measure transmission at a wavelength of 550 nm of the various suspensions with a geometrical pathlength of 10 mm (see also RTE and transmission measurement). They arrived at the following formula describing the variation of transmission of a suspension with the particle mass concentration, nm [kg m -3]:
| logT = az n log[(nm z f / z + b) / b] | (1) |
where parameters a, f, n (all nondimensional), and b [kg m -3] depend on the particle type and the pathlength of light in the suspension, and z is the nondimensional volume-weighed median diameter of the particles formed by dividing the median diameter in µm by a value of 1 µm. Note that the particle mass concentration, nm, equals to ρnv, where ρ is the particle material density, and nv is the particle volume concentration.
Liang and Shinohara found parameters a, b, f, and n to equal respectively to: 0.808, 0.698, 0.0681, and -0.175, for the silica particles, while for the polyethylene particles these parameters were found to equal respectively to: 0.633, 17.8, 11.1, and 0.164 (note significant differences in the values between the two particle types).
If the concentration nm is small (nm << b / z f / z), Eq. 1 reduces to the Lambert-Beer law (see Beer's law):
| logT = nm loge (az n + f / z) / b | (2) |
The relative difference between the transmission, T, values calculated with these two formulas is on the order of 0.5% for a 1 µm silica microspheres suspension with an optical thickness of ~10 [see also comments on related work of Swanson et al (1999) in Lambert's law]. That difference is nearly an order of magnitude higher (~3%) for the polyethylene dispersion of similar optical thickness.
See also Dick (1988) for a brief review of older experimental results regarding the applicability of the Lambert and Beer laws to dense dispersions.
| CITATION: Jonasz M. 2006. Beer's law for dense aquatic dispersions (www.tpdsci.com/Tpc/BeerLwHD.php). In: Top. Part. Disp. Sci. (www.tpdsci.com). |
HISTORY: Published: 18-Jan-2006 Modified: 06-Dec-2006 Peer-reviewed: 06-Dec-2006 |
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