Topics in Particle and Dispersion Science

Power-law PSD and the fractal dimension of particles

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The power-law function is frequently called scale-free or self-similar, because its shape is invariant under scale transformations of the independent variable (for example, Newman MEJ 2005). Thus, the power-law distribution is an indication that objects whose properties are governed by the power law have fractal geometry.

Jiang Q and Logan 1991 showed by dimensional analysis that, under certain assumptions, the slope, s, of the power law PSD in dispersions that are formed by aggregation is a function of the two- and three-dimensional fractal dimensions of the aggregates, d₂ and d₃:

sBr = 1 + d3 / 2

(1)

ssf = (d3 + 5) / 2

(2)

sds = {3 + d3 + (2 + d3 - d2) / [2 - f(Re)]} / 2

(3)

where suffices Br, sf, and ds denote the brownian, shear flow, and differential settling aggregation modes respectively, and Re is the Reynolds number.

Fractal dimensions are defined by the following relationships:

G ≈ D d2

(4)

where G is the projected area of the aggregate, D is the aggregate "size", and

V ≈ D d3

(5)

where V is the aggregate volume. The two fractal dimensions are related as follows: if d₃ < 2 then d₂ = d₃, otherwise, d₂ = 2. Bushell GC et al 2002 review methods of determining the fractal dimensions of aggregates.

The assumptions made by Jiang Q and Logan include:

• The particle size distribution is in steady state, i.e. particles are created and removed (by sedimentation) at a constant rate.
• A single process (brownian, shear flow, or differential settling aggregation) governs aggregation in a given non-overlapping particle size range; particles that are larger than the upper particle size limit of all these mechanisms settle out and are not involved in aggregation. As noted by Burd AB and Jackson 2002, this assumption implies that particles coagulate with similarly-sized particles only.
• The probability that two particles will adhere once they have collided is constant.

Burd AB and Jackson 2002 note that these assumptions may not necessarily all hold and show by numerical simulation for d₃ = 3 (solid particles) that the fractal dimension of aggregates obtained in this manner may be in error, except that obtained for size distributions resulting from the brownian aggregation mode.

CITATION: Jonasz M. 2006. Power-law PSD and the fractal dimension of particles (www.tpdsci.com/Tpc/PsdPwLwFct.php). In: Top. Part. Disp. Sci. (www.tpdsci.com).

HISTORY: Published: 13-Jan-2006 Modified: 29-May-2006 Peer-reviewed: PENDING

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