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Mathematical Description of Diffusion Processes in the Soil-Plant System¹

ABSTRACT

A mathematical treatment of the process of diffusion of ions through a porous medium to plant roots was derived by assuming the root acts as a cylindrical sink in an infinite, unstirred medium. The concentration dependence of the ion uptake process was coupled to diffusion in the medium by assuming transfer across the root surface was directly proportional to the concentration at the surface. The apporximate value of the proportionality constant for a limited range of concentration was derived from published results of uptake of phosphorus by excised roots in aerated, stirred phosphate solutions. Solutions of the diffusion equation were obtained for values of the diffusion coefficient from 10^-5 to 10^-9 cm.² per sec.

The results indicate flux per unit of root surface area may increase tenfold as the radius of the roots is decreased from 5 x 10^-2 cm. (approximate radius of root cylinder) to 7.5 x 10^-4 cm. (approximate radius of root hairs). Estimates of root hair dimensions and incidence obtained by microscopic examination of roots removed from greenhouse pots indicate that the total flux across root hair surfaces may be 3 to 10 times greater than the flux across the surface of the central root cylinder.

¹ Contribution from the Soils and Fertilizer Research Branch, Division of Agricultural Relations, Tennessee Valley Authority, Wilson Dam, Ala. Presented before Div. II, Soil Science Society of America, Dec. 5, 1960, at Chicago, Ill.

² Research Chemist.

Received for publication December 5, 1960. Accepted for publication July 26, 1961.