E21 Diffusion - Salted and Marinated

Topic

During the salting and marinating of the food, solutes from the surrounding brine or marinade—such as common salt, sugar, an acid, or a dissolved spice—penetrate the food via molecular diffusion. This process is driven by the concentration gradient between the high-concentration brine or marinade and the food's interior, where the concentration is initially low. Fick's Second Law governs how the solute's concentration profile within the food evolves over time, while the solute's diffusion coefficient—a value specific to the interaction between the solute and the food's matrix—determines the rate of that evolution. Because the food is meat, its protein matrix offers considerable resistance to the solute; consequently, the solute's diffusion coefficient is much lower than it would be in a free aqueous solution unhindered by a protein matrix.

The depth to which the solute penetrates the food increases not in proportion to the elapsed time, but rather in proportion to the square root of that time. Doubling the penetration depth therefore requires quadrupling the marinating time needed to achieve it. This square-root dependency explains the empirical rule stating that marinating time increases quadratically—rather than proportionally—with the thickness of the piece of food. A thicker piece of food requires a greater penetration depth to reach the center; consequently, the necessary marinating time increases—in accordance with this quadratic empirical rule—at a rate that is more than proportional to the thickness. A higher temperature for the brine or marinade increases the solute's diffusion coefficient—following the Arrhenius relationship; consequently, marinating at this higher temperature (while remaining within food safety limits) can significantly reduce the time required for marination. In practice, rapid marination at a moderate temperature, maintained for several hours, produces a reasonably uniform flavor distribution between the surface and the interior of a food item of moderate thickness.

Therefore, this model—characterized by the diffusing solute and the penetration depth introduced earlier—uses the complementary error function to describe the sigmoidal shape of the solute concentration profile within the food: the solute concentration drops sharply (following that same complementary error function) from the saturation level at the surface to the initially low level in the interior, while the diffusion front advances deeper into the food, progressing according to the square root of time (a key factor in the model's initial formulation). The fact that the solute concentration profile advances according to the square root of time—exactly like the temperature profile described in greater detail by another model in this series on heat conduction—means that both phenomena (solute diffusion and heat conduction) fundamentally share the same mathematical family of diffusion equations, even though they progress through the food at vastly different characteristic speeds.

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