Diffusion

Diffusion is the process by which individual molecules of a substance move from an area of high concentration to an area of low concentration due to random molecular motion, driven by a concentration gradient.

Diffusion holds significant importance in pharmaceutical operations. Additionally, it serves as a crucial component in a wide range of applications. These include aiding drug release from various delivery systems, facilitating drug absorption and elimination processes within the body. Furthermore, diffusion plays an essential role in procedures like dialysis, osmosis, and ultrafiltration, which are indispensable in medical treatments and research.

Fick’s First Laws of Diffusion

The chemical potential of a solute in solution is given by:
$$ \mu=\mu^0+RT\ln\left(c\right) $$
where:
µ: The chemical potential.
µ°: The chemical potential in the standard state.
R: The universal gas constant.
T: The absolute temperature.
C: The concentration of the solute.

The change in chemical potential dµ due to a concentration gradient dC is then.

$$ \frac{d\mu}{dc}=\frac{d(\mu^0+RT\ln\left(c\right))}{dc} $$

Or

$$ d\mu=RT\left(\frac{dc}c\right) $$

The  chemical potential difference (dµ) represents the work done on the system when transferring a mole of solute from concentration C + dC to concentration C. This work can be expressed as the force (F) multiplied by the distance (dx), thus yielding dµ = -Fdx. So:

$$ F=-\frac{d\mu}{dx}=-\frac{RT}c(\frac{dc}{dx}) $$

The negative sign arises because the concentrations (C) and distances (x) increase in opposite directions.

When a molecule (or particle) experiences a driving force, its velocity increases until the frictional force acting on it balances the driving force. This frictional force (Ff) is directly proportional to the molecule’s velocity, with the constant of proportionality (f) termed the frictional coefficient, denoted as:

$$ F_f=Nvf $$

Where v represents the velocity in the x direction and N is the Avogadro’s number. Hence, we have:

$$ Nvf=-\frac{RT}c(\frac{dc}{dx}) $$

Or.

$$ cv=-\frac{RT}{Nf}(\frac{dc}{dx})=-\frac{kT}f(\frac{dc}{dx}) $$

where k is Boltzmann’s constant and Cv is the flux (J).

$$ J=-\frac{kT}f(\frac{dc}{dx}) $$

When the temperature is maintained constant, the proportionality (kT/f) constant relies on the molecular quantity . This constant equates to the diffusion coefficient (D), which is a macroscopic quantity measurable through experimentation.

$$ J=-D\frac{dc}{dx} $$

This is the one-dimensional form of Fick’s first law, which states that the flux of a substance is proportional to the negative of the concentration gradient. This negative sign signifies that when the concentration gradient is positive, the flux moves in the direction of decreasing concentration.

Fick’s Second Law of Diffusion

Fick’s second law of diffusion serves as the foundation for many mathematical models of diffusion processes. It addresses the rate of change of diffusant concentration at a specific point within the system. Unlike Fick’s first law, which focuses on the mass diffusing across a unit area of barrier in unit time, Fick’s second law emphasizes the change in concentration with time at a definite location.

Derivation of Fick’s second law

Consider a particular volume element within the system. The concentration (c) within this volume element changes over time (t) due to the net flow of diffusing molecules into or out of the region. Any difference in concentration arises from disparities between input and output (see the Fig).

Fig. : illustrate the diffusion through a membrane with thickness dx. J in is the flux into the membrane and J out is flux out of membrane

Therefore, the concentration of diffusant in the volume element changes with time, denoted as Δc/Δt , as the flux or amount diffusing changes with distance, ΔJ/Δx, in the x direction, or:

$$ \frac{\partial c}{\partial t}=-\frac{\partial J}{\partial x}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left(1\right)\\ $$

Upon differentiation of the expression for Fick’s first law with respect to x, we obtain:

$$ -\frac{\partial J}{\partial x}=D\frac{\partial^2c}{\partial x^2}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left(2\right)\\ $$

Substituting the expression for ∂c/∂t from equation (1) into equation (2) results in another expression, which is a formulation of Fick’s second law.

$$ \frac{\partial c}{\partial t}=D\frac{\partial^2c}{\partial x^2}\\ $$

This equation specifically accounts for diffusion occurring along the x direction. To capture concentration changes of a diffusant in three dimensions, Fick’s second law is expressed in its general form:

$$ \frac{\partial c}{\partial t}=D(\frac{\partial^2c}{\partial x^2}+\frac{\partial^2c}{\partial y^2}+\frac{\partial^2c}{\partial z^2})\\ $$

In pharmaceutical diffusion problems, the general form of Fick’s second law is often unnecessary as movement in a single direction suffices for most scenarios. This law states that the rate of concentration change over time within a particular region is proportional to the concentration gradient’s change at that point in the system.

Steady State and Sink Conditions

The steady state condition in diffusion occurs when the rate of diffusion remains constant over time. In this state, there is no net accumulation or depletion of the diffusing substance within the system. This condition is described by Fick’s first law, which gives the flux (or rate of diffusion through unit area) at equilibrium.

In contrast, the sink condition is established in diffusion experiments to maintain a low concentration of the diffusing substance in the receptor compartment. This is achieved by constantly removing and replacing the solution in the receptor compartment with fresh solvent. By doing so, any diffusing substance reaching the receptor compartment is rapidly diluted, preventing its accumulation and maintaining a constant concentration gradient for continued diffusion. The donor compartment serves as the source of the diffusing substance, while the receptor compartment acts as the sink, effectively removing the diffusing substance from the system.

Reference:

  • Sinko, P. (2011). Martin’s Physical Pharmacy and Pharmaceutical Sciences. Baltimore, : Lippincott Williams & Wilkins, a Wolters Kluwer business.
  • Robinson, J. Lee, V. (1997). Controlled drug delivery fundamentals and applications. New york, NY: Taylor & Francis Group, LLC