Debye Length - Physical Origin

Physical Origin

The Debye length arises naturally in the thermodynamic description of large systems of mobile charges. In a system of different species of charges, the -th species carries charge and has concentration at position . According to the so-called "primitive model", these charges are distributed in a continuous medium that is characterized only by its relative static permittivity, . This distribution of charges within this medium gives rise to an electric potential that satisfies Poisson's equation:

,

where is the electric constant.

The mobile charges not only establish but also move in response to the associated Coulomb force, . If we further assume the system to be in thermodynamic equilibrium with a heat bath at absolute temperature, then the concentrations of discrete charges, may be considered to be thermodynamic (ensemble) averages and the associated electric potential to be a thermodynamic mean field. With these assumptions, the concentration of the -th charge species is described by the Boltzmann distribution,

,

where is Boltzmann's constant and where is the mean concentration of charges of species .

Identifying the instantaneous concentrations and potential in the Poisson equation with their mean-field counterparts in Boltzmann's distribution yields the Poisson-Boltzmann equation:

.

Solutions to this nonlinear equation are known for some simple systems. Solutions for more general systems may be obtained in the high-temperature (weak coupling) limit, by Taylor expanding the exponential:

\exp\left(- \frac{q_j \, \Phi(\mathbf{r})}{k_B T} \right) \approx 1 - \frac{q_j \, \Phi(\mathbf{r})}{k_B T}.

This approximation yields the linearized Poisson-Boltzmann equation

\nabla^2 \Phi(\mathbf{r}) = \left(\sum_{j = 1}^N \frac{n_j^0 \, q_j^2}{\varepsilon_r \varepsilon_0 \, k_B T} \right)\, \Phi(\mathbf{r}) - \frac{1}{\varepsilon_r \varepsilon_0} \, \sum_{j = 1}^N n_j^0 q_j

which also is known as the Debye-Hückel equation: The second term on the right-hand side vanishes for systems that are electrically neutral. The term in parentheses has the units of an inverse length squared and by dimensional analysis leads to the definition of the characteristic length scale

\lambda_D = \left(\frac{\varepsilon_r \varepsilon_0 \, k_B T}{\sum_{j = 1}^N n_j^0 \, q_j^2}\right)^{1/2}

that commonly is referred to as the Debye-Hückel length. As the only characteristic length scale in the Debye-Hückel equation, sets the scale for variations in the potential and in the concentrations of charged species. All charged species contribute to the Debye-Hückel length in the same way, regardless of the sign of their charges.

The Debye-Hückel length may be expressed in terms of the Bjerrum length as

\lambda_D = \left(4 \pi \, \lambda_B \, \sum_{j = 1}^N n_j^0 \, z_j^2\right)^{-1/2},

where is the integer charge number that relates the charge on the -th ionic species to the elementary charge .

Read more about this topic:  Debye Length

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