Diffusion coefficient

The diffusion coefficient , also called diffusion constant or diffusivity , is a transport coefficient and is used in Fick's laws to calculate the thermally induced transport of a substance due to the random movement of the particles . These can be individual atoms in a solid or particles in a gas or liquid . The diffusion coefficient is therefore a measure of the mobility of the particles and can be determined according to the Einstein-Smoluchowski equation from the average square of the distance covered per time : ${\displaystyle D}$ ${\displaystyle \langle x^{2}\rangle }$ ${\displaystyle t}$

${\displaystyle D={\frac {\langle x^{2}\rangle }{2t}}}$(or alternatively using the Green-Kubo relations ).

The SI unit of the diffusion coefficient is therefore . The specification of the diffusion coefficient always includes the specification of which substance diffuses in which substance , and the most important influencing variable is temperature . ${\displaystyle \mathrm {m^{2}/s} }$

In gases

Examples of diffusion coefficients in gases (at 1 atm )

system	Temperature in ° C	Diffusion coefficient in m² / s
Air - oxygen	0	1.76 × 10 −5
Air - carbon dioxide	8.9	1.48 × 10 −5
	44.1	1.77 × 10 −5
Hydrogen - nitrogen	24.1	7.79 × 10 −5

Diffusion coefficients in gases are strongly dependent on temperature and pressure . As a first approximation, doubling the pressure results in halving the diffusion coefficient.

According to the Chapman - Enskog theory, the diffusion coefficient follows the following equation for two gaseous substances (indices 1 and 2):

${\displaystyle D_{12}={\frac {3}{8}}\left({\frac {N}{\pi }}{\frac {M_{1}+M_{2}}{2M_{1}M_{2}}}\right)^{\frac {1}{2}}{\frac {\left(k_{\mathrm {B} }T\right)^{\frac {3}{2}}}{p\sigma _{12}^{2}\Omega _{12}^{*}}}}$

with the physical quantities

• ${\displaystyle D_{12}}$ - diffusion coefficient
• N - Avogadro's constant
• M _1.2 - molar masses of the substances
• k _B - Boltzmann constant
• T - temperature
• p - pressure
• ${\displaystyle \sigma _{12}={\frac {\sigma _{1}+\sigma _{2}}{2}}}$ - (mean) collision diameter (values ​​tabulated)
• ${\displaystyle \Omega _{12}}$ - Collision integral, depending on temperature and substances (values ​​tabulated, order of magnitude: 1).

For self-diffusion (i.e. in the event that only one type of particle is present), the above is simplified. Related to:

${\displaystyle D={\frac {1}{3}}{\bar {v}}\,l={\frac {2}{3}}{\frac {1}{n\,d^{2}}}{\sqrt {\frac {k_{\mathrm {B} }\,T}{\pi ^{3}\,m}}}}$

With

• ${\displaystyle {\bar {v}}}$ - mean thermal velocity of the particles
• l - mean free path
• n - particle number density
• d - particle diameter
• k _B - Boltzmann constant
• m - molecular mass

In liquids

Examples of diffusion coefficients in water (at infinite dilution and 25 ° C)

material	Diffusion coefficient in m² / s
oxygen	2.1 × 10 −9
sulfuric acid	1.73 × 10 −9
Ethanol	0.84 × 10 −9

Temperature dependence of the diffusion coefficient of 100 nm spheres ( R ₀ = 50 nm) in water

Diffusion coefficients in liquids are usually around one ten-thousandth of the diffusion coefficients in gases. They are described by the Stokes-Einstein equation :

${\displaystyle D={\frac {k_{\mathrm {B} }\,T}{6\,\pi \,\eta \,R_{0}}}}$

With

• k _B - Boltzmann constant
• T - temperature
• η - dynamic viscosity of the solvent
• R ₀ - hydrodynamic radius of the diffusing particles

Many empirical correlations are based on this equation .

Since the viscosity of the solvent is a function of the temperature, the dependence of the diffusion coefficient on the temperature is non-linear.

In solids

Examples of diffusion coefficients in solids

system	Temperature in ° C	Diffusion coefficient in m² / s
Hydrogen in iron	10	1.66 × 10 −13
	50	11.4 × 10 −13
	100	124 × 10 −13
Carbon in iron	800	15 × 10 −13
	1100	450 × 10 −13
Gold in lead	285	0.46 × 10 −13

Diffusion coefficients in solids are usually several thousand times smaller than diffusion coefficients in liquids.

For diffusion in solids, jumps between different lattice sites are required. The particles have to overcome an energy barrier E, which is easier at a higher temperature than at a lower temperature. This is described by the context:

${\displaystyle D=D_{0}\,\exp \left(-{\frac {E}{R\,T}}\right)}$

With

• E - energy barrier
• R - general gas constant
• T - temperature.

D ₀ can be calculated approximately as:

${\displaystyle D_{0}\approx \alpha _{0}^{2}\,N\,\omega }$

With

• α ₀ - atomic distance
• N - proportion of vacant grid positions
• ω - hop frequency

However, it is advisable to determine diffusion coefficients in solids in particular experimentally.

Effective diffusion coefficient

The effective diffusion coefficient describes diffusion through the pore space of porous media . Since it does not consider individual pores , but the entire pore space, it is a macroscopic quantity: ${\displaystyle D_{e}}$

${\displaystyle D_{e}={\frac {\varepsilon _{t}\,\delta }{\tau }}\,D}$

With

• ε _t - porosity available for transport; it corresponds to the total porosity minus pores, which are not accessible to the diffusing particles due to their size, and minus dead-end and blind pores (pores without connection to the rest of the pore system)
• δ - constrictivity ; it describes the slowing down of diffusion due to an increase in viscosity in narrow pores as a result of the greater average proximity to the pore wall and is a function of the pore diameter and the size of the diffusing particles.
• τ - tortuosity  ("tortuousness")

Apparent diffusion coefficient

The apparent (apparent) diffusion coefficient extends the effective diffusion coefficient to include the influence of sorption .

For linear sorption it is calculated as follows:

${\displaystyle D_{a}={\frac {D_{e}}{1+{\frac {K_{d}\,\rho }{\epsilon }}}}}$

With

• K _d - linear sorption coefficient
• ρ - bulk density
• ε - porosity

In the case of non-linear sorption isotherms , the apparent diffusion coefficient is always a function of the concentration , which makes the calculation of the diffusion considerably more difficult.

See also

• Apparent Diffusion Coefficient (magnetic resonance imaging)
• Anomalous diffusion

Individual evidence

1. ^ ^{A b c} E. L. Cussler: Diffusion - Mass Transfer in Fluid Systems . Cambridge University Press, Cambridge / New York, 1997, ISBN 0-521-56477-8 .
2. ^ TR Marrero, EA Mason: Gaseous Diffusion Coefficients . In: Journal of Physical and Chemical Reference Data . tape 1 , no. 1 , January 1, 1972, ISSN  0047-2689 , pp. 3–118 , doi : 10.1063 / 1.3253094 ( nist.gov [PDF; accessed October 8, 2017]).
3. ↑ ^{a b c} J. Hirschfelder, CF Curtiss, RB Bird: Molecular Theory of Gases and Liquids . Wiley, New York, 1954, ISBN 0-471-40065-3
4. ↑ Franz Durst: Fundamentals of fluid mechanics: An introduction to the theory of the flow of fluids . Springer, Berlin, 2006, ISBN 3-540-31323-0 .
5. ↑ A. Einstein: About the movement of particles suspended in liquids at rest, required by the molecular kinetic theory of heat (PDF; 733 kB), Annalen der Physik . 17, 1905, pp. 549ff.
6. ↑ W. Jost: Diffusion in solids, liquids and gases . Academic Press Inc., New York, 1960.
7. ↑ ^{a b} P. Grathwohl: Diffusion in natural porous media: Contaminant transport, sorption / desorption and dissolution kinetics . Kluwer Academic Publishers, 1998, ISBN 0-7923-8102-5 .