The Use of a Magnetic Field and Magnetic Force as a Means to Improve the Quality of Protein Crystals - Molecules: Nucleation, Aggregation and Crystallization - page 198

The Gibbs free energy representation requires that when the free energy
changes in the presence of the magnetic field, some force can be gener-
ated to minimize the free energy. When the strength of the magnetic field
1 H is chosen as an independent variable, the free energy representation is

… ),

d G =− S d T + V d P − ∇ µ 0 M d H ( +

where d G is the free energy change, S is entropy, T is temperature, V is
volume, P is pressure, and M is magnetization. µ 0 is the permeability

of the vacuum, and “ … ” means the extra and unnecessary terms coming
from electric field, gravitational field, stress field, addition/deletion of
masses, and so on.
For all feeble magnetic substances, M can be expressed as

M = χ H ,

where is the magnetic susceptibility of the corresponding substance. For

paramagnetic substances,

χ

χ > 0, and for diamagnetic substances, χ < 0.

The value is regarded as a constant as long as the substance is feebly

magnetic. Then it comes that

χ

… …

d G = −∇ µ 0 χ H d H or G = −∇ µ 0 χ (1/2) H 2 .

Since the negative of the spatial derivative of the free energy is a force,
the magnetic force F that results is

F = µ 0 χ H grad H .

0, i.e. if the magnetic field is
homogeneous and if it does not depend on position. However, if H is
nonzero and inhomogeneous, then F will have a nonzero value.
An intuitive explanation of the above description is as follows. All
the substances are more or less magnetized when placed in a magnetic field.
The Gibbs free energy change coming from the magnetization is positive for

This force F becomes zero if (grad H ) =