Untangling Complex Systems, page 13
µ k, Adink, A µ k, Bdin
d
k, B
iS = −
−
[3.19]
T
T
The Conservation Law of mass allows us to write: − dink, A = dink, B = dink. Therefore, the entropy production is
(µ , − µ , )
diS
k A
k B
dink
=
[3.20]
dt
T
dt
The entropy production, as shown in equations [3.20], is again the product of two terms: (1) the rate
of exchange of k-th moles, dink
( ); (2) the gradient of chemical potential divided by the temperature,
dt
(µ k, A−µ k, B) , which drives the direction and the extent of flow of matter. If the gradient is positive, that is T
µ k >
>
, A
µ k, B, it means that Ck, A Ck, B and the k-th molecules diffuse from the A to the B region. On the other hand, if µ k <
, A
µ k, B, the k-th molecules diffuse from the B to the A compartment. When the chemi-
cal potential of the k-th species is uniform, there is no net diffusion, and the system is at the equilibrium.
3.2.4 migraTion
When a chemical species is within a conservative vector field,4 its molar Gibbs free energy is
expressed through the extended chemical potential, µ k :
µ
0
k = µ k + τ ψ
k
= µ k + RTln k
C +τ ψ
k
[3.21]
A
dink,A
dink,B
B
z-axis
FIGURE 3.5 Tank divided into two portions (A and B) containing the k-th species at two different
concentrations.
3 Coupled reactions are important in biochemistry, where the system hosting the reactions could be a cell or an organelle.
4 A conservative vector field has the property that the value of the line integral from one point to another is path independent; it depends just on the position of the starting and final points enclosed into the conservative vector field.
48
Untangling Complex Systems
where µ0 k is the chemical potential in standard conditions, and τ ψ
k represents the molar potential
energy of the k-th species immersed inside the conservative vector field of potential ψ . In the case
of charged atoms or molecules within an electric field ( E), we have
τ ψ
k
= zk ψ
F with E = −∇ψ [3.22]
where zk is the charge of the k-th molecule, and F is the Faraday constant (i.e., the charge per mole of electrons: 96,485 C/mol).
In the case of dipolar molecules embedded in an electric field, we have
τ ψ
k
= − NAv k
p E cosθ [3.23]
wherein N Av is the Avogadro number, pk is the dipole moment of the k-th molecule and θ is the angle included between the direction of the electric field and that of the dipole moment.
In case of a gravitational field generating a force FG per unit of mass, we have
τ ψ
k
= (
) k
MW ψ with G
F = −∇ψ [3.24]
where MWk is the molecular weight of k- th molecule.
In the previous paragraph we learned that when inside a system there is a gradient of chemical
potential due to an inhomogeneous distribution of a compound, there will be diffusion of its mol-
ecules. The diffusion tends to reduce the gradient until it becomes null. When it is null, the system
is at equilibrium. If this system is immersed inside a vector field exerting a force on the k-th spe-
cies, the k-th extended chemical potential will not be uniform. The vector field pushes the system
out-of-equilibrium. For a matter of simplicity, let us imagine that the vector field generates forces
along just one spatial coordinate: the z-axis. The extended chemical potential will be a function of
the z coordinate, and between two points separated by an infinitesimal distance dz, there will be a
gradient: µ ( )
(
)
k z
−
+
0
µ k z 0 dz (see Figure 3.6a). This gradient induces the migration, i.e., an ordered motion of molecules of k-th species along the direction of the vector field.
The internal energy variation inside the volume dV = ( Ar) dz will be
dU
dS
= du = T
+ (µ ( 0
) µ 0 )
k z + dz −
( kz dC [3.25]
dV
dV
k
Ar
z 0
z 0+ dz
(a)
⎞ dμ ⎞
⎞ dμ
μ
k ⎞
k
>
=
0
0
k
⎞ dμ ⎞
( z)
μk ( z)
k
μk ( z)
<
⎠
⎠
0
dz
⎠
⎠
⎠
dz ⎠
dz
Jk
Jk
Jk = 0
(b)
z
z
z
FIGURE 3.6 Migration of a species contained in a cylinder of section Ar in the presence of a vector field directed along the z-axis (graph a). The relationship between the gradient of the extended chemical potential dµ k
( ) and the flow of matter J
dz
k (graphs labeled as b).
Out-of-Equilibrium Thermodynamics
49
The term µ (
)
k z +
0
dz can be expressed through the Taylor series expansion arrested at the first-order
term:
d
µ ( z + ) ≈ ( ) +
0
dz
µ
µ k
k
k z 0
dz
+… [3.26]
dz z 0
It derives that the entropy production per unit of volume is
d
*
µ
iS
P
*
1
d k
dC
=
= p = −
dz
k
dVdt
dV
T dz
dt
z 0
[3.27]
1 dµ
dn
k
µ
= −
k
1 d k
J z
0
T
k ( ) ≥
dz
= −
Ar dt
T
dz
z 0 (
)
z 0
wherein Jk ( z) is the flow of the k-th species along the z-axis. The entropy production per unit of volume is the product of a flow ( Jk ( z)) and the gradient of extended chemical potential divided by
T. The latter term represents the cause of the flow; it is also defined as the thermodynamic force
responsible of J
µ k
( )
k ( z). Based on equation [3.27], we can claim that when d
is positive, the flow
dz
z 0
is towards the negative direction of z; when dµ k
( ) is negative, the flow is towards the positive direction
dz
z 0
of z (see figure 3.6b). Finally, when dµ k
( ) is null, the flow of migration is null. If our system starts
dz
z 0
from a uniform distribution of k-th species, whereby µ k does not depend on z, introducing equation
[3.21] into equation [3.27], we achieve that
τ
ψ
*
k
d
p = −
J
0 [3.28]
T
≥
dz
k
z 0
The term −( dψ represents the force due to the vector field. The flow of migration ceases when the
dz ) z 0
extended chemical potential has the same value everywhere into space.
Equation [3.28] can be applied to define the entropy production for the process of electrical con-
duction inside a mono-dimensional electrical wire (extended along the z-axis). The electrical wire
is a conductor wherein we can assume that the electron density and the temperature have spatially
uniform values. Therefore, the electron chemical potential is constant along z, and the entropy pro-
duction per unit of length reduces to:
ψ
*
F d
dne
Ez
I
p =
0 [3.29]
T
≥
dz
=
0 ( Ar ) dt
T Ar
z
where E is the electrical field along z, and I is the electrical current.
z
If we integrate equation [3.29] along the entire length of the wire ( L), we obtain
L
1
(∆ )
*
diS
V I
P =
= EzIdz =
≥
∫
0 [3.30]
dt
T
T
0
The term Δ V represents the potential difference across the entire wire: if Δ V > 0, the electrons flow towards the positive direction of z. On the other hand, if Δ V < 0, the electrons migrate in the opposite direction. Finally, if Δ V = 0, there is no electron migration.
TRY EXERCISES 3.1 AND 3.2
50
Untangling Complex Systems
3.2.5 generalizaTion
The thermodynamic analysis of heat conduction, chemical reaction, diffusion, and migration, has
revealed that entropy production is always the product of two terms: a gradient (divided by T) and a
flow. The gradient, divided by T, is a thermodynamic force ( F). It drives a thermodynamic flow ( J).
The force is the cause of the flow. In the case of more than one force and hence of more than one
flow, the entropy production is given by the summation
diS
=
Fk J ≥
∑
0 [3.31]
dt
k
k
If at least one force is not null, the system is out-of-equilibrium. On the other hand, when all the
forces are null, the system is at the equilibrium. At the equilibrium, all the thermodynamic flows
vanish. Remember that when a system evolves up to reach an equilibrium state, the entropy of the
universe (i.e., the entropy of the system and environment) reaches a maximum, whereas the Gibbs
free energy of the system reaches a minimum
TRY EXERCISE 3.3
The out-of-equilibrium systems can operate in two distinct regimes: (1) in the linear regime,
when they are not so far-from-equilibrium, and (2) in the non-linear regime when they are very
far-from-equilibrium. In the next paragraphs of this chapter, we are going to discover the properties
of out-of-equilibrium systems in the two different regimes.
3.3 NON-EQUILIBRIUM THERMODYNAMICS IN LINEAR REGIME
When a system is out-of-equilibrium, but it is close to the equilibrium, the relationships linking the
flows to the forces are linear. This statement grounds on few empirical laws, such as the Fourier’s
Law for heat conduction, Ohm’s Law for electrical conduction, Fick’s Law for diffusion, and the
Poiseuille’s Law for the laminar flow of fluids.
3.3.1 fourier’s law: The law of heaT conducTion
In the first half of the nineteenth century, the French mathematician and physicist Joseph Fourier
formulated the empirical law of heat conduction, named after his surname as Fourier’s Law:
q
Jq =
∂
k T
t
∂ ∂( Ar) = − ∇ [3.32]
The heat, which flows per unit time and per unit of area ( Ar), the latter being disposed perpendicular
to the direction of the flow, is directly proportional to the gradient of temperature:
T
T
T
∇ T = ∂
i
+ ∂
j
+ ∂
k
[3.33]
∂ x
∂ y
∂ z
where i, j, k are the unit vectors of the three Cartesian coordinates. The higher the temperature gradient, the larger the heat flow. The proportionality constant, k, represents the thermal conductivity,
which is a peculiar property of every material. It is larger in metals than in non-metallic materials.
In metals, the valence electrons are not tightly bound to the reticular ions. Therefore, they move
Out-of-Equilibrium Thermodynamics
51
quite easily. The thermal conductivity can be increased by cooling down the metal because at low
temperature the random thermal motion of electrons is damped.
In case of anisotropic solids, we do not have just one k, but many. Fourier’s law for such systems
has the following form:
T
T
T
Jq, x =
−
∂
kxx
−
∂
kxy
−
∂
k
x
∂
y
xz
∂
z
∂
T
T
T
Jq, y =
−
∂
kyx
−
∂
kyy
−
∂
k
[3.34]
x
∂
y
yz
∂
z
∂
T
T
T
Jq, z =
−
∂
kzx
−
∂
kzy
−
∂
k
x
∂
y
zz
∂
z
∂
The thermal conductivity is a tensor. If we consider the expression of the thermal force proposed in
equation [3.9], i.e., as ∇ ( 1 , equation [3.32] can be rearranged in the following linear form:5
T )
q
2
1
1
Jq =
∂
kT
= L ∇
[3.35]
t
∂ ∂( Ar) =
∇
T
q
T
where L
2
q = kT is named as the phenomenological coefficient.
TRY EXERCISE 3.4
3.3.2 ohm’s law: The law of elecTrical conducTion
In the first half of the nineteenth century, the German mathematician and physicist Georg Simon
Ohm, drawing inspiration from Fourier’s work on heat conduction, formulated the empirical law of
electrical conduction. When an electric potential difference (Δ V) is applied between two points inside
a conductor, the charges start to migrate orderly. The unidirectional migration of charges corresponds
to electrical current. The intensity of the current is related to Δ V through the following equation:
∆ V = Vi − Vf = RI [3.36]
The constant of proportionality R is the resistance of the conductor. For a conductor of length l and cross-sectional area Ar (see Figure 3.7), the resistance R is proportional to ρ∙l/Ar, where ρ is the resistivity. Introducing the definition of R in equation [3.36], we obtain
∆ V =
I
σ
e E =
J
[3.37]
l
( Ar) = e
ρ
wherein σe = 1/ ρ is the electrical conductivity and Je is the flow of charges.
Vi
Vf
l
→
→
Ar
E
Je Vf + Vi > 0
FIGURE 3.7 Block of a conductor of length l and section Ar crossed by a flow of electric current ( J
e )
generated by the electric field E. Note that Je and E have the same direction and orientation.
1
1
5 Note that d = −
dT .
T
T 2
52
Untangling Complex Systems
Pi
Pf
L
r
FIGURE 3.8 A tube of length L and radius r crossed by the fluid that flows due to a pressure gradient.
