Untangling Complex Systems, page 14
3.3.3 Poiseuille’s law: The law of laminar flow of fluids
The flow of fluid in parallel layers with no disruption between layers, termed laminar flow, is
described by a law that is formally equivalent to that describing an electrical conduction. Instead of
having the migration of charges due to an electrical potential difference, we have the non-turbulent
movement of liquid due to a pressure gradient. The flow of the volume of liquid ( dV/dt) is linearly
dependent on the pressure gradient (Δ P), and the proportionality constant is the inverse of the resis-
tance to flow R′, which depends on the length L of the tube (see Figure 3.8), the fourth power of its radius r, and the viscosity η of the fluid:
dV
1
r 4
π
=
( P
)
i − Pf
=
∆ P [3.38]
dt
R′
8η L
This law was formulated by the French physicist and physiologist Jean Leonard Poiseuille in the
first half of the nineteenth century.
The physical meaning of viscosity η can be understood if we consider layers of liquid in contact
with each other moving at different speeds (see Figure 3.9). If we apply a shear force on layer 1 of the liquid along the x-direction, a linear momentum will propagate along the z-axis from one layer
to the other. The flow of linear momentum along z is linearly dependent on the gradient of v along
x
the z-axis:
dv
J
x
p = −η k
[3.39]
z
dz
The proportionality constant is the viscosity, whose unit is the poise in the centimeter–gram–second
(cgs) system, and the Poiseuille in the SI (International System of Unit). 1 poseuille = 10 poise.
x
1 2 3 4 5 6 7
z
FIGURE 3.9 Laminar flow of a fluid partitioned in ideal seven layers. A shear force (represented by the black arrow) along the x-axis is applied to layer 1. A linear momentum propagates along the positive direction of z.
Out-of-Equilibrium Thermodynamics
53
TRY EXERCISE 3.5
3.3.4 fick’s law: The law of diffusion
In the second half of the nineteenth century, the German physiologist Adolf Fick, investigating the
fluxes of salts diffusing between two reservoirs through tubes of water, formulated a law that gov-
erns the transport of mass through any diffusive means (in the absence of bulk fluid motion). The
flow of moles of the k-th solute ( Jk ) depends on the gradient of its concentration (∇ Ck):6
Jk = − Dk∇ Ck [3.40]
The constant of proportionality is the diffusion coefficient D . If we consider an ideal solution,
k
wherein the intermolecular forces are not relevant, the chemical potential of the k-th component can
be expressed as µ
0
1
k = µ k + RTln k
C . Knowing that dlnCk = ( ) dC , we can rearrange equation [3.40]
C
k
k
and write the Fick’s law as a function of the thermodynamic force (−∇µ k/ T ):
DkC
∇
J
k
k
k = − DkCk∇ lnCk = −
∇µ
µ
k = − L
[3.41]
RT
k, d
T
wherein
DkC
L
k
k, d =
[3.42]
R
is the phenomenological coefficient for the diffusion.
In the presence of a conservative vector field exerting a force on the k-th species, equation [3.41]
can be rewritten in terms of the extended chemical potential µ k (see equation [3.21]). The flow of the
k-th moles due to its concentration gradient and the vector field is given by two distinct contributions:
∇µ
k
L τ
J
k k
k = − Lk
= − LkR∇ lnC −
∇ψ [3.43]
T
k
T
The first contribution is the diffusion, which is described by equation [3.40]. The second contribution
represents the migration. The vector field exerts a force drawing the k-th molecules towards the region
where the potential is lower. Against this drift motion proceeding at velocity vk, the surrounding
medium exerts a frictional force (see Figure 3.10), given by Ffr = −γ kvk ( γ is the frictional coefficient).
k
→
Ffield
ψ
→
Ffri
v→ k
x
FIGURE 3.10 A conservative field exerts a force ( Ffield ) drawing a molecule (black circle) in the region where the potential ( ψ) decreases. A frictional force, having opposite direction, hampers this motion.
6 The concentration C is expressed in moles/volume units.
k
54
Untangling Complex Systems
When the two forces balance, we obtain
Ffield = − Ffr [3.44]
−τ k∇ψ = γ kvk
[3.45]7
The migration produces a flow given by:
τ kC
J
k
k
=
= −
∇ψ
, field
vkCk
[3.46]
γ k
The ratio τ / is named as the mobility of k, driven by a force field in a specific medium. By merg-
k γk
ing equations [3.46] and [3.43], we obtain another definition of the phenomenological coefficient L :
k
TC
L
k
k =
[3.47]
N Avγ k
Combining the two definitions of L (equations [3.42] and [3.47]), we achieve an expression of the
k
diffusion coefficient, known as Stokes–Einstein relation:
k T
D
B
k =
[3.48]
γ k
Note that both the diffusion coefficient and the mobility are inversely proportional to the friction
coefficient. The Irish physicist Stokes (1819–1903) derived an expression for the frictional coef-
ficient valid in case of small spherical molecules (having radius r ) moving inside a continuous
k
viscous fluid (with viscosity η):
γ k = 6πη kr [3.49]
Combining equations [3.48] and [3.49], we obtain a relationship linking the microscopic parameters
D and r with the macroscopic properties T and
k
k
η.
TRY EXERCISE 3.6 AND 3.7
3.3.5 generalizaTion: symmeTry PrinciPle and onsager reciProcal relaTions
In the previous paragraphs, we have ascertained that the phenomena of heat and electrical current
conduction, laminar flow in fluids, diffusion, and migration, are all based on linear relationships
between the flows and the forces: J ∝ LF . One force may give rise to more than one flow. For
example, a thermal gradient in a solution can promote heat flow and diffusion: this phenomenon
is known as thermal diffusion, or Soret effect (see Section 3.3.8). On the other hand, a chemical
potential gradient drives a matter flow and a heat flow, known as the Dufour effect (Section 3.3.8). It may occur that when a thermal gradient is applied between two metal junctions, it induces not only
a thermal flow but also a flow of charges: this is the so-called Seebeck effect (see Section 3.3.6). On the other hand, when an electrical potential difference is applied between the two metal junctions,
both a flow of charges and a thermal flow are originated according to the phenomenon known as
Peltier effect (see Section 3.3.6). These are examples of cross effects in the linear regime. In case of cross effects, we can write the following relations:
7 Note that τ N = .
k
Av
τ k
Out-of-Equilibrium Thermodynamics
55
J j =
LjkFk
∑
[3.50]
k
diS
=
J jFj =
LjkFkF ≥
∑
∑
0 [3.51]
dt
j
j
j, k
Equation [3.50] shows that the j-th flow is linearly dependent on k forces. The terms L are named jk
as “proper phenomenological coefficients” when j = k, whereas they are named as “cross phenom-
enological coefficients” when j ≠ k.
Now a question arises: Is it possible to observe any combination of flows and forces in linear
regime? To answer this question, we need (1) the “symmetry principle,” and (2) the Onsager recip-
rocal relations.
1. The “symmetry principle” was formulated by the French physicist Pierre Curie (1894)
before working on radioactivity with his wife, Marie. The symmetry principle states that
when certain causes produce certain effects, the symmetry elements of the causes must be
found in the produced effect. In other words, when certain effects show a certain asymme-
try, such asymmetry must be found also in the causes that generated them. The symmetry
principle asserts that the effects have the same or larger number of symmetry elements
than their causes. In other words, macroscopic causes have a degree of symmetry always
equal or lower than the effects they induce.8 From this principle, we can infer that an iso-
tropic force can originate an isotropic flow, but not an anisotropic flow. In other words, a
scalar force gives rise just to a scalar flow (see Figure 3.11).
An example of scalar force is A/T, i.e., the force for a reaction performed inside a
thermo-stated well-stirred tank reactor. Such a force has a spherical symmetry, meaning
it has all the symmetry elements and it is perfectly isotropic. It cannot originate an aniso-
tropic flow, like a unidirectional thermal flow. Therefore, if we consider a system where
a chemical reaction close to the equilibrium9 is going on and the heat flows in one spatial
direction, we can write
A
1
v = Lcc
+ L ∇
T
cq
T
[3.52]
A
1
Jq = Lqc
+ L ∇
T
T
8 The properties of symmetry for flows and forces may be described in terms of the presence of symmetry elements and their associated symmetry operations.
Symmetry Element
Symmetry Operation
Symbol
Identity
Identity
E
Proper axis
Rotation by (360/ n)°
Cn
Plane
Reflection
σ
Inversion center
Inversion of a point x, y, z to − x,− y,− z
i
Improper axis
Rotation by (360/ n)° followed by reflection in
Sn
a plane perpendicular to the rotation axis
9 In this chapter, we will learn that the relation between the chemical flow and force is non-linear unless we are close to equilibrium.
56
Untangling Complex Systems
Scalar
Scalar
force
flow
Isotropic
Anistropic
force
flow
FIGURE 3.11 Schematic representation of the symmetry principle proposed by Pierre Curie.
The symmetry principle allows us to add that Lqc = 0. The scalar chemical force cannot
be the cause of the thermal flow that is a vector. To know if the thermal gradient can be a
cause of the chemical flow v, we need the Onsager reciprocal relations.
2. The “reciprocal relations” were first noticed by William Thomson (Lord Kelvin) in the
nineteenth century, and then theoretically demonstrated by Lars Onsager in the first half
of the twentieth century:10
Ljk = Lkj [3.53]
If we use equation [3.53] for the case indicated by equation [3.52], we infer that L
cq = Lqc = 0. This
result can be generalized. If we consider a system hosting three different kinds of flows and forces
of tensorial ( ten),11 vectorial ( vec), and scalar ( sc) features, respectively, we can expect to write the following equations in linear regime:
Jten = Lten te, nFten + Lten, vecFvec + Lten, scFsc
Jvec = Lvec te, nFten + Lvec, vecFvec + Lvec, scFsc
[3.54]
Jsc = Lsc te, nFten + Lsc, vecFvec + Lsc, sc s
F c
The symmetry principle along with the Onsager reciprocal relations allow us to infer that
Lten
= 0 =
= 0 =
= 0 =
, vec
Lvec, ten; Lten, sc
Lsc, ten; Lvec, sc
Lsc, vec. Therefore,
Jten = Lten te, nFten
Jvec = Lvec, vecFvec [3.55]
Jsc = Lsc, scFsc
10 The demonstration of the “reciprocal relations” by Onsager is based on the principle of detailed balance or microscopic reversibility that is valid for systems at equilibrium. For the interested reader, I recommend the original papers published by Onsager (1931a, 1931b) in the Physical Reviews journal.
11 To have an idea of what a tensorial force is, imagine a tensorial force applied in one point along the x-axis and directed towards the positive direction of x. Such tensorial force can induce effects not only along x-axis, but also along y-axis and z-axis.
Out-of-Equilibrium Thermodynamics
57
It derives that only forces and flows having the same degree of symmetry can interact mutually. In a
system where there are tensorial, vectorial and scalar forces and flows, the total entropy production
will be given by
diS
=
JscFsc +
JvecFvec +
JtenF
∑
∑
∑
[3.56]
dt
ten
sc
vec
ten
and for each term
Jsc sc
F
∑ ≥0
sc
Jvec vec
F
∑
≥ 0 [3.57]
vec
Jten t
Fen
∑
≥ 0
ten
3.3.6 an exPerimenTal Proof of The reciProcal relaTions
An experimental proof of the validity of the reciprocal relations (Miller 1960) regarding cross phe-
nomenological coefficients is offered by the thermoelectric phenomena.
Suppose to have a thermocouple, i.e., two rods of different metals soldered together and to have
the two soldered points at two different temperatures (see Figure 3.12). If the temperature gradient Δ T is maintained constant, it gives rise to an electromotive force ( emf). Both a thermal and an electrical flow cross the two rods when the circuit is closed. This phenomenon is known as the Seebeck
effect after the German physicist who discovered it in 1821. The emf is measured by a potentiometer
when no current is permitted to flow:
x 2 dϕ
emf = −
dx
∫
[3.58]
dx
x 1
where φ is the electrical potential. Its derivative, with respect to T
d ( emf )
dϕ
= −
[3.59]
dT
dT
T
T + Δ T
x 1
x 2
FIGURE 3.12 Sketch of a thermocouple. A potentiometer (in the case of Seebeck effect) or a battery (in the
case of Peltier effect) are bound to the circuit at x and .
1
x 2
58
Untangling Complex Systems
is called the thermoelectric power. Its value is small, of the order of a few μV/K. It can be exploited
just to devise thermometers. However, arranging several thermocouples in a series, a thermopile is
obtained that generates an appreciable emf.
The reverse of the Seebeck effect, which is called the Peltier effect (after the French physicist
who discovered it in 1834), occurs when an electric potential is applied to the two junctions, kept
at the same temperature. An electrical current will pass through the two rods and heat will be pro-
duced at one of the two junctions and absorbed at the other.12 The ratio between the heat current ( Iq) and the electrical current ( I) is defined as Peltier heat (Π):
Π = Iq [3.60]
I
The Peltier effect can be used as a refrigerator.
For simplicity, we assume that our circuit is mono-dimensional. The entropy production per unit
of length generated by the two irreversible processes, i.e., heat and electrical conduction, is given
