5.2 Nucleation Rate of Gas Hydrates
123
have been many orders of magnitude larger than any experimentally determined
nucleation rates reported in the literature [51–53].
Given the unique attributes of clathrate hydrates in the context of nucleation, it is
pertinent to re-examine the applicability of classical nucleation theory to nucleation
of clathrate hydrates [54]. A good starting point is Eq. (14) of Ref. [54]:
J = z f C 0 exp
−W
∗
/kT
(5.2.2)
where J is the nucleation rate, z is the so-called Zeldovich factor that can take a value
somewhere between 0.01 and 1, f is the attachment frequency of building units to an
emerging nucleus, C 0 is the concentration of nucleation sites in the system, W * is the
nucleation work that is required to surmount the activation barrier, k is the Boltzmann
constant, and T is the absolute temperature [54]. Simply put, the nucleation rate is a
product of (1) how often the building units attach to an emerging nucleus, (2) how
many such potential nucleation sites exist in the system, and (3) the probability of
realizing a system energy that is greater than the activation barrier in the Arrhenius
form, with a Zeldovich factor z. Nucleation rates are often reported in the form of
(J/C 0 ) in the literature.
For three-dimensional nucleation, the Zeldovich factor is z = (W */3π kTn*
2 )
1/2
where n* is the number of building blocks in the smallest supernucleus [18]. Even
for a clathrate hydrate for which nucleation sites are limited to next to an interface, the thickness of such interfacial regions is much larger than the size of the
molecules. Whether such interfacial regions are substantially thicker than the size of a
supernucleus and hence would allow a treatment using three-dimensional nucleation
remains to be seen. We follow Ref. [54] for now and assume that C 0 = 1/v w where
v w is the molecular volume of water for homogeneous nucleation (which provides
the minimum nucleation rate) and that C 0 is proportional to the concentration of
active nucleation sites in the system, N p (whose identities are unknown) for heterogeneous nucleation. The attachment frequency f itself is a function of the driving
force or supersaturation, μ, and is related to the attachment–detachment frequencies at equilibrium, f e , through f = f e exp(μ/kT ) [54]. This last point follows
from the principle of detailed balance we detailed in Chap. 1. We do not quite know
which physical factors will dominate f or f e for the nucleation of clathrate hydrates,
however, the expression shows that f approaches f e in the limit of μ → 0 (no
subcooling at all) and f approaches zero at an infinite subcooling, which partially
accounts for viscous slowdowns (in reality, viscous slowdowns should render f zero
long before the system approaches 0 K).
For analyses, it is convenient to lump all the kinetic factors into one “kinetic
parameter”, A, in the form; A = zf e C 0 . Then, noting f = f e exp(μ/kT ), Eq. (5.2.2)
can be rewritten as [54]
J = A exp(μ/kT ) exp
−W
∗
/kT
(5.2.3)
The nucleation work, W *, corresponds to the activation barrier in the Arrhenius
equation and can be found from the calculus of variations that renders ∂W /∂n = 0.
123
have been many orders of magnitude larger than any experimentally determined
nucleation rates reported in the literature [51–53].
Given the unique attributes of clathrate hydrates in the context of nucleation, it is
pertinent to re-examine the applicability of classical nucleation theory to nucleation
of clathrate hydrates [54]. A good starting point is Eq. (14) of Ref. [54]:
J = z f C 0 exp
−W
∗
/kT
(5.2.2)
where J is the nucleation rate, z is the so-called Zeldovich factor that can take a value
somewhere between 0.01 and 1, f is the attachment frequency of building units to an
emerging nucleus, C 0 is the concentration of nucleation sites in the system, W * is the
nucleation work that is required to surmount the activation barrier, k is the Boltzmann
constant, and T is the absolute temperature [54]. Simply put, the nucleation rate is a
product of (1) how often the building units attach to an emerging nucleus, (2) how
many such potential nucleation sites exist in the system, and (3) the probability of
realizing a system energy that is greater than the activation barrier in the Arrhenius
form, with a Zeldovich factor z. Nucleation rates are often reported in the form of
(J/C 0 ) in the literature.
For three-dimensional nucleation, the Zeldovich factor is z = (W */3π kTn*
2 )
1/2
where n* is the number of building blocks in the smallest supernucleus [18]. Even
for a clathrate hydrate for which nucleation sites are limited to next to an interface, the thickness of such interfacial regions is much larger than the size of the
molecules. Whether such interfacial regions are substantially thicker than the size of a
supernucleus and hence would allow a treatment using three-dimensional nucleation
remains to be seen. We follow Ref. [54] for now and assume that C 0 = 1/v w where
v w is the molecular volume of water for homogeneous nucleation (which provides
the minimum nucleation rate) and that C 0 is proportional to the concentration of
active nucleation sites in the system, N p (whose identities are unknown) for heterogeneous nucleation. The attachment frequency f itself is a function of the driving
force or supersaturation, μ, and is related to the attachment–detachment frequencies at equilibrium, f e , through f = f e exp(μ/kT ) [54]. This last point follows
from the principle of detailed balance we detailed in Chap. 1. We do not quite know
which physical factors will dominate f or f e for the nucleation of clathrate hydrates,
however, the expression shows that f approaches f e in the limit of μ → 0 (no
subcooling at all) and f approaches zero at an infinite subcooling, which partially
accounts for viscous slowdowns (in reality, viscous slowdowns should render f zero
long before the system approaches 0 K).
For analyses, it is convenient to lump all the kinetic factors into one “kinetic
parameter”, A, in the form; A = zf e C 0 . Then, noting f = f e exp(μ/kT ), Eq. (5.2.2)
can be rewritten as [54]
J = A exp(μ/kT ) exp
−W
∗
/kT
(5.2.3)
The nucleation work, W *, corresponds to the activation barrier in the Arrhenius
equation and can be found from the calculus of variations that renders ∂W /∂n = 0.