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Phys. Chem. Glasses: Eur. J. Glass Sci. Technol. B, February 2007, 48 (1), 61–63
Viscosity activation energy
Isak Avramov
Institute of Physical Chemistry, 1113 Sofia, Bulgaria
Accepted 6 October 2006
The standard procedure of determining the activation energy for shear viscosity from the slope L of the Arrhenius plot,
lgη against 1/T leads frequently to overestimated values. The article discusses the significance of L and shows that an
important information can be retrieved from it value as the fragility parameter can be formulated as α=0·007Lg/Tg.
Fragility parameter α=1 is an indication for a strong glass with constant activation energy. It is demonstrated that the
average jump frequency model describes fairly the dependence of viscosity on chemical composition. On the contrary,
models based on mean activation energy considerations fail completely.
1. Introduction
Most of the models of shear viscosity assume that it is
a thermally activated process. To move, the molecules
have to overcome activation energy E(T) barriers(1)
created by the resistance of the surrounding building
units. Therefore, the jump frequency, is related to the
vibration frequency ν∞ according to equation
Ê E (T ) ˆ
n = n • exp Á Ë RT ˜¯
(1)
where R is the ideal gas constant. On the other hand,
viscosity η is related to the jump frequency ν of the
building units according to the Frenkel equation
h=
RT 1
3pVm n
(2)
where Vm is the molar volume. Maxwell gives an
alternative expression in the form
h=
G•
n
(3)
G∞ being the shear modulus. Therefore, viscosity is
also thermally activated
Ê E (T ) ˆ
h = h• exp Á
Ë RT ˜¯
(4)
There is a fundamental question: how to determine
experimentally the activation energy and how to
interpret it. Although the answer seems trivial, the
situation is quite tricky and misleading. The aim of
the present article is elucidate this problem.
2. Activation energies problem
The conventional procedure of determining the activation energy is to plot experimental data in Arrhenius
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coordinates, logη against 1/T. In the high viscosity
region, experimental data give a straight line. Therefore, a widespread fallacy is that the slope determines
directly the activation energy according to
L0 =
1 E
2 ◊3 R
The provisional slope Lo, given by Equation (5),
should be equal to the experimentally determined
slope L if the activation energy was constant. Since the
experimental data resemble a straight line, it seems
that this assumption is right. However, every curve
resemble to a straight line if the investigated interval
is sufficiently short (even the Equator looks like a
straight line to the horizon limits). When viscosity
is studied in wider interval, the declination from
a straight line is becoming evident. The situation
is more complicated with relaxation experiments
because, in this case, it is much more difficult to
widen the experimental interval. Hopefully, it was
proven(2–6) that relaxation process and shear viscosity are controlled by the jump frequency of the same
building units, i.e. the relaxation time (reciprocal of
the jump frequency) is proportional to viscosity η. If
the activation energy depends on temperature the
slope L is much larger the value of the provisional
slope L0
Ê
ˆ
1 Á ∂
˜
L = L0 +
E (T )˜ 2 ◊ 3RT Á ∂ 1
ÁË
˜¯
T
(6)
Unreasonably high values of E(T) are obtained if the
second term, in brackets, of Equation (6) is neglected.
At any temperature T, the absolute value of the activation energy can be determined from Equation (4)
provided the preexponential constant η∞ is estimated
properly. Taking into account typical values of the
shear modulus of glasses, the expected value is about
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I. Avramov: Viscosity activation energy
logη∞≈−1. Earlier,(7) on the basis of experimental data
of more then 100 substances, we have demonstrated
that logη∞ varies in the limits −2≤logη∞≤1·5 [dPa s]. Let
us determine the activation energy at temperature
Tg at which ηg=1013·5 [dPa s]. Equation (4) permits the
determination of the activation energy at Tg in the
following way
E(Tg)=2·3R(13·5−lgη∞)Tg=εRTg
(7)
where ε is a dimensionless constant having a value
e = 2 ◊ 3 log
hg
h•
=
E (Tg )
RTg
ª 32 ± 10% (8)
Earlier, Bartenev has shown(8,9) that, for relaxation of
polymers ε≈31, i.e. this parameter is about the average
value reported here. With these considerations, the
activation energy for viscous flow is given by
E(Tg)=εRTg =265Tg±10% [J/mol]
(9)
In the literature, there is no unified definition
of the glass transition temperature. Here we adopt
ηg=1013·5 [dPa s] as a suitable definition of Tg. In
this way Tg is always in the glass transition region
and will coincide, within 2% of accuracy, with the
value determined according to other methods.
Additional advantage of this definition is that the
relaxation time at this temperature is estimated to
be logτg≈2.
The comparison of Equations (5) and (8) determines the provisional slope Lo as
L0 =
e
Tg 2 ◊3
(10)
The dimensionless ratio α
a=
( ) = 2 ◊3 L
eT
L (T )
L Tg
0
g
g
g
ª 0 ◊ 07
Lg
Tg
± 10% (11)
is a measure of the “fragility” of the system. In other
words, it determines how fast the activation energy
is changing with temperature. In Equation (11) Lg
stands for slope at the glass transition temperature,
Lg≡L(Tg).
3. Influence of chemical composition on
viscosity
Data on fragility α on more then 100 substances
are summarised in Ref. 7. It was shown experimentally (for alkaline and for most of the alkaline earth
silicates), that α depends on composition x in the
following way
α(x)=1+6x
(12)
The chemical composition is expressed by the molar
fraction x of the modifying oxides, respectively,
1−x is the molar fraction of SiO2. The same result,
Equation (12), is derived theoretically(7) within the
framework of the “jump frequency” model. A brief
comparison between the “jump frequency” model
and the “average activation energy” models is given
in the Discussion part.
Equation (11) is based only on the assumption
that viscous flow is a thermally activated process.
Therefore, the definition of the dimensionless fragility parameter α is universal. It accounts for the
rate at which the activation energy changes with
temperature. By means of Equations (5) and (6) α is
expressed as follows
a = 1+
˘
∂
E (T )˙
∂ (1/T )
˚T =T
( )
g
Tg E Tg
(13)
It is seen there is relationship between composition
x and sensitivity of the activation energy to temperature changes in the corresponding glass transition
interval
˘
∂
E (T )˙
∂ (1 /T )
˚T =T
g
( )
Tg E Tg
= 6x (14)
4. Discussion
There are several models(7,10–12) capable of describing the temperature dependence of viscosity with
sufficient accuracy. To distinguish which of them is
better one has to test their possibility to describe with
no new assumptions, or new adjustable parameters,
the dependence of viscosity on other chemical composition. Here we discuss viscosity of SiO2, when
low concentration of modifying oxides is added. In
this case, a tiny fraction Qi(x) of the SiO4 tetrahedra
is expected to have i broken Si–O–Si bridges (i varies
between 1 and 4). Detailed investigation on Qi(x) is
given in Ref. 13
Qi =
4!
(1 - p)4- i pi i ! ( 4 - i )!
(15)
The mobility of SiO4 tetrahedra is connected with
much lower activation energy in the presence of
broken bridges. If E0 is the activation energy of SiO4
tetrahedra with no broken bonds and Ei(T) is the
corresponding value of SiO4 tetrahedra with i broken
bonds, the average activation energy is given approximately as follows
E =
4
ÂQ (x) E (T ) i
i
(16)
i=0
It is seen that at low concentrations of the modifying
oxides the average value of the activation energy
remains almost unchanged. However, experimental
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I. Avramov: Viscosity activation energy
H
solid line in Figure 1 is drawn according to Equation
(17), for Ei≈Ei−1/2. The same values are applied in calculating the dotted line according to <E> model. The
superior accuracy of the jump frequency approach
is readily seen.
5. Conclusions
Figure 1. Dependence of the viscosity on composition at
1473 K. Solid line is according to the jump frequency model
while the dotted line is according to the mean activation
energy approach. The parameters used are as follows:
Eo=380 kJ/mol; Ei=Ei−1/2, i=1 to 4. The solid points are
experimental results from Refs 14–18
It is fallacious to determine the activation energy E(T)
from the slope L of the Arrhenius plot of viscosity
against reciprocal temperature. In the glass transition
interval the slope Lg plays an important role as the
ratio α=0·07(Lg/Tg)±10% determines the “fragility”
parameter, indicating the sensibility of the activation
energy on temperature. In other words, determining
how “short” is the glass.
There is a strong experimental indication that the
average jump frequency model is capable to describe
with superior accuracy viscosity as compared to the
mean activation energy approaches.
Acknowledgment
data(14–18) show that viscosity drops fast as illustrated
by the solid points in Figure 1. It gives viscosity at
1473 K in dependence of the molar fraction x of the
modifying oxides. Most of the approaches are based
on sophisticated models determining the average
activation energy <E>. Afterwards, the viscosity is
determined by introducing the value of <E> into
Equation (4). The result is illustrated by a dotted
line in Figure 1. The complete failure of the “average
energy” models is evident. Once the effective activation energy E0 for motion of SiO4 tetrahedra with no
broken bond is fixed, it is not possible to find a set of
Ei values suitable to describe the experimental data
by means of <E> models.
The alternative approach (see Refs 7,13,19) is first
to determine the mean jump frequency <ν> according to
n =
4
ÂQ (x)n (T ) i
i=0
i
(17)
where the jump frequencies νi(T) are determined by
the corresponding activation energies Ei according to
Equation (1). Than viscosity is determined by introducing the average jump frequency <ν> in Equation
(2) (or alternatively in Equation (3)). It is seen that the
mean jump frequency is not determined by the mean
the activation energy, i.e. <ν>≠ν∞ exp(−<E>/RT). The
The author appreciates support of the Project INTERCONY.
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