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THE JOURNAL OF CHEMICAL PHYSICS 136, 074514 (2012)
Viscosity of liquid mixtures: The Vesovic-Wakeham methodfor
chain molecules
Astrid S. de Wijn,1,2 Nicolas Riesco,2,3 George Jackson,4 J. P.
Martin Trusler,4
and Velisa Vesovic2,a)1Institute for Molecules and Materials,
Radboud University Nijmegen, Heyendaalseweg 135,NL-6525 AJ
Nijmegen, The Netherlands2Department of Earth Science and
Engineering, South Kensington Campus, Imperial College
London,London SW7 2AZ, United Kingdom3Qatar Carbonates and Carbon
Storage Research Centre (QCCSRC), Department of Earth Science
andEngineering, South Kensington Campus, Imperial College London,
London SW7 2AZ, United Kingdom4Department of Chemical Engineering,
South Kensington Campus, Imperial College London,London SW7 2AZ,
United Kingdom
(Received 4 October 2011; accepted 30 January 2012; published
online 21 February 2012)
New expressions for the viscosity of liquid mixtures, consisting
of chain-like molecules, are derivedby means of Enskog-type
analysis. The molecules of the fluid are modelled as chains of
equally sized,tangentially joined, and rigid spheres. It is assumed
that the collision dynamics in such a fluid canbe approximated by
instantaneous collisions. We determine the molecular size
parameters from theviscosity of each pure species and show how the
different effective parameters can be evaluated by ex-tending the
Vesovic-Wakeham (VW) method. We propose and implement a number of
thermodynam-ically consistent mixing rules, taking advantage of
SAFT-type analysis, in order to develop the VWmethod for chain
molecules. The predictions of the VW-chain model have been compared
in the firstinstance with experimental viscosity data for
octane-dodecane and methane-decane mixtures, thus,illustrating that
the resulting VW–chain model is capable of accurately representing
the viscosity ofreal liquid mixtures. © 2012 American Institute of
Physics. [http://dx.doi.org/10.1063/1.3685605]
I. INTRODUCTION
Understanding the relationship between the macroscopictransport
properties of fluids and the interactions amongindividual molecules
is the ultimate goal of kinetic theory.The last decade has
witnessed great advances in our ability tocalculate the transport
properties of fluids directly from inter-molecular forces.1–10 Such
calculations do not only improveour insight into the dominant
microscopic processes, but alsoallow us to develop more accurate
and reliable methods forthe prediction of transport properties.
Although it is essentialto validate such methods against a bank of
high-qualityexperimental data, the reliance purely on experimental
dataand empirical correlations based on them is not
sufficient,especially as there is an urgent need to facilitate a
reliableprediction of the viscosity of liquid mixtures over
wideranges of temperature, pressure and composition.
At present, there is no rigorous kinetic theory that allowsfor
the calculation of the viscosity of a dense fluid from arealistic
intermolecular potential. The lack of a general so-lution of the
formal Boltzmann integro-differential equationis still a
fundamental unresolved problem. So far the onlytractable solutions
have been based on simplifying the in-termolecular interaction by
assuming that molecules in thefluid interact as hard spheres and
that molecular collisions are
a)Author to whom correspondence should be addressed. Electronic
mail:[email protected].
uncorrelated. For such a system it is possible, throughEnskog’s
analysis,11, 12 to derive a relationship betweenthe viscosity of
the fluid and molecular parameters. TheEnskog equation, though
approximate in nature, has neverthe-less provided a useful
theoretical basis for both understandingand predicting the
viscosity of fluids.13, 14 Notwithstandingthe recent advances in
molecular dynamics3–6, 10 and density-fluctuation theory15, 16 all
indications are that it will remain acornerstone for the
development of viscosity models based onkinetic theory.
Recently, Enskog’s analysis has been extended to incor-porate
molecular shape (size asymmetry) in the expressionsfor the
self-diffusion coefficient17–19 and the viscosity.20
Molecules were modelled as chains formed from equallysized hard
spheres. Chain models provide a very usefullink, at the molecular
level, with the Wertheim TPT1(Refs. 21–23) and statistical
associating fluid theory (SAFT)(Ref. 24 and 25) that has proved to
be very successful indescribing the thermodynamic properties of a
wide variety offluids and fluid mixtures. In principle,
representing moleculesas chains provides a further degree of
realism and shouldallow for a more accurate description of the
viscosity ofthe fluid. However, the resulting viscosity model is
stillbased on Enskog-type collision dynamics and the postulatesof
instantaneous collisions and uncorrelated molecularmotion.12 It
therefore suffers from the same deficienciesas the original
hard-sphere (HS) model. This renders itunsuitable for a priori
predictions of viscosity or any othertransport properties. For the
original hard-sphere model
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of Physics136, 074514-1
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074514-2 de Wijn et al. J. Chem. Phys. 136, 074514 (2012)
this can be circumvented by using effective hard-spherediameters
that are weakly temperature dependent.13, 14 Aproper choice of the
effective diameter is paramount forsuccess in representing the
viscosity of the fluid. For fluidmixtures, the problem of choosing
the appropriate effectivediameters is compounded by the presence of
more thanone species. In order to address this problem, Vesovicand
Wakeham (VW) (Ref. 26) proposed that the effectiveparameters for a
mixture are obtained from the viscosity ofpure species. This choice
of effective parameters is at theheart of the development of the VW
method26, 27 that canbe used to predict accurately the viscosity of
a variety ofdifferent mixtures, including natural gas,28
refrigerant,29 andsupercritical fluid mixtures.26, 27 The accuracy
can be retainedwhen predicting the viscosity of liquid mixtures27,
29 provid-ing the systems contain molecules of similar molecular
massand size, thus allowing for a representation of each moleculeby
an effective hard sphere. If the molecules are differentin size the
hard-sphere representation is no longer adequateand a chain
representation becomes more appropriate, if theaccuracy is to be
retained.
In our current work we first present expressions for
theviscosity of liquid mixtures consisting of chain-like
moleculesthat are derived with an Enskog-type analysis. We then
showhow the different effective parameters can be evaluated
byextending the VW method. Finally, we illustrate throughtwo
examples that the resulting VW–chain model is capa-ble of
accurately representing the viscosity of real liquidmixtures.
The development of the VW-chain model is primarilydriven by the
needs of the petroleum industry, where theknowledge of oil
viscosity is essential for optimal exploita-tion of oil reservoirs.
Reservoir fluids are complex mixturesconsisting of a large number
of hydrocarbons, predominantlychain molecules. In order to develop
accurate and reliable pre-dictions of viscosity of such mixtures it
is essential to take aproper account of the shape of the molecules
making up themixture.
II. MODEL AND THEORY
In this section, we present an expression for the viscos-ity of
a liquid mixture that consists of molecules representedas chains of
hard spheres. The new expression is derived bycombining the
Enskog-like analysis for hard-spheres11, 12 andour recent work20 on
chain molecules.
A. The viscosity of a pure chain-fluid
A hard-sphere fluid consists of spheres of diameter σthat
interact only on contact.30, 31 If we assume that the colli-sions
between the hard spheres are uncorrelated (i.e., molec-ular chaos)
then the shear viscosity, η, of such a fluid can bedescribed with
Enskog’s expression:12
η = η(0)[
1
χ+ αρ + 1
βα2ρ2χ
], (1)
where ρ is the molar density, η (0) is the viscosity in the
limitof zero density, and β is a constant equal to (1/4 + 3/π
)−1.
The quantity χ is the radial distribution function at
contact,while α is a parameter proportional to the excluded
volumeper molecule, Vexcl,
α = 815
NAπσ3 = 2
5NAVexcl, (2)
where NA is Avogadro’s constant.Considerable effort has been
made to extend the hard-
sphere model to fluids of non-spherical molecules. One wayof
including the non-sphericity is to model the molecules
astangentially bonded chains consisting of equally-sized,
hard,spherical segments. Such a representation of real fluids
hasbeen very successful for the description of
thermodynamicproperties and has recently been extended by
ourselves20 totreat the system’s viscosity. With this type of
approach, theviscosity of a fluid consisting of N chains, each made
up of msegments, can be approximated in the dense region by that
ofa fluid consisting of mN hard-spheres. We refer to this fluidas a
segment fluid. In such a fluid, the collision dynamics isgoverned
principally by collisions between segments and onecan make use of
Enskog’s approach. However, the collisionrate is still affected by
the neighbouring segments in the chain,and the resulting viscosity
expression,
η = η̃(0)[
1
χ̃+ α̃ρ̃ + 1
βα̃2ρ̃2χ̃
], (3)
now involves quantities defined on a per segment basis,
indi-cated here by the tilde. Unlike in Ref. 20, in order to
avoidconfusion here between the usual symbol for viscosity in
thezero-density limit, η(0), and the corresponding parameter inthe
segment fluid, η̃(0), a tilde is used to indicate a quan-tity
defined for segments. The segment density ρ̃ is given byρ̃ = ρm,
while α̃ is a parameter proportional to the excludedvolume of a
segment in the presence of another. As the seg-ments in the same
chain screen each other from collisions, theexcluded volume of each
segment still corresponds to the ex-cluded volume of a chain.20
Hence, the parameter α̃ can beapproximated as
α̃
α̃segment= 1 + 3
2(m − 1) + 3
8(m − 1)2, (4)
where α̃segment is the excluded volume of the free segment,4πσ
3/3.
The zero-density viscosity of the segments η̃(0) is relatedto
the zero-density viscosity of the fluid by the expression,20
η̃(0) = η(0)χ̃ (0) = η(0)(
1 − 58
(m − 1
m
)). (5)
We refer the reader to Ref. 20 for the details of the
derivationof Eqs. (4) and (5), but also point out that Eq. (4) is a
well-known result by Onsanger32 for the excluded volume of
hardspherocylinders, while Eq. (5) is consistent with the
correla-tion hole effect.33
B. The viscosity of chain-fluid mixture
In the present paper, we extend the Enskog-Thorneapproach12, 34
for evaluating the viscosity of mixtures of hardspheres to mixtures
where the molecules are represented as
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074514-3 The VW method for chain molecules J. Chem. Phys. 136,
074514 (2012)
chains made up of hard, spherical segments. We consider
themolecules of component i as chains consisting of mi segmentswith
a mass given by M̃i = Mi/mi , where Mi is the molecularmass.
We are primarily interested in developing a model thatallows for
the prediction of the viscosity of liquid mixtures.Hence we
constrain both the model and the discussion toliquid-like
densities. In the dense fluid, the collision rate ishigh and in
general the mean-free path between the collisionsis smaller than
the size of the segments. It is thus reasonableto assume that a
particular segment will undergo a numberof collisions before the
effects of the initial collision are feltfurther down the chain. We
therefore postulate that, at liquid-like densities, as far as the
collision dynamics is concerned,a fluid consisting of chain
molecules can be described asan analogous fluid consisting of
unbound or weakly-boundsegments. The viscosity of such a mixture
consisting of chainmolecules can then be obtained by following
Enskog-Thorneapproach and is given by:
η = K̃mix −
∣∣∣∣∣∣∣∣∣∣∣∣
H̃11 . . . H̃1N Ỹ1
.... . .
......
H̃N1 . . . H̃NN ỸN
Ỹ1 . . . ỸN 0
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
H̃11 . . . H̃1N
.... . .
...
H̃N1 . . . H̃NN
∣∣∣∣∣∣∣∣∣
, (6)
K̃mix = 3ρ̃2
π
∑i,j
x̃i x̃j χ̃ij α̃2ij η̃
(0)ij , (7)
Ỹi = x̃i⎡⎣1 + ρ̃ ∑
j
M̃j
M̃i + M̃jx̃j α̃ij χ̃ij
⎤⎦, (8)
H̃ij = −x̃i x̃j χ̃ij2A∗ij η̃
(0)ij
M̃iM̃j
(M̃i + M̃j )2[
20
3− 4A∗ij
], (9)
H̃ii = x̃2iχ̃ii
η̃(0)i
+∑j �=i
x̃i x̃j χ̃ij
2A∗ij η̃(0)ij
M̃iM̃j
(M̃i + M̃j )2[
20
3+ 4M̃j
M̃iA∗ij
],
(10)
where ρ̃ = (∑i ximi)ρ is the segment density, xi is the
molefraction of component i, and x̃i = ximi/(
∑j xjmj ) is the
segment fraction. The quantities η̃(0)ij and A∗ij are the
segment
interaction viscosity and ratio of collision integrals,35,
36
respectively, for the i-j pair in the limit of zero density.
Theparameter α̃ij is the excluded volume of a segment of a chainof
species i in the presence of a segment of a chain of speciesj while
χ̃ij represents the segment-segment radial distributionfunction at
contact for the species i and j in the presenceof all other species
in the mixture. In Sec. II C we examinehow to obtain the relevant
pure species properties, in order to
combine them using mixing rules discussed in the subsequentSec.
II D.
C. The VW method for chain molecules
In principle, knowledge of the excluded volume and theradial
distribution function at contact, both of which can beobtained from
thermodynamic considerations, together withthe pure component
viscosities in the limit of zero density,would be sufficient to
evaluate the viscosity of any pure fluidor fluid mixture. However,
if Enskog’s theory is used in itsoriginal form then generally the
predicted viscosity will bemuch higher than that observed
experimentally. There arenumber of ways of modifying the Enskog
expressions in orderto predict the behaviour of real fluids.14 In
our current workwe focus on the solution successfully used as part
of the VWmethod26–29 and extend the VW method to mixtures
modelledas chains formed from hard segments.
The crux of VW method is to obtain the effective
radialdistribution function at contact from the
experimentallydetermined viscosity of each pure species, thus
ensuring that,in the limit of each pure species, viscosity of the
mixturetends to a correct value. This is achieved by inverting Eq.
(3)in quadratic form
χ̃±i =β
2ρ̃i α̃i
⎡⎢⎣
(η
ρ̃i α̃i η̃(0)i
− 1)
±
√√√√( ηρ̃i α̃i η̃
(0)i
− 1)2
− 4β
⎤⎥⎦.
(11)
To ensure realistic physical behaviour, it is necessary toswitch
from the χ̃−i branch to the χ̃
+i branch of the solution
at some particular density, ρ̃∗i = miρ∗i , at which the two
rootsare equal. This “switch-over density” can be obtained37
fromthe solution of (
∂ηi
∂ρ
)= η
∗i
ρ∗i. (12)
The use of Eq. (12) ensures a unique value of parameter α̃i
,namely,
α̃i = η∗i
ρ̃∗i η̃(0)i
(1 + 2√
β
) . (13)It is important to stress that although α̃i and χ̃i
determinedin this fashion are unique, they are effective
parameters.In the process of using them to reproduce the
viscosityof pure species, the link between the two, in terms of
thehard-sphere diameter, has been broken. What this confirmsis that
Enskog’s expression, Eq. (1), does not adequatelydescribe the
viscosity of a real fluid, even if we allow thehard-sphere diameter
to become an effective parameterdependent on temperature. There is
no reason to believethat a single effective diameter can correctly
account forthe simplifications to both the dynamics and the
geometryof the molecular interactions that Enskog introduced.
Inessence the VW method postulates that in order to reproducethe
experimental viscosity by means of a hard-sphere fluidone needs to
use one effective size of the molecule for theexcluded volume and
another for the collisional dynamics.
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074514-4 de Wijn et al. J. Chem. Phys. 136, 074514 (2012)
Hence, we differentiate between the two diameters, σ̃α,i
andσ̃χ,i , using the subscript α to indicate that it was
obtainedfrom the parameter α by means of Eq. (13) and the
subscriptχ to indicate that it has been obtained from the radial
distri-bution function at contact, Eq. (11). These effective
diametersare distinguished further by the subscript “i” to indicate
thatthey can take a different value for different species.
In order to be able to use Eqs. (6)–(10) to calculatethe
viscosity of a mixture, we need to relate the proper-ties of the
pure species segments to those for the i-j binaryinteraction.
D. Estimating the mixture interaction parameters
1. Estimating the effective mutual excludedvolume, α̃ij
It is important to stress that the mutual excluded volumeof a
segment of a chain of species i in the presence of a seg-ment of a
chain of species j is different to the excluded volumeof two free
segments. The excluded volume of two segments,belonging to two
chains, is in fact equal to the excluded vol-ume of the two chains.
The chain excluded volume can beapproximated by the mutual excluded
volume of two sphero-cylinders of the same lengths as the
chains,20
α̃ij
α̃segment,ij= 1 + 3
2
⎛⎜⎜⎜⎝
(α̃segment,i
α̃segment,ij
)1/3(mα,i − 1) +
(α̃segment,j
α̃segment,ij
)1/3(mα,j − 1)
2
⎞⎟⎟⎟⎠
+ 38
(α̃segment,i
α̃segment,ij
)1/3(mα,i − 1)
(α̃segment,j
α̃segment,ij
)1/3(mα,j − 1), (14)
where mα,i is the number of segments of chain of species i.We
obtain the expression for the unlike i-j interaction of
segments by simply invoking the arithmetic result that the
ex-cluded volume of two spheres of unequal diameter is relatedto
that with the average diameter (additive spheres),
α̃1/3segment,ij =
1
2
(α̃
1/3segment,i + α̃1/3segment,j
)∝ σ̃α,ij = 1
2(σ̃α,i + σ̃α,j ). (15)
2. Estimating the effective radial distributionfunction,
χ̃ij
For a pure fluid the effective radial distribution functionat
contact of two segments of equal size, χ̃i , can be foundfrom
Wertheim’s first-order thermodynamic perturbation the-ory
(TPT1),38, 39 and can be written as the sum of a hard-sphere
contribution and a chain contribution,
χ̃i = χ̃HS,i + χ̃chain,i . (16)The chain contribution, χ̃chain,i
, arises due to the segments inthe same chain screening each other
from collisions. In theCarnahan and Starling40 treatment of the
hard-sphere systemthese contributions can be expressed as20
χ̃HS,i =1 − 12 ỹχ,i(1 − ỹχ,i)3 , (17)
χ̃chain,i = −58
(mχ,i − 1
mχ,i
)1 − 25 ỹχ,i(
1 − 12 ỹχ,i)
(1 − ỹχ,i), (18)
where ỹχ,i = (π/6) σ̃ 3χ,imχ,iNAρ is the segment packing
frac-tion. The radial distribution function does not go to unity
inthe low-density limit, as segments on the same chain screeneach
other from collisions with other segments, even at lowdensity.
In order to estimate a segment diameter σ̃χ,i and a chainlength
mχ , i consistent with Eq. (16) an additional constraint isneeded.
To this end we impose the constraint that the distancebetween the
end segments calculated using σ̃α,i and mα, i, andσ̃χ,i and mχ , i
are equal, namely,
σ̃α,i(mα,i − 1) = σ̃χ,i(mχ,i − 1). (19)
This constraint ensures that the length of the backbone of
thechain remains constant and that taking mα, i = 1, in the limitof
a spherical molecule, ensures mχ , i = 1.
In order to infer the mixing rule for the interaction
pa-rameters χ̃ij , we have followed the approach described
inSAFT-HS,23, 41, 42 here generalised to describe mixtures ofchains
differing in the number and size of the hard spheres.The resulting
expressions are given by
χ̃ij = χ̃CS,ij(χ̃
(0)ij + F (ρ̃)
)(20)
χ̃CS,ij = 11 − ξ̃3
+ 3(
σ̃χ,i σ̃χ,j
σ̃χ,i + σ̃χ,j
)ξ̃2
(1 − ξ̃3)2
+ 2(
σ̃χ,i σ̃χ,j
σ̃χ,i + σ̃χ,j
)2ξ̃ 22
(1 − ξ̃3)3, (21)
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074514-5 The VW method for chain molecules J. Chem. Phys. 136,
074514 (2012)
χ̃(0)ij = 1 −
1
8
[(mχ,i − 1
mχ,i
) (σ̃ 3χ,j + 32 σ̃χ,i σ̃ 2χ,j( σ̃χ,i+σ̃χ,j
2
)3)
+(
mχ,j − 1mχ,j
) (σ̃ 3χ,i + 32 σ̃χ,j σ̃ 2χ,i( σ̃χ,i+σ̃χ,j
2
)3)]
, (22)
F (ρ̃) =∑
i,j x̃i x̃j(π6
( σ̃χ,i+σ̃χ,j2
)3)χ̃CS,ij
(1−χ̃ (0)ij
)+ Z̃chain4ρ̃∑i,j x̃i x̃j
(π6
( σ̃χ,i+σ̃χ,j2
)3)χ̃CS,ij
(23)
Z̃chain = −∑
i
x̃i
(mχ,i − 1
mχ,i
) ⎛⎝ ξ̃3(1 − ξ̃3) + 32 σ̃χ,i ξ̃2(1 + ξ̃3) + 12 σ̃ 2χ,i ξ̃ 22(
2+ξ̃3
1−ξ̃3)
(1 − ξ̃3)2 + 32 σ̃χ,i ξ̃2(1 − ξ̃3) + 12 σ̃ 2χ,i ξ̃ 22
⎞⎠ , (24)
where the moment densities are defined as ξ̃m = (π/6)NA
∑i σ̃
mχ,imχ,ixiρ. More details are given in the Appendix,
together with various mixing rules that are considered.
3. Estimating the zero-density parameters
In order to calculate the zero-density limit of the viscos-ity
of free segments, η̃(0)i , we make use of Eq. (5) for eachpure
species. The interaction viscosity in the zero-densitylimit,
η̃(0)ij , for each binary pair, is then given by
η̃(0)ij = η(0)ij χ̃ (0)α,ij , (25)
where χ̃ (0)α,ij is given by Eq. (22) using σ̃α,i and mα, i
forconsistency with Eq. (5).
E. Application to real mixtures
To perform an initial assessment of the accuracy of thenewly
developed VW-chain method, we limit our investi-gation to liquid
mixtures consisting of n-alkane molecules.Although in the VW model
different species are repre-sented by homonuclear chains, whose
segments can havedifferent diameters, in this particular example,
we representeach alkane molecule as a chain made up of equally
sized“methane-like” segments. For this purpose, the diameterof a
segment, σ̃α , is taken to be the effective diameter ofmethane at a
given temperature, where methane is modelledas a single segment
molecule. We have successfully used thisconcept in our previous
work20 to analyse the viscosity ofpure normal alkanes. The
effective diameter of methane, σ̃α ,is obtained from Eq. (2), where
the parameter α is evaluatedfrom Eq. (13), which requires knowledge
of the viscosity ofpure methane at a given temperature. A figure
showing σ̃αas a function of temperature can be found in Ref. 20.
For ann-alkane of carbon number C, the number of segments, mα ,
iscalculated from the formula mα = 1 + (C − 1)/3 developedfrom
consideration of the equilibrium thermodynamics,41–43
that was also shown to be valid when analysing viscosity.20
Once σα and mα are known one can calculate η̃(0)i , χ̃
(0)i , and
α̃i for each alkane by means of Eqs. (5), (11), and
(13),respectively. The calculation of σ̃χ and mχ for each alkane
isslightly more intricate and it involves a simultaneous solutionof
Eqs. (16) and (19), where the value of effective radialdistribution
function is obtained from Eq. (11).
It is interesting to note that, unlike σ̃α which is onlya
function of temperature, σ̃χ is also a weak function ofdensity.
This is not surprising since σ̃χ is evaluated fromthe effective
hard-core radial distribution function, χ̃ , (seeEq. (16)), which
is a function of density. This raises aninteresting question, at
what density should one evaluate σ̃χ?Or more to the point, given
the molar density of the mixture,ρ, at what density should one
evaluate the pure speciesparameters, so that they are
representative of the interactionsthat the pure species undergo in
a mixture? In the originalVW method,26, 27 based on hard spheres,
the mixing rule forthe effective radial distribution function was
written in termsof radial distribution functions for pure species
and hence themolar density was an implied choice. However, this
choice isunsuitable at liquid-like densities for mixtures of
moleculesthat are very different in size. This difference in size
makesthe packing fraction very different for each species at
thesame molar density. Hence, such a pure fluid does not providea
good representation of the interactions of that particularspecies
in the mixture. For this reason and given that the crit-ical volume
of a fluid is usually regarded as a measure of thehard-core volume,
we have chosen to evaluate the requiredproperties of the pure
species at the same reduced density,(ρr = ρ/ρcritical), as that of
the mixture. For the purposes ofthis paper the critical density of
the mixture was estimated bymeans of ρc,mix. = [
∑xi/ρc,i]−1. We will further examine
the consequence of this density choice in Sec. III.Once the pure
species parameters, σ̃α,i , mα, i, σ̃χ,i , and
mχ , i have been evaluated one can evaluate all the
mixtureparameters by means of Eqs. (20)–(25) and subsequently
themixture viscosity by means of Eqs. (6)–(10). Therefore,
toevaluate the viscosity of a liquid mixture of n-alkanes withthe
VW-chain method one only requires a knowledge of theviscosity of
pure species and two temperature-dependent,dilute-gas binary
parameters, η(0)ij and A
∗ij . For the pur-
pose of this work η(0)ij and A∗ij are obtained from standard
references,35, 36 while the sources of pure species viscosityare
discussed in Sec. III.
III. RESULTS AND DISCUSSION
In order to illustrate the predictive power of the VW-chain
method and to investigate some of the assumptionsmade in deriving
it, we examine two examples in detail.One, a (n-octane +
n-dodecane) mixture made up of long
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074514-6 de Wijn et al. J. Chem. Phys. 136, 074514 (2012)
chain-like molecules and the other, a (methane +
n-decane)mixture consisting of spherical and long chain-like
molecules.For both of these mixtures the original VW method,
whichis based on representing molecules as effective hard
spheres,failed to provide an accurate description of the
liquidviscosity.
There exist accurate sets of experimental data, for
bothmixtures, that in this study we use as a benchmark. In hisPh.D.
thesis, Caudwell44 reported viscosity and density mea-surements for
two liquid mixtures of (n-octane + n-dodecane)(xoctane = 0.434 and
xoctane = 0.743), at three temperaturesfrom 323.15 to 423.15 K and
pressures up to 200 MPa with aquoted uncertainty of 2%. Audonnet
and Padua45 have mea-sured the viscosities and densities of
(methane + n-decane)mixtures using a vibrating-wire technique with
a quoted un-certainty of 3%. These measurements cover a
temperaturerange from 303.15 to 393.15 K and pressures up to 75
MPa.
The VW-chain method requires knowledge of the pure-species
viscosity. The viscosity of pure methane is obtainedfrom
Quinones-Cisneros et al.46 as implemented in REF-PROP V8. In the
temperature and density range of interest inour work the claimed
uncertainty of the correlation rangedfrom 2% to 5%. The
correlations of Huber et al.47, 48 areused to estimate the
viscosity of the n-octane, n-decane, andn-dodecane. The uncertainty
of these correlations is between2% and 3%. The correlations
reproduce Caudwell’s pure-species viscosity data with deviations
ranging from −4.2%to +0.6% for n-octane and from −1.5% to +4.9%
forn-dodecane, while the pure n-decane data of Audonnet andPadua is
reproduced with a deviation ranging from −2.5% to0.7%.
The percentage deviations of the VW-chain pre-dicted data from
the experimental data for the (n-octane+ n-dodecane) mixture is
illustrated in Fig. 1. The experi-mental data are reproduced with
an absolute average deviation(AAD) of 1.3% and maximum absolute
deviation of 3.5%. Notrends in temperature or density could be
discerned. Takinginto account the uncertainty of pure species
correlations andthe quoted experimental uncertainty of the data,
the agree-ment can be deemed to be very good.
The percentage deviation for the (methane + n-decane)mixture is
shown in Fig. 2. The deviations are larger thanfor the (n-octane +
n-dodecane) mixture, with an AAD of5.4% and maximum absolute
deviation of 14%. There is astrong trend with density which
indicates that further refine-ment of VW-chain model is necessary
for highly asymmetricalkane mixtures of this type. However, it
should be pointedout that the viscosity of this mixture exhibits a
very strong,non-linear increase with increasing composition of
n-decanewhich makes accurate predictions rather difficult. For
instanceat a mixture density of 742.4 kg/m3 the viscosity of pure
n-decane is approximately five times larger than that of
puremethane, at the same reduced density.
A number of assumptions have been made in develop-ing the
VW-chain method. We investigate the influence ofsome of the
assumptions on the overall agreement betweenthe predicted and
experimental data and report the resultsfor the (methane +
n-decane) mixture only, as the (n-octane+ n-dodecane) system
appeared to exhibit similar qualitative
FIG. 1. Viscosity deviations obtained with the VW method for
chainmolecules developed here from the experimental data for
(n-octane+ n-dodecane) of Caudwell (Ref. 44).
behaviour. An assessment of the use of different mixing
rules(see Appendix) on predictive capability of VW-model is
illus-trated in Fig. 3. Although there are differences of the order
of5% between different sets of mixing rules the general trendis to
shift the deviations, but not affect the density trend al-ready
observed for this mixture. The difference between thepredictions
with mixing rules 1–3 and 4–5 appears to decreasewith a decrease in
the asymmetry of the mixture and for the(n-octane + n-dodecane)
mixture it is less than 1%.
The effect of evaluating the pure species properties at
themolar, mass, and reduced density of the mixture is demon-strated
in Fig. 4. The results for molar density indicate a twoorder of
magnitude over-prediction. Furthermore, evaluatingpure species
properties at the molar density of the mixture re-quires properties
of the heavier species at unrealistically highdensities, where
either of the pure species is solid or there areno viscosity
correlations available. Although the deviations
FIG. 2. Viscosity deviations obtained with the VW method for
chainmolecules developed here from the experimental data for
(methane+ n-decane) of Audonnet and Padua (Ref. 45).
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074514-7 The VW method for chain molecules J. Chem. Phys. 136,
074514 (2012)
FIG. 3. Viscosity deviations obtained with the VW method for
chainmolecules developed here from the experimental data for
(methane+ n-decane) of Audonnet and Padua (Ref. 45) using: Eq.
(A10), Ansatz 1;Eq. (A11), Ansatz 2; Eq. (A12), Ansatz 3; Eq.
(A13), Ansatz 4; andEq. (A14), Ansatz 5.
decrease with a decrease in the asymmetry of the mixture,this
choice of density is unsuitable for application in the liq-uid
state. This is not surprising since the molar density willresult in
a large overestimation of collision rate for heavierspecies, that
unduly influences the collision rate between un-like species. The
choice of either mass or reduced densitywould appear to offer a
better description of the overall col-lision rate. As illustrated
in Fig. 4, the choices of mass andreduced density result in similar
deviations. However, evalu-ating pure species properties at the
mass density of the mix-ture requires properties of the lighter
species at unrealisticallyhigh densities, thus limiting its
applicability.
We have refrained in this work from examining the rolethat the
connectivity of the segments in a chain plays in de-
FIG. 4. Viscosity deviations of obtained with the VW method
forchain molecules developed here from the experimental data for
(methane+ n-decane) of Audonnet and Padua (Ref. 45) when the
properties ofthe pure species are evaluated at the molar, mass, and
reduced density ofthe mixture.
termining the viscosity. Although one could, for this
purpose,form a fluid of disconnected segments, the VW model, in
itspresent form, cannot be used to estimate the viscosity of sucha
fluid. In the VW method, one evaluates the effective size(and
shape) of the species from the viscosity of each purespecies. In
doing so, one postulates that the molecules canbe represented as
chains of connected segments whose ef-fective parameters, σ̃χ and
mχ , are obtained from the viscos-ity. Although for alkane
mixtures, presented in this work, onerepresents a segment as having
a “methane-like” effective di-ameter σ̃α , the mass of segment is
given by M̃ = Malkane/mα .Hence, if we were to break up the chain
to form a fluid con-sisting of disconnected segments, there is no
equivalent purefluid to be used as the source of viscosity for σ̃χ
and mχ . Nev-ertheless the comparison of the high-density limit of
the ra-dial distribution function at contact for chains with that
forspheres, Eqs. (16)–(18), does confirm the intuitive expecta-tion
that the connectivity of chain molecules has less impactat high
density.
IV. CONCLUSIONS
The VW method, that used to predict the viscosity ofdense fluid
mixtures made up of molecules represented ashard spheres, has been
extended in this work to predictthe viscosity of liquid mixtures
consisting of chain-likemolecules. This was achieved by postulating
that themolecules can be represented as chains made up of
hard,spherical segments that undergo instantaneous collisions.The
new expressions for the viscosity of liquid mixtureswere
subsequently derived by extending the Enskog-Thorneapproach to
chain-like molecules. For realistic fluids atliquid-like densities,
the resulting description suffers fromthe same deficiencies as the
original Enskog’s theory. Inparticular, it cannot be used to
predict a priori the viscosityfrom the knowledge of the size and
shape of the molecules.However, following the original VW method,
we showed inthe present work that it is possible to assign the
molecularsize and shape to each species in the mixture from
knowledgeof its viscosity. One of the consequences of using the
effectivemolecular parameters is that one needs to distinguish
betweeneffective size of the molecule for the collisional
dynamicsand that for the excluded volume. By making an
additionalconstraint that ensures that the length of the backbone
ofthe chain remains constant we can describe the molecules ofeach
pure species by three effective parameters; namely twodiameters,
one for collision dynamics and one for excludedvolume, and the
number of segments in the chain.
In order to calculate the viscosity of a mixture, we needto
relate the effective parameters of the pure species to thosefor the
like and unlike binary interactions. We have chosen todo so at the
level of excluded volumes and radial distributionfunctions and
consequently we have developed mixing rulesfor these two
quantities. The mixing rule for excluded vol-ume is relatively
straightforward and is based on geometricconsiderations that
include the mutual excluded volume oftwo spherocylinders and the
assumption that the excludedvolume of two spheres can be obtained
by using an averagediameter. In the limit of zero density it is
possible to derive
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074514-8 de Wijn et al. J. Chem. Phys. 136, 074514 (2012)
the thermodynamically consistent mixing rule for the
radialdistribution function and we have done so by following
theSAFT approach. At finite density the present theory does
notallow us to uniquely determine the chain contribution. Hence,we
have postulated a number of possible Ansätze regardingthe
approximate density behaviour of the chain contribution.At present,
it is only at the level of validation of the VW-chain method where
one could distinguish between differentpossibilities.
The newly developed VW-chain model has been pre-sented for the
prediction of the viscosity of liquid mixtures. Itis founded on the
kinetic theory, modified to take into accountthe behaviour of real
fluids, and on a set of thermodynam-ically consistent mixing rules.
The model has no adjustableparameters, and requires no dense
mixture viscosity data. TheVW-chain model has been tested by
comparing its predic-tions with the experimental viscosity data for
the (n-octane+ n-dodecane) and the (methane + n-decane) mixtures,
i.e.,mixtures made up of long, chain-like molecules and
mixturesconsisting of spherical and long chain-like molecule.
Theexperimental data for the (n-octane + n-dodecane) mixtureare
reproduced with an AAD of 1.3% and maximum abso-lute deviation of
3.5%, while for the (methane + n-decane)mixture the AAD was 5.4%
and maximum absolute deviationwas 14%. This illustrates that the
newly developed VW–chainmodel is capable of accurately representing
the viscosity ofreal liquid mixtures.
We are currently undertaking a more encompassing vali-dation of
the VW-chain method and will shortly report on itsability to
predict the viscosity of a plethora of n-alkane mix-tures. In
future work we intend to examine the explicit effectof attractive
interactions on the viscosity of chain moleculeswithin a full
SAFT-VR treatment43, 49 for systems with hard-core segments
interacting via variable range square-well,43
Yukawa50 or soft-core51, 52 interactions.
ACKNOWLEDGMENTS
The authors acknowledge a grant from the Engineer-ing and
Physical Sciences Research Council (EP/E007031)for partial support
of this work and a travel grant from theBritish Council Partnership
Programme in Science to A.S.W.A.S.W.’s work is financially
supported by a Veni grant ofNetherlands Organisation for Scientific
Research (MWO).
APPENDIX: THE RADIAL DISTRIBUTION FUNCTIONOF CHAIN MIXTURES
In this appendix, we derive a mixing rule for the
segmentcollision rate parameters, χ̃ij . The factor χ was
originally in-troduced by Enskog11, 12 to correct the probability
of colli-sion in dense fluids made up of hard spheres. From the
Clau-sius virial expression for the pressure, it is possible to
provethat, in the thermodynamic limit, χ converges to the
radialdistribution function at contact. Here, following the
approachpresented in our previous work,20 we assume this link is
stillvalid for chain molecules and define an effective radial
distri-bution function at contact per segment, χ̃ij . As discussed
inSec. II B, within the dense region, a fluid consisting of a
mix-
ture of chain molecules is modelled as a fluid of hard spheresof
various diameters. By means of the compressibility fac-tor, an
equation is obtained for a sum of all χ̃ij and furtherequations are
found for the zero-density limits of χ̃ (0)ij . Fi-nally, a simple
Ansatz is made in order to infer the density-dependence of χ̃ij
.
The compressibility factor of a chain molecule mixture,Z̃, can
be used to define an effective radial distribution func-tion, χ̃ij
, using the pressure equation,53
Z̃ ≡ 1 + 4ρ̃∑i,j
x̃i x̃j
(π
6
(σ̃i + σ̃j
2
)3)χ̃ij . (A1)
Furthermore, the Helmholtz free energy can be expressed as
A = AHS + Achain = AHS +∑
i
Ni (mi − 1) aii , (A2)
where the index i runs over all the species. AHS is theHelmholtz
free energy of the hard-sphere contribution to themixture. Ni and
mi are the number of molecules and segmentsper molecule for species
i, and aii is the free energy changedue to the bonding of two
adjacent segments belonging to agiven molecule of species i.
By differentiating Eq. (A2) with the respect to the volumewe can
obtain the compressibility factor. Hence, we definethat
Z̃ = Z̃HS + Z̃chain, (A3)χ̃ij ≡ χ̃HS,ij + χ̃chain,ij , (A4)
where Z̃HS is the compressibility factor for a mixture of
freehard spheres and χ̃HS,ij is the radial distribution function
atcontact of free hard spheres of species i and j.
Z̃chain is the contribution due to the segment bonding inthe
chains and can be written as23
Z̃chain = ρ̃ÑkBT
(∂Achain
∂ρ̃
)T
= ρ̃kBT
∑i
x̃i
(mi − 1
mi
) (∂aii
∂ρ̃
)T
, (A5)
where Ñ = ∑i Nimi is the number of segments in the mix-ture.
Z̃HS is defined as
Z̃HS = 1 + 4ρ̃∑i,j
x̃i x̃j
(π
6
(σ̃i + σ̃j
2
)3)χ̃HS,ij . (A6)
In order to estimate the free energy contribution due to
bond-ing, aii, one can consider an associating mixture of
monomersin the limit of complete association, corresponding to
theTPT1 approximation.23, 38 The chemical potential due to
theformation of each bond at infinite dilution is given exactly54,
55
by −kBT ln χ̃HS,ij , and, as is common in SAFT approaches,the
cost in free energy per bond per molecule for the fullybonded chain
fluid can thus be approximated as
aii = −kBT lnχ̃HS,ii , (A7)which essentially defines χ̃HS,ii =
exp (−aii/kBT ) as thecontact value of the potential of mean force,
first introducedby Kirkwood.56
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074514-9 The VW method for chain molecules J. Chem. Phys. 136,
074514 (2012)
Combining Eqs. (A1) and (A3)–(A7) yields an equationfor a
weighted sum of χ̃chain,ij in terms of chain lengths andhard-sphere
parameters, namely
∑i,j
x̃i x̃j
(π
6
(σ̃i + σ̃j
2
)3)χ̃chain,ij
= −14
∑i
x̃i
(mi − 1
mi
) (∂lnχ̃HS,ii
∂ρ̃
). (A8)
χ̃HS,ii can be estimated using the extensions of the Carnahanand
Starling expression40 to mixtures57, 58 which is given byEq. (21).
In order to solve Eq. (A8) for χ̃chain,ij we first ex-amine the
zero-density limit.
When two segments are on trajectories that lead to a col-lision,
a third segment near one of the segments may collidewith it before,
thus screening the original collision. χ̃chain,ijincorporates the
effect of this screening of collisions betweenspecies i and j by
other segments, including those of thesame species and the same
chain. The probability of find-ing two segments near to each other,
so that one can screenthe collision of the other, is non-zero even
in the low-densitylimit, because the neighbouring segments in the
same chainare always close. Screening, therefore, also happens in
thelow-density limit, and χ̃ (0)chain,ij = lim
ρ→0χ̃chain,ij , is non-zero.
However, as the probability of finding two chains in
closeproximity does vanish in the low-density limit, screening
ofcollisions between segments of two species can only occur inthe
low-density limit due to segments of either of the sametwo species,
and not those of any third species. This meansthat all other
species can be ignored and in order to determineχ̃
(0)chain,ij one can simply consider a binary mixture of
species
i and j.For collisions between segments of species i with
other
segments of species i, the radial distribution function at
con-tact is equal to that of a pure fluid of species i,
χ̃(0)chain,ii = χ̃ (0)chain,i , (A9)
which can be found from Eq. (18). Furthermore, for a
binarymixture, χ̃chain,ij = χ̃chain,j i , and thus χ̃ (0)chain,ij
is uniquely de-termined by Eq. (A8) for a binary mixture of species
i and j.By substituting Eq. (A7), and rearranging terms, one thus
ob-tains Eq. (22) that is independent of segment fractions. Thisis
consistent with a low-density virial expansion where onlypair terms
will contribute to the pressure of the system. Byanalogy with the
virial expansion, we can also obtain Eq. (22)by simply assuming
that χ̃ (0)chain,ij is not a function of compo-sition. Equation
(22) obtained in this manner is not limited tobinary mixtures, but
it is valid for any multicomponent mix-ture.
However, the behaviour of χ̃chain,ij at finite density is
lesseasily understood; Eq. (A8) does not contain enough
informa-tion to determine χ̃chain,ij for higher densities. We
thereforepropose a simple Ansatz regarding the approximate
behaviourof χ̃chain,ij with density. Several possibilities are
assessed
χ̃chain,ij = χ̃ (0)chain,ij + F (ρ̃), (A10)
χ̃chain,ij = χ̃ (0)chain,ij + F (ρ̃) χ̃HS,ij , (A11)
χ̃chain,ij = χ̃ (0)chain,ij + F (ρ̃)(χ̃HS,ij − χ̃ (0)HS,ij
), (A12)
χ̃chain,ij = χ̃ (0)chain,ij χ̃HS,ij + F (ρ̃), (A13)
χ̃chain,ij =(χ̃
(0)chain,ij + F (ρ̃)
)χ̃HS,ij , (A14)
where F (ρ̃) is a function of segment density only which canbe
determined from Eq. (A8). The results in the present workcorrespond
to Ansatz in Eq. (A14) as shown in Eq. (20) andsubsequent
expressions. Additionally, at high densities thechain contributions
tend to be relatively small compared tothe hard-sphere terms. This
means that for well-behaved sys-tems, these Ansätze all produce
very similar results.
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