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)
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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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 σ
that 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 = 8
15NAπσ 3 = 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 = η(0)
[1
χ+ αρ + 1
βα2ρ2χ
], (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, η(0), a tilde is used to indicate a quan-tity defined for segments. The segment density ρ is given byρ = ρm, while α 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 α can beapproximated as
α
αsegment
= 1 + 3
2(m − 1) + 3
8(m − 1)2, (4)
where αsegment is the excluded volume of the free segment,4πσ 3/3.
The zero-density viscosity of the segments η(0) is relatedto the zero-density viscosity of the fluid by the expression,20
η(0) = η(0)χ (0) = η(0)
(1 − 5
8
(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 Mi = 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 = Kmix −
∣∣∣∣∣∣∣∣∣∣∣∣
H11 . . . H1N Y1
.... . .
......
HN1 . . . HNN YN
Y1 . . . YN 0
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
H11 . . . H1N
.... . .
...
HN1 . . . HNN
∣∣∣∣∣∣∣∣∣
, (6)
Kmix = 3ρ2
π
∑i,j
xi xj χij α2ij η
(0)ij , (7)
Yi = xi
⎡⎣1 + ρ
∑j
Mj
Mi + Mj
xj αij χij
⎤⎦, (8)
Hij = −xi xj χij
2A∗ij η
(0)ij
MiMj
(Mi + Mj )2
[20
3− 4A∗
ij
], (9)
Hii = x2i
χii
η(0)i
+∑j �=i
xi xj χij
2A∗ij η
(0)ij
MiMj
(Mi + Mj )2
[20
3+ 4
Mj
Mi
A∗ij
],
(10)
where ρ = (∑
i ximi)ρ is the segment density, xi is the molefraction of component i, and xi = ximi/(
∑j xjmj ) is the
segment fraction. The quantities η(0)ij and A∗
ij are the segmentinteraction viscosity and ratio of collision integrals,35, 36
respectively, for the i-j pair in the limit of zero density. Theparameter αij is the excluded volume of a segment of a chainof species i in the presence of a segment of a chain of speciesj while χij 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
χ±i = β
2ρi αi
⎡⎢⎣
(η
ρi αi η(0)i
− 1
)±
√√√√(η
ρi αi η(0)i
− 1
)2
− 4
β
⎤⎥⎦.
(11)
To ensure realistic physical behaviour, it is necessary toswitch from the χ−
i branch to the χ+i branch of the solution
at some particular density, ρ∗i = miρ
∗i , at which the two roots
are 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 αi ,namely,
αi = η∗i
ρ∗i η
(0)i
(1 + 2√
β
) . (13)
It is important to stress that although αi and χi 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, σα,i andσχ,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, αij
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
αij
αsegment,ij
= 1 + 3
2
⎛⎜⎜⎜⎝
(αsegment,i
αsegment,ij
)1/3
(mα,i − 1) +(
αsegment,j
αsegment,ij
)1/3
(mα,j − 1)
2
⎞⎟⎟⎟⎠
+ 3
8
(αsegment,i
αsegment,ij
)1/3
(mα,i − 1)
(αsegment,j
αsegment,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),
α1/3segment,ij = 1
2
(α
1/3segment,i + α
1/3segment,j
)∝ σα,ij = 1
2(σα,i + σα,j ). (15)
2. Estimating the effective radial distributionfunction, χij
For a pure fluid the effective radial distribution functionat contact of two segments of equal size, χi , 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,
χi = χHS,i + χchain,i . (16)
The chain contribution, χchain,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
χHS,i = 1 − 12 yχ,i
(1 − yχ,i)3, (17)
χchain,i = −5
8
(mχ,i − 1
mχ,i
)1 − 2
5 yχ,i(1 − 1
2 yχ,i
)(1 − yχ,i)
, (18)
where yχ,i = (π/6) σ 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 σχ,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 σα,i and mα, i, andσχ,i and mχ , i are equal, namely,
σα,i(mα,i − 1) = σχ,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 χij , 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
χij = χCS,ij
(χ
(0)ij + F (ρ)
)(20)
χCS,ij = 1
1 − ξ3+ 3
(σχ,i σχ,j
σχ,i + σχ,j
)ξ2
(1 − ξ3)2
+ 2
(σχ,i σχ,j
σχ,i + σχ,j
)2ξ 2
2
(1 − ξ3)3, (21)
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074514-5 The VW method for chain molecules J. Chem. Phys. 136, 074514 (2012)
χ(0)ij = 1 − 1
8
[(mχ,i − 1
mχ,i
) (σ 3
χ,j + 32 σχ,i σ
2χ,j( σχ,i+σχ,j
2
)3
)+
(mχ,j − 1
mχ,j
) (σ 3
χ,i + 32 σχ,j σ
2χ,i( σχ,i+σχ,j
2
)3
)], (22)
F (ρ) =∑
i,j xi xj
(π6
( σχ,i+σχ,j
2
)3)χCS,ij
(1−χ
(0)ij
)+ Zchain
4ρ∑i,j xi xj
(π6
( σχ,i+σχ,j
2
)3)χCS,ij
(23)
Zchain = −∑
i
xi
(mχ,i − 1
mχ,i
) ⎛⎝ ξ3(1 − ξ3) + 3
2 σχ,i ξ2(1 + ξ3) + 12 σ 2
χ,i ξ22
( 2+ξ3
1−ξ3
)(1 − ξ3)2 + 3
2 σχ,i ξ2(1 − ξ3) + 12 σ 2
χ,i ξ22
⎞⎠ , (24)
where the moment densities are defined as ξm = (π/6)NA
∑i σ
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, η
(0)i , we make use of Eq. (5) for each
pure species. The interaction viscosity in the zero-densitylimit, η
(0)ij , for each binary pair, is then given by
η(0)ij = η
(0)ij χ
(0)α,ij , (25)
where χ(0)α,ij is given by Eq. (22) using σα,i and mα, i for
consistency 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, σα , 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, σα ,
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 σα
as 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 η(0)i , χ
(0)i , and
αi for each alkane by means of Eqs. (5), (11), and (13),respectively. The calculation of σχ 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 σα which is onlya function of temperature, σχ is also a weak function ofdensity. This is not surprising since σχ is evaluated fromthe effective hard-core radial distribution function, χ , (seeEq. (16)), which is a function of density. This raises aninteresting question, at what density should one evaluate σχ?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, σα,i , mα, i, σχ,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 standardreferences,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, σχ 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 σα , the mass of segment is given by M = 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 σχ 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, χij . 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, χij . 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 χij and furtherequations are found for the zero-density limits of χ
(0)ij . Fi-
nally, a simple Ansatz is made in order to infer the density-dependence of χij .
The compressibility factor of a chain molecule mixture,Z, can be used to define an effective radial distribution func-tion, χij , using the pressure equation,53
Z ≡ 1 + 4ρ∑i,j
xi xj
(π
6
(σi + σj
2
)3)
χij . (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
Z = ZHS + Zchain, (A3)
χij ≡ χHS,ij + χchain,ij , (A4)
where ZHS is the compressibility factor for a mixture of freehard spheres and χHS,ij is the radial distribution function atcontact of free hard spheres of species i and j.
Zchain is the contribution due to the segment bonding inthe chains and can be written as23
Zchain = ρ
NkBT
(∂Achain
∂ρ
)T
= ρ
kBT
∑i
xi
(mi − 1
mi
) (∂aii
∂ρ
)T
, (A5)
where N = ∑i Nimi is the number of segments in the mix-
ture. ZHS is defined as
ZHS = 1 + 4ρ∑i,j
xi xj
(π
6
(σi + σj
2
)3)
χHS,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 χHS,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 lnχHS,ii , (A7)
which essentially defines χHS,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 χchain,ij in terms of chain lengths andhard-sphere parameters, namely
∑i,j
xi xj
(π
6
(σi + σj
2
)3)
χchain,ij
= −1
4
∑i
xi
(mi − 1
mi
) (∂lnχHS,ii
∂ρ
). (A8)
χHS,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 χchain,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. χchain,ij
incorporates 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 χ
(0)chain,ij = lim
ρ→0χchain,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 determineχ
(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,
χ(0)chain,ii = χ
(0)chain,i , (A9)
which can be found from Eq. (18). Furthermore, for a binarymixture, χchain,ij = χchain,j i , and thus χ
(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 χ
(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 χchain,ij at finite density is lesseasily understood; Eq. (A8) does not contain enough informa-tion to determine χchain,ij for higher densities. We thereforepropose a simple Ansatz regarding the approximate behaviourof χchain,ij with density. Several possibilities are assessed
χchain,ij = χ(0)chain,ij + F (ρ), (A10)
χchain,ij = χ(0)chain,ij + F (ρ) χHS,ij , (A11)
χchain,ij = χ(0)chain,ij + F (ρ)
(χHS,ij − χ
(0)HS,ij
), (A12)
χchain,ij = χ(0)chain,ij χHS,ij + F (ρ), (A13)
χchain,ij = (χ
(0)chain,ij + F (ρ)
)χHS,ij , (A14)
where F (ρ) 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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