Theory and Experiment of Binary Diffusion Coefficient of n-Alkanes in

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Theory and Experiment of Binary Diffusion Coefficient of n-Alkanes in Dilute Gases Changran Liu, William Sean McGivern, Jeffrey A Manion, and Hai Wang J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b08261 • Publication Date (Web): 27 Sep 2016 Downloaded from http://pubs.acs.org on October 1, 2016

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The Journal of Physical Chemistry

Theory and Experiment of Binary Diffusion Coefficient of n-Alkanes in Dilute Gases Changran Liu1, W. Sean McGivern2, Jeffrey A Manion2,*, Hai Wang1,* 1 Stanford University, Stanford, CA 94304, United States, 2 National Institute of Standards and Technology, Gaithersburg, Maryland 20899, United States Abstract Binary diffusion coefficients were measured for n-pentane, n-hexane and n-octane in helium and of n-pentane in nitrogen over the temperature range of 300 to 600 K, using reversed-flow gas chromatography. A generalized, analytical theory is proposed for the binary diffusion coefficients of long-chain molecules in simple diluent gases, taking advantage of a recently developed gas-kinetic theory of the transport properties of nano-slender bodies in dilute free-molecular flows. The theory addresses the long-standing question about the applicability of the Chapman-Enskog theory in describing the transport properties of non-spherical molecular structures, or equivalently, the use of isotropic potentials of interaction for a roughly cylindrical molecular structure such as large normal alkanes. An approximate potential energy function is proposed for the intermolecular interaction of long-chain n-alkane with typical bath gases.

Using this potential and the

analytical theory for nano slender bodies, we show that the diffusion coefficients of n-alkanes in typical bath gases can be treated by the resulting analytical model accurately, especially for compounds larger than nbutane.

* Corresponding authors Jeffrey A Manion ([email protected]) Hai Wang ([email protected])

Submitted to Journal of Physical Chemistry B August 2016 Revised September 19, 2016

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I. Introduction Gas-phase transport properties are critical components of chemically reacting flow simulations. Among these properties, the mass diffusivity of species in dilute gases is important to a range of problems, spanning gas-phase combustion,1-3 heterogeneous catalysis,4 aerosol and heterogeneous atmospheric chemistry,5,6 fuel cells7-9 and chemical vapor deposition.8 In general, the mass diffusivity of a particular species in a reacting flow is approximated from knowledge of the mutual or binary diffusion coefficients.10-13 Despite the need, measurements of binary diffusion coefficients, especially for larger hydrocarbon molecules, have been rare, and these coefficients have been typically determined at low temperatures because of the pyrolysis of these molecules at elevated temperature. Such diffusion coefficients are usually estimated under reactive conditions using isotropic, Lennard-Jones (LJ) 12-6 self-interaction potentials and simple mixing rules for extrapolation to high temperatures. The Chapman-Enskog (CE) theory14 is by far the most successful theory for calculating the gas-phase mutual diffusion coefficient D12. By applying the LJ potential function, Hirschfelder et al.10 obtained the Hirschfelder-Bird-Spotz (HBS) equation given as D12 =

3 kT 8 2π mr

1

2 ( Nσ 12 Ω

1,1)

*

,

(1)

where k, T, mr and N are the Boltzmann constant, temperature, reduced mass and gas number density, *

respectively. In eq 1, σ 12 is the collision diameter and Ω(1,1) is the reduced collision integral which is a *

function of the temperature and the potential well depth ε12 . Ω(1,1) is usually tabulated for the LJ 12-6 potential as a function of kT ε12 . The isotropic LJ potential is rigorous only when both interacting molecules are spherically symmetric. Given that small molecules can undergo rapid rotation, the HBS equation usually gives good predictions for such molecules as methane and ethane in N2 or noble gases. For large molecules, e.g., long-chain alkanes, the significant deviation from sphericity, even under reactive conditions, would be expected to limit the applicability of an isotropic potential. This is especially true when the potential parameters are estimated from self-collision parameters typically derived from low-temperature viscosity data using empirical mixing

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rules: σ 12 = (σ 1 + σ 2 ) 2 and ε12 = ε1ε 2 . Indeed, Molecular Dynamic (MD) simulations of Violi and coworkers15,16 have shown that the diffusion coefficients of a long-chain alkane molecule in a dilute gas, e.g., N2, can deviate from the HBS equation quite significantly. Jasper et al.17 proposed that for a non-isotropic potential that can better model elongated species, effective σ 12 and ε12 parameters can be estimated by fitting eq 1 against results of classical trajectories based on ab

initio potential energy. Specifically, the effective isotropic collision diameter and potential well depth can be extracted by averaging the minimum collision distances over a range of incident angles. To better capture the non-isotropic potential characteristics, Jasper and Miller18 also proposed that collision integrals can be obtained from full-dimensional collision trajectories. Clearly, both of these methods require trajectory calculations using pair-specific ab initio potential function and are computationally demanding considering the vast number of pairs that must be calculated for a typical multi-component reacting flow problem. The purpose of the present work is to propose a general expression for the binary diffusion coefficient involving a long-chain molecule and a roughly spherical mass (e.g., N2 and He), taking advantage of a recently derived, generalized transport theory of aerodynamic drag, electric mobility and diffusion on the basis of a rigorous gas-kinetic theory analysis for slender bodies in dilute gases and the free molecule regime.19 The theory was tested against experimental electric mobility of carbon nanotubes over wide ranges of tube diameter and length. In the current work, we examine the applicability of this theory to binary diffusion coefficients of several n-alkane compounds in He and N2. For this purpose, experiments were carried out to determine the diffusion coefficients of n-CmH2m+2 (m = 5, 6, and 8) in He and n-pentane in N2 over the temperature range of 300 to 600 K using reversed-flow gas chromatography.20,21 The theory is tested using these data, and a general analytical approach is proposed to predict the binary diffusion coefficients of nalkanes in typical diluent gases.

II. Theory The diffusion coefficient D can be equivalently expressed in the Einstein relation,22 D = kT c ,

(2)

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where c is the drag coefficient or the proportionality between the drag force on a particle in a bath of small molecules as it undergoes drift relative to the background gas. Previous studies23,24 have employed a gaskinetic theory analysis of drag on spherical particles in the free molecule regime and demonstrated that Eq 2 is identical to the HBS equation. The derivation considered the intermolecular potential of interaction and assumed that the drag force is caused by momentum exchange due to gas-particle collisions. Following a similar approach, we recently derived a generalized expression for the drag coefficient of a small cylinder in free molecular flow, considering detailed momentum transfer and the potential energy of interactions between the cylinder and gas, integrated over the Boltzmann energy distribution and a full-range of impact parameters.19 The theory is shown to predict the experimental measurements of the electric mobility of carbon nanotubes accurately. To our knowledge, this approach is the most rigorous found in the literature. According to Liu et al.,19 the drag force, F⊙ , on a freely rotating, perfect cylinder of the aspect ratio

L

R ≫ 1 , where L is the

cylinder length and R is the cylinder radius, is F⊙ =

{

}

1 2 ⎡ϕ c + (1 − ϕ ) c s ,⊥ ⎤⎦ + ϕ c d ," V 3 ⎣ d ,⊥

(3)

where V is the drift velocity, c’s are the drag coefficients and ϕ is the momentum accommodation function.23,24 In the above equation, the subscripts s and d denote the limiting specular elastic and diffuse inelastic collisions, respectively, and

⊥

and

!

indicate whether the cylinder axis is perpendicular or parallel to

the drift velocity, respectively. Following the Chapman-Enskog assumption,14 we assume the collision is specular elastic locally and thus ϕ ≡ 0 . This assumption is consistent with a series of analyses we conducted on the collision of typical bath gases with small particles.23-26 The orientation-averaged drag coefficient is therefore 2 c cyl = c s ,⊥ . 3

(4)

The above equation considers only the momentum transfer on the side of the cylinder body. If the cylinder ends can be each approximated by a half-sphere, the overall drag coefficient of a cylinder with two hemispherical ends of length

L

+ 2R is

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c = c cyl + c sph .

(5)

c sph is the drag coefficient on a sphere, given as23 c sph =

8 (1,1)* 2 2π mr ,sphkT Nσ sph Ω s ,sph . 3

(6)

( ) The reduced collision integral Ω s ,sph is identical to what is given in Hirschfelder et al.10 for binary diffusion 1,1 *

coefficients (e.g., Table 4a of Ref. 27). It has been shown19 that the drag coefficient on the side wall of a cylinder is c s ,⊥ =

4 2mr kT (1,1)* N L RΩ s ,⊥ . π π

(7)

(1,1)* is the reduced collision integral of the cylindrical section in the specular scattering limit with the

where Ω s ,⊥

cylinder axis perpendicular to the drift velocity. In the above equation, the radius R will be replaced by an effective collisional diameter after a potential function is defined, as discussed below. The treatment of the reduced masses mr,sph (Eq 6) and mr (Eq 7) requires some special consideration due to the partitioning of momentum exchange into angular and translational momenta. This partitioning is dependent on, among other things, the mass and length of the chain molecule and the impact parameter. In the current work, we propose to use mr,sph = msphmM / (msph + mM) for Eq 6 and mr = mcylmM/(mcyl+mM) for Eq 7, where mM is the mass of the bath gas M, msph is the mass of the equivalent end groups, and mcyl is the mass of the entire n-alkane molecule. Combing Eq 2 with Eqs 4 thru 7, we obtain 1 8 = D 3π

2 (1,1)* (1,1)* ⎞ ⎛ 2 N ⎜ m L RΩ s ,⊥ + mr ,sph π 2σ sph Ω s ,sph ⎟ ⎠ π kT ⎝ r

(8)

(1,1)* may be given by the following equation:

Liu et al.19 showed that Ω s ,⊥

(1,1)*

Ω s ,⊥ =

where γ = g

( )

∞ 2π π 1 γ 5 exp −γ 2 Qs ,⊥ cos2 φ sin φdφdθ dγ , 2R L 0 0 0

∫ ∫ ∫

(9)

2kT mr is the reduced velocity, and g is the velocity of the cylinder relative to gas, and Qs ,⊥ is

the collision cross section:

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Qs ,⊥ = L sin 2 φ cos2 θ + cos2 φ

∫

∞

−∞

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(1 − cos χ ) db

(10)

where b is the impact factor, and χ is the scattering angle given as

χ = π − 2b

∫

∞

rm

⎡ 2 2Φ ( r ) ⎛b⎞ −2 ⎢ r 1− − ⎢ ⎜⎝ r ⎟⎠ 2 2 mr g sin φ cos2 θ + cos2 φ ⎢⎣

(

)

⎤ ⎥ ⎥ ⎥⎦

−1 2

dr .

(11)

We treat Φ ( r ) based on a simple pair interaction potential of n-alkane molecules of arbitrary size with

(1,1)* are

typical bath gases, including He and N2, as will be discussed in section III. The values of Ω s ,⊥

(1,1)* is dependent on temperature,

numerically calculated and tabulated in the same section. The resulting Ω s ,⊥

and two generic, pseudo-potential energy parameters—the collision diameter and well depth of the –CH2– group, that are applicable to all n-CmH2m+2 with m ≥ 4 . The latter parameters are fitted to the binary diffusion coefficient data of n-pentane and n-hexane in He. The accuracy of the potential parameter values is verified against the data of n-butane in He and N2, of n-octane in He and of n-pentane in N2.

III. Potential function For simplicity, the bath gas molecule is considered as spherical with self-interaction potential well depth εM and collision diameter σM. The n-alkane molecule is approximated as a rigid cylinder defined by a zigzag chain of methylene groups (Fig. 1) capped by two methyl groups at the two ends. Internal rotations and conformational isomers are not considered. In C4 to C7 n-alkanes, the fully anti conformer ground states were found in early work28 to be about 2.55 kJ mol–1 lower in energy than species with a single gauche interaction, with a more recent experimental determination29 setting the respective conformational energy differences in n-butane and n-pentane to be (2.761 ± 0.092) kJ mol–1 and (2.586 ± 0.025) kJ mol–1. Results from computational chemistry are in accord, although for high accuracy Balabin30 notes a requirement to account for basis set superposition errors, particularly for larger n-alkanes. Neglecting symmetry, the ratio of a single gauche to anti conformer is 0.6 and 0.8 at 700 and 1500 K, respectively. However, computations show that conformations with multiple gauche interactions, which are those that can lead to the most globular shapes, are progressively disfavored. In pentane, for example, Salam and Deleuze31 compute the TG, G+G+, and G+G- conformers to be respectively 2.60 kJ/mol, 4.46 kJ/mol, and 12.2 kJ/mol higher in energy than the

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fully anti configuration, where T is the anti conformation, and G(+ or

-)

represent the gauche configurations.

Gruzman et al.32 report analogous trends for C6 to C8 alkanes. Hence, one expects a relatively modest impact on the experimental diffusion coefficient due to the presence of conformers. The impact of conformers is expected to be larger towards higher temperatures, but this effect is not expected to be strong in the limit of random orientation of the n-alkane molecule with respect to the trajectory of the colliding bath gas and the fact that collisions lead to localized interactions between the bath and n-alkane molecules.

Molecular

dynamics results of Chae and Violi15 support this view. For example, the diffusion coefficients of n-heptane and 2-methylhexane were found to be nearly the same at 500 K. The difference increases to about 5% at 1000 K. We shall treat the methylene chain and the methyl groups separately. For the methylene chain, we obtain an equivalent potential function for gas-cylinder interaction as a function of one single independent variable, namely the shortest distance from the gas molecule to the cylinder axis, by summing up the contributions from the methylene groups. Moreover, we seek a generalized potential function that can be used to predict the binary diffusion coefficient of n-alkane molecule of an arbitrary size. We first consider a simple case between a bath molecule and one half of the methylene groups along the l = 1 or l = 2 lines with φ = 0 (see Fig. 1). For an infinitely long normal alkane molecule, the chain of the methylene groups may be approximated as a cylinder. In this case, the potential function depends on the shortest distance of a bath molecule to the cylinder axis only. The total potential can be approximated by summing up the LJ 12-6 interactions between the bath molecule and each of the methylene groups

( )

Φ l rl = 4ε

∑

⎡⎛ ⎞ 12 ⎛ ⎞ 6 ⎤ σ ⎢ σ −⎜ ⎟ ⎥, ⎜ ⎟ ⎢ i=−∞ ⎝ rl ,i ⎠ ⎝ rl ,i ⎠ ⎥⎥ ⎢⎣ ⎦ i=∞

(12)

where rl,i is the distance between the molecule and the ith methylene group on the lth line, σ (σ-CH2-,M) and ε (ε-CH2-,M) are the collision diameter and well depth of the pseudo-potential interaction of the methylene group with a bath molecule M. We note that in principle, intermolecular potential is not linearly additive,33 but the higher order terms are generally small since both dispersion interaction and electrostatic repulsion tend to be localized.

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Knowing that rl2,i = rl2 + ( iδ ) , where δ is the distance between two methylene groups that are one 2

removed from their immediate neighbors and rj is the shortest distance between the bath molecule to the lth line, the total potential can be expressed as

( )

Φ l rl∗

⎡ = 4ε ⎢rl∗−12 ⎢ ⎣

∑

(

)

2⎤ ⎡ 1 + iδ ∗ rl∗ ⎥ ⎢ i=−∞ ⎣ ⎦ i=∞

−6

− rl∗−6

∑

−3 ⎤ 2 ⎡ ∗ ∗ ⎤ ⎥ , ⎢1 + iδ rl ⎥ i=−∞ ⎣ ⎦ ⎥⎦ i=∞

(

)

(13)

where δ ∗ = δ σ and rl∗ = rl σ . We will transform the above equation into an approximate yet generalized potential equation. First, noting that the summation terms are each a function of δ ∗ rl∗ , we propose an appropriate equation in the form of

( )

Φ l rl∗ =

4ε

δ

∗

(c r

∗−11 − c 3rl∗−5 6l

),

(14)

to represent Eq 13, where c3 and c6 are constants. The exponents –11 and –5 were chosen to best represent Eq 13, though it may be shown that the potential function values in the well regime and the repulsive part of the function relevant to collisions interest are not very sensitive to the exponent values. A similar approach has been used by Rudyak et al.34,35 in their approximation of the potential function of gas-particle interactions. Second, we sum up the potential energy contributions from both the l = 1 and l = 2 lines. In the special case where φ = 0 , we have

( )

(

)

(

)

Φ r ∗ = Φ1 r ∗ + R ∗ + Φ 2 r ∗ − R ∗ ,

(15)

where r ∗ = r σ and R ∗ = R σ are all normalized with respect to σ . For an arbitrary angle φ , the above equation becomes

( )

(

)

(

)

1 2⎤ 1 2⎤ ⎡ ⎡ ∗2 ∗ ∗ ∗2 Φ r ∗ , φ = Φ1 ⎢ r ∗2 + 2r ∗R ∗ cosφ + R ∗2 ⎥ + Φ 2 ⎢ r − 2r R cosφ + R ⎥. ⎣ ⎦ ⎣ ⎦

(16)

According to the intermediate value theorem, there exists an η for a given φ value such that

( )

Φ r ∗ , φ = 2Φ j ⎡r * − η (φ ) σ ⎤ . Through a second transformation, we propose a potential function of the form ⎣ ⎦

( )

Φ r∗ =

−11 −5 8ε ⎡ ⎛ ∗ η ⎞ η⎞ ⎤ ⎛ ⎢c 6 ⎜ r − ⎟ − c3 ⎜ r ∗ − ⎟ ⎥ , ⎝ σ⎠ σ⎠ ⎥ δ ∗ ⎢⎣ ⎝ ⎦

(17)

8


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where η is a displacement factor that gives the potential function averaged over 0 ≤ φ ≤ 2π . In the reduced form, we have η ∗ = η σ . Among parameters of Eq 17, δ ( = δ ∗σ ) is determined from standard n-alkane

{

geometry data, i.e. δ = l C−C 2 ⎡⎣1 − cos (ϕ ) ⎤⎦

}

12

, where l C−C is the C–C bond length (1.54 Å) and ϕ is the

dihedral angle (109.54°). η , c3 and c6, are chosen to best represent the actual potential function over a relevant range of collision diameter ( 3.5 ≤ σ ≤ 4.5 Å). Their values are listed in Table I. The values of σ and ε may be determined by fitting the experimental diffusion coefficients. Details will be given in section V. In Fig 2, we examine the approximate potential function given by Eq 17 by comparing it with the “actual” potential functions of chains of methylene groups with respect to the effects of chain lengths (Fig. 2a), z-axis displacement (Fig. 2b), and angle φ (Fig. 2c). The actual potential functions are calculated using Eq 13 with the summations over finite numbers of methylenes. For illustration purposes, a σ value of 3.8 Å is used for all cases. We can see that Eq 17 with constants of Table I gives a better approximation as the length of the methylene chain increases, as expected. Shorter methylene chains show greater deviations but the discrepancy in the well depth is not as consequential to calculating diffusion coefficient because of its weak sensitivity to the potential function, as will be discussed later. Also, Eq 17 was derived from a zero z-axis displacement for the bath molecule. The impact of this assumption is seen to be rather small. As shown in Fig. 2(b), the potential function of a {CH2}n=13 methylene chain varies little when the bath molecule is displaced from z = 0 to δ /6 and δ /3. The well depth does drop appreciably when the bath molecule is placed at the end of the {CH2}n=13 chain, as shown in Fig. 2(b). The net effect of reduced well depth is small on the diffusion coefficient, as a sensitivity analysis will demonstrate in a later part of this paper. In any case, the well depth value is treated as a constant in our fit to the experimental diffusion coefficients. It represents an average across the methylene chain. The accuracy of the treatment will be examined by addressing the question whether a uniform, constant well depth value can reconcile a wide range of data, an issue to be discussed in length later. Lastly, the angle-specific potential function is sensitive to the angle value, but the angle-averaged potential function matches closely the function at φ = π 4 . In all cases, Eq 17 is accurate as long as the chain length is not too small.

(1,1)* values are tabulated in Table II as a

Using Eq 17, we integrated Eq 9 numerically. The resulting Ω s ,⊥

function of two reduced parameters: a combined temperature and geometric parameter θ ∗ = ( kT ε )δ ∗ and a 9



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displacement factor η ∗ . Both parameters are based on Eq 17. The choice of the R value can be arbitrary, since R in Eqs 7 and 9 cancel out each other.

(1,1)* is a strong function of θ ∗ , but is a weak function of the displacement factor η ∗ . (1,1)* (1,1)* Over the η ∗ range of interest, the Ω s ,⊥ changes by less than 2% for a given θ ∗ value. The Ω s ,⊥ can be It is seen that Ω s ,⊥

parameterized closely by the following equation:

(1,1)* π 2 η ∗ + ⎛ 3.441 − 0.975 + 4.955 ⎞ + ⎛ 0.002236 − 0.00739 + 0.004487 ⎞ 1 ⎜ *1 4 *1 2 ⎟ ⎜ *1 4 *1 2 ⎟ ∗

Ω s ,⊥ =

2

⎝

θ

θ

⎠ ⎝

θ

θ

⎠η

(18)

(1,1)* may also be parameterized as a function of θ ∗

with a maximum fitting error of 0.8%. Alternatively, Ω s ,⊥ only with a maximum error of 1.1%:

(1,1)*

Ω s ,⊥ =3.959 −

1.489

θ

*1 4

+

5.281

(19)

θ *1 2

IV. Experimental Diffusion measurements were accomplished by reversed-flow gas chromatography, as detailed in previous publications.20,21 A schematic of the apparatus is shown in Figure 3. Briefly, it consists of an electropolished 316 stainless steel tube of 6.35 mm o.d. (3.2 mm i.d.) and length of 60.9 cm (61.28 cm effective length20), called the diffusion tube, that is capped at one end. At the opposite end, a second “sampling” tube of the same material is attached perpendicularly with an orbitally welded tee. The diffusion cell is contained in the oven of a gas chromatograph for temperature control. Temperatures are monitored with calibrated high-accuracy platinum wirewound resistance temperature detectors. A flow of the bath gas of interest is established through the sampling tube to a flame ionization detector (FID). This flow is kept laminar so that the bath gas in the diffusion tube remains stagnant. Pressures are maintained at (1.90 to 1.96) bar, and the experiment is initiated using an autoinjection system to introduce a small amount of the organic analyte precisely at the capped end of the diffusion tube. Sample diffuses through the length of the diffusion tube and is entrained into the sampling tube flow to be detected by the flame ionization detector (FID). The concentration of analyte exiting the diffusion tube and the corresponding FID response slowly increase to a maximum at anywhere from 10 to 90 min after injection, depending on the binary diffusion coefficient and

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the diffusion tube pressure, followed by a long, slow decrease. However, to account for possible drift of the baseline signal in such long experiments, a software-controlled valve was introduced that allowed the sampling flow to be reversed for 6 seconds every 2 min for the first 1 h and every 3 min thereafter. Each reversal allows sample to pass the outlet of the diffusion tube twice, effectively doubling the analyte concentration. As shown in the inset of Figure 3, the reversal and subsequent return to the original flow direction leads to first a negative peak (clean bath gas flowing into the FID during the reversal) followed by a positive peak (the doubly concentrated sample) that is superimposed on top of the original FID response 36. The heights of these peaks above the standard diffusion signal envelope were taken as the signal for analysis and diffusion coefficient determination. By taking a Laplace transform of Fick’s law and applying several simplifying assumptions unique to the reversed-flow apparatus, the resulting signal, s, is related the binary diffusion coefficient of the analyte in the bath gas by36 ⎛ L2 ⎞ s ∝ τ −3 2 exp ⎜ − ⎟, ⎝ 4Dτ ⎠

(20)

where L is the length of the diffusion tube, and D is the binary diffusion coefficient. τ is the adjusted diffusion time and is calculated as τ = t − t h − t r 2 , where t is diffusion time (time since analyte injection), th is the holdup time required for a sample parcel to pass from the outlet of the diffusion tube to the detector, and tr is the time length of the reversal. Plotting ( ln s )τ 3 2 vs 1 τ leads to a linear region with a slope of − L2 ( 4D ) . Experimental diffusion coefficients reported herein are derived from such plots.

V. Results Measured binary diffusion coefficients for dilute alkanes in He and dilute pentane in N2 are presented in Table III. Values have been scaled linearly from the measurement pressures of about 2 bar to 1.013 bar (1 atm), as is customary. Deviations from non-linear scaling are expected to be very small under our conditions,21 but raw diffusion coefficients and measurement pressures are reported in the supporting information to allow alternative treatments if desired.

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Uncertainties in the n-alkane-He data were determined from multiple samples taken hours or days apart and are reported as two-standard deviations throughout this study. The n-pentane-N2 data are single-point measurements, and we have elected to present these data with conservative uncertainty estimates. Based on typical scatter between runs at each temperature and the consistency in the temperature dependence with previous diffusion measurements on the same apparatus,21 we assign a two-standard deviation uncertainty of ±8 %, which is approximately twice the worst-case scatter in other measurements. Significant difficulties in measurement were observed at the temperature extremes, particularly for low vapor-pressure analytes (n-octane) or under experimental conditions where diffusion becomes slow (e.g. pentane-N2 at low temperatures). At 300 K, the pentane-N2 reversal peaks showed significant tailing at longer experimental times, and the analyzed spectrum showed dramatically larger scatter than those from analogous studies of pentane diffusion in He at 300 K or previous measurements of higher-vapor pressure components.21 The value of D was also anomalously low, relative to values derived using the highertemperature measurements and assumption of a typical temperature dependence of the diffusion constant. A similar outcome was observed at 300 K for octane-He. We attributed both of these artifacts to sticking of the analyte to the walls of the diffusion and sampling tubes at low temperatures, and we have elected to omit both measurements. Higher-temperature measurements (up to 723 K) were also attempted for n-octane-He, but extremely large variations in the observed D were found from run to run, and the average value of repeated measurements was found to be anomalously high. We attribute these artifacts to surface-induced decomposition of the analyte in the diffusion tube and consider this a limitation in the technique itself for high-temperature measurements, at least when using the present diffusion tube. The measured diffusion coefficients are plotted in Figs. 4 and 5 for He and N2 as the mutual diffusion partners, respectively. As discussed before, a CmH2m+2 n-alkane molecule is treated here as a cylinder containing a chain of m–1 methylene groups and 2× one half of one methane molecule to account for the end cap effect. The length of the cylinder is thus

L

= ( m − 1)δ , and the cylinder radius may be replaced by the

collision diameter σ . In this way Eq 8 may be written as 1 8 2 kT ⎛ (1,1)* (1,1)* ⎞ 2 2 = ⎜⎝ mr ( m − 1)δσΩ s ,⊥ + mr ,sph π σ sph Ω s ,sph ⎟⎠ . D 3π π kT p

12


(21)

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The term

(1,1)*

2 mr ,sph π 2σ sph Ωs,sph is modeled by the diffusion coefficient of methane DCH ,M in a particular bath 4

gas M directly, giving an equivalent expression as 1 8 = D 3π

(1,1)*

2 kT π kT p

mr ( m − 1)δσΩ s ,⊥ +

1 DCH ,M 4

.

(22)

For m = 1, i.e., methane, and the first term vanishes as expected. For the current analysis, we take the diffusion coefficient of methane in He and N2 directly from literature data.20,37 The conventional mixing rule is used for the potential parameters for the cylinder part of the molecule:

σ=

(σ

M

)

+ σ −CH − 2 , 2

ε = ε M ε −CH

2

−

(23)

,

(24)

The potential parameters for the self-interaction of He and N2 are taken from the literature38,39 and provided in Table 1. We found the values of σ −CH − and ε −CH 2

2

−

(listed in Table I) by fitting against the experimental

data for n-pentane and n-hexane in He only. The fits are then verified against the data of n-butane in helium (He) and nitrogen (N2) from an earlier study,21 of n-octane in He and n-pentane in He from the current work. The results are presented in Table III and Figs. 4 and 5. Clearly the current model reproduces the measurements rather well. As shown in Fig. 5, the predictions are in good agreement with the n-pentane-N2 data, but the model appears to over predict the diffusion coefficient of n-butane above 500 K. The causes for the discrepancy include the accuracy of the empirical mixing rule and the fact that both the drag force formulation and the potential function become less accurate as the aspect ratio of the molecule decreases. Nevertheless, the comparisons shown in Figs. 4 and 5 suggest that the current theory is accurate for predicting diffusion coefficients of n-alkane molecules of sizes equal to or larger than n-pentane. The sensitivity of the diffusion coefficient to the well depth and collision diameter is analyzed and illustrated in Fig. 6. It shows that the diffusion coefficient is highly dependent on σ-CH2-,M but it is only weakly dependent on ε-CH2-,M. The weak dependency on the well depth explains why the current model (using the infinitely long chain approximation for a finite chain length) is able to reconcile with the data well despite the fairly strong dependency of potential well depth on the chain length, as seen in Fig. 2a.

13



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Page 14 of 30

To facilitate reacting flow computations, we examined whether the diffusion coefficients can still be fitted into the HBS expression using effective or pseudo LJ 12-6 potential parameters. Despite the fundamental deficiency of the HBS expression associated with long-chain molecules, the expression can be used to represent the mutual diffusion coefficients on a pair-by-pair basis across a relevant range of temperature. We calculated the binary diffusion coefficients over a wide range of temperature and determined the effective, pair-wise potential parameter values for the HBS expression. The results are tabulated in the supporting information from pentane to hexadecane in several common bath gases (N2, O2, Ar, and He).

The

parameters may be used almost directly in ChemKin-based computations. Additionally, various ratios of the collision integrals are also given to allow for computations using the multicomponent transport formulation,11 as discussed earlier.2,40

VI. Discussion Diffusion coefficients calculated using other models and methods should be discussed. Figure 7 shows comparisons of the binary diffusion coefficients of n-pentane in N2 from 300 to 1600 K calculated using Eq 21, the HBS expression with effective LJ 12-6 potential parameters from Jasper et al.18 and from the estimates of JetSurF,41 the result of MD simulations15 and the experimental data of the current study. JetSurF41 used the empirical Tee-Gotoh-Stewart correlations42 of the LJ 12-6 potential parameters for the self-interaction of n-alkane molecules with their critical pressure and temperature—a method employed earlier for estimating the transport parameters of polycyclic aromatic hydrocarbons.43 These self-interaction potential parameters can give a reasonable prediction for the viscosity of vapor-phase, single component n-alkanes,42 but there is no fundamental reason why they can be used to predict mutual diffusion coefficients through the mixing rule given by Eqs 23 and 24. Somewhat surprisingly, all models and methods reproduce the experimental data in N2 closely, as shown in Fig. 7. Not surprisingly, the theoretical result of Jasper et al.,18 the estimates of JetSurF,41 the MD results15 diverge from each other above 600 K. In the case of n-pentane in He, both the theoretical result and JetSurF estimate show marked deviation from the current experimental data, as shown in Fig. 8. The overlapping diffusion coefficient values between Jasper et al.’s classical trajectory calculations18 and the JetSurF estimates41 is only fortuitous. 14


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In principle, the inability of the HBS expression to predict the long-chain normal alkanes is due to its isotropic potential assumption, although, as noted by Jasper et al.,18 the dependency of the diffusion coefficients on temperature can be represented by the HBS expression rather well. Yet Eq 1 gives the diffusion coefficient a σ –2 dependency, while the first term of Eq 22 gives a σ –1 dependency (i.e., for the cylinder part of the expression). This discrepancy essentially renders the mixing rule (Eqs 23 and 24) meaningless at the molecular level. Such an effect is in fact the cause for the discrepancy between the JetSurF estimate and experimental data as shown in Fig. 8. The smaller collision diameter of He gives a smaller σ . The σ –2 scaling in the HBS expression naturally produces a diffusion coefficient that increases faster as σ is decreased than the σ –1 scaling given by Eq 22. This observation would suggest that the reasonably good agreement for n-pentane-N2 between the JetSurF estimate and the current theory, as seen in Fig 7, can be only fortuitous. We note that full dimensional trajectory calculations17 have been carried out to compute the collision integral of alkane molecules (n-CmH2m+2, m=2 to 4 ) in N2. Good agreement was demonstrated with the experiment results (around 1% error). These trajectory calculations are extremely useful in that they could be used to verify the accuracy of the current theoretical model beyond n-octane.

VII. Conclusions Mutual diffusion coefficients of n-pentane, n-hexane and n-octane in helium and of n-pentane in nitrogen were determined experimentally for the first time in the temperature range of 300 to 600 K and 2 bar pressure, using reversed-flow gas chromatography (RF-GC). Low-temperature measurements of low vapor pressure species were found to exhibit sticking to the analysis tube walls, and thermal decomposition of the larger alkanes was observed at higher temperatures, limiting the temperature range observable with the RFGC technique. Nonetheless, uncertainties in the n-alkane-He data reported here are well quantified from multiple measurements to typically be less than 2% or were conservatively estimated where multiple points are unavailable. In an attempt to address the long-standing question about the applicability of the Chapman-Enskog theory in describing the transport properties of non-spherical molecules and more specifically, the use of 15



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isotropic potentials of interaction for an otherwise non-spherical molecular structure, a generalized, analytical model is proposed for the binary diffusion coefficients of long-chain molecules in simple diluent gases. The model is based on a recently developed gas-kinetic theory of the transport properties of nano-slender bodies in dilute, free-molecular flows. We proposed an approximate potential energy function for the intermolecular interaction of long-chain n-alkane molecules with typical bath gases (N2 and He). We demonstrate that the diffusion coefficients of n-alkanes equal to or larger than n-pentane through at least n-octane can be treated by the resulting analytical model accurately, and we expect modeling to continue to be accurate to longer chain molecules.

ASSOCIATED CONTENT Supporting information Raw diffusion coefficients measured at 2 bar, effective Lennard-Jones 12-6 potential parameters for evaluating the binary diffusion coefficients of n-alkane in typical bath gases, and coefficients of Aij* , Bij* and

Cij* for multicomponent transport formulation. AUTHOR INFORMATION Corresponding Authors *E-mail: [email protected]. Phone (301) 975-3188. *E-mail: [email protected]. Phone (650) 497-0433. Funding The work at Stanford University was supported by the U.S. Air Force Office of Scientific Research (AFOSR) under contractor numbers FA9550-14-1-0235 and FA9550-16-1-0051. The work at NIST was partially supported by the U.S. Air Force Office of Scientific Research (AFOSR) under contract number FIAT06004G003.

VIII. References (1) Miller, J. A.; Kee, R. J.; Westbrook, C. K., Chemical kinetics and combustion modeling. Ann. Rev. Phys. Chem. 1990, 41 (1), 345-387. (2) Dong, Y.; Holley, A. T.; Andac, M. G.; Egolfopoulos, F. N.; Davis, S. G.; Middha, P.; Wang, H., Extinction of premixed H2/air flames: Chemical kinetics and molecular diffusion effects. Combust. Flame 2005, 142 (4), 374-387. (3) Middha, P.; Yang, B.; Wang, H., A first-principle calculation of the binary diffusion coefficients pertinent to kinetic modeling of hydrogen/oxygen/helium flames. Proc. Combust. Inst. 2002, 29 (1), 1361-1369.

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(4) Satterfield, C. N., Mass transfer in heterogeneous catalysis. The MIT Press: Cambridge, MA, 1970. (5) Marcolli, C.; Luo, B.; Peter, T.; Wienhold, F., Internal mixing of the organic aerosol by gas phase diffusion of semivolatile organic compounds. Atmos. Chem. Phys. 2004, 4 (11/12), 2593-2599. (6) Liu, Y.; Cain, J. P.; Wang, H.; Laskin, A., Kinetic study of heterogeneous reaction of deliquesced NaCl particles with gaseous HNO3 using particle-on-substrate stagnation flow reactor approach. J. Phys. Chem. A 2007, 111 (40), 10026-10043. (7) O'Hayre, R. P.; Cha, S.-W.; Colella, W.; Prinz, F. B., Fuel Cell Fundamentals. John Wiley & Sons: New York, 2006. (8) Todd, B.; Young, J., Thermodynamic and transport properties of gases for use in solid oxide fuel cell modelling. J. Power Sources 2002, 110 (1), 186-200. (9) Zhu, H.; Kee, R. J., A general mathematical model for analyzing the performance of fuel-cell membraneelectrode assemblies. J. Power Sources 2003, 117 (1), 61-74. (10) Hirschfelder, J. O.; Curtiss, C. F.; Bird, R. B., Molecular Theory of Gasses and Liquids. Wiley: New York, 1954. (11) Kee, R. J.; Dixon-Lewis, G.; Warnatz, J.; Coltrin, M. E.; Miller, J. A., A Fortran computer code package for the evaluation of gas-phase, multicomponent transport properties. Sandia National Laboratories: Albuquerque, NM, 1988. (12) Ern, A.; Giovangigli, V., Multicomponent transport algorithms. Springer -Verlag: Berlin, 1994; Vol. 24. (13) Ern, A.; Giovangigli, V., Fast and accurate multicomponent transport property evaluation. J. Comput. Phys. 1995, 120 (1), 105-116. (14) Chapmann, S.; Cowling, T. G., The Mathematical Theory of Non-Uniform Gasses. Cambridge University Press: Cambridge, 1970. (15) Chae, K.; Elvati, P.; Violi, A., Effect of molecular configuration on binary diffusion coefficients of linear alkanes. J. Phys. Chem. B 2010, 115 (3), 500-506. (16) Chae, K.; Violi, A., Mutual diffusion coefficients of heptane isomers in nitrogen: a molecular dynamics study. J. Chem. Phys. 2011, 134 (4), 044537. (17) Jasper, A. W.; Miller, J. A., Lennard–Jones parameters for combustion and chemical kinetics modeling from full-dimensional intermolecular potentials. Combust. Flame 2014, 161 (1), 101-110. (18) Jasper, A. W.; Kamarchik, E.; Miller, J. A.; Klippenstein, S. J., First-principles binary diffusion coefficients for H, H2, and four normal alkanes + N2. J. Chem. Phys. 2014, 141 (12), 124313. (19) Liu, C.; Li, Z. G.; Wang, H., Drag force and transport property of small cylinder in free molecule flow: A gas-kinetic theory analysis. Phys. Rev. E 2016, in press. (20) McGivern, W. S.; Manion, J. A., Extending reversed-flow chromatographic methods for the measurement of diffusion coefficients to higher temperatures. J. Chromatogr. A 2011, 1218, 8432-8442. (21) McGivern, W. S.; Manion, J. A., Hydrocarbon binary diffusion coefficient measurements for use in combustion modeling. Combust. Flame 2012, 159, 3021-3026. (22) Einstein, A., Zur theorie der brownschen bewegung. Ann. Phys. 1906, 324 (2), 371-381. (23) Li, Z.; Wang, H., Drag force, diffusion coefficient, and electric mobility of small particles. I. Theory applicable to the free-molecule regime. Phys. Rev. E 2003, 68 (6), 061206. (24) Li, Z.; Wang, H., Drag force, diffusion coefficient, and electric mobility of small particles. II. Application. Phys. Rev. E 2003, 68 (6), 061207. (25) Li, Z.; Wang, H., Gas-nanoparticle scattering: A molecular view of momentum accommodation function. Physical review letters 2005, 95 (1), 014502.

17



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(26) Wang, H., Transport properties of small spherical particles. Annals of the New York Academy of Sciences 2009, 1161 (1), 484-493. (27) Klein, M.; Smith, F. J., Tables of collision integrals for the (m,6) potential function for 10 values of m. J. Res. Natl. Bur. Stand. Sec. A 1968, 72A, 359-423. (28) Bartell, L. S.; Kohl, D., Structure and rotational isomerization of free hydrocarbon chains. J. Chem. Phys. 1963, 39 (11), 3097-3105. (29) Balabin, R. M., Enthalpy difference between conformations of normal alkanes: Raman spectroscopy study of n-pentane and n-butane. J. Phys. Chem. A 2009, 113 (6), 1012-1019. (30) Balabin, R. M., Enthalpy difference between conformations of normal alkanes: effects of basis set and chain length on intramolecular basis set superposition error. Mol. Phys. 2011, 109 (6), 943-953. (31) Salam, A.; Deleuze, M., High-level theoretical study of the conformational equilibrium of n-pentane. J. Chem. Phys. 2002, 116 (4), 1296-1302. (32) Gruzman, D.; Karton, A.; Martin, J. M., Performance of Ab Initio and Density Functional Methods for Conformational Equilibria of CnH2n+ 2 Alkane Isomers (n= 4− 8)†. J. Phys. Chem. A 2009, 113 (43), 1197411983. (33) Farina, C.; Santos, F.; Tort, A., A simple way of understanding the nonadditivity of van der Waals dispersion forces. Am. J. Phys. 1999, 67 (4), 344-349. (34) Rudyak, V. Y.; Krasnolutski, S., Kinetic description of nanoparticle diffusion in rarefied gas. Dokl. Phys. 2001, 46 (12), 897-899. (35) Rudyak, V. Y.; Krasnolutskii, S.; Nasibulin, A.; Kauppinen, E., Methods of measuring the diffusion coefficient and sizes of nanoparticles in a rarefied gas. Dokl. Phys. 2002, 47 (10), 758-761. (36) Katsanos, N. A.; Karaiskakis, G., Measurement of diffusion coefficients by reversed-flow gas chromatography instrumentation J. Chromatogr. A 1982, 237 (1), 1-14. (37) Marrero, T. R.; Mason, E. A., Gaseous diffusion coefficients. J. Phys. Chem. Ref. Data 1972, 1 (1), 3-118. (38) Trengove, R. D.; Robjohns, H. L.; Dunlop, P. J., Diffusion coefficients and thermal diffusion factors for five binary systems of methane with noble gases. Ber. Bunsenges. Phys. Chem. 1982, 86, 951-955. (39) Boushehri, A.; Bzowski, J.; Kestin, J.; Mason, E., Equilibrium and transport properties of eleven polyatomic gases at low density. J. Phys. Chem. Ref. Data 1987, 16 (3), 445-466. (40) Middha, P.; Wang, H., First-principle calculation for the high-temperature diffusion coefficients of small pairs: the H–Ar Case. Combust. Theor. Model. 2005, 9 (2), 353-363. (41) Sirjean, B.; Dames, E.; Sheen, D.; You, X.; Sung, C.; Holley, A.; Egolfopoulos, F.; Wang, H.; Vasu, S.; Davidson, D. et al., A high-temperature chemical kinetic model of n-alkane oxidation. In JetSurF version 1.0, 2009. (42) Tee, L. S.; Gotoh, S.; Stewart, W. E., Molecular parameters for normal fluids. Lennard-Jones 12-6 Potential. Ind. Eng. Chem. Fundam. 1966, 5 (3), 356-363. (43) Wang, H.; Frenklach, M., Transport properties of polycyclic aromatic hydrocarbons for flame modeling. Combust. Flame 1994, 96 (1-2), 163-170.

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Table 1. Constants and parameters. Constant

Value

Comments /References

δ

2.514 Å

η

0.12 Å

c3

1.031

Determined from molecular geometry See text. Applicable for 3.5 ≤ σ ≤ 4.5 Å See text

c6

0.676

See text

Potential parameters σ (Å) Species He 2.64 N2 3.65 –CH2– 4.52

ε k (K) 10.8 98.4 48.1

References Ref 38 Ref 39 This work

19



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Page 20 of 30

(1,1)* as a function of the reduced temperature and

Table 2. Values of reduced collision integral Ωs,⊥ reduced collision diameter.

η*×100 T* 0.84 1.05 1.40 2.80 4.20 5.60 7.00 8.40 9.80 11.20 12.60 14.00 16.80 19.60 22.4000 25.2000 28.0000 30.8000 33.6000 36.4000 39.2000 42.0000

2.70 7.9926 7.4208 6.7892 5.7533 5.2762 4.9984 4.8156 4.6847 4.5852 4.5063 4.4415 4.3869 4.2991 4.2302 4.1739 4.1265 4.0855 4.0496 4.0175 3.9886 3.9624 3.9383

2.90 8.0001 7.4284 6.7974 5.7632 5.2864 5.0086 4.8257 4.6948 4.5953 4.5163 4.4515 4.3969 4.3090 4.2401 4.1838 4.1363 4.0953 4.0593 4.0273 3.9984 3.9721 3.9480

3.08 8.0152 7.4423 6.8098 5.7737 5.2963 5.0182 4.8351 4.7041 4.6045 4.5255 4.4606 4.4060 4.3180 4.2491 4.1927 4.1452 4.1042 4.0682 4.0362 4.0073 3.9810 3.9569

3.28 8.0284 7.4548 6.8217 5.7848 5.3071 5.0289 4.8457 4.7146 4.6149 4.5358 4.4709 4.4162 4.3282 4.2592 4.2029 4.1553 4.1143 4.0783 4.0462 4.0173 3.9910 3.9669

3.51 8.0506 7.4752 6.8399 5.7994 5.3204 5.0415 4.8580 4.7267 4.6269 4.5477 4.4827 4.4280 4.3399 4.2708 4.2144 4.1669 4.1258 4.0898 4.0577 4.0288 4.0025 3.9784

20

4.00 8.0702 7.4961 6.8631 5.8261 5.3469 5.0676 4.8838 4.7522 4.6522 4.5728 4.5077 4.4529 4.3647 4.2955 4.2391 4.1914 4.1504 4.1143 4.0822 4.0532 4.0269 4.0028

4.76 8.1040 7.5309 6.9005 5.8679 5.3884 5.1084 4.9239 4.7919 4.6916 4.6120 4.5467 4.4917 4.4032 4.3339 4.2773 4.2296 4.1885 4.1523 4.1202 4.0912 4.0648 4.0406


5.56 8.1049 7.5419 6.9247 5.9085 5.4304 5.1502 4.9654 4.8330 4.7324 4.6526 4.5871 4.5320 4.4433 4.3739 4.3171 4.2693 4.2281 4.1919 4.1597 4.1307 4.1043 4.0801

6.67 8.1812 7.6130 6.9904 5.9688 5.4894 5.2085 5.0231 4.8903 4.7894 4.7094 4.6437 4.5884 4.4994 4.4298 4.3729 4.3250 4.2836 4.2473 4.2148 4.1853 4.1579 4.1320

7.81 8.2623 7.6871 7.0581 6.0318 5.5509 5.2690 5.0829 4.9496 4.8483 4.7680 4.7020 4.6466 4.5573 4.4874 4.4304 4.3823 4.3409 4.3046 4.2722 4.2431 4.2166 4.1923

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Table 3. Binary diffusion coefficients measured for dilute alkanes in He and N2 at 1.013 bar. D (cm2/s)b T (K) Expt n-pentane-He 300 0.297±0.005 350 0.384±0.006 400 0.475±0.003 450 0.575±0.002 500 0.679±0.009 550 0.786±0.004 600 0.906±0.005 n-hexane-He 300 0.246±0.004 350 0.342±0.001 400 0.427±0.002 450 0.512±0.001 500 0.607±0.001 550 0.703±0.004 600 0.804±0.002 n-octane-He 350 0.247±0.009 400 0.336±0.001 450 0.421±0.002 500 0.500±0.001 550 0.581±0.002 600 0.662±0.003 n-pentane-N2d 350 0.118±0.009 400 0.150±0.012 450 0.184±0.015 500 0.220±0.018 550 0.256±0.020 600 0.298±0.024

Model

Errorc

0.294 0.381 0.475 0.578 0.689 0.807 0.932

–1.0% –0.8% 0.0% 0.5% 1.5% 2.7% 2.9%

0.257 0.333 0.415 0.505 0.602 0.706 0.815

4.5% –2.6% –2.8% –1.4% –0.8% 0.4% 1.4%

0.266 0.332 0.403 0.481 0.563 0.650

7.7% –1.2% –4.3% –3.8% –3.1% –1.8%

0.118 0.149 0.183 0.220 0.259 0.301

0.0% –0.7% –0.5% 0.0% 1.2% 1.0%

a

Number of experiments. b The uncertainty values quoted are 2-standard deviations. Error between model and experiment. d Uncertainty assumed to be ±8 %. See text for details.

21



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FIG 1. Schematic of the coordinate system and distance parameters.

22


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Page 23 of 30

0.50

(a)

Φ(r*)/(8ε/δ*)

0.25

0.00

n=3 5

-0.25

7 13

(eq 17)

8

-0.50

Φ(r*)/(8ε/δ*)

1.0

1.5

2.0

0.2

z = (1/3)δ

0.1

z = (1/6)δ

2.5

3.0

(b) z = L/2

0.0 z=0

-0.1 -0.2 -0.3 -0.4

eq 17

-0.5 1.0

1.5

2.0

(c)

0.2

Φ(r*)/(8ε/δ*)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.1

[CH2]13

0.0

φ=0 φ = π/4

-0.1

φ = π/2

-0.2 -0.3 -0.4

eq 17

-0.5 1.0

1.5

2.0

r* ∗ FIG 2. Various potential functions normalized by 8ε δ . Unless explicitly indicated, all potential functions are averaged over angle φ . (a) Comparison of Eq 17 with those of a methylene chain {CH2}n with n = 3, 5, 7 and 13; (b) Effect of z-axis displacement of the bath molecule for a {CH2}n=13 methylene chain, compared to Eq 17 (see Fig. 1). The Z = L/2 line is calculated for the bath gas placed at the end of the methylene chain; (c) Eq 17 compared with angle-specific potential functions and an angle-averaged potential function of a {CH2}n=13 methylene chain. 23



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FIG 3. Schematic diagram of diffusion apparatus. The inset at the lower left shows an idealized detector response in the absence of a reversal (dotted line) and with a single reversal (solid line).

24


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n-C4H10

1.5

n-C5H12

D (cm2/s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


n-C6H14

1.0 n-C8H18

0.5

0.0

300

400

500

600

700

800

T (K) FIG 4. Binary diffusion coefficients of n-butane, n-pentane, n-hexane, and n-octane in He at 1 atm pressure. Open symbols: experimental data of n-butane21 n-pentane (this work), n-hexane (this work) and n-octane (this work); lines: model predictions. The error bars are two-standard deviations.

25



0.5 n-C4H10

0.4

D (cm2/s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 26 of 30

n-C5H12

0.3 0.2 0.1 0.0

300

400

500

600

700

T (K) FIG 5. Binary diffusion coefficients of n-butane and n-pentane in N2 at 1 atm pressure. Open symbols: experimental data of n-butane21 and n-pentane (this work); lines: model predictions. The error bars are twostandard deviations.

26


Page 27 of 30

1.0

1.0

(a)

(b)

0.8

0.8

D (cm2/s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


n-C8H18 in He

0.9xσ

0.6

n-C8H18 in He

0.5xε

0.6

2xε

1.1xσ 0.4

0.4 n-C5H12 in N2

0.9xσ

0.2

0.5xε 0.2

2xε

1.1xσ 300

400

500

600

700

800

0.0

n-C5H12 in N2

300

400

T (K)

500

600

700

800

T (K)

FIG 6. Sensitivity of the binary diffusion coefficients of n-C8H18 in He and n-C5H12 in N2 to the potential parameters: (a) collision diameter, (b) well depth.

27



HBS with σ and ε

1.0

from Jasper et al.17 HBS with σ and ε

D (cm2/s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 28 of 30

of JetSurF41 This work (eq 22) MD (Chae et al.15)

0.1 500

1000

1500

T (K) FIG 7. Comparison of binary diffusion coefficients of n-pentane in N . Symbols are experimental data; lines are from this work (Eq 22), the HBS theory with potential parameters from Jasper et al.18 and JetSurF,41 and MD simulation.15

28


Page 29 of 30

7 5

HBS with σ and ε from Jasper et al.17

3 D (cm2/s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


HBS with σ and ε

1 0.8 0.6

of JetSurF41 This work (eq 22)

0.4 0.2

500

1000

1500

T (K) FIG 8. Comparison of binary diffusion coefficients of n-pentane in He. Symbols are experimental data; lines are from this work (Eq 22), the HBS theory with potential parameters from Jasper et al.18 and JetSurF.41

29



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

TOC graphic

30


Page 30 of 30

Theory and Experiment of Binary Diffusion Coefficient of n-Alkanes in

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