New Thermal Diffusion Coefficient Measurements for Hydrocarbon

Apr 26, 2008 - Measurement of Thermodiffusion Coefficient in n-Alkane Binary Mixtures: Composition Dependence. J. A. Madariaga , C. Santamaría , M...
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J. Phys. Chem. B 2008, 112, 6442–6447

New Thermal Diffusion Coefficient Measurements for Hydrocarbon Binary Mixtures: Viscosity and Composition Dependency Alana Leahy-Dios,* Lin Zhuo, and Abbas Firoozabadi* Department of Chemical Engineering, Mason Lab, Yale UniVersity, New HaVen, Connecticut 06520-8286 ReceiVed: NoVember 21, 2007; ReVised Manuscript ReceiVed: February 22, 2008

New thermal diffusion coefficients of binary mixtures are measured for n-decane-n-alkanes and 1-methylnaphthalene-n-alkanes with 25 and 75 wt % at 25 °C and 1 atm using the thermogravitational column technique. The alkanes range from n-pentane to n-eicosane. The new results confirm the recently observed nonmonotonic behavior of thermal diffusion coefficients with molecular weight for binary mixtures of n-decane-n-alkanes at the compositions studied. In this work, the mobility and disparity effects on thermal diffusion coefficients are quantified for binary mixtures. We also show for the binary mixtures studied that the thermal diffusion coefficients and mixture viscosity, both nonequilibrium properties, are closely related. I. Introduction Thermal diffusion is the coupled effect between gradients of composition and temperature. It has various potential applications, including drug delivery,1 optical tweezers,2 and polymer separation.3 It has proven applications in areas such as ice core measurements for abrupt climate changes4 and in various aspects of petroleum engineering (wax deposition in pipelines5 and species distribution in recovery hydrocarbon reservoirs).6,7 Various research groups have investigated thermal diffusion experimentally; Wiegand3 and Platten8 review experimental results for different binary mixtures at various conditions. We have previously9 provided a list of references to earlier measurements in mixtures similar to those in this work. Although thermal diffusion coefficients, represented by DT, can be accurately determined experimentally, there is limited microscopic understanding, even for simple fluids. Prigogine et al.10,11 considered thermal diffusion to be a stepwise activated process and defined the activation energy as the potential energy to break cohesive bonds. Denbigh12 provided a physical interpretation for the heat of transport, which is intimately related to thermal diffusion, as the net amount of energy transported across a reference plane per mole in the absence of a temperature gradient. Recently, on the basis of experimental work by Duhr and Braun,2 Astumian13 described two generic classes of mechanisms by which thermal diffusion can occur (based on fluid dynamics and on equilibrium thermodynamics) and provided the conditions when each mechanism is activated during thermal diffusion. A specific microscopic explanation for thermal diffusion is still lacking. In a recent work,9 we used concepts of component mobility (related to infinite-dilution diffusion coefficients) and similarity between components (associated with similarity in response to force fields) to explain the variation of DT in 1-methylnaphthalene-n-alkane and n-decane-n-alkane binary mixtures with 50 wt % of each component. For such mixtures, DT increases with component mobility and decreases with similarity between components. In this work, we quantify the mobility and similarity effects and show that the same observations as those in ref 9 hold true for mixtures of different compositions. These * To whom correspondence should be addressed. E-mail: alana.leahy-dios@ yale.edu (A.L.-D.); [email protected].

concepts have been used by other authors to describe DT in polymer-solvent mixtures; the similarity concept has been represented by the Hildebrand solubility parameter,1 and the mobility concept has been represented by segmental diffusion of polymer beads14 in terms of the inverse of a “friction coefficient” of the entire polymer15 (or a local friction coefficient)16 and of the inverse of the local effective viscosity.17,18 In a binary mixture, the thermal diffusion coefficient DT of component 1 is related to its mass diffusive flux J1

J1 ) -F(DM∇ω1 + ω1(1 - ω1)DT∇T)

(1)

where F is the mixture mass density, is the molecular diffusion coefficient, and ω1 is the mass fraction of component 1. The Soret coefficient ST is defined as the ratio of thermal to molecular diffusion coefficients, which can be related to the temperature and composition difference at steady state in one dimension, from eq 1 DM

ST )

∆ω1 DT 1 ) – ω1(1 - ω1) ∆T DM

(2)

There have been some attempts to provide physical interpretation to ST using, for example, solubility parameters.3 We believe that ST merely determines the ratio of the imposed temperature gradient to the separation gradient at steady state (in one dimension), with no unique physical meaning; the indirect molecular interactions embodied in ST are based on the independent molecular mechanisms related to DT and DM. Therefore, it may be more fruitful to investigate the microscopic nature underlying DT and DM separately. Thermal diffusion, like viscosity, is a nonequilibrium quantity. As such, DT cannot be described using only equilibrium thermodynamics.19 However, recent work,20 shows that there is still some debate whether only equilibrium properties are sufficient to model DT. In this work, we investigate experimentally the relationship between DT and viscosity to demonstrate the need for nonequilibrium properties to accurately describe thermal diffusion. The main objectives of this work are (a) to study the composition dependency of DT for binary mixtures of n-decane-nalkane and 1-methylnaphthalene-n-alkane; (b) to investigate the relationship between DT and mixture viscosity; and (c) to determine the mobility and similarity effects for such mixtures

10.1021/jp711090q CCC: $40.75  2008 American Chemical Society Published on Web 04/26/2008

Diffusion Coefficients for Hydrocarbon Binary Mixtures

J. Phys. Chem. B, Vol. 112, No. 20, 2008 6443 TABLE 1: Composition ω (for MN in SET 1 and nC10 in SET 2), Density G, Compositional Coefficient γ, Thermal Expansion Coefficient r, and Viscosity µ for the Mixtures in SET 1 and SET 2 at 25 °C and 1 atm; Viscosity Shows Typical Accuracy Greater Than 99% mixture

Figure 1. Sketch of the thermogravitational column.

and their dependencies on mixture viscosity. This paper is organized as follows. In section II, we present the mixtures studied, the equipment used, and a brief description of the thermogravitational technique. We present results in section III, discuss the results in section IV, and conclude the work in section V. II. Experimental Methods Thermal Diffusion Coefficients. We measure DT using the thermogravitational column technique. A schematic of the column is given in Figure 1. In this technique, we measure the vertical composition gradient of a mixture inside the column, which is submitted to a linear horizontal temperature gradient. Details of the experimental setup and technique are given elsewhere.9 The temperature gradient across the column is 10 °C, with an average temperature of 25 °C. The dimensions of the column are Lz ) 46.7 ( 0.1 cm, Ly ) 4.7 ( 0.1 cm, and Lx ) 1.60 ( 0.02 mm. The following equation gives the thermal diffusion coefficient for component 1 in a binary mixture21,22

DT )

F0gRL4x ∆ω1 504µω1(1 - ω1) ∆z

(3) -F0-1(∂F/

where R is the thermal expansion coefficient R ) ∂T), g is gravity acceleration, µ is the dynamic viscosity, and ∆ω1/∆z is the composition gradient of component 1 at steady state. The vertical separation in the column is independent of the imposed horizontal temperature gradient. Calibration Curve. To obtain the vertical composition gradient of a binary mixture in the thermogravitational column within the necessary accuracy, we measure mixture density using a precise densitometer and relate it to mixture composition by a calibration curve. We assume that within a small composition range, the mixture density at a constant temperature is a linear function of the composition and can be expressed as a firstorder Taylor series expansion, with constant derivative ∂F/∂ω. For small composition changes

F ) a0 + a1ω1

(4)

where a0 and a1 are constant parameters. To construct the calibration curve, we determine a0 and a1 for each mixture; we first measure the density of seven mixtures of composition around that of the reference and then fit eq 4 to composition and density data. We obtain the compositional coefficient γ ) (1/F0)(∂F/∂ω) from a1 ) γF0, where (∂F/∂ω) is given for the reference component. Linearity is observed for every mixture.

ω

F (g/cm3)

γ

µ R (10-4 K-1) (10-3 Pa · s)

SET 1: MN-nCi MN-nC5 25 wt % 0.695111 0.4618 MN-nC6 0.723889 0.4107 MN-nC7 0.744131 0.3784 MN-nC8 0.760863 0.3543 MN-nC10 0.784017 0.3171 MN-nC12 0.800080 0.2927 MN-nC14 0.811099 0.2713 MN-nC16 0.820836 0.2624 0.5287 MN-nC5 75 wt % 0.891458 MN-nC6 0.903160 0.4644 MN-nC7 0.909855 0.4246 MN-nC8 0.919091 0.4044 MN-nC10 0.926536 0.3583 MN-nC12 0.935678 0.3345 MN-nC14 0.937928 0.3134 MN-nC16 0.944236 0.2987 MN-nC18 0.946824 0.2904 MN-nC20 0.949186 0.2773

-13.58 -12.13 -11.21 -10.55 -9.73 -9.23 -8.91 -8.67 -9.24 -8.91 -8.65 -8.47 -8.19 -8.03 -7.92 -7.83 -7.77 -7.72

0.335 0.417 0.545 0.660 1.014 1.483 2.096 2.849 0.964 1.109 1.291 1.463 1.814 2.108 2.421 2.750 3.105 3.468

SET 2: nC10-nCi nC10-nC5 25 wt % 0.646302 0.1575 nC10-nC6 0.672303 0.1041 nC10-nC7 0.690897 0.0666 nC10-nC12 0.740410 -0.0260 nC10-nC14 0.750736 -0.0443 nC10-nC16 0.758836 -0.0576 nC10-nC18 0.765013 -0.0678 nC10-nC5 75 wt % 0.698808 0.1554 nC10-nC6 0.707924 0.1033 nC10-nC7 0.714243 0.0661 nC10-nC8 0.719067 0.0371 nC10-nC12 0.730879 -0.0272 nC10-nC14 0.734219 -0.0443 nC10-nC16 0.736832 -0.0587 nC10-nC18 0.738807 -0.0705 nC10-nC20 0.740421 -0.0800

-14.58 -12.88 -11.94 -9.89 -9.55 -9.30 -9.13 -11.57 -11.15 -10.86 -10.68 -10.23 -10.11 -10.01 -9.94 -9.89

0.333 0.395 0.507 1.249 1.652 2.246 2.859 0.581 0.644 0.743 0.775 0.997 1.076 1.152 1.235 1.328

Mixtures, Materials, and Equipment. In this work, we measure DT for binary mixtures of 1-methylnaphthalene-nalkanes (SET 1) and binary mixtures of n-decane-n-alkanes (SET 2) at 25 °C and 1 atm. We measure DT at 25 and 75 wt % of MN and nC10 for SETs 1 and 2, respectively. We denote 1-methylnaphthalene as MN and n-alkanes as nCi, where i is the number of carbon atoms. The n-alkanes used for both sets range from nC5 to nC20. We choose MN and nC10 as the fixed components in each set, respectively, because they have similar molecular weights but very different shapes and physical properties such as density and viscosity; the differences in DT between the two sets provide insight into the different thermodynamic forces important to thermal diffusion. We did not conduct measurements for mixtures with 75 wt % of MN (SET 1) and with nC10 (SET 2) for nCi > nC16 because of the formation of a solid phase. Measured properties of each mixture (density F, compositional coefficient γ, thermal expansion coefficient R, and viscosity µ) are given in Table 1. In a previous work,9 we reported data for the same sets with a 1:1 mass ratio (that is, 50 wt %). We use reagents from Acros Organics with a purity of 99% or higher, except for MN (97%), and without further refining. The thermogravitational column was constructed in our laboratory. We measured density using the Anton PAAR

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Figure 2. Density variation with temperature for binary mixtures for SET 1 with 25 wt % MN at 1 atm: MN-nC5 (b), MN-nC6 (2), MN-nC7 (f), MN-nC8 (9), MN-nC10 (+), MN-nC12 (side triangle), MN-nC14 (O), MN-nC16 (4), MN-nC18 (0).

Figure 4. Viscosity (µ) variation with normalized molecular weight of normal alkanes (MWn) for binary mixtures of SETs 1 and 2 with 25 wt % (O), 50 wt % (f),9 and 75 wt % (∆) MN and nC10, respectively, at T ) 25 °C and P ) 1 atm. The solid symbols represent data for SET 1; the open symbols represent data for SET 2. The solid lines simply connect the data points. The horizontal dotted line shows the viscosity of pure MN23 and pure nC10.

Figure 3. Density variation with composition for binary mixtures of SET 1 with an average composition of 25 wt % MN: MN-nC5(b), MN-nC6 (2), MN-nC7 (f), MN-nC8 (9), MN-nC10 (+), MN-nC12 (side triangle), MN-nC14 (O), MN-nC16 (∆), MN-nC18 (0) at T ) 25 °C and P ) 1 atm.

densimeter DMA 500, with a temperature fluctuation of (0.005 °C and accuracy of (2 × 106 g/cm3. We weighed the samples on a digital scale with a precision of (0.0001 g. We used a Haake falling ball viscometer to measure the dynamic viscosity, with a typical accuracy better than 99%. The water bath temperature control has an accuracy of (0.1 °C. III. Results Figures 2 and 3 show the density variation with temperature and composition, respectively, for SET 1 with 25 wt % of MN. Thermal expansion and compositional coefficients (given in Table 1) can be calculated from the slopes in Figures 2 and 3, respectively. Similar linearity is obtained for all mixtures in both sets. Figure 4 shows the viscosity dependency on the molecular weight of n-alkanes, normalized by the molecular weight of MN (or nC10), for SETs 1 and 2. From here on, the normalized molecular weight of n-alkane will be referred to as MWn. The two plots include previously measured9 viscosities for SETs 1 and 2 with 50 w t% MN and nC10. In the top plot in Figure 4, the horizontal dotted line represents the viscosity of pure MN (µ ) 2.77 × 10-3 Pa · s).23 It is interesting to notice that the viscosity for the three different compositions of SET 1 intersect

close to the viscosity of pure MN. In the bottom plot in Figure 4, the intercept of the dotted lines shows the viscosity of pure nC10 (µ ) 0.84 × 10-3 Pa · s),24 where MWn ) 1. All three composition lines should cross at that point. The behavior observed in this plot is as expected for MWn < 1; the viscosity of mixtures with 25 wt % nC10 should be less than that for mixtures with 50 and 75 wt % nC10 since those mixtures (25 wt % nC10) contain more of the low viscosity alkanes (nC5-nC8). For MWn > 1, the trend is reversed; the viscosity of mixtures with 25 wt % nC10 should be greater than those with 50 and 75 wt %. The 25 wt % nC10 mixtures contain more components with higher viscosity (nC12-nC20). Table 2 lists composition gradient at steady state and DT for all mixtures in SETs 1 and 2, along with experimental error. The slopes shown in Table 2 are averages of 3 runs, all of which had a minimum coefficient of determination (R2) of 0.95; the error shown is the maximum deviation between slopes. Figure 5 shows the variation of DT with MWn for SETs 1 and 2. Results for 50 wt % in both figures are from ref 9. In SET 1, DT decreases monotonically with increasing MWn. Interestingly, DT data for the three different compositions of SET 1 converge to the same value at the same MWn as the viscosity data intersect (MWn ) 1.6), shown as a dotted line. For SET 1, DT decreases with MN concentration, irrespective of the n-alkane used. However, the change in DT with concentration decreases with MWn. In SET 2, the intercept of the dotted line with the x axis gives DT for pure nC10 (which is zero); all three compositions should cross the x axis at MWn ) 1, compatible with viscosity plots in Figure 4. Due to small errors in the measured DT, there is a small deviation from the same intercept for all three plots in SET 2. For all mixture compositions in SET 2, we observe a nonmonotonic behavior with a point of minimum around MWn of 1.6 and 1.8, depending on the mixture

Diffusion Coefficients for Hydrocarbon Binary Mixtures TABLE 2: Steady-State Composition Gradient (∆ω/∆z) and Thermal Diffusion Coefficient at 25.0 °C and 1 atm; Reference Components Are MN (SET 1) and nC10 (SET 2) mixture MN-nC5 MN-nC6 MN-nC7 MN-nC8 MN-nC10 MN-nC12 MN-nC14 MN-nC16 MN-nC5 MN-nC6 MN-nC7 MN-nC8 MN-nC10 MN-nC12 MN-nC14 MN-nC16 MN-nC18 MN-nC20

ω 25 wt %

75 wt %

nC10-nC5 25 wt % nC10-nC6 nC10-nC7 nC10-nC12 nC10-nC14 nC10-nC16 nC10-nC18 nC10-nC5 75 wt % nC10-nC6 nC10-nC7 nC10-nC8 nC10-nC12 nC10-nC14 nC10-nC16 nC10-nC18 nC10-nC20

J. Phys. Chem. B, Vol. 112, No. 20, 2008 6445 the magnitude of DT increases with nC10 concentration. This behavior is in accordance with the viscosity dependency on composition, as shown in Figure 4.

∆ω/∆z (10-2 m-1) DT (10-12 m2 s-1 K-1)

IV. Discussion

SET 1: MN-nCi -1.79 ( 0.03 -1.81 ( 0.05 -1.87 ( 0.03 -1.73 ( 0.03 -1.92 ( 0.04 -2.23 ( 0.07 -2.32 ( 0.08 -2.41 ( 0.05 -3.17 ( 0.03 -3.18 ( 0.02 -3.00 ( 0.06 -2.69 ( 0.03 -2.69 ( 0.07 -2.42 ( 0.02 -2.48 ( 0.04 -2.28 ( 0.03 -2.28 ( 0.03 -2.41 ( 0.04

34.38 ( 0.57 25.89 ( 0.67 19.41 ( 0.30 14.33 ( 0.21 9.82 ( 0.19 7.54 ( 0.24 5.45 ( 0.18 4.10 ( 0.08 18.43 ( 0.19 15.69 ( 0.08 12.44 ( 0.26 9.72 ( 0.11 7.67 ( 0.21 5.88 ( 0.04 5.18 ( 0.09 4.18 ( 0.06 3.68 ( 0.04 3.47 ( 0.05

SET 2: nC10-nCi -0.70 ( 0.02 -0.65 ( 0.01 -0.62 ( 0.04 0.25 ( 0.02 0.62 ( 0.02 1.22 ( 0.01 0.92 ( 0.01 -0.77 ( 0.07 -0.66 ( 0.01 -0.52 ( 0.03 -0.36 ( 0.04 0.37 ( 0.04 0.58 ( 0.03 0.71 ( 0.02 0.95 ( 0.01 0.87 ( 0.01

13.40 ( 0.31 9.72 ( 0.22 6.92 ( 0.39 -1.00 ( 0.07 -1.84 ( 0.05 -2.62 ( 0.03 -1.53 ( 0.01 7.28 ( 0.64 5.53 ( 0.08 3.67 ( 0.23 2.42 ( 0.29 -1.89 ( 0.18 -2.73 ( 0.15 -3.20 ( 0.09 -3.85 ( 0.02 -3.28 ( 0.04

Thermal and molecular diffusion coefficients in a binary mixture are related25 to the thermal diffusion factor RT

DT DM

RT ) T

(5)

In a binary mixture, RT is related to the net heat of transport12,19 of components 1 and 2, Q*1 and Q*2

RT )

(Q*1 - Q*2) (∂µ1 ⁄ ∂ ln x1)T,P

(6)

where µ1 is the chemical potential of component 1 and x1 is the mole fraction of component 1. The derivative (∂µ1/∂ ln x1)T,P can be calculated from an equation of state (EOS). At the critical point, the derivative(∂µ1/∂ ln x1)T,P goes to zero, and RT diverges, as has been observed experimentally.26,27 Combining eqs 5 and 6, we obtain

DT )

* * DM (Q1 - Q2) T (∂µ1 ⁄ ∂ ln x1)T,P

(7)

We can relate DM to the binary Maxwell-Stefan diffusion coefficients, ÐMS 28

DM ) ÐMS(∂ ln f1 ⁄ ∂ ln x1)T,P

(8)

where f1 is the fugacity of component 1. The derivative (∂ ln f1/∂ ln x1)T,P can be calculated from an EOS and gives a measure of mixture nonideality. Substituting eq 8 into eq 7

DT )

(Q*1 - Q*2)ÐMS (∂ ln f1 ⁄ ∂ ln x1)T,P T (∂ ln µ1 ⁄ ∂ ln x1)T,P

(9)

We can simplify eq 9, by noting that29 observed9

composition; this behavior has been for the first time with mixtures with 50 wt % nC10 as is confirmed here. For SET 2, for a mixture with nCi < nC10 (MWn < 1), the magnitude of DT decreases with an increase of nC10 concentration. For a mixture with nCi > nC10 (MWn > 1),

µ1(T, P, x) ) µ01(T, P) + RT

f1(T, P, x) f 01(T, P)

(10)

Taking the derivative of eq 10 with respect to ln x1 at constant temperature and pressure yields

∂ ln f1(T, P, x) ∂µ1(T, P, x) T,P ) RT T,P ∂ ln x1 ∂ ln x1

(11)

Substituting eq 11 into eq 9, we obtain a simple expression for DT in terms of Maxwell-Stefan diffusion coefficients and heats of transport

DT )

Figure 5. Variation of DT with the normalized molecular weight of normal alkanes (MWn) for binary mixtures of SETs 1 and SET 2 with 25 wt % (O), 50 wt % (0),9 and 75 wt % (∆) MN and nC10, respectively, at 25 °C and 1 atm. We use closed symbols for SET 1 and open symbols for SET 2.

(Q*1 - Q*2) 2

RT

ÐMS

(12)

In ref 9, we identified the differences in DT between the SETs 1 and 2 based on two opposing effects in a binary mixture, mobility and similarity. An increase of molecular mobility increases DT, and an increase of similarity between components decreases DT. From eq 12, we can quantify the mobility and similarity effects on DT. The mobility effect is proportional to ÐMS; the greater ÐMS, the greater the mobility and the greater the DT. A derivation of the ÐMS is based on a simple momentum balance;28,30 the physical significance of ÐMS is of an inverse drag coefficient. We emphasize that neither Fickian (DM) nor

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Maxwell-Stefan ÐMS diffusion coefficients are directly related to molecular interaction between different components. In a ternary ideal gas mixture, where molecular interactions are considered to be absent, the existence of cross-effect is in support of our understanding.31 A well-known relation between molecular diffusion coefficients and viscosity for ideal liquids is the Stokes-Einstein equation.32 We have previously shown9 that the DM of binary mixtures of n-alkanes and aromatics represents the component mobilities and does not depend on the similarity (molecular interactions) between components. We may also relate ÐMS to the infinite dilution diffusion coefficients of the binary mixture, D∞12 and D∞21, using, for example, the Vignes relation33

ÐMS ) (D∞12)x2(D∞21)x1

(13)

Other mixing rules28,34,35 can be used to relate ÐMS to infinite dilution and self-diffusion diffusion coefficients, without changing our conclusions. In eq 12, the similarity effect is inversely proportional to the difference in the net heat of transport (Q*1 - Q*2). Here, it is more convenient to define the inverse of similarity as the disparity between components in a mixture since, for a pure component, Q*1 ) Q*2; thus, the similarity is infinite, but the disparity is finite (zero). The greater the disparity between the components in a binary mixture, the greater the difference between Q*1and Q*2 and the greater the DT. Some authors have proposed to use solubility parameters to quantify disparity between molecules in a mixture;1,36 however, the values of the solubility parameters are dependent upon the method of calculation.1 It has also been verified that various mixtures, including relatively simple mixtures such as benzene-carbon tetrachloride and benzene-cyclohexane,36 do not obey the solubility parameter rule. Using (Q*1 - Q*2) to quantify the disparity (or similarity) between components in a mixture may be more reliable than using solubility parameters. In a binary mixture, Q*1 does not only depend on thermodynamic properties of pure component 1 but also on properties of component 2 since Q*i is primarily related to the partial molar volumes and residual partial molar internal energies of both components in the mixture, which are calculated from an appropriate EOS (for example, the Peng-Robinson EOS for nonassociating19 fluids and other approaches for associating fluids).37 If a component in a binary mixture is changed to a new component, both Q*1 and Q*2 will change. The fact that Q*1 and Q*2 are not independent (and their dependency is nonadditive) might be an indication that simple additive interactions are not sufficient to understand thermal diffusion. In fact, Wiegand3 has examined literature data and suggests that there is no simple relation between thermal diffusion and other physical properties. We can rewrite eq 12 as

DT )

b·ζ RT2

(14)

where b, a measure of mobility, is defined by the measurable quantity ÐMS and ζ, a measure of disparity, is defined by (Q*1 - Q*2). Rauch et al.16,18 have recently suggested that DT of polymer-solvent mixtures could be described by the product ∆T × (1/ηs), where ∆T is a constant that depends only on the polymer and not on polymer-solvent interactions, and ηs is the solvent viscosity. Their expression for DT in polymer-solvent systems could have a similar interpretation as our eq 14. However, the concept of disparity between components is not taken into account in their work since the nonmeasurable

Figure 6. Variation of DT, mobility (b), and disparity (ζ) with mixture viscosity for SETs 1 (closed symbols) and 2 (open symbols) with 25 wt % (O), 50 wt % (0),9 and 75 wt% (∆) of MN and nC10, respectively, at 25 °C and 1 atm.

constant ∆T depends only on the polymer and not on the interactions between the polymer and solvent. In Figure 6, we show the relationship between DT, mobility (b), and disparity (ζ) with mixture viscosity; b is calculated using our recent work,38 and ζ is calculated from DT and b using eq 14. The mobilities of the two sets have very similar functional dependencies on viscosity (center plot in Figure 6); in fact, the two sets have very similar mobility for the same viscosity. This result confirms that mobility should be independent of molecular interactions between the components in a mixture. Results for DT versus viscosity show that both sets have a similar functional dependency on viscosity (top plot in Figure 6). However, the disparity between molecules for the two sets shows a very different functional dependency on viscosity (bottom plot in Figure 6). As expected, disparity cannot be accurately described by mixture viscosity, and other thermodynamic properties need to be taken into account. Also, disparity values are very different for the 2 sets; SET 1 has a much larger ζ than SET 2. For SET 1, there is a continuous decrease in ζ (molecules with increasing MWn are becoming gradually similar to MN molecules). For SET 2, however, ζ shows a minimum at the viscosity of pure nC10 (µ ) 0.84 × 10-3 Pa · s), as expected, then increases and reaches a maximum at MWn ) 1.6 or 1.8 (depending on mixture composition), and decreases again. A similar trend is observed for all compositions studied. We may point out that our measurements for higher molecular weight are limited to the formation of a solid phase beyond the data shown in Figures 4 and 5.

Diffusion Coefficients for Hydrocarbon Binary Mixtures Further analysis of eq 12 allows us to make an interesting theoretical observation concerning the behavior of DT at the critical point. Haase39 suggested that at the critical point, DT should go to infinity to explain RT approaching infinity. Other earlier authors have used different approaches to conclude that DT could diverge close to the critical point, even if only weakly.40,41 However, it was later shown by various authors that this is not the case.25,42,43 It has also been verified experimentally 44,45 that DT remains finite as RT diverges at the mixture critical point. It is clear from eq 12 that DT has a finite value at the critical point and from eq 8 that DM approaches zero at the critical point because the thermodynamic quantity (∂ ln f1/∂ ln x1)T,P becomes zero. Therefore, it follows from eq 5 that RT approaches infinity because DM is approaching zero toward the critical point, not because DT is diverging, as suggested by Haase. In our analysis, we rely on the argument that the quantities (Q*1 - Q*2) and ÐMS have nonzero finite values at the critical point. There is no empirical or theoretical evidence that the net heat of transport should diverge at the critical point; Q*i is mainly governed by the residual partial molar internal energy and by the partial molar volume of the components, as mentioned before. These thermodynamics quantities do not exhibit unexpected behavior close to or at the critical point. The Maxwell-Stefan diffusion coefficient ÐMS is not related to the mixture critical point and retains an approximately constant value at the critical point, neither diverging nor approaching zero.46 Similarly, D∞12 and D∞21 are not related to the mixture critical point and do not show any discontinuity at the critical point of the binary mixture.47–49 V. Conclusions In this work, we provide new measurements for DT of two very different sets of binary mixtures (1-methylnaphthalene-nalkanes and n-decane-n-alkanes) at two different concentrations. We confirm the previously observed nonmonotonic behavior of DT with the n-alkane molecular weight for binary mixtures of n-decane-n-alkanes. For the mixtures studied, thermal diffusion has a very similar behavior with n-alkane molecular weight as mixture viscosity, which is expected since thermal diffusion and viscosity share the same nonequilibrium nature. The results shown in Figures 4 and 5 reveal that modeling of thermal diffusion by equilibrium thermodynamics alone may not be an appropriate approach. On the basis of a simple expression derived for the thermal diffusion coefficients, we quantify the previously proposed mobility and similarity (given in terms of disparity) effects; the mobility effect is related to the Maxwell-Stefan diffusion coefficients and therefore viscosity; the disparity between components is related to the difference in the net heat of transport of the components in the binary mixture. The same simple expression allows us to verify theoretically that DT of binary mixtures is finite at the critical point. Acknowledgment. This work was supported by the member companies of the Reservoir Engineering Research Institute (RERI) in Palo Alto, CA, and by the Petroleum Research Fund Grant PRF 45927-AC9 of the American Chemical Society to Yale University.

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