Mass transport and structure of liquid n-alkane mixtures in the vicinity of α-quartz substrates

Abstract

The molecular-scale structure and in-plane self-diffusion coefficients of several binary n-alkane mixtures in the vicinity of the (001) crystal surface of an α-quartz substrate terminated with H-groups have been examined in this study. The molar fraction of hexane has almost negligible influence on molecular structure of hexane–hexadecane liquid mixture near the α-quartz substrate, mainly because of strong molecular interaction between the α-quartz substrate and the alkane molecules. The in-plane self-diffusion coefficients of alkane molecules in their mixture in the vicinity of α-quartz have been studied. The self-diffusion coefficient becomes considerably lower in the solid–liquid interface region than in the bulk liquid region. The self-diffusion coefficients of hexane and hexadecane in the mixture are higher in the solid–liquid interface region as well as in the bulk liquid region with an increase in the molar fraction of hexane, primarily due to the increase in total free volume with an increase in molar fraction of hexane. The self-diffusion coefficient of alkane species in the mixture exhibits anisotropic characteristics in the solid–liquid interface region, which is mostly caused by the topology of the α-quartz substrate surface.

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1. Introduction

A molecular level knowledge of the dynamic and structural properties of liquid mixtures in the vicinity of solid surfaces is important for a wide range of technological applications, including lubrication, adhesion, catalysis, and chromatography. It is well known that liquids form molecular level layers near the solid surfaces^1–9 and heat and mass transport properties are significantly different in the layered region than in the bulk liquid region.^10–12 For example, the viscosity of liquid is higher¹³ and the molecular mobility is reduced next to the solid surface.³ The structure and transport properties in the interface region are influenced by several parameters such as thermodynamic conditions, physical and chemical nature of the substrate.^14,15 In addition to the above parameters, in liquid mixtures, the asymmetry in chain length of molecules might affect the interfacial properties at the solid–liquid interfaces. Study on this kind of liquid mixtures in contact with the solid substrate has relevance to coating technology and oil industry.^16,17

Experimental investigations on alkane mixtures in contact with graphite have observed preferential adsorption of longer alkane molecules onto the substrate.^18,19 Castro et al.^20–23 have examined the molecular adsorption structure of several binary alkane mixtures in the vicinity of graphite using differential scanning calorimetry and incoherent elastic neutron scattering. They noticed a crystallized monolayer for both pure liquid and liquid mixture next to the solid substrate. The structure of interface layer formed between C₃₆H₇₄–C₁₇H₃₆ mixture and graphite has been examined using scanning tunneling microscopy by Cousty and Van.²⁴ They detected a partially demixed solution monolayer with C₁₇H₃₆ and C₃₆H₇₄ molecules.

Molecular dynamics simulations have been used to investigate structure of pure alkane liquids and their binary mixtures in the vicinity of solid substrates.^25,26 Vasilyuk and Bell²⁵ have studied equimolar mixture of hexane and perfluorohexane liquid film on silica-like substrate using MD simulations and reported that perfluorohexane molecules are preferentially adsorbed in the solid–liquid interface region, they believed that this behavior was due to the shape of the perfluorohexane molecule. The molecular-scale structure of alkane mixtures involving comparable length of alkanes (e.g., hexane–octane mixture) in the vicinity of a structureless Au substrate have been examined by Wenzel et al.²⁶ They observed that longer molecules were preferentially adsorbed at the solid–liquid interfaces even though the chains differ only by one methylene unit and the longer molecule was minor component, which was due to the favorable interaction between longer molecules and substrate. A few researchers have measured the self-diffusion coefficient of alkane mixtures in the bulk liquid phase from the MD simulations.^27,28 Although knowledge of diffusion characteristics of mixtures in the vicinity of solid surfaces has great significance in many engineering processes, limited efforts have been made to understand the mass transport phenomenon. Xia and Landman³ have determined the self-diffusion coefficient variation in hexane–hexadecane liquid film on Au(001) surface (computational system has liquid–vapor and solid–liquid interfaces). They observed that the self-diffusion coefficient of hexane and hexadecane in the mixture was substantially quenched next to the substrate and increases monotonically towards the liquid–vapor interface. It is to be noted that self-diffusion coefficient is not constant in the bulk-like region away from both the interfaces in their study. Smith et al.¹⁰ measured the self-diffusion coefficient of butane–octane mixture in the vicinity of wax-like substrate and examined the effect of temperature on diffusion coefficient. The results reported in their study were suffering from small sampling time. More work is desired to accurately predict and understand the mass transport characteristics of binary alkane mixtures in the vicinity of solid substrates. Furthermore, to the best of our knowledge, there exists no report concerning the influence of molar fraction and lattice scale roughness of the solid surface on the self-diffusion coefficient of alkane mixtures in the solid–liquid interface region.

This paper reports molecular-scale structure and mass transport properties of three liquid alkane mixtures in the vicinity of (001) α-quartz crystal surface, one of the frequently used substrates in micro- and nanotechnology applications. Structural quantities such as density, orientation order parameter, radius of gyration, and gauche fraction have been examined in the solid–liquid interface region. The mobility of molecules in the vicinity of α-quartz substrate was studied by measuring the self-diffusion coefficient. The effect of a molar fraction on the above mentioned quantities was examined in the hexane–hexadecane mixture. The properties of three types of liquid mixtures have been compared to examine the influence of chain length of alkanes.

2. Simulation details

The molecular-scale structure and mass transport properties of binary n-alkane liquid mixtures in contact with the α-quartz substrate were investigated by constructing a computational system as illustrated in Fig. 1. Three alkane mixtures, viz., hexane (C₆H₁₄)–hexadecane (C₁₆H₃₄), hexane–dodecane (C₁₂H₂₆), and hexane–octane (C₈H₁₈) were examined in this work. The alkane molecules were modeled using a united atom model, in which each of CH₃ and CH₂ groups is treated as a pseudoatom positioned at the location of carbon atom. Based on our previous studies^11,12,29 and other reports,²⁷ it is evident that the NERD potential³⁰ can be utilized to reproduce physical and transport properties of alkanes and their mixtures in the liquid state. Therefore, in this study also the molecular interaction between the pseudoatoms was represented with the NERD potential. This force field consists of bond stretching, bond bending, torsional motion and Lennard-Jones (LJ) interaction between pseudoatoms. An atomistic description of α-quartz wall was used and the molecular interaction between the atoms was given by Lopes et al.³¹ An artificial wall was located on the right side of the computational system as shown in Fig. 1. The artificial wall was modeled by explicit atoms and the atoms were interacted via the Morse potential with force field parameters of gold³² and have the mass of gold. The molecular interaction between α-quartz and alkane molecules, and α-quartz and artificial wall was taken into account by Lennard-Jones (LJ) potential. The artificial wall and alkane molecules are also interacted through the LJ potential with energy parameters ε_CH3–AW and ε_CH2–AW = 0.9ε_AW and size parameters σ_CH3–AW and σ_CH2–AW = 1.16σ_AW, where the energy and size parameters for artificial wall are 2.7109 × 10⁻²¹ J and 2.9337 Å, respectively,³³ so that there is no adsorption of alkane molecules at the artificial wall. The LJ interactions beyond a cut-off radius of 16 Å were neglected. The interaction parameters between dissimilar interaction sites were attained using the Lorentz–Berthelot mixing rules. The SPME method³⁴ was utilized to calculate electrostatic interaction between atoms in the α-quartz substrate.

Fig. 1 Schematic of the α-quartz (001)/hexane–hexadecane mixture computational system. The hexane and hexadecane molecules are given in red and cyan colors, respectively. The α-quartz and artificial walls are located at the left and right side of the computational system, respectively.

The time integration of atomic equations of motion was employed by r-RESPA method.³⁵ The simulation system had a thin alkane liquid mixture film on an α-quartz substrate in a coexistence state with its own vapor as shown in Fig. 1. In order to apply periodic boundary conditions in all three directions, an artificial wall was constructed in the computational system. The α-quartz substrate and artificial wall were located on the left and right side in the computational system. The α-quartz substrate was created with the lattice parameters give in ref. 31. The artificial wall was constructed with slightly altered unit cell parameters of gold so that the lengths of the artificial wall in x and y directions were equal to the lengths of the α-quartz substrate in the x and y directions. The α-quartz surface was terminated with H-groups on one side, which was positioned in such a manner that it was perpendicular to the z-axis and was in contact with the liquid mixture. The dimensions of the computational systems used in the simulation of hexane–hexadecane were 57.59 × 49.88 × 225 ų and for hexane–dodecane and hexane–octane were 43.19 × 41.56 × 200 ų. The thickness of the liquid film along the z-direction is considered in such a manner that a bulk-like phase is formed at the center of the film away from both solid–liquid and liquid–vapor interfaces. The distance between the liquid–vapor interface and the artificial wall was selected in such a way that the saturated vapor density of the mixture obtained in the present simulation was almost equal to the saturated vapor density of the mixture obtained from the liquid–vapor simulations, which can be performed as described in ref. 29. The distances between the artificial wall and the liquid–vapor interface used in this work were in the range of 30 Å to 40 Å.

The initial configuration was constructed by randomly placing two chosen species of alkane molecules to form a liquid film in between the α-quartz substrate and the artificial wall at such a composition after the equilibration of the system desired molar fraction was attained in the bulk-like region far away from interfaces. The liquid film was positioned close to the α-quartz substrate. After positioning the liquid film, the system temperature was incrementally raised to the desired temperature using the velocity scaling method. Afterwards, the system was equilibrated in the NVT ensemble, where the system temperature was kept constant using the Nosé–Hoover chain thermostat.³⁶ The equilibration time periods for the systems were in the span of 5 ns to 30 ns, which strongly depend on the molar fraction of the mixture and chain length difference of the alkanes in the mixture. After the equilibration, the data were accumulated for 15 ns for the analysis.

3. Results and discussion

3.1. Structure in the solid–liquid interface region

In order to confirm the accuracy of alkane mixture properties obtained from the present simulation system, pure liquid hexane was simulated in the computational system as described previously. It was verified that the obtained densities of hexane in the bulk liquid and bulk vapor regions were in excellent agreement with the saturated vapor and liquid densities obtained from the liquid–vapor simulations, which were performed as explained in ref. 29. Fig. 2(a) and (b) show variation of total, hexane, hexadecane mass densities along the z direction in hexane–hexadecane mixture at T = 400 K for three molar fractions of hexane in the bulk-like liquid region, x_C6 = 0.46, 0.64, and 0.85. The density distributions were plotted by slicing the simulation system into rectangular bins with a thickness of 0.1 Å along z-axis. The density profiles were plotted by considering the average position of surface layer silicon atoms as the z coordinate origin. The density profiles exhibit oscillations near the α-quartz substrate, which is a characteristic feature of liquid density in the solid–liquid interface region. It suggests that liquid has layered structure in the interface region, which occurs due to the packing of molecules, mainly because of the surface forces.³⁷ These characteristics are consistent with the results reported for alkane mixtures in contact with the solid substrate from simulations^25,26 and experiments.^20–23 The oscillations in the density profile disappear within 20 Å from the α-quartz substrate and density reaches a uniform value representing the bulk-like liquid region. After the bulk-like liquid region, density decreases through the liquid–vapor region and becomes a constant value in the bulk-like vapor region. Thus the distance between the liquid–vapor and solid–liquid interfaces is far enough that the interaction between the interfaces was insignificant. Hexane, hexadecane, and total density profiles show similar characteristics in the solid–liquid interface region and no obvious changes are observed in their characteristics with an increase in the molar fraction of hexane, which is mainly due to the strong molecular interaction between α-quartz substrate and liquid molecules. The total density of hexane–hexadecane mixture noticeably decreases with an increase in x_C6 in the bulk region as well as in the solid–liquid interface region. Wenzel et al.²⁶ reported that composition has minor effect on total density in the bulk liquid region for octane–nonane mixture. The total density dependence on the molar fraction observed in our results can be attributed to the larger difference in chain length between hexane and hexadecane. This tendency is more obviously noticeable for hexane–tetracosane mixture in the bulk liquid region.²⁹ The number density profiles of CH₃ for each molecule in hexane–hexadecane mixture are shown in Fig. 2(c). As both hexane and hexadecane molecules have two CH₃ groups, the number density of CH₃ allows for a direct comparison of two types of species in the solid–liquid interface region. For x_C6 = 0.46, hexadecane molecules are present in excess amounts in all layers of the interface region. On the other hand, the interface region is dominated with hexane molecules when x_C6 = 0.85. This suggests that the preferential adsorption of longer molecules in the interface region decreases as the molar fraction of smaller molecule goes higher in the mixture. Although the number of hexadecane molecules is considerably lower in the first adsorption layer at higher values of x_C6, the peak mass density value of hexadecane is higher than hexane value in the first adsorption layer for all three values of x_C6, because of larger molecular weight of hexadecane. At T = 400 K and x_C6 = 0.64, density profiles of hexane–hexadecane, hexane–dodecane, and hexane–octane mixtures are shown in Fig. 3. Hexane density profiles demonstrate similar type of characteristics in all three mixtures. Although number density profiles of CH₃ in hexane–hexadecane, hexane–dodecane and hexane–octane mixtures are not presented here, it is observed that the adsorption of hexane molecules is higher over the other type of molecule in the interface region when the chain length difference between the two molecule types in the mixture is small.

Fig. 2 Hexane–hexadecane mixture for x_C6 = 0.46, 0.64, 0.85 at T = 400 K (a) total mass density distributions; the zoomed area of the interface region is shown in the inset (b) hexane and hexadecane mass density distributions (c) number density distributions of CH₃ for hexane and hexadecane.

Fig. 3 Density distributions for hexane–hexadecane, hexane–dodecane, and hexane–octane mixtures at T = 400 K and x_C6 = 0.64. Only the left side portion of the computational system consisting of the α-quartz/alkane interface region is shown in the figure.

In order to assess how the interface has influenced the average alignment of molecules in the solid–liquid interface region, the orientation order parameter was examined in this study, which is given as follows^38,39

where δ is the angle between the interface normal, z-axis and a line connecting two united atoms separated by two bonds in a molecule. The orientation order parameter values of −0.5 and 1.0 imply that molecules are entirely parallel and perpendicular to the interface, respectively. Preferential alignment of molecules in the perpendicular and parallel directions to the interface is indicated by positive and negative values of the orientation order parameter, respectively. Random arrangement of molecules is represented by a zero value. All structural quantities studied below were presented by taking the average position of surface layer silicon atoms as the z-axis coordinate origin. The orientation order parameter as a function of distance from the surface silicon atoms for hexane and hexadecane in their mixture for three values of x_C6 at T = 400 K are illustrated in Fig. 4. Since our main interest is in the solid–liquid interface region in this paper, only the left side portion of the computational system consisting of the α-quartz/alkane interface region is shown for all the quantities studied below. The molecular-scale structure and self-diffusion coefficient of several alkane mixtures in the liquid–vapor interface region have been investigated in the authors' previous report.²⁹ The orientation order parameter for hexane and hexadecane changes between positive and negative values in an oscillating manner similar to the oscillations in the density profile in the solid–liquid interface region. This implies that molecules are arranged in the form of layers in the interface region. Similar features have been observed from MD simulations and experiments for alkane mixtures in the vicinity of solid substrate.^3,20–22,26 In the first adsorption layer, the magnitude of the orientation order parameter for hexane and hexadecane is close to −0.5, which indicates that both molecules lie mostly parallel to the interface on the solid substrate. This kind of orientation for liquid molecules in the first adsorption layer maximizes the molecular interaction between alkane molecules and the quartz substrate.²⁶ The orientation order parameter exhibits positive values in between the negative minimums, which suggest that some segments of molecules in a layer are bent in the perpendicular direction and align preferentially parallel to the interface in the subsequent layer, which is known an interlayer interdigitation.⁴⁰ The present distributions indicate high degree of interdigitation in between the layers in the interface region for all three values of x_C6. It is to be noted that the hexane and hexadecane in their binary mixture have similar orientation tendencies to those that of their pure alkane liquids in the solid–liquid interface region.¹¹ The molecular arrangement of hexane and hexadecane in their mixture in the first adsorption layer is not changed with an increase in a molar fraction of hexane. The molecular ordering for both hexane and hexadecane marginally decreases with an increase in x_C6 in the remaining layers. It is observed that the ordering of hexane molecules is not noticeably changed in the interface region with the length of the other alkane molecule present in the mixture by studying hexane–hexadecane, hexane–dodecane and hexane–octane mixtures at T = 400 K and x_C6 = 0.64 in Fig. 5.

Fig. 4 Orientation order parameter profiles for (a) hexadecane and (b) hexane molecules in hexane–hexadecane mixture for x_C6 = 0.46, 0.64, 0.85 at T = 400 K. Density distributions are depicted in dashed lines.

Fig. 5 Orientation order parameter profiles for hexane–hexadecane, hexane–dodecane, and hexane–octane mixtures at T = 400 K and x_C6 = 0.64. Density distributions are depicted in dashed lines.

To understand the molecular shape and how it is changed in the alkane liquid mixtures adjacent to the solid substrate, radius of gyration profiles along the z-axis were studied, whose calculation procedure was described elsewhere.³⁹ Fig. 6 shows distributions of total mean-squared radius of gyration for hexane and hexadecane in the mixture for three values of x_C6 at T = 400 K. It is observed that total radius of gyration increases noticeably in the first adsorption layer and is constant in the bulk liquid region for both hexane and hexadecane for all three values of x_C6. This suggests that both molecules are elongated in the first adsorption layer and the molecular shape is not changed in the bulk liquid region. The molecular shape of hexane and hexadecane in the bulk liquid region as well as in the interface region does not change with an increase in x_C6. It is observed that hexane molecule shape is not altered in the interface region as well as in the bulk liquid region with the chain length of other molecule present in the mixture, although the profiles of total radius of gyration for hexane in hexane–hexadecane, hexane–dodecane and hexane–octane mixtures at T = 400 and x_C6 = 0.64 are not presented here. The radius of gyration was resolved into components in x, y and z directions. The variation of components of radius of gyration in the x and z directions along the z-axis are given in Fig. 7. It is noticed that for both hexane and hexadecane molecules, component of radius of gyration in the z direction decreases whereas the component of radius of gyration in the x direction increases noticeably in the first adsorption layer. From the orientation order parameter profiles it is known that both hexane and hexadecane molecules are almost parallel to the interface in the first adsorption layer. From these two observations it can be inferred that both hexane and hexadecane molecules are flattened in the x and y directions and elongated in the z direction, in other words molecules have rod-like structure, in the first adsorption layer.

Fig. 6 Total radius of gyration profiles for (a) hexadecane and (b) hexane molecules in hexane–hexadecane mixture for x_C6 = 0.46, 0.64, 0.85 at T = 400 K. Density distributions are depicted in dashed lines.

Fig. 7 Components of radius of gyration profiles for (a) hexadecane and (b) hexane molecules in hexane–hexadecane mixture for x_C6 = 0.46, 0.64, 0.85 at T = 400 K. Density distributions are depicted in dashed lines.

Gauche fraction was obtained to examine the molecular configuration, which was calculated by taking the ratio of total number of gauche angles in a slab region to total dihedral angles in that slab over the entire production run. The details of the calculation method were described elsewhere.²⁹ Fig. 8 illustrates distributions of gauche fraction for hexane and hexadecane in their mixture as a function of the distance from the surface silicon atoms of the substrate for three values of x_C6. Gauche fraction changes in a fluctuating manner in the interface region and is constant in the bulk liquid region. The gauche fraction value next to the substrate is almost equal to 1, which suggests that portions of the molecules tend to contact with the substrate in all-gauche state. From both distributions of gauche fraction and distributions of radius of gyration, it can be interpreted that tendency of hexane and hexadecane molecules in the mixture to contact with the α-quartz substrate have helix shape in the first adsorption layer. There is no visible effect on the distributions of the gauche fraction for hexane and hexadecane in the bulk liquid region and in the interface region with an increase in a molar fraction of the hexane. By comparing the profiles of hexane gauche fraction in hexane–hexadecane, hexane–dodecane and hexane–octane mixtures at T = 400 K and x_C6 = 0.64 (see ESI Fig. 1†), it is observed that the gauche fraction of hexane does not change in the interface region as well as in the bulk liquid region with the length of the other component present in the binary mixture.

Fig. 8 Gauche fraction as a function of distance along z-axis for (a) hexadecane and (b) hexane molecules in hexane–hexadecane mixture for x_C6 = 0.46, 0.64, 0.85 at T = 400 K. Density distributions are depicted in dashed lines.

3.2. In-plane self-diffusion coefficient in the solid–liquid interface region

The self-diffusion coefficient concerning the migration of molecules in the parallel direction to the interface in the solid–liquid interface region was obtained by utilizing the modified form of the Einstein relation as follows^41–43where τ is the entire time interval over which the molecules are continuously present in a particular slab (CV_k), S(τ) is survival probability of molecules in a slab and Δx(τ)² is mean square displacement (MSD) of center of mass of molecules in a slab. In our previous studies, this modified Einstein relation has been used to investigate the variation of self-diffusion coefficient of pure liquid alkanes in the vicinity of α-quartz substrate as well as in the liquid–vapor interface region.^11,39 To obtain the variation of in-plane self-diffusion coefficient along the z-axis in the layered interface region the computational system was divided into slabs and the value of self-diffusion coefficient in each slab was measured from the slope value of MSD versus time curve in the linear region. The thickness of slabs in the bulk liquid region was set to be 8 Å and in the interface region the slab thickness was adjusted so that the consecutive minimums in the density profile were located in a slab.

The distributions of the in-plane self-diffusion coefficients in x and y directions for hexane and hexadecane in hexane–hexadecane mixture at T = 400 K for three molar fractions of hexane are illustrated in Fig. 9. Density distributions are also given in the figure. The zero point on the z-axis is set at the average position of surface silicon atoms. In order to obtain uncertainties shown in the figure, the sampling run of 15 ns was divided into 5 subgroups. The self-diffusion coefficient in each slab from each subgroup was determined and its variance, σ, in each slab was calculated using the values obtained from five subgroups. ±σ is shown as uncertainty in the figure. The in-plane self-diffusion coefficient becomes lower in the vicinity of the α-quartz surface, and about 25 Å away from the wall it is constant, which is considered to be the bulk-like liquid region. As expected, the self-diffusion coefficients in the x and y directions are equal in the bulk liquid region within the statistical uncertainty. On the other hand, the self-diffusion coefficients of hexane and hexadecane in the x direction are significantly higher than the self-diffusion coefficient values in the y direction in the layered region for all three molar fractions of hexane. It can be clearly observed that the lattice scale roughness is relatively small in the x direction than that in the y direction on the H-terminated (001) crystal surface of α-quartz substrate (see ESI Fig. 2†). We believe that the anisotropic characteristics of self-diffusion coefficient in the interface region might be due to the difference in the lattice scale roughness in the x and y directions of the α-quartz substrate. The difference between the self-diffusion coefficients in the x and y directions is significantly noticeable in the first adsorption layer, and gradually decreases when moving away from the wall, and completely disappears in the bulk-like liquid region. As the x_C6 increases, the self-diffusion coefficient of hexane and hexadecane in the mixture goes higher in the bulk liquid region as well as in the interface region. In the bulk liquid region, this tendency is mainly attributed to the increase in total free volume in the mixture as the x_C6 goes higher.^27,29 From the mass density profiles it is evident that hexane density goes higher and hexadecane density goes lower in the layered region with an increase in a molar fraction of hexane in the mixture. Similar to the discussion in the bulk liquid region, we can believe that the availability of total free volume increases in the interface region with an increase in a molar faction of hexane, which causes the increase in the self-diffusion coefficient of hexane and hexadecane in their mixture in the solid–liquid interface region. It is to be noted that the magnitude of increase in the self-diffusion coefficients of hexane and hexadecane with an increase in a molar fraction of hexane is comparably small in the interface region than that in the bulk liquid region. This might be due to the adsorption tendencies of two species in the interface region with an increase in x_C6.

Fig. 9 Self-diffusion coefficients D_xx and D_yy as a function of distance for (a) hexadecane and (b) hexane in hexane–tetracosane mixture for x_C6 = 0.46, 0.64, 0.85 at T = 400 K; error bars were calculated as described in the text. Density distributions are depicted in dashed and dashed-two dotted lines for hexadecane and hexane, respectively. Self-diffusion coefficient profiles are shown in solid lines with rectangular and circular symbols.

Castro et al.^20–23 experimentally studied binary liquid mixture of octane and decane adsorbed on graphite and reported that a solid monolayer was observed next to the substrate. However, in this work, it is observed that the self-diffusion coefficients for alkane molecules have some significant non-zero values in the solid–liquid interface region, although they are considerably smaller as compared with that in the bulk liquid region. Even though the molecular migration is low in the interface region, it does not vanish, which indicates that the layers are behaving as liquids rather than being frozen like solids. Similar tendencies have been reported for alkane mixtures in the vicinity of wax-like substrate using MD simulations.¹⁰ The solid-like region was observed in the interface region in our previous study, where pure liquid methane is in contact with the α-quartz substrate.¹¹ The self-diffusion coefficient of methane in a few layers from the (100) crystal surface of the substrate were found to be extremely low (almost equal to zero). At the same reduced temperature, liquid decane and tetracosane exhibit a certain self-diffusion coefficient value next to the α-quartz substrate.¹¹ This suggests that the tendency of finding a solid-like region in the interface region is higher when smaller length liquids are in contact with the α-quartz substrate. It is important to note that the temperature and atomic topology of the substrate surface have a considerable effect on migration of molecules in the solid–liquid interface region.

The self-diffusion coefficients in the x and y directions of hexane and hexadecane in their mixture in the first adsorption layer are shown in Fig. 10 as a function of a molar fraction of hexane. It is observed that the in-plane self-diffusion coefficients of both hexane and hexadecane in their mixture increases proportional to the increase in a molar fraction of hexane in the examined range. The self-diffusion coefficients of both hexane and hexadecane in the x direction are approximately 40 percent higher than those in the y direction for all the three values of x_C6. It suggests that the degree of anisotropy induced by the solid wall in the self-diffusion coefficients of hexane and hexadecane in the mixture is not affected by the molar fraction of hexane.

Fig. 10 Self-diffusion coefficient of hexane and hexadecane in the first adsorption layer as a function of molar fraction of hexane (x_C6) in hexane–hexadecane mixture at T = 400 K; error bars were calculated as described in the text.

The self-diffusion coefficients of hexane in the x and y directions in hexane–hexadecane, hexane–dodecane, and hexane–octane mixtures at T = 400 K and x_C6 = 0.64 are shown in Fig. 11. The diffusion coefficient of hexane in the liquid mixture with the hexadecane molecules has lower magnitude than that in other two mixtures. This is mainly due to the lower total free volume in the mixture due to the presence of longer hexadecane molecules.²⁹ The self-diffusion coefficients of hexane in the first adsorption layer in the x direction are nearly 40 percent higher than those in the y direction in all three mixtures. This implies that the degree of anisotropy occurred in hexane self-diffusion coefficient due to the lattice-scale roughness of the substrate surface is not influenced by the chain length of the other alkane molecule present in the mixture.

Fig. 11 Self-diffusion coefficients D_xx and D_yy of hexane as a function of distance in hexane–hexadecane, hexane–dodecane, and hexane–octane mixtures at T = 400 K and x_C6 = 0.64; error bars were calculated as described in the text. Density distributions are depicted in dashed lines and self-diffusion coefficient profiles are depicted in solid lines with rectangular and circular symbols.

4. Conclusions

In this paper, molecular dynamics simulations have been used to investigate the molecular-scale structure and self-diffusion coefficient of various binary alkane mixtures near (001) α-quartz surface terminated with H-groups. As expected, the solid–liquid interface region is characterized by oscillations in the liquid density profile. In hexane–hexadecane mixture the adsorption of hexadecane molecules in the vicinity of α-quartz substrate decreases with an increase in a molar fraction of hexane in the tested range. The alkane molecules in their mixture are preferentially aligned in the parallel direction to the interface and the molecules are elongated in the first adsorption layer. It is observed that the molecular-scale structure of hexane–hexadecane mixture in the solid–liquid interface region is not significantly influenced by the molar fraction of hexane in the tested range, which is due to the strong molecular interaction between the α-quartz substrate and liquid mixture. The molecular migration in the parallel direction to the interface in the solid–liquid interface region was studied by examining the self-diffusion coefficient. The in-plane self-diffusion coefficient is considerably lower in the layered region next to the substrate than that in the bulk liquid region. The self-diffusion coefficients of hexane and hexadecane in their mixture goes higher in the bulk liquid region and in the interface region with an increase in a molar fraction of hexane, which is mostly due to the increase in local free volume. It is found that the self-diffusion coefficient of alkane molecules in their mixture exhibits anisotropic characteristics in the in-plane direction in the interface region, which is mostly due to the lattice scale roughness of the α-quartz substrate surface.

Acknowledgements

This work was supported by the Grant-in-Aid for Scientific Research by the Japan Society for the Promotion of Science (JSPS). All numerical simulations were performed on the SGI Altix UV1000 and UV2000 at the Advanced Fluid Information Research Center, Institute of Fluid Science, and Tohoku University.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c6ra22398b

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