Fluid-solid phase transition of n-alkane mixtures: Coarse-grained molecular dynamics simulations and diffusion-ordered spectroscopy nuclear magnetic resonance

Wax appearance temperature (WAT), defined as the temperature at which the first solid paraffin crystal appears in a crude oil, is one of the key flow assurance indicators in the oil industry. Although there are several commonly-used experimental techniques to determine WAT, none provides unambiguous molecular-level information to characterize the phase transition between the homogeneous fluid and the underlying solid phase. Molecular Dynamics (MD) simulations employing the statistical associating fluid theory (SAFT) force field are used to interrogate the incipient solidification states of models for long-chain alkanes cooled from a melt to an arrested state. We monitor the phase change of pure long chain n-alkanes: tetracosane (C24H50) and triacontane (C30H62), and an 8-component surrogate n-alkane mixture (C12-C33) built upon the compositional information of a waxy crude. Comparison to Diffusion Ordered Spectroscopy Nuclear Magnetic Resonance (DOSY NMR) results allows the assessment of the limitations of the coarse-grained models proposed. We show that upon approach to freezing, the heavier components restrict their motion first while the lighter ones retain their mobility and help fluidize the mixture. We further demonstrate that upon sub-cooling of long n-alkane fluids and mixtures, a discontinuity arises in the slope of the self-diffusion coefficient with decreasing temperature, which can be employed as a marker for the appearance of an arrested state commensurate with conventional WAT measurements.

where "# is the centre-to-centre distance between segments k and l, "# is the potential energy depth, "# is the size scale, defined loosely as the segment diameter, and "# 6 and "# 7 are the 2 repulsive and attractive exponents, respectively. The latter one has been fixed in this work to a value of 6, to represent London dispersion forces. The constant "# is defined as . (2) The SAFT-γ Mie parameters are fitted to vapor pressure and saturated liquid density data properties of small alkanes (C 6 to C 12 ). The unlike interactions for the end (E) and middle (M) beads are obtained from combining rules for the unlike diameters and repulsive exponent, while the unlike energy parameter was directly tuned in the development of the model. In Table   S1, the intermolecular force-fields obtained via the SAFT-γ equation of state, and the unlike parameters, are provided. The intramolecular interactions were determined using atomistic simulations as benchmarks, adopting a united atom (UA) representation, in which hydrogens are implicitly considered in the model. Monodisperse systems of n-alkanes containing 3n carbons (where n = 2, 3, and 4) were simulated to obtain the target bond-length and angular distributions. The intramolecular CG potentials were expressed as (1) where OPQR and 7QUV& are the stretching and bending constants respectively, and correspond to the bond length and angle, while the subscript o denotes the equilibrium values.

Surrogate model mixture
Waxy crude oil originating from the Asia Pacific region under PETRONAS Operations was analysed by High-Temperature Gas Chromatography (HTGC) and SARA analysis to obtain its compositional properties. The compositional distribution analysis was performed in accordance to ASTM D7169 3 analyzing up to a carbon number of C 120 . The chromatographic spectrum is then compared with reference standard hydrocarbons for component identification.
Weight percent (wt %) of carbon distribution is calculated from the area count of each component. The HTGC analysis resulted in a carbon number distribution in the range of C 5 to C 66. Carbon numbers between C 18 and C 32 are the major components with distribution between 3.2 to 8.8 wt %. The heavy compounds were less than 1.0 wt %, while those lower than C 18 were less than 2.0 wt %. Upon the presumption that the carbon distribution of the saturate fraction follows that of the entire crude, a discrete model consisting of eight representative components is employed. Several alkane cuts were grouped together to represent the short, middle and long chained alkanes. The composition was fixed mimicking the compositional distribution obtained from the HTGC analysis. Table S2 below includes the alkane components and their respective composition. The chemicals were purchased from Sigma Aldrich and were used without further purification. In Table S3, the composition details of the model mixture studied in the MD simulations is presented.

Molecular Simulation Details
Molecular dynamics simulations in this work are performed using HOOMD-blue (Highly Optimized Object-oriented Many-particle Dynamics) 4  K in the NPT and NVT ensembles, as described above.

8-alkane Mixture Model System
The simulation box for the 8-alkane mixture was generated randomly with the composition given in NPT and NVT ensembles, as described above.

Diffusion coefficients from MD simulations
The diffusion coefficient is employed as a measure of the inherent mobility of the species in solution. It is directly related to the mean square displacement (MSD), which is the average distance that a given particle in a system travels, In this work, the diffusion coefficient is calculated from the slope in the long-time  Figure S2. In contrast, when a single starting point is taken, the last histogram contains just one value: r(t tot ) -r(0). This is also presented schematically in Figure S2

End-to-end distances and radius of gyration from MD simulations
Order parameter analysis is also applied to complement the study of the changes that occur with the alkanes during the transition process in the MD simulations. The end-to-end distance and radius of gyration of the n-alkanes in neat and mixture form were analysed to evaluate the molecule's flexibility. In Figure S3, the end-to-end distance and radius of gyration for C 12 H 26 , C 24 H 50 , and C 30 H 62 in the 8-alkane mixture are presented as a function of temperature. Both properties increase with the carbon number, as expected, and both properties increase as the temperature is decreased. The general trends are consistent with the ones reported in the literature 1314 . The chains appear to be more stretched or extended at a lower temperature, which would improve the alignment between chains in the system towards the glassy state. It is interesting to note that C 12 H 26 , being the smallest alkane in the system, has a minimum variation in the structural properties as a function of the temperature, while the change in longer chains, C 24 H 50 and C 30 H 62 , is more evident. Further comparison of the simulation results is made for the end-to-end distance and the radius of gyration of n-alkanes in their neat form and in a mixture, as shown in Figure S4. It is found that there is no significant difference in both systems, suggesting that the structural changes of the n-alkanes are not governed by the mixture.   Figure S5 shows the melting point for C 24 H 50 and C 30 H 62 determined by DSC, which is consistent with the melting behavior reported elsewhere 18 . Two peaks were recorded for both alkanes, and this observation is also consistent with other works in the literature, where higher alkanes were generally reported to have two peaks recorded. The true melting point is recorded at the main peak, which is usually at a higher temperature than the second peak. The appearance of the second peak corresponds to a solid-solid transition between a rotator phase and the more common crystal 19 . The melting curve for the 8-component alkane mixture is shown in Figure S6, where there is evidence of a

Melting point for C-24
For the results shown in figure 1 ( melting of a solid ), we chose to construct the crystal by aligning stretched-out molecules packed in a primitive orthorhombic structure. The resulting solid is equilibrated at 200 K and a 1 bar in the NσΤ ensemble, allowing the different dimensions of the solid to accommodate to the closest packing. The resulting structure is temporally stable, displays translational order and has a higher density than its liquid counterpart.

Experimental diffusion data
Raw NMR data is reported in table S4. All measurements are repeated three times. Error bars are calculated based on the NMR average values.