Molecular dynamics simulations of the crystallization process of n

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Molecular dynamics simulations of the crystallization process of n-alkane mixtures and the resulting thermal conductivity Yi Zeng, and J.M. Khodadadi Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.8b02500 • Publication Date (Web): 27 Sep 2018 Downloaded from http://pubs.acs.org on September 28, 2018

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Revised Manuscript (ID: ef-2018-02500s) Submitted to the Energy & Fuels September 25, 2018

Molecular dynamics simulations of the crystallization process of nalkane mixtures and the resulting thermal conductivity Yi Zeng and J. M. Khodadadi* 1418 Wiggins Hall, Department of Mechanical Engineering, Auburn University Auburn University, AL 36849-5341, USA Abstract To elucidate the underlying mechanism of crystallization processes of n-alkane mixtures, molecular dynamics (MD) simulation is used to assess the effect of the morphology on thermal conductivity. Binary mixtures of n-eicosane and n-triacontane with different molecular number ratios are solidified following two approaches, the surface potential route and the free surface route. There exists a marked discrepancy of the orientation factor of the resulting solid n-alkane molecules following the imposed surface potential and the free surface routes corresponding to observed layered systems and random ordering of molecular chains, respectively. In addition, the thermal conductivities of the n-alkane mixtures are evaluated by using the non-equilibrium MD. The results suggest that the orientation factor of the molecular system greatly impacts the thermal transport of n-alkane mixtures at the nanometer scale, with a negligible influence of the number ratio of n-triacontane. Moreover, the solid ratio plays a significant role on the thermal conductivity for low orientation factor systems by adjusting the number of thermal interfaces of the entire molecular system. * Corresponding [email protected].

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Keywords: Crystallization, molecular dynamics, n-alkane mixtures, thermal conductivity. 1/26 ACS Paragon Plus Environment

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1. Introduction Existence of n-paraffins wax in crude oil constitutes as the primary source of difficulty for oilfield operators due to their low insolubility at low temperatures, leading the undesirable wax deposition on the inner walls of pipelines and processing units, thus narrowing flow passageways and eventually impeding processing of crude oil.1,2 Straight-chain n-alkanes CnH2n+2 (henceforth denoted as Cn) with heavy molecular weight crystallizing at low temperatures are the primary components forming the wax deposits, which involves nucleation of liquid molecules and thermal transport through the thin wax deposits layer.3,4 Crystallization of n-alkanes has been widely investigated by experimental methods, identifying the respective phases and transition temperatures.5 During the crystallization processes, the rotator phases of n-alkanes are observed in-between the liquid and the crystalline solid phase, presenting noticeable variation of properties dependent on the n-alkanes’ molecular length and the even-odd effect.5-7 In comparison to a great number of studies on pure single n-alkanes, investigations on n-alkane mixtures are more relevant and of greater importance in industrial applications. The thermodynamic behavior of binary mixtures of n-alkanes mainly depends on the difference of the chain-length of the constituent n-alkanes, ∆nc.6,8-10 A deeper understanding of crystallization of n-alkanes is explored at the nanoscale by performing molecular dynamics (MD) simulations with various empirical and semi-empirical potentials. The all-atoms force fields, such as OPLS11, COMPASS12 and ReaxFF13, have apparent success in describing the thermodynamics properties of organic molecular systems. The united-atoms force field, approximately treating the methylene and the methyl as groups of atoms, also provide robust macroscopic thermodynamics properties of n-alkane molecules.14-16 The effect of the chain-length, crystallization temperature and surface potential on the dynamic characteristics and quantitative nucleation rate of n-alkane molecules are elucidated by MD 2/26 ACS Paragon Plus Environment

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simulations.17-23 In the presence of a surface potential, the crystal of n-eicosane in a hexagonal phase grows on the surface potential from an amorphous phase as the temperature decreases.18 For shorter chain n-alkanes, the strength of the surface potential is highly crucial to the orientation of the molecules in the crystal,19 adjusting the orientation of the lamella structure to be parallel or perpendicular to the surface. At the microscopic level, the end-to-end alignment of n-alkanes has a sensitive connection to the thermal conductivity and about three times enhancement of thermal conductivity can be achieved by adjusting the orientation factor of the molecular system from 0 to 1.24 In addition, the study on the thermal conductivity of n-alkanes with various chain lengths exhibited the weak dependence of the thermal conductivity on the chain length of n-alkanes due to the existence of huge discrepancy between the intra-molecular and inter-molecular interactions. The low thermal conductivity of n-alkanes is mainly contributed by the value of the thermal interfacial conductance and the number density of the inter-molecule interfaces.25 In addition, the interfaces between the chain-like organic systems and nanoscale structures also play an important role on thermal transport26, 27 that could be utilized in practical applications such a thermal management of electronic systems. The multiple components of n-paraffin in petroleum exhibits a high degree of complexity on both conceptual and numerical comprehension of the crystallization of the n-alkanes mixtures. MD simulations28,29 reveal that the crystallization of the n-alkane mixtures is not only determined by the kinetics of each individual component in the mixture but also by the interactions among molecules with different chain lengths. The response of the molecules with different chain-lengths to the equilibrium temperature leads to the phenomena of segregation of molecules. Shorter chain n-alkanes have higher mobility in comparison to the longer ones at the same temperature, which leads to a strong molecular length-dependent segregation in the binary mixtures of n-alkanes (C10 + C20 and C10 + C30). Smaller difference of the chain length of 3/26 ACS Paragon Plus Environment

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molecules in a polydisperse mixture exhibits a weaker segregation than the binary case.28 In addition, the existence of the surface potential drives the long-chain molecules in the multicomponent mixture of n-alkanes forming a lamellar structure and leads to a rapid crystallization process. Despite all the existing knowledge on the topic, research focusing on the underlying mechanisms at the molecular level of n-alkane mixtures is still lacking. The ensuing thermal transport characteristics of n-alkane mixtures is directly related to the particular problem of wax deposition and hydrocarbon fuel engineering in general. In the present study, the crystallization processes of the binary n-alkane mixtures based on the free surface and the imposed surface potential are firstly discussed. The thermal conductivity of these mixture are then evaluated and analyzed in relation to the orientation factor and the solid ratio of the mixtures.

2. Methods 2.1. Molecular system and cooling method In this work, all the MD simulations are performed utilizing the large-scale atomic/molecular massively parallel simulator (LAMMPS).30 As listed in Table 1, the simulation box contains a total number of 1200 n-alkane molecules, a combination of n-eicosane (C20) and n-triacontane (C30), at different values of the number ratio of C30 (nC30). The united-atoms potential is applied to describe the n-alkanes’ molecules interaction instead of the all-atoms model due to limited contributions from the high-frequency vibrations of the C-H bonds to the macroscopic heat conduction flux.24 The methylene group (-CH2-) and the methyl group (CH3-) are treated as the interaction site in the united-atoms potential. The intramolecular bonding, angular and dihedral interactions are described by the NERD (after the authors’ names) force field14 and the summary of the pertinent parameters is listed in Table 2. 4/26 ACS Paragon Plus Environment

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Table 1 Details of the simulated n-alkanes mixtures at different number ratios of C30 (nC30)

Table 2

nC30

C20

C30

0

1200

0

0.17

1000

200

0.33

800

400

0.5

600

600

0.67

400

800

0.83

200

1000

1

0

1200

Summary of the NERD united-atoms potential parameters

Interaction

Form

Parameters

Bond

U (r ) K r (r − beq ) 2 = kB 2

Angle

U (θ ) Kθ (θ − θ eq ) 2 = kB 2

Torsional

U (φ ) = U1 (1 + cos φ ) + U 2 (1 + cos 2φ ) + U 3 (1 + cos 3φ ) kB

Kr = 96,500 K/Å2 beq = 1.53 Å Kθ= 62,500 K/rad2 θeq = 114° U1= 355.04 K U2= -68.19 K U3= 701.32 K

beq: Equilibrium bond length -23

kB: The Boltzmann constant 1.3781×10

J/K

r: Distance between atoms

θ: Bond angle

θeq: Equilibrium angle

ψ: Torsional angle

In addition, considering the rapid decrease of the van der Waals force with the interatomic spacing increasing, a cut-off length is applied to reduce the required calculations. This non-bonded interaction potential is calculated by the Lennard-Jones (LJ) potential with a 10 Å (~2.5 σ) cut-off distance. The summary of the pertinent LJ parameters is listed in Table 3.

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Table 3

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Parameters of the Lennard-Jones model for various pair potentials

Interaction

Form CH2 – CH2

Nonbonded

 σ 12  σ 6  U (r ) = 4ε   −     r   r  

CH3 – CH3

Parameters ε = 0.0907 kcal/mol σ = 3.93 Å ε = 0.2059 kcal/mol σ = 3.91 Å

ε = (ε CH + ε CH )1/2 σ CH + σ CH σ= 2

CH3 – CH2

3

2

3

2

Surface Potential

12 6  σ  σ   U surface (r ) = 4ε surface  surface  −  surface    r   r  

εsurface = 10 kcal/mol σsurface = 3 Å

In MD simulation, the cooling processes (i) on the free surface and (ii) through imposition of the surface potential can be considered as two limiting crystallization schemes due to the marked discrepancy on the structure of the ensuing crystallized n-alkane mixture. The nalkane mixture is crystallized isotropically on the free surface from a liquid phase to a solid phase. The molecular system is cooled under the isothermal-isobaric ensemble (NPT) conditions with the periodic boundary condition applied in all three directions. In presence of an imposed surface potential, the periodic boundary condition cannot be applied to all three directions. As illustrated in Figure 1, an attractive substance modeled by a virtual LJ potential is introduced on the bottom surface of the simulation box, while the periodic boundary conditions are applied in the x- and y-directions. In comparison to the ε value of the LJ potential between molecules, a much greater value of εsurface is assumed to form a relatively strong attractive potential well. In order to minimize the ensuing effect on thermalization of the system, all the molecules are initially placed above a finite distance σsurface from the LJ well and a 5 Å cut-off distance of this 6/26 ACS Paragon Plus Environment

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potential is set to make sure that this potential can only interact with the bottom-most molecules. A repulsive wall is set on the top surface of the simulation box to prevent escape of molecules during the simulation. The canonical ensemble (NVT) are conducted for the cooling processes of the n-alkanes mixture with the imposed surface potential approach. Both cooling methods are run at a cooling rate 1 K/ns. The value of the time step used in numerical integration of the governing equations was 0.5 fs in all simulations. 2.2. Calculation of the thermal conductivity The calculation of the thermal conductivity of the n-alkanes mixture is performed by using the non-equilibrium MD (NEMD) simulation. Through the Fourier’s equation, q = -k dT / dx

(1)

the thermal conductivity, k, of the molecular system can be determined by the known constant heat flux, q, and the temperature gradient, dT/dx, fitted from the temperature profile. The resulting temperature gradient in the simulation box is realized by exchanging a constant kinetic energy (0.01 kcal/(mol fs)) between the atoms in the regions of the heat sink and heat sources which are set in the center and two sides of the simulation box, respectively (Figure 2). The solid structure crystallized from the liquid phase is firstly equilibrated under NPT at 270 K and atmospheric pressure for 3 ns (6 million steps). The molecular system is then applied the heat flux under microcanonical ensemble (NVE) for 3 ns (6 million steps) to reach the steady-state. The mean temperature profile is averaged for another 2 ns (4 million steps) of simulation. In order to weaken the simulation box length dependence on the thermal conductivity of the mixture of n-alkanes, the thermal conductivities of the mixture of n-alkanes for different simulation box lengths (L) are calculated by using the NEMD method. The values of the thermal conductivities corresponding to different simulation box lengths are extrapolated as a plot of 1/k vs. 1/L, thus the macroscopic thermal conductivity can be determined by extrapolating the values 7/26 ACS Paragon Plus Environment

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to 1/L = 0. 2.3. Orientation factor and the solid ratio Due to the randomness of the distribution of the orientation of the molecular chains, an “orientation factor” is defined to quantify the whole molecular system’s orientation, S = ⟨| cos(θi)|⟩

(2)

where θi is the angle between the end-to-end vector of each individual molecule i and the heat flux direction (Figure 3), with | | indicating the absolute value and ⟨ ⟩ standing for the ensemble average for all the molecules. The orientation factor can be used to analyze the collective effect of the whole molecular system’s orientation on the value of thermal conductivity. In addition, a crystallization criterion determined by the ratio of end-to-end distance over the fully-stretched length is defined to provide a quantitative measure of the solid morphology.28 For a single nalkane chain, if the ratio between the end-to-end distance and full-stretched length is greater than 0.9, it is defined as a straight molecule (all-trans-conformation) and is crystallized. Moreover, a crystallized chain must be found in a bundle with a minimum 3 nearby straight chains. The “solid ratio” representing crystallinity is calculated simply as the ratio of the number of straight molecule over the number of all molecules.

3. Results and discussion 3.1. Phase diagram of the n-alkane mixture As illustrated in Figure 4, the reduced non-bounded potential energy of n-alkane mixtures at different nC30 is changing with temperature. Each data point is averaged from 10 million steps (5 ns). It is shown clearly that n-alkane mixtures exhibit a reduction of the degree of supercooling during crystallization following the imposed surface potential route. In effect, the existence of the imposed surface potential acting as a nucleation plane decreases the energy barrier for crystallization. In Figure 5, several snapshots of the solid molecules during respective 8/26 ACS Paragon Plus Environment

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crystallization schemes are picked to illustrate the instantaneous state of the molecular systems following the free surface and the surface potential routes. As illustrated in Figure 5 (a) for the free surface approach, the onset of solidification is visible as seen at three sites of the simulation box when small molecular blocks are formed by several solid n-alkanes aggregating and pointing in the same direction. This is followed by more solid molecules starting to aggregate on the initially-formed molecular blocks and the size of the molecular blocks increases. In effect, several molecular blocks aligned in different directions are formed in the simulation box as the temperature goes down slowly. When the temperature of the molecular system reaches to the target cooling temperature, a solid n-alkanes mixture is formed with a certain orientation factor. In Figure 5 (b), the evolution of the mixture in response to the imposed surface potential route during the crystallization processes is presented. At an early time instant of the top row, several molecules form a recognizable lamellar structure and are constrained within a small distance from the bottom surface due to the strongly attractive potential well. Gradually, the molecules of C20 and C30 “join” on the lamellar structure and the lamella structure grows.

The final

structures ensuing from the free surface and imposed surface potential routes shown on the bottom rows of Figure 5 are distinctly different. The values of the orientation factor of the molecular systems crystallized following both routes are illustrated in Figure 6. There is a clear discrepancy of the values of the orientation factors following these two cases. The surface potential treatment forces the molecules “laying down” and orienting randomly in the horizontal direction on the x-y planes. As a result of the existence of the imposed surface potential during crystallization of the mixtures, the orientation factor of the molecular system should be close to zero in the z-direction. The values being around 0.1 are the result of incomplete stretching for several molecules. With the free surface treatment of the system, the values of the orientation factor of each molecular block in an infinitely large 9/26 ACS Paragon Plus Environment

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simulation box should be equally distributed due to the stochasticity of the formation of the initial molecular bulks. Hence, the ensemble average orientation factor in an n-alkane molecular system is equal to 0.5. During freezing, the solid ratio is increasing as cooling proceeds (Figure 7). The solid ratio of the n-alkane exhibits rapid increase when it is cooled from 310 K to 280 K following the imposed surface potential route, illustrating crystallization of the mixture from a liquid phase to a solid phase. However, after the temperature reaches 280 K, the solid ratio of the mixture is kept around 0.75 although the temperature keeps decreasing to 240 K. Following the free surface approach, crystallization process mainly happens in the range 275 K to 260 K and a lower solid ratio is formed in the final structure. 3.2. Thermal conductivity of the n-alkanes mixtures As illustrated in Figure 8, the inverse of the thermal conductivity of the molecular system is plotted against the inverse of the simulation box length in the imposed heat flux direction (zdirection). The extrapolation of these values to 1/L = 0 gives the macroscopic thermal conductivity with the consideration of the size of the simulation box. In the meantime, the orientation factor is extracted from the final structure of the molecular systems. Since the mobility of molecular chains at 270 K is higher than that at the lower temperature, the orientation factors of the n-alkane mixtures while performing the calculation of thermal conductivity are higher than that for the mixture with the same nC30 system crystallized immediately from the liquid phase. In addition, replication of the simulation box during performing the calculation of thermal conductivity provides extra space in the z-direction for the cases crystallized following the imposed surface potential approach, which leads to an extra increase of the orientation factor. The macroscopic thermal conductivities of the n-alkane mixtures for different values of nC30 are illustrated as a function of the orientation factor in Figure 9. It is clearly observed that the 10/26 ACS Paragon Plus Environment

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orientation factor is still the main contribution to the values of the thermal conductivity of the molecular system over the wide range of the number ratio of C30 covered. Higher orientation factor in a system always leads to a high thermal conductivity of the n-alkane mixture, being consistent with the findings for molecular system of pure n-alkanes of ideal crystals. The thermal conductivity of the pure n-alkane ideal crystal systems, e.g. C20 and C30,25 are also included in Figure 9. The single crystal system of the pure n-alkanes can provide molecular structures with the maximum and minimum orientation factors by replicating the molecular structure in a certain direction. The differences in the thermal conductivities reflect the geometrical characteristics of intermolecular interactions occurring in individual single crystals. The combination of the weak van der Waals force between the methylene groups in adjoining molecules and the low packing density on the plane transverse to the direction of molecular chains in the crystal, exhibiting a low thermal interfacial conductance, accounts for the small value of the thermal conductivity transverse to the direction of molecular chains. Along the molecular chain axis, although similar weak van der Walls forces are responsible for the methyl group layer between molecular blocks, the much longer molecular dimension in the chain direction markedly decreases the number of the thermal interfaces, allowing fewer sites of weak interaction per unit length to occur than for the transverse direction. This anisotropic property of single crystal provides the theoretical highest and lowest thermal conductivities attainable. Difference of thermal conductivity between the n-alkane mixtures with small orientation value (~0.1) and the stacked lamellae suggests that the solid ratio of the molecular system also plays an important role on thermal transport. The n-alkane mixtures crystallized following the imposed surface potential route have low solid ratio value even though they are at low temperature, suggesting that not all the molecules are in the fully-stretched form and contributing to thermal transport while the molecular system is in the solid state. These folded chains, being 11/26 ACS Paragon Plus Environment

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in a liquid-like state, mainly reside in the small regions between the stacked lamellae structures in the n-alkanes mixtures crystallized following the free surface approach (Figure 10). The stacked lamellas, based on the initial fully-stretched molecules, are growing as the temperature is lowered. Therefore, the molecules in the regions between those stacked lamellas have a probability to be constrained from becoming fully-stretched due to the low mobility of these chains at low temperatures. These folded chains locally restrain thermal transport across the interface between the stacked lamella structures and increase the number of the effective thermal interfaces of the entire system.

4. Conclusions In summary, the effect of the morphology of the n-alkane mixtures at different nC30 on their thermal conductivity are studied by performing MD simulations. The n-alkane mixtures are crystallized from liquid phase to solid phase following the free surface and the imposed potential surface routes. Comparing to the random ordering of molecular chains resulting from the free surface approach during crystallization, a dominant layering of molecules can be observed for the system undergoing freezing following the imposed potential surface route. Determination of thermal conductivity based on those solid structures of the n-alkane mixtures suggests the strong correlation between the orientation factor and the thermal conductivity of the mixtures but a negligible influence of the number ratio of C30 on thermal conductivity of the mixtures. In addition, the solid ratio also plays a significant role on the thermal conductivity of n-alkane mixtures with small orientation factor values by adjusting the number of thermal interfaces of the entire molecular system.


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Acknowledgment This material is based upon work supported by the US Department of Energy under Award

Number

DE-SC0002470

(http://www.eng.auburn.edu/nepcm).

The

first

author

acknowledges support from the Alabama Commission on Higher Education’s Graduate Research Student Program Fellowships (Rounds 10, 11 and 12). The authors also acknowledge the Alabama Supercomputer Center (http://www.asc.edu/) and Auburn University Hopper Cluster (http://www.hpcportal.auburn.edu/) for providing computational resources for performing simulations.

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Rastgarkafshgarkolaei, R.; Zeng, Y.; Khodadadi, J. M.; A molecular dynamics study of the effect of thermal boundary conductance on thermal transport of ideal crystal of nalkanes with different number of carbon atoms. J. Appl. Phys. 2016, 119(20), 205107. 15/26 ACS Paragon Plus Environment

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Figure 1

Schematic diagram of the molecular system cooling on the surface potential (z =

constant). The periodic boundary condition is applied in the x and y directions. A potential well acting as an attractive substance is set on the bottom of the simulation box and a repulsive wall is set on the top. (Red and blue spheres represent the C20 and C30 atoms, respectively)


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Figure 2

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Temperature profile of the n-alkane mixture under an assigned constant heat flux

(0.01 kcal/(mol fs)).


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Energy & Fuels

Figure 3

Schematic diagram of the end-to-end vector angle for a single molecule used in

determining the orientation factor


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Figure 4

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The reduced non-bonded potential energy of the n-alkane mixtures at different

nC30 following the imposed surface potential and the free surface schemes during cooling processes (1 K/ns) from the liquid phase to the respective solid phase.


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Figure 5

280 K

315 K

275 K

311 K

272 K

306 K

270 K 293 K (a) (b) Snapshots of the n-alkanes mixtures (nC30 = 0.83) during the cooling processes

following (a) the free surface and (b) the imposed surface potential. (C20 is represented by red spheres and C30 is represented by blue spheres) 21/26 ACS Paragon Plus Environment

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Figure 6 Orientation factor of each component in the n-alkane mixtures resulting from the free surface and the imposed surface potential routes. (Only the orientation factor in the z-direction are presented for the cases with the imposed surface potential.)


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Figure 7 Solid ratio of the n-alkane mixture (nC30 = 0.5) during cooling process (1 K/ns) following the free surface and the imposed surface potential routes


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Figure 8

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Inverse of the thermal conductivity in the z-direction versus inverse of the

multiple lengths of stacked layers for the n-alkane mixtures in the solid phase crystallized following the free surface and the imposed potential surface approaches.


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Energy & Fuels

Figure 9

Correlation between the orientation factor and the thermal conductivity of the n-

alkane mixtures at different nC30 (solid ratio of each case is labelled next to each symbol)


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Figure 10

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A top view of the n-alkane mixtures (the blue lines represent the C30 molecules

and the red lines represent the C20 molecules; the green boxes are regions where the molecules are folded)


Molecular dynamics simulations of the crystallization process of n

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