# Diffusion and Friction Dynamics of Probe Molecules in Liquidn-Alkane Systems: A Molecular Dynamics Simulation Study

Authors: Song Hi Lee
Journal: Journal of Chemistry
 (2019)

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## Abstract

We present a molecular dynamics simulation study of the probe diffusion and friction dynamics of Lennard-Jones particles in a series of liquidn-alkane systems from C12 up to C400 at 318 K, 418 K, 518 K, and 618 K, to investigate the power law dependence of self-diffusion of polymer liquids on their molecular weights. Two LJ particles MY1 with a mass of 114 g/mol and MY2 with a mass of 225 g/mol are used as probes to model methyl yellow. We observed that a clear transition in the power law dependence of n-alkane self-diffusion on the molecular weight (M) of n-alkane, Dself∼M−γ, occurs in the range C120∼C160 at temperatures of 318 K, 418 K, and 518 K, corresponding to a crossover from the “oligomer” to the “Rouse” regime. We also observed that a clear transition in the power law dependence of the diffusion coefficient DMY2 on the molecular weight (M) of n-alkane, DMY2∼M−γ, occurs at low temperatures. The exponent γ for DMY2 shows a sharp transition from 1.21 to 0.52 near C36 at 418 K and from 1.54 to 0.60 near C36 at 318 K. However, no such transition is found for the probe molecule MY2 at temperatures of 518 K and 618 K and for MY1 probe at temperatures of 418 K, 518 K, and 618 K, but the power law exponent γ for MY1 at 318 K shows instead a linear or a rather slow transition. The dependence of the probe diffusion (DMY2) on the matrix molecular weight (M) reflects a significant change of the matrix dynamics associated with the probe diffusion: a crossover from the “solvent-like” to the “oligomer” regime. As the molecular weight of n-alkane increases, the ratio of Dself/DMY2 becomes less than 1 and the probe molecules encounter, in turn, two different microscopic frictions depending on MMY/Mmatrix and the temperature. It is believed that a reduction in the microscopic friction on the probe molecules that diffuse at a rate faster than the solvent fluctuations leads to large deviations of slope from the linear dependence of the friction of MY2 on the chain length of the n-alkane at 318 K and 418 K.


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## Body

## 1. Introduction

The study for the power law dependence of self-diffusion of pure polymer liquids on the molecular weight of polymers has a long history: self-diffusion in polymer melts (and solution) can be described by the theory of Bueche [1] and by the reptation model of de Gennes [2, 3]. The latter depicts the diffusion process of a molecule of molecular weight M as wiggling through a tube confined by the neighboring molecules which for linear molecules predicts Dself∼M−2 for M > MC, where MC is the entanglement coupling molecular weight [2, 3].In 1959, self-diffusion investigations of polymer melts were made by McCall et al. [4] who measured diffusion of two fractions of polyethylene (M = 4100 and 5800 g/mol) besides higher alkanes; they found the power law Dself∼M−5/3. Detailed investigation by Klein [5] of the self-diffusion of melts of polyethylene fractions from M = 3600 up to 23000 g/mol in matrices of comparable and higher molecular weight polyethylenes confirmed the reptation model down to the lowest molecular weight. Pulsed field gradient NMR experiments for Dself of melts of four monodisperse molecular weight polystyrenes and nine fractions of linear and branched polyethylene [6] revealed that the Dself ∼ n−2, where the number of monomeric units holds in the range of n from about 20 up to 1500, i.e., the diffusion occurs via reptation.BelowMC, the polymer chain dynamics of untangled chains is commonly described by the Rouse model [7] when Dself ∼ M−1 for M < MC, which describes the conformational dynamics of an ideal chain. In this model, the single chain diffusion is represented by Brownian motion of beads connected by harmonic springs. There are no excluded volume interactions between the beads, and each bead is subjected to a random thermal force and a drag force as in Langevin dynamics. This model was proposed by Prince E. Rouse in 1953. An important extension to include hydrodynamic interactions mediated by the solvent between different parts of the chain was worked out by Bruno Zimm in 1956. [8] While the Rouse model overestimates the decrease of the diffusion coefficient D with the number of beads N as N −1, the Zimm model predicts D∼N−ν (where ν is the Flory exponent), which is consistent with the experimental data for dilute polymer solutions. In a polymer melt, the Rouse model correctly predicts long-time diffusion only for chains shorter than the entanglement length. For long chains with noticeable entanglement, the Rouse model holds only up to a crossover time τe. For longer times, the chain can only move within a tube formed by the surrounding chains. This slow motion is usually approximated by the reptation model.There are relatively few molecular dynamics (MD) studies in the molecular weight region from small molecules to the Rouse regime. MD studies of this molecular weight region will shed light on understanding the dynamics of a technologically important class of molecules, oligomers. At molecular weights below the Rouse regime,Dself of n-alkanes again show power law behavior. For example, Dself of n-alkanes from n-octane to polyethylene of the molecular weight of several thousands was reported to follow the relation Dself ∼ M−γ, in which the exponents γ are in the range of 2.72∼1.75 depending on temperature. [6, 9, 10] Although the power law dependence has the same functional form as in the Rouse and reptation models, the origin is totally different. The molecular weight dependence of Dself in the Rouse and reptation models is attributed to the topological entanglement effect not the segmental friction, while the exponent γ found below the Rouse regime reflects the molecular weight dependence of the local friction and not the topological effect.The solvent-oligomer transition has been observed for the first time in a recent study, [11] in which diffusion of methyl yellow (MY) in the oligomeric host of n-alkanes and n-alcohols was studied by forced Rayleigh scattering as a function of the molecular weight and the viscosity of the medium. It was observed that the diffusion coefficient of the probe molecule DMY follows a power law dependence on the molecular weight of the oligomers, with DMY∼M−γ. As the molecular weight of the oligomers increases, the exponent γ shows a sharp transition from 1.88 to 0.91 near docosane (C22) in the n-alkanes and from 1.31 to 0.60 near 1-hexadecanol (C16OH) in the n-alcohols at 45°C. A similar transition is also found in a molecular dynamics (MD) simulation for the diffusion of a Lennard-Jones particle of a size similar to MY in n-alkane systems [11]. This transition reflects a change of the dynamics of oligomeric chain molecules in which the motion of the segments, not the entire molecules, becomes responsible for the transport of the probe molecule as the molecular weight of the oligomer increases.In this article, we use molecular dynamics (MD) simulations to study the probe diffusion and friction dynamics of Lennard-Jones particles in liquidn-alkane systems up to C400 at temperatures of 318 K, 418 K, 518 K, and 618 K and extend our previous work on the subject [12, 13]. We have chosen 14 liquid n-alkane systems of various chain lengths, 12 ≤ N ≤ 400. Our aim is to analyze the effect of diffusion and friction dynamics on probe molecules by extending the length of n-alkanes. We investigate the exponent dependence on the molecular size of the probe relative to the molecular size of the diffusing media. In the primary MD simulation for diffusion of a probe molecule [11], the system considered was at the temperature of 318 K only. We also try to investigate the exponent dependence of the probe diffusion in liquid n-alkane systems on the temperature.In short, compared with our previous studies for C12∼C80 [12] and C12∼C200 [13], extending the length of n-alkane up to C400, the first transition (“solvent-like” to “oligomer”) in DMY1∼M−γ is newly observed at 318 K near C80 with ambiguity, but no transition at 418 K, 518 K, and 618 K is reconfirmed in this work. That is, DMY2∼M−γ is also reconfirmed at 318 K and 418 K near C36 but not near C32 [12]. The second transition (“oligomer” to “Rouse”) in Dself∼M−γ is newly observed at 318 K, 418 K, and 518 K near C120∼C160 and at 618 K near C160 with ambiguity, which is undetectable in the previous studies due to the short length of n-alkanes only for C12∼C80 [12] and C12∼C200 [13]. In addition, the first transition is explained in terms of “retardation” in reducing diffusion of the probe molecules, and the investigation for the second transition is under work for pure n-alkanes, up to C400.


## 2. Molecular Models and Molecular Dynamics Simulation

We have chosen two kinds of probe molecules to study the effect of the molecular size of the probe molecule—MY1 with a molecular weight of 114 g/mol and MY2 with a molecular weight of 225 g/mol. MY2 is modeled for the real probe molecule, methyl yellow (MY), which interacts with the individual sites ofn-alkanes with Lennard-Jones (LJ) potential parameters σ = 6.0 A and ε = 0.6 kJ/mol, while MY1 interacts with LJ potential parameters of σ = 4.0 A and ε = 0.4 kJ/mol. For small n-alkane systems (N≤80), only one probe molecule was inserted at the center of the simulation box, and for large systems (N≥120), 8 probe molecules are inserted at the centers of 8 equally divided sub-boxes of the simulation box without any interaction between the probe molecules. We observed that the probe molecules come very close to each other after long times in our MD simulation. At this time, we then started the MD simulation over again with 8 probe molecules at the centers of 8 sub-boxes of the simulation box.14 systems of liquidn-alkane systems are selected—C12H26, C20H42, C28H58, C36H74, C44H90, C80H162, C120H242, C160H322, C200H402, C240H482, C280H562, C320H642, C360H722, and C400H802—for which a united atom (UA) model was employed, in which the methyl and methylene groups are considered as spherical interaction sites centered at each carbon atom [14–18]. The interaction between the sites on different n-alkane molecules and between the sites separated by more than three bonds in the same n-alkane molecule was described by a Lennard-Jones (LJ) potential. All the sites in a chain have the same LJ size parameter σi ≡ σii = 3.93. The well depth parameters were εi ≡ εii = 0.94784 kJ/mol for interactions between the end sites and εi = 0.39078 kJ/mol for interactions between the internal sites. The Lorentz-Berthelot combining rules (εij ≡ (εiεj)1/2, σji ≡ (σi + σj)/2) were used for interactions between an end site and an internal site and between the probe LJ particle and all the sites of n-alkanes. A cutoff distance of 2.5σi was used for all the LJ interactions.The C-C bond length of 1.54 Å was fixed by a constraint force in our simulations with the RATTLE algorithm. [19] The bond bending interaction was also described by a harmonic potential with an equilibrium angle of 114o and a force constant of 0.079187 kJ/mol·degree2. The torsional interaction was described by the potential developed by Jorgensen et al. [20]:(1)Utorsionϕ=a0+a1cosϕ+a2cos2ϕ+a3cos3ϕ,where ϕ is the dihedral angle, and a0  = 8.3973 kJ/mol, a1  = 16.7862 kJ/mol, a2  = 1.1339 kJ/mol, and a3. = −26.3174 kJ/mol. For the time integration of the equations of motion, we adopted Gear’s fifth-order predictor-corrector algorithm [21] with a time step of 5 femtoseconds for all the systems.Each simulation was carried out in an NpT ensemble with probe molecule(s) inN=100 n-alkanes to determine the volume of each system at the given temperatures, and after a total of 2,000,000 time steps (10 nanoseconds) for each system, the equilibrium density and hence the length of cubic simulation box were obtained. New NVT MD simulations were then performed for each system to store the configurations of the probe molecule(s), and N = 100 n-alkanes for later analyses. The usual periodic boundary condition in the x-, y-, and z-directions and the minimum image convention for pair potential were applied. Gaussian isokinetics was used to keep the temperature of the system constant. [22, 23] After a total of 1,000,000 time steps (5 nanoseconds) for equilibration, the equilibrium properties were averaged over 5 blocks of 200,000 time steps (1 nanosecond). The configurations of all the molecules for further analyses were stored every 10 time steps (0.05 picoseconds) which is small enough to determine any of the time auto-correlation functions.The self-diffusion coefficient (Dself) of liquid n-alkane and diffusion coefficient (DMY) of the probe LJ particle can be obtained through the Green–Kubo formula from velocity autocorrelation (VAC) function:(2)D=13∫0∞dtvi0⋅vit.A microscopic expression for the friction coefficient has been obtained through a Green–Kubo formula by Kirkwood [24] from the time integral of the force auto-correlation (FAC) function [25, 26] in the following form:(3)ζ=13kT∫0τdtfi0⋅fit,where fit=Fit−Fit, Fit is the total force exerted on molecule i.During our MD simulations for the short chains ofn-alkanes, we observed the “evaporation” of the systems, which indicates that the volumes of the systems increase infinitely as the temperature increases. No other phase transition is observed.


## 3. Results and Discussion

### 3.1. Log(D) vs Log(M)

The self-diffusion coefficients (Dself) of liquid n-alkanes and diffusion coefficients of the probes MY1 (DMY1) and MY2 (DMY2) in liquid n-alkane systems are easily obtained from velocity auto-correlation (VAC) function through the Green–Kubo formula, equation (2). The logarithm of diffusion coefficients D (10−6 cm2/s) of n-alkanes, MY1, and MY2 in n-alkanes at T = 318, 418, 518, and 618 K are listed in S Table 1 of Supplementary Materials. The calculated Dself are almost equal for both the n-alkane systems with MY1 and MY2 probes embedded in the n-alkane systems. The log-log plots of the self-diffusion coefficients (Dself) of liquid n-alkanes and those of the diffusion coefficients of MY1 (DMY1) in liquid n-alkane systems vs the molecular weight (M) of liquid n-alkanes at four different temperatures are shown in Figure 1.Figure 1
Log-log plots forDself (10−6 cm2/s) of n-alkanes vs M (g/mol) at T = 618 K (△), 518(◇), 418 (□), and 318 K(○), respectively, and those for DMY1 (10−6 cm2/s) in n-alkane systems vs M (g/mol) at T = 618 K (▲), 518 (◆), 418 (■), and 318 (•).The log(DMY1)−log(M) plot shows a linear behavior from which the slopes give the exponent γ = 0.28 at 618 K (▲), γ = 0.47 at 518 K (◆), and γ = 0.80 at 418 K (■) from DMY1∼M−γ. It is generally accepted that DMY in the low-molecular-weight matrix appears to follow the general behavior found in the probe diffusion in small molecular liquids. DMY decreases with the molecular weight of the matrix according to the power law originates from the Einstein relation D = kBT/ζ for a Brownian particle with the friction coefficient ζ, the Boltzmann constant kB, and the Stokes law ζ = Cπηd from hydrodynamic arguments for a quasimacroscopic size where d is the radius of the probe, η the viscosity of the solvent molecules, and C the hydrodynamic boundary condition which 4 for slip and 6 for “stick.” [27, 28] The time-correlation function formula [29] gives η∼M1/2 and so D∼M−1/2, which gives the exponent γ = 0.5, and this value is close to γ = 0.47 at 518 K. There should exist the temperature effect on the exponent γ. However, the linearity of the log(DMY1)−log(M) plot is quite ambiguous at the temperature of 318 K: linear or transitional. The exponent obtained assuming linear behavior was γ = 1.14, or the assumed transition occurs for C80 from the exponent γ = 1.27 to 0.95 at 318 K (•).In Figure2, for the probe MY2, the log(DMY2)−log(M) plot also shows a linear behavior at high temperatures, and the exponent γ calculated from the slope is 0.47 at 618 K (▲) and 0.68 at 518 K (◆). However, the clear transition in the power law dependence of DMY2 on the molecular weight (M) of n-alkanes occurs near C36 from γ = 1.21 to 0.52 at 418 K (■) and near C36 from γ = 1.54 to 0.60 at 318 K (•). Note that γ = 0.47 at 618 K is close to 0.5 predicted from the Einstein–Stokes (ES) equation. In addition, the transition of the probe diffusion (DMY) on the matrix molecular weight (M) reflects a significant change of the matrix dynamics associated with the probe diffusion. For example, the diffusion behaviors of MY2 in the whole n-alkane matrix at 618 K and 518 K and those of MY2 up to C36 n-alkane matrix at 418 K and 318 K originate from the ES equation but with the increasing exponent 0.47, 0.68, 1.21, and 1.54, which is interpreted as the n-alkane matrix being “solvent-like.” On the contrary, the diffusion behavior of MY2 in the C36 n-alkane matrix at 418 K and 318 K shows that the exponent 0.52 increases to 0.60, which corresponds to a reduction in the microscopic friction for the probe molecules that diffuse at a rate faster than the solvent fluctuations. [30] The friction of the n-alkane solvent for MY diffusion mainly comes from the segmental motion of the chain and not from the motion of the whole chain. We believe that this is the origin of the transition, and we henceforth call the matrix molecular weight regime at each side of the transition point as the “solvent” and “oligomer” regimes for convenience. Note that the transition is determined by the ratio of MMY/Mmatrix and the temperature.Figure 2
Log-log plots forDself (10−6 cm2/s) of n-alkanes vs M (g/mol) at T = 618 K (△), 518 (◇), 418 (□), and 318 (○), respectively, and those for D MY2 (10−6 cm2/s) in n-alkane systems vs M (g/mol) at T = 618 K (▲), 518 (◆), 418 (■), and 318 (•).At this point, it is interesting to compareDMY1 and DMY2 in n-alkane systems with Dself of n-alkane as plotted in Figures 1 and 2. DMY1 (the black symbols in Figure 1) in n-alkane systems are greater than DMY2 (the black symbols in Figure 2) at all temperatures since MY1 with a molecular weight of 114 g/mol in the n-alkane matrix diffuses more easily than MY2 with a molecular weight of 225 g/mol in the same matrix. DMY1 are also greater than Dself (the white symbols in Figures 1 and 2) at all the temperatures. However, DMY2 are greater than Dself at 618 K and 518 K and become greater than Dself near C28 at 418 K and near C44 at 318 K. However, DMY2 is slightly smaller than Dself for low-molecular-weight n-alkanes due to the relatively large molecular size of MY2. As the molecular weight of n-alkane increases, Dself decreases much faster than DMY2, and the two diffusion coefficients become similar around C24 at 418 K and C40 at 318 K, which can be interpreted as occurring when the hydrodynamic radius of MY2 becomes comparable to that of n-alkane constituting the diffusing media. At higher molecular weights of n-alkanes, DMY2 does not decrease much as the molecular weight of the n-alkane matrix increases, while Dself decreases continuously and becomes smaller than DMY2 near C28 at 418 K and near C44 at 318 K. In the context of the Brownian motion that is behind any diffusive process, processes slower than or comparable to solvent fluctuations will be affected by the full spectrum of the solvent fluctuations and experience the full shear viscosity of the medium. On the contrary, processes much faster than the solvent fluctuation do not experience the Brownian fluctuating force and are not viscously damped. Thus, one expects a reduction in the microscopic friction for the probe molecules that diffuse at a rate faster than the solvent fluctuations. [30] Therefore, for small solute molecules which diffuse in a time scale shorter than the solvent fluctuations, longer chain n-alkane solvents offer a reduced friction relative to the shorter chain n-alkanes. We call this “retardation” in reducing diffusion due to the reduction in the microscopic friction on the probe molecules.Again, in Figure1 or Figure 2, plots of log(Dself)−log(M) at 618 K shows an ambiguous behavior (△): linear or transitional, in which the exponent, Dself∼M−γ, obtained assuming linear behavior was γ = 1.52 or the assumed transition occurs for C160 from the exponent γ = 1.57 to 1.28. However, a clear transition in the power law dependence of Dself of n-alkane on the molecular weight (M) of n-alkanes occurs at low temperatures. As the molecular weight of n-alkanes increases, the exponent γ shows a sharp transition near C160 from 1.65 to 1.08 at 518 K (◇), near C160 from 1.74 to 0.89 at 418 K (□), and most dramatically near C120 from 2.28 to 0.78 at 318 K (○).Probably the clear transition in the power law dependence ofDself of n-alkane on the molecular weight (M) of n-alkane systems at low temperatures may correspond to crossing over from the “oligomer” to the “Rouse” regimes as discussed in “Introduction”. The “oligomer” regime is characterized by the exponents γ in the range of 2.72–1.75 for self-diffusion Dself∼M−γ depending on the temperature [6, 9, 10] and the “Rouse” regime [7] when γγ = 1 (Dself∼M−1) for M < MC. Our results for the exponents γ of 1.65, 1.74, and 2.28 at 518 K, 418 K, and 318 K in the oligomer region are larger than those of 1.08, 0.89, and 0.78 (roughly γ ≈ 1) at the same temperature ranges in the Rouse regime. Note that the transitions occur near C120∼C160, the exponent γ in the oligomer region increases, and the one in the Rouse region decreases as the temperature decreases.As the molecular weight of the matrix is increased further, we can expect a transition from the oligomer to the “polymer” (Rouse) regime where the molecular weight dependence of the probe diffusion disappears. It has been shown experimentally that the diffusion of small molecules in polymer solutions of sufficiently large molecular weight is nearly independent of the molecular weight or viscosity of the polymers [31]. Therefore, DMY in the high-molecular-weight polymer must be larger than that extrapolated from the oligomer regime. The last point for n-alkanes in Figure 2 of [11] is the diffusion of MY in the polyethylene of Mw = 4,000, which clearly departs from the extrapolated line of the oligomer regime. Chu and Thomas measured the diffusion of pyrene and phthalic anhydride in poly(dimethylsiloxane) of various molecular weights (from 237 to 423,000). [32] They observed a single transition at which the power law exponent, γ, changed from 0.4 to 0.04. Their transition molecular weight was 3,780 (average degree of polymerization; 48.8) and interpreted it as the change of the matrix property from “solvent-like” to “polymer-like.”As the temperature decreases, no clear transition in the power law dependence of diffusion (DMY1) of probe molecule (MY1) on the molecular weight (M) of n-alkanes (DMY1 ∼ M−γ) is observed except the ambiguity in linear or transitional at 318 K, but two clear transitions in DMY2 ∼ M−γ are observed at 418 K and 318 K for the short chain of n-alkane. In the case of Dself ∼ M−γ, all the transitions are observed for the long chain of n-alkane except the ambiguity in linear or transitional at 618 K. From this point of view, the temperature seems to be an important effect to change the diffusion and friction dynamics of the probe molecules and n-alkane matrix to let occur the two kinds of transition in the power law dependence of diffusion (D) of n-alkanes and probe molecules (MY1 and MY2) in n-alkanes on the molecular weight (M) of n-alkanes.


### 3.2. Log(ζ) vs Log(M)

Logarithm of friction coefficientsζ (g/ps·mol) of n-alkanes, MY1, and MY2 in n-alkanes at T = 318, 418, 518, and 618 K are listed in S Table 2 of Supplementary Materials. The expression in the integral of equation (3) is the auto-correlation functions of the force exerted on the center of n-alkane or on the probe molecule(s) by the liquid n-alkane. This expression for ζ vanishes if the upper bound in the integral is set to infinity [33]. The introduction of a cutoff time τo was the solution to this problem given by Kirkwood [24], who assumed that the integral of the force auto-correlation (FAC) function vs the upper bound presented a plateau value of τo. The friction coefficient could then be evaluated from this plateau region. Lagr’kov and Sergeev [34] proposed to choose τo as the first zero in the auto-correlation function. We were unable to get the plateau value in the running time integral of the force auto-correlation function, but we could obtain the friction coefficients from the method proposed by Lagr’kov and Sergeev [34].In Figure3, we show the log-log plot of friction coefficients (ζ) of n-alkanes vs the molecular weight (M) of n-alkanes at all temperatures, and the slopes are almost linear at given temperatures except for a dent for n-alkanes over C200 at 318 K. This small reduction in ζ from the assumed linear behavior of the log-log plot of ζ vs M is quite ambiguous. In order to confirm the reduction in ζ, MD simulations for much longer n-alkane chains such as C1000 at low temperatures are required. The restricted linear behaviors at given temperatures indicate that the behavior ζ vs M are well described by ζ∼Mβ. The obtained exponents β are 0.98, 1.01, 1.03, and 1.06 at 318 K, 418 K, 518 K, and 618 K. Roughly, β ≈ 1 which holds for the Rouse model of Dself ∼ M−1 and ζ ∼ η ∼ M1 for M < MC [7] as discussed in “Introduction”. However, the ES equation predicts that ζ ∼ η ∼ M1/2.Figure 3
Log-log plots forζ (g/ps·mol) of n-alkanes vs M (g/mol). From top, T = 318 K (○), 418 (□), 518 (◇), and 618 (△), respectively.We show the log-log plot of friction coefficient (ζMY) vs the molecular weight (M) of n-alkane in Figure 4. For the smaller probe molecule, MY1, the slope from ζMY1 ∼ M−β increases almost linearly up to C44, and the increment of the slope decreases from C80 to C400 at all the temperatures as the chain length of n-alkane increases. The values of β for ζMY1 are 0.28, 0.23, 0.21, and 0.20 at 618 K, 518 K, 418 K, and 318 K in the shorter n-alkane systems and are almost equal to 0.11 at the same temperature ranges for the longer n-alkane systems. These smaller values of β in the longer chains compared to the shorter chains for small solute molecules reflect a more reduced friction in the longer chains, which indicates that MY1 diffuses at a rate much faster than the solvent fluctuations in the longer chains as discussed above. However, the reduction in the microscopic friction for the probe molecules is not large enough to be abrupt, and the transitions are rather smooth. For the larger probe molecule, MY2, the friction coefficients are much larger than those of MY1 at all the temperatures. At high temperatures of 618 and 518 K, the behavior of the slope in the log-log plot of ζMY2 vs M is very similar to that of MY1 at all temperatures, with β = 0.25 and 0.22 at 618 K and 518 K, respectively, in the shorter chains and β = 0.10 and 0.11 at 618 K and 518 K, respectively, in the longer chains, except the smooth change of slope from the assumed linear dependence of the friction of MY2 on the chain length of n-alkane starts at C44 instead of C80 in the case of MY1.Figure 4
Log-log plots forζ (g/ps·mol) of MY1 and MY2 in n-alkane systems vs M (g/mol). From top, T = 318 K (•), 418 (■), 518 (◆), and 618 (▲) for ζMY2 and the corresponding plot for ζMY1 (the white symbols).At low temperatures of 418 and 318 K, Figure4 shows a rapid change of slope from the linear dependence of the friction of MY2 on the chain length of n-alkane starting at C36, β = 0.044 and 0.033 at 418 K and 318 K, respectively, but the initial slopes in the log-log plot of ζMY2 vs M, 0.21 and 0.20, are almost equal to those at high temperatures. This tells us that solute molecules are slower than or comparable to solvent fluctuations and will be affected by the full spectrum of the solvent fluctuations and experience the full shear viscosity (friction) of the medium. As discussed above, Dself of n-alkanes decreases much faster than DMY2 as the molecular weight of n-alkane increases, and at the higher molecular weights of n-alkane, MY2 diffuses faster than the solvent fluctuations. Therefore, there is a large reduction of friction in longer chains compared to the shorter chains, which enhances the diffusion of the probe molecules, MY2. We believe that this is the origin of the “solvent-oligomer” transition as seen in Figure 1. The “retardation” in reducing diffusion due to the reduction in the microscopic friction on the probe molecules is demonstrated.


## 3.1. Log(D) vs Log(M)

The self-diffusion coefficients (Dself) of liquid n-alkanes and diffusion coefficients of the probes MY1 (DMY1) and MY2 (DMY2) in liquid n-alkane systems are easily obtained from velocity auto-correlation (VAC) function through the Green–Kubo formula, equation (2). The logarithm of diffusion coefficients D (10−6 cm2/s) of n-alkanes, MY1, and MY2 in n-alkanes at T = 318, 418, 518, and 618 K are listed in S Table 1 of Supplementary Materials. The calculated Dself are almost equal for both the n-alkane systems with MY1 and MY2 probes embedded in the n-alkane systems. The log-log plots of the self-diffusion coefficients (Dself) of liquid n-alkanes and those of the diffusion coefficients of MY1 (DMY1) in liquid n-alkane systems vs the molecular weight (M) of liquid n-alkanes at four different temperatures are shown in Figure 1.Figure 1
Log-log plots forDself (10−6 cm2/s) of n-alkanes vs M (g/mol) at T = 618 K (△), 518(◇), 418 (□), and 318 K(○), respectively, and those for DMY1 (10−6 cm2/s) in n-alkane systems vs M (g/mol) at T = 618 K (▲), 518 (◆), 418 (■), and 318 (•).The log(DMY1)−log(M) plot shows a linear behavior from which the slopes give the exponent γ = 0.28 at 618 K (▲), γ = 0.47 at 518 K (◆), and γ = 0.80 at 418 K (■) from DMY1∼M−γ. It is generally accepted that DMY in the low-molecular-weight matrix appears to follow the general behavior found in the probe diffusion in small molecular liquids. DMY decreases with the molecular weight of the matrix according to the power law originates from the Einstein relation D = kBT/ζ for a Brownian particle with the friction coefficient ζ, the Boltzmann constant kB, and the Stokes law ζ = Cπηd from hydrodynamic arguments for a quasimacroscopic size where d is the radius of the probe, η the viscosity of the solvent molecules, and C the hydrodynamic boundary condition which 4 for slip and 6 for “stick.” [27, 28] The time-correlation function formula [29] gives η∼M1/2 and so D∼M−1/2, which gives the exponent γ = 0.5, and this value is close to γ = 0.47 at 518 K. There should exist the temperature effect on the exponent γ. However, the linearity of the log(DMY1)−log(M) plot is quite ambiguous at the temperature of 318 K: linear or transitional. The exponent obtained assuming linear behavior was γ = 1.14, or the assumed transition occurs for C80 from the exponent γ = 1.27 to 0.95 at 318 K (•).In Figure2, for the probe MY2, the log(DMY2)−log(M) plot also shows a linear behavior at high temperatures, and the exponent γ calculated from the slope is 0.47 at 618 K (▲) and 0.68 at 518 K (◆). However, the clear transition in the power law dependence of DMY2 on the molecular weight (M) of n-alkanes occurs near C36 from γ = 1.21 to 0.52 at 418 K (■) and near C36 from γ = 1.54 to 0.60 at 318 K (•). Note that γ = 0.47 at 618 K is close to 0.5 predicted from the Einstein–Stokes (ES) equation. In addition, the transition of the probe diffusion (DMY) on the matrix molecular weight (M) reflects a significant change of the matrix dynamics associated with the probe diffusion. For example, the diffusion behaviors of MY2 in the whole n-alkane matrix at 618 K and 518 K and those of MY2 up to C36 n-alkane matrix at 418 K and 318 K originate from the ES equation but with the increasing exponent 0.47, 0.68, 1.21, and 1.54, which is interpreted as the n-alkane matrix being “solvent-like.” On the contrary, the diffusion behavior of MY2 in the C36 n-alkane matrix at 418 K and 318 K shows that the exponent 0.52 increases to 0.60, which corresponds to a reduction in the microscopic friction for the probe molecules that diffuse at a rate faster than the solvent fluctuations. [30] The friction of the n-alkane solvent for MY diffusion mainly comes from the segmental motion of the chain and not from the motion of the whole chain. We believe that this is the origin of the transition, and we henceforth call the matrix molecular weight regime at each side of the transition point as the “solvent” and “oligomer” regimes for convenience. Note that the transition is determined by the ratio of MMY/Mmatrix and the temperature.Figure 2
Log-log plots forDself (10−6 cm2/s) of n-alkanes vs M (g/mol) at T = 618 K (△), 518 (◇), 418 (□), and 318 (○), respectively, and those for D MY2 (10−6 cm2/s) in n-alkane systems vs M (g/mol) at T = 618 K (▲), 518 (◆), 418 (■), and 318 (•).At this point, it is interesting to compareDMY1 and DMY2 in n-alkane systems with Dself of n-alkane as plotted in Figures 1 and 2. DMY1 (the black symbols in Figure 1) in n-alkane systems are greater than DMY2 (the black symbols in Figure 2) at all temperatures since MY1 with a molecular weight of 114 g/mol in the n-alkane matrix diffuses more easily than MY2 with a molecular weight of 225 g/mol in the same matrix. DMY1 are also greater than Dself (the white symbols in Figures 1 and 2) at all the temperatures. However, DMY2 are greater than Dself at 618 K and 518 K and become greater than Dself near C28 at 418 K and near C44 at 318 K. However, DMY2 is slightly smaller than Dself for low-molecular-weight n-alkanes due to the relatively large molecular size of MY2. As the molecular weight of n-alkane increases, Dself decreases much faster than DMY2, and the two diffusion coefficients become similar around C24 at 418 K and C40 at 318 K, which can be interpreted as occurring when the hydrodynamic radius of MY2 becomes comparable to that of n-alkane constituting the diffusing media. At higher molecular weights of n-alkanes, DMY2 does not decrease much as the molecular weight of the n-alkane matrix increases, while Dself decreases continuously and becomes smaller than DMY2 near C28 at 418 K and near C44 at 318 K. In the context of the Brownian motion that is behind any diffusive process, processes slower than or comparable to solvent fluctuations will be affected by the full spectrum of the solvent fluctuations and experience the full shear viscosity of the medium. On the contrary, processes much faster than the solvent fluctuation do not experience the Brownian fluctuating force and are not viscously damped. Thus, one expects a reduction in the microscopic friction for the probe molecules that diffuse at a rate faster than the solvent fluctuations. [30] Therefore, for small solute molecules which diffuse in a time scale shorter than the solvent fluctuations, longer chain n-alkane solvents offer a reduced friction relative to the shorter chain n-alkanes. We call this “retardation” in reducing diffusion due to the reduction in the microscopic friction on the probe molecules.Again, in Figure1 or Figure 2, plots of log(Dself)−log(M) at 618 K shows an ambiguous behavior (△): linear or transitional, in which the exponent, Dself∼M−γ, obtained assuming linear behavior was γ = 1.52 or the assumed transition occurs for C160 from the exponent γ = 1.57 to 1.28. However, a clear transition in the power law dependence of Dself of n-alkane on the molecular weight (M) of n-alkanes occurs at low temperatures. As the molecular weight of n-alkanes increases, the exponent γ shows a sharp transition near C160 from 1.65 to 1.08 at 518 K (◇), near C160 from 1.74 to 0.89 at 418 K (□), and most dramatically near C120 from 2.28 to 0.78 at 318 K (○).Probably the clear transition in the power law dependence ofDself of n-alkane on the molecular weight (M) of n-alkane systems at low temperatures may correspond to crossing over from the “oligomer” to the “Rouse” regimes as discussed in “Introduction”. The “oligomer” regime is characterized by the exponents γ in the range of 2.72–1.75 for self-diffusion Dself∼M−γ depending on the temperature [6, 9, 10] and the “Rouse” regime [7] when γγ = 1 (Dself∼M−1) for M < MC. Our results for the exponents γ of 1.65, 1.74, and 2.28 at 518 K, 418 K, and 318 K in the oligomer region are larger than those of 1.08, 0.89, and 0.78 (roughly γ ≈ 1) at the same temperature ranges in the Rouse regime. Note that the transitions occur near C120∼C160, the exponent γ in the oligomer region increases, and the one in the Rouse region decreases as the temperature decreases.As the molecular weight of the matrix is increased further, we can expect a transition from the oligomer to the “polymer” (Rouse) regime where the molecular weight dependence of the probe diffusion disappears. It has been shown experimentally that the diffusion of small molecules in polymer solutions of sufficiently large molecular weight is nearly independent of the molecular weight or viscosity of the polymers [31]. Therefore, DMY in the high-molecular-weight polymer must be larger than that extrapolated from the oligomer regime. The last point for n-alkanes in Figure 2 of [11] is the diffusion of MY in the polyethylene of Mw = 4,000, which clearly departs from the extrapolated line of the oligomer regime. Chu and Thomas measured the diffusion of pyrene and phthalic anhydride in poly(dimethylsiloxane) of various molecular weights (from 237 to 423,000). [32] They observed a single transition at which the power law exponent, γ, changed from 0.4 to 0.04. Their transition molecular weight was 3,780 (average degree of polymerization; 48.8) and interpreted it as the change of the matrix property from “solvent-like” to “polymer-like.”As the temperature decreases, no clear transition in the power law dependence of diffusion (DMY1) of probe molecule (MY1) on the molecular weight (M) of n-alkanes (DMY1 ∼ M−γ) is observed except the ambiguity in linear or transitional at 318 K, but two clear transitions in DMY2 ∼ M−γ are observed at 418 K and 318 K for the short chain of n-alkane. In the case of Dself ∼ M−γ, all the transitions are observed for the long chain of n-alkane except the ambiguity in linear or transitional at 618 K. From this point of view, the temperature seems to be an important effect to change the diffusion and friction dynamics of the probe molecules and n-alkane matrix to let occur the two kinds of transition in the power law dependence of diffusion (D) of n-alkanes and probe molecules (MY1 and MY2) in n-alkanes on the molecular weight (M) of n-alkanes.


## 3.2. Log(ζ) vs Log(M)

Logarithm of friction coefficientsζ (g/ps·mol) of n-alkanes, MY1, and MY2 in n-alkanes at T = 318, 418, 518, and 618 K are listed in S Table 2 of Supplementary Materials. The expression in the integral of equation (3) is the auto-correlation functions of the force exerted on the center of n-alkane or on the probe molecule(s) by the liquid n-alkane. This expression for ζ vanishes if the upper bound in the integral is set to infinity [33]. The introduction of a cutoff time τo was the solution to this problem given by Kirkwood [24], who assumed that the integral of the force auto-correlation (FAC) function vs the upper bound presented a plateau value of τo. The friction coefficient could then be evaluated from this plateau region. Lagr’kov and Sergeev [34] proposed to choose τo as the first zero in the auto-correlation function. We were unable to get the plateau value in the running time integral of the force auto-correlation function, but we could obtain the friction coefficients from the method proposed by Lagr’kov and Sergeev [34].In Figure3, we show the log-log plot of friction coefficients (ζ) of n-alkanes vs the molecular weight (M) of n-alkanes at all temperatures, and the slopes are almost linear at given temperatures except for a dent for n-alkanes over C200 at 318 K. This small reduction in ζ from the assumed linear behavior of the log-log plot of ζ vs M is quite ambiguous. In order to confirm the reduction in ζ, MD simulations for much longer n-alkane chains such as C1000 at low temperatures are required. The restricted linear behaviors at given temperatures indicate that the behavior ζ vs M are well described by ζ∼Mβ. The obtained exponents β are 0.98, 1.01, 1.03, and 1.06 at 318 K, 418 K, 518 K, and 618 K. Roughly, β ≈ 1 which holds for the Rouse model of Dself ∼ M−1 and ζ ∼ η ∼ M1 for M < MC [7] as discussed in “Introduction”. However, the ES equation predicts that ζ ∼ η ∼ M1/2.Figure 3
Log-log plots forζ (g/ps·mol) of n-alkanes vs M (g/mol). From top, T = 318 K (○), 418 (□), 518 (◇), and 618 (△), respectively.We show the log-log plot of friction coefficient (ζMY) vs the molecular weight (M) of n-alkane in Figure 4. For the smaller probe molecule, MY1, the slope from ζMY1 ∼ M−β increases almost linearly up to C44, and the increment of the slope decreases from C80 to C400 at all the temperatures as the chain length of n-alkane increases. The values of β for ζMY1 are 0.28, 0.23, 0.21, and 0.20 at 618 K, 518 K, 418 K, and 318 K in the shorter n-alkane systems and are almost equal to 0.11 at the same temperature ranges for the longer n-alkane systems. These smaller values of β in the longer chains compared to the shorter chains for small solute molecules reflect a more reduced friction in the longer chains, which indicates that MY1 diffuses at a rate much faster than the solvent fluctuations in the longer chains as discussed above. However, the reduction in the microscopic friction for the probe molecules is not large enough to be abrupt, and the transitions are rather smooth. For the larger probe molecule, MY2, the friction coefficients are much larger than those of MY1 at all the temperatures. At high temperatures of 618 and 518 K, the behavior of the slope in the log-log plot of ζMY2 vs M is very similar to that of MY1 at all temperatures, with β = 0.25 and 0.22 at 618 K and 518 K, respectively, in the shorter chains and β = 0.10 and 0.11 at 618 K and 518 K, respectively, in the longer chains, except the smooth change of slope from the assumed linear dependence of the friction of MY2 on the chain length of n-alkane starts at C44 instead of C80 in the case of MY1.Figure 4
Log-log plots forζ (g/ps·mol) of MY1 and MY2 in n-alkane systems vs M (g/mol). From top, T = 318 K (•), 418 (■), 518 (◆), and 618 (▲) for ζMY2 and the corresponding plot for ζMY1 (the white symbols).At low temperatures of 418 and 318 K, Figure4 shows a rapid change of slope from the linear dependence of the friction of MY2 on the chain length of n-alkane starting at C36, β = 0.044 and 0.033 at 418 K and 318 K, respectively, but the initial slopes in the log-log plot of ζMY2 vs M, 0.21 and 0.20, are almost equal to those at high temperatures. This tells us that solute molecules are slower than or comparable to solvent fluctuations and will be affected by the full spectrum of the solvent fluctuations and experience the full shear viscosity (friction) of the medium. As discussed above, Dself of n-alkanes decreases much faster than DMY2 as the molecular weight of n-alkane increases, and at the higher molecular weights of n-alkane, MY2 diffuses faster than the solvent fluctuations. Therefore, there is a large reduction of friction in longer chains compared to the shorter chains, which enhances the diffusion of the probe molecules, MY2. We believe that this is the origin of the “solvent-oligomer” transition as seen in Figure 1. The “retardation” in reducing diffusion due to the reduction in the microscopic friction on the probe molecules is demonstrated.


## 4. Conclusion

We have observed two kinds of transition in the power law dependence of diffusion (D) of n-alkanes and probe molecules (MY1 and MY2) in n-alkanes on the molecular weight (M) of n-alkanes. The first type of transition in the diffusion coefficient as a function of molecular weight DMY2∼M−γ occurs at low temperatures of 418 K and 318 K where the exponent γ shows a sharp transition from 1.21 to 0.52 near C36 at 418 K and from 1.54 to 0.60 near C36 at 318 K. However, no such transition is found for the larger and heavier probe MY2 at temperatures of 518 K and 618 K and for the lighter probe MY1 at temperatures of 418 K, 518 K, and 618 K with an ambiguous behavior at 318 K. Hence, the ratio MMY/Mmatrix and the temperature are important factors for this transition which reflect a significant change of the matrix dynamics associated with the probe diffusion: crossing over from the “solvent-like” to the “oligomer” regime. It is believed that a reduction in the microscopic friction on the probe molecules that diffuse at a rate faster than the solvent fluctuations results in the large deviation of slope from the linear dependence of the friction of MY2 on the n-alkane chain length at 318 K and 418 K. We call this as “retardation” in reducing diffusion due to the reduction in the microscopic friction on the probe molecules. The second kind of a clear transition in the exponent of Dself ∼ M−γ occurs near C120∼C160 at temperatures of 318 K, 418 K, and 518 K with an ambiguous linearity at 618 K. It is believed that this transition corresponds to crossing over from the “oligomer” to the “Rouse” regime. The exponents γ obtained from our simulations in the oligomer region are 1.57, 1.65, 1.74, and 2.28 at 618 K, 518 K, 418 K, and 318 K compared to the exponents γ in the range of 2.72–1.75 obtained experimentally, while those in the Rouse regime are 1.28, 1.08, 0.89, and 0.74 (roughly γ ≈ 1) at the same temperatures characterized by Dself∼M−1 for M < MC.



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*Source: 8134904-2019-02-04.xml*