We present a molecular dynamics simulation study of the probe diffusion and friction dynamics of Lennard-Jones particles in a series of liquid _{12} up to C_{400} 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 _{self}∼M^{−γ}, occurs in the range C_{120}∼C_{160} 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 _{MY2} on the molecular weight (_{MY2}∼M^{−γ}, occurs at low temperatures. The exponent _{MY2} shows a sharp transition from 1.21 to 0.52 near C_{36} at 418 K and from 1.54 to 0.60 near C_{36} 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 _{MY2}) on the matrix molecular weight (_{self}/_{MY2} becomes less than 1 and the probe molecules encounter, in turn, two different microscopic frictions depending on M_{MY}/M_{matrix} 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

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 [_{self}∼M^{−2} for _{C}, where _{C} is the entanglement coupling molecular weight [

In 1959, self-diffusion investigations of polymer melts were made by McCall et al. [_{self}∼M^{−5/3}. Detailed investigation by Klein [_{self} of melts of four monodisperse molecular weight polystyrenes and nine fractions of linear and branched polyethylene [_{self} ∼ n^{−2}, where the number of monomeric units holds in the range of

Below _{C}, the polymer chain dynamics of untangled chains is commonly described by the Rouse model [_{self} ∼ M^{−1} for _{C}, 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. [^{−1}, the Zimm model predicts ^{ −ν} (where _{e}. 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, _{self} of _{self} of _{self} ∼ M^{−γ}, in which the exponents _{self} in the Rouse and reptation models is attributed to the topological entanglement effect not the segmental friction, while the exponent

The solvent-oligomer transition has been observed for the first time in a recent study, [_{MY} follows a power law dependence on the molecular weight of the oligomers, with _{MY}∼M^{−γ}. As the molecular weight of the oligomers increases, the exponent _{22}) in the _{16}OH) in the

In this article, we use molecular dynamics (MD) simulations to study the probe diffusion and friction dynamics of Lennard-Jones particles in liquid _{400} at temperatures of 318 K, 418 K, 518 K, and 618 K and extend our previous work on the subject [

In short, compared with our previous studies for C_{12}∼C_{80} [_{12}∼C_{200} [_{400}, the first transition (“solvent-like” to “oligomer”) in _{MY1}∼M^{−γ} is newly observed at 318 K near C_{80} with ambiguity, but no transition at 418 K, 518 K, and 618 K is reconfirmed in this work. That is, _{MY2}∼M^{−γ} is also reconfirmed at 318 K and 418 K near C_{36} but not near C_{32} [_{self}∼M^{−γ} is newly observed at 318 K, 418 K, and 518 K near C_{120}∼C_{160} and at 618 K near C_{160} with ambiguity, which is undetectable in the previous studies due to the short length of _{12}∼C_{80} [_{12}∼C_{200} [_{400}.

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 of

14 systems of liquid _{12}H_{26}, C_{20}H_{42}, C_{28}H_{58}, C_{36}H_{74}, C_{44}H_{90}, C_{80}H_{162}, C_{120}H_{242}, C_{160}H_{322}, C_{200}H_{402}, C_{240}H_{482}, C_{280}H_{562}, C_{320}H_{642}, C_{360}H_{722}, and C_{400}H_{802}—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 [_{i} ≡ _{ii} = 3.93. The well depth parameters were _{i} ≡ _{ii} = 0.94784 kJ/mol for interactions between the end sites and _{i} = 0.39078 kJ/mol for interactions between the internal sites. The Lorentz-Berthelot combining rules (_{ij} ≡ (_{i}_{j})^{1/2}, _{ji} ≡ (_{i} + _{j})/2) were used for interactions between an end site and an internal site and between the probe LJ particle and all the sites of _{i} 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. [^{o} and a force constant of 0.079187 kJ/mol·degree^{2}. The torsional interaction was described by the potential developed by Jorgensen et al. [

Each simulation was carried out in an NpT ensemble with probe molecule(s) in

The self-diffusion coefficient (_{self}) of liquid _{MY}) of the probe LJ particle can be obtained through the Green–Kubo formula from velocity autocorrelation (VAC) function:

A microscopic expression for the friction coefficient has been obtained through a Green–Kubo formula by Kirkwood [

During our MD simulations for the short chains of

The self-diffusion coefficients (_{self}) of liquid _{MY1}) and MY2 (_{MY2}) in liquid ^{−6} cm^{2}/s) of _{self} are almost equal for both the _{self}) of liquid _{MY1}) in liquid

Log-log plots for _{self} (10^{−6} cm^{2}/s) of _{MY1} (10^{−6} cm^{2}/s) in

The log(_{MY1})−log(_{MY1}∼M^{−γ}. It is generally accepted that D_{MY} in the low-molecular-weight matrix appears to follow the general behavior found in the probe diffusion in small molecular liquids. D_{MY} decreases with the molecular weight of the matrix according to the power law originates from the Einstein relation _{B}T/_{B}, and the Stokes law ^{1/2} and so ^{−1/2}, which gives the exponent _{MY1})−log(M) plot is quite ambiguous at the temperature of 318 K: linear or transitional. The exponent obtained assuming linear behavior was _{80} from the exponent

In Figure _{MY2})−log(_{MY2} on the molecular weight (_{36} from _{36} from _{MY}) on the matrix molecular weight (_{36} _{36} _{MY}/M_{matrix} and the temperature.

Log-log plots for _{self} (10^{−6} cm^{2}/s) of _{MY2} (10^{−6} cm^{2}/s) in

At this point, it is interesting to compare _{MY1} and _{MY2} in _{self} of _{MY1} (the black symbols in Figure _{MY2} (the black symbols in Figure _{MY1} are also greater than _{self} (the white symbols in Figures _{MY2} are greater than _{self} at 618 K and 518 K and become greater than _{self} near C_{28} at 418 K and near C_{44} at 318 K. However, _{MY2} is slightly smaller than _{self} for low-molecular-weight _{self} decreases much faster than _{MY2}, and the two diffusion coefficients become similar around C_{24} at 418 K and C_{40} at 318 K, which can be interpreted as occurring when the hydrodynamic radius of MY2 becomes comparable to that of _{MY2} does not decrease much as the molecular weight of the _{self} decreases continuously and becomes smaller than _{MY2} near C_{28} at 418 K and near C_{44} 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. [

Again, in Figure _{self})−log(_{self}∼M^{−γ}, obtained assuming linear behavior was _{160} from the exponent _{self} of _{160} from 1.65 to 1.08 at 518 K (_{160} from 1.74 to 0.89 at 418 K (□), and most dramatically near C_{120} from 2.28 to 0.78 at 318 K (○).

Probably the clear transition in the power law dependence of _{self} of _{self}∼M^{−γ} depending on the temperature [_{self}∼M^{−1}) for _{C}. Our results for the exponents _{120}∼C_{160}, the exponent

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 [_{MY} in the high-molecular-weight polymer must be larger than that extrapolated from the oligomer regime. The last point for _{w} = 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). [

As the temperature decreases, no clear transition in the power law dependence of diffusion (_{MY1}) of probe molecule (MY1) on the molecular weight (_{MY1} ∼ M^{−γ}) is observed except the ambiguity in linear or transitional at 318 K, but two clear transitions in _{MY2} ∼ M^{−γ} are observed at 418 K and 318 K for the short chain of _{self} ∼ M^{−γ}, all the transitions are observed for the long chain of

Logarithm of friction coefficients _{o} was the solution to this problem given by Kirkwood [_{o}. The friction coefficient could then be evaluated from this plateau region. Lagr’kov and Sergeev [_{o} 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 [

In Figure _{200} at 318 K. This small reduction in _{1000} at low temperatures are required. The restricted linear behaviors at given temperatures indicate that the behavior ^{β}. The obtained exponents _{self} ∼ M^{−1} and ^{1} for _{C} [^{1/2}.

Log-log plots for

We show the log-log plot of friction coefficient (_{MY}) vs the molecular weight (_{MY1} ∼ M^{−β} increases almost linearly up to C_{44}, and the increment of the slope decreases from C_{80} to C_{400} at all the temperatures as the chain length of _{MY1} are 0.28, 0.23, 0.21, and 0.20 at 618 K, 518 K, 418 K, and 318 K in the shorter _{MY2} vs _{44} instead of C_{80} in the case of MY1.

Log-log plots for _{MY2} and the corresponding plot for _{MY1} (the white symbols).

At low temperatures of 418 and 318 K, Figure _{36}, _{MY2} vs _{self} of _{MY2} as the molecular weight of

We have observed two kinds of transition in the power law dependence of diffusion (_{MY2}∼M^{−γ} occurs at low temperatures of 418 K and 318 K where the exponent _{36} at 418 K and from 1.54 to 0.60 near C_{36} 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 M_{MY}/M_{matrix} 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 _{self} ∼ M^{−γ} occurs near C_{120}∼C_{160} 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 _{self}∼M^{−1} for _{C}.

The data used to support the findings of this study are included within _{self}) and the friction coefficients (_{self}) of liquid _{MY1} and _{MY1}) and MY2 (_{MY2} and _{MY2}) in liquid

The author declares that there are no conflicts of interest regarding the publication of this paper.

The author gratefully thanks Prof. J. C. Rasaiah for his criticism and comments on this paper.

The data used to support the findings of this study are included, which provide the self-diffusion coefficients (_{self}) and the friction coefficients (_{self}) of liquid _{MY1} and _{MY1}) and MY2 (_{MY2} and _{MY2}) in liquid