Abstract
Physical arguments and comparisons with published experimental data suggest that in simple liquids: (i) singlemoleculescale viscous forces are produced by temperaturedependent London dispersion forces, (ii) viscosity decay with increasing temperature reflects electron cloud compression and attendant suppression of electron screening, produced by increased nuclear agitation, and (iii) temperaturedependent selfdiffusion is driven by a narrow band of phonon frequencies lying at the lowfrequency end of the solidstatelike phonon spectrum. The results suggest that collisioninduced electron cloud distortion plays a decisive role in single molecule dynamics: (i) electron cloud compression produces shortlived repulsive states and single molecule, selfdiffusive hops, while (ii) shearinduced distortion generates viscosity and singlemoleculescale viscous drag. The results provide new insight into nonequilibrium molecular dynamics in nonpolar, nonmetallic liquids.
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Introduction
Gaining a deeper understanding of single molecule dynamics in liquids bears on fundamental problems in chemical reaction kinetics^{1,2}, subcellular water transport^{3,4} and biomass transfer^{5,6}, detection of cosmic particles and radiation^{7}, dark matter detection^{8,9,10}, detection of collision products in high energy physics^{7}, corrosion^{11}, and weathering of terrestrial and extraterrestrial surface rock^{12}. The problem has attracted the attention of luminaries like Einstein^{13,14}, Perrin^{15}, Laundau^{16}, Prigogine^{17}, and Feynman^{18}. Nevertheless, the physical mechanisms that determine single molecule motion in liquids remain poorly understood.
A variety of experimental and theoretical approaches have been developed for studying molecular dynamics in liquids. Experimental techniques include light and particle scattering^{19,20,21,22,23} which probes dynamic responses over single to multiplemolecule length scales, and subcollision and longer \( \left( t \ge \mathrm {O} \left( 10^{14} \ \mathrm {s} \right) \right) \) time scales. Photonic techniques^{24,25,26,27,28} are capable of exposing intramolecular dynamics on femtosecond time scales \( \left( \mathrm {t} = \mathrm {O} \left( 10^{15} \ \mathrm {s} \right) \right) . \) Molecular dynamics simulations provide a computational approach for probing each of these scales^{29,30,31}.
Theoretical modeling drives and allows interpretation of typically complicated experimental observations. Over multimolecule length scales and multiplecollision time scales, molecular hydrodynamics^{20,21,22,31,32,33,34} successfully connects observed spectral responses of dense fluids to the continuum NavierStokes (NS) equations. However, on singlemolecule length scales and collision and subcollision time scales, mapping molecularscale response into generalized NS models requires time and spacedependent transport coefficients^{20,22,33,34,35}, revealing our poor understanding of singlemolecule liquidstate dynamics.
For atomic, and small polyatomic, nominally spherical, nonpolar liquids, Langevin (LE) models^{22,33,34} provide a powerful, particlebased framework for tackling molecular dynamics problems, both under classical conditions—where the dynamical processes of interest take place on time scales exceeding the ’dispersion time scale’, \( \tau _d = O \left( 10^{16} \ \mathrm {s} \right) ,\) see below—and under conditions where quantum smearing of the dynamics becomes important^{33,36}. Generally speaking, LE models are suitable for particle dynamics problems characterized by shorttime scale random forcing and longertime scale nonrandom dynamical dissipation, as well as by possible external forcing.
This paper presents three results, which together, provide new insight into the dynamics of single molecules in nonpolar, nonmetallic liquids:
(a) A simple physical model is proposed which suggests that: (i) liquidstate viscosity is produced by temperaturedependent London dispersion forces, and (ii) viscosity decay with increasing temperature reflects decreased electron screening of nuclear charge. Comparison of predicted and experimentally observed viscosities^{22,37} for liquid Ne, Ar, Kr, Xe, \( \mathrm {N_2},\) \( \mathrm {O_2} ,\) and \( \mathrm {CH_4} ,\) support the proposed physical picture.
(b) A Langevin model of subcollision time scale, single molecule dynamics, which explicitly accounts for solidstatelike phonon modes, leads to a physically consistent explanation for self diffusion coefficients measured in liquid Ar, Kr, and Xe^{38}. The model suggests that on time scales ranging from the Frenkel scale, \( \tau _F = O \left( 10^{14} \ \mathrm {s} \right) , \) down to the fast dispersion scale, \( \tau _d = O \left( 10^{16} \ \mathrm {s} \right) , \) molecular dynamics in simple liquids is solidlike, and thus dominated by phonon modes, consistent with the equilibrium statistical mechanics picture presented by^{39,40}. In addition, the model indicates that the random diffusional hopping of individual molecules is produced by a narrow band of phonon modes lying near the lowfrequency end of the phonon spectrum, \( \omega _c \sim \omega _F = 2 \pi / \tau _F .\)
[Note, for context, \( \tau _F \) is approximately an order of magnitude shorter than the characteristic intermolecular collision time scale, \( \tau _c = O \left( 10^{13} \ \mathrm {s} \right) . \) In order to provide a physical feel for the important time scales in this problem, we will often use those associated with liquid Ar. In addition, the term ’molecular’ will refer to monatomic as well as small, polyatomic liquids.]
(c) A set of time scaledependent Langevin equations are proposed for describing single molecule dynamics in simple, nonpolar liquids. The equations apply over the poorly characterized subcollision time scale, \( \tau _d \lesssim t \lesssim \tau _c , \) incorporate the above results, and represent bestguess extrapolations of wellestablished dynamics on longer time scales.
As a consequence of the modeling, experimental comparisons, and consistency checks that are presented, we arrive at a preliminary picture of the decisive role apparently played by collisioninduced electron cloud distortion in single molecule dynamics. Specifically, arguments and evidence are presented suggesting that phononinduced electron cloud compression forces colliding molecular pairs into shortlived repulsive states, producing, in turn, single molecule, selfdiffusive hops. In addition, we propose that nonequilibrium, shearinduced, ’tangential’ electron cloud distortion generates viscosity and single molecule scale, resistive viscous forces. In order to support the hypothesis that viscosity arises on single molecule scales (SMS), the paper first derives SMS Navier–Stokes equations, describing the ensemble average, fieldbased dynamics of (Newtonian) SMS fluid systems. Given these, a scaling estimate of SMS viscosity, based on the GreenKubo viscosity relation^{35}, then leads to a parametrically correct solution for SMS viscosity, obtained in terms of the SMS shear stress (i.e., the ensemble average, SMS transverse current correlation^{22,34}).
Results, methods and discussion
Dispersion forces and electron screening determine temperaturedependent dynamic viscosity
As a measurable property determined by molecularscale processes, viscosity provides a window into molecular dynamics. Here, we study temperaturedependent viscosities observed in liquid Ne, Ar, Kr, Xe, \( \mathrm {N_2} ,\) \( \mathrm {O_2} ,\) and \( \mathrm {CH_4 }, \) at fixed pressures, over the temperature ranges on which each specie exists as a liquid^{22,37}.
The corresponding states principle^{22,41,42} provides the basis for our argument. The simplest form of the principle postulates that viscosity is determined by a characteristic intermolecular potential energy, \( \epsilon , \) a characteristic intermolecular length scale, \( \sigma , \) the molecular mass, M, and a speciedependent temperaturescale, \( \epsilon / k_B : \)
where, on dimensional grounds, M can be grouped with \( \epsilon . \) Dimensional analysis allows restatement of (1) in nondimensional form:
where \( {\tilde{\mu }} = \mu / \sqrt{M \epsilon } / \sigma ^{2} , \) \( {\tilde{T}} = T / \left( \epsilon / k_B \right) , \) \( k_B \) is Boltzmann’s constant, and \( {\tilde{f}} \) represents the experimentally determined correlation. The principle holds nominally well^{22,41} in nonpolar atomic and diatomic liquids that are wellmodeled by the Lennard–Jones potential^{22,41}. More comprehensive correlations incorporating quantum (low temperature and small mass) effects and information on the shape of the intermolecular potential have been proposed^{42}.
In order to derive what turns out to be a simple physical model for predicting viscosity in nonpolar liquids, we proceed in three steps. First, a scaling argument is used to place the corresponding states principle on a physical basis, leading to an approximate relationship for \( \mu : \)
Focusing on nonpolar liquids subject to London dispersion forces, we then follow^{41} and state the intermolecular energy, \( \epsilon ,\) in terms of specie polarizability, \( \alpha ,\) and the separation, \( r_{ab} , \) between colliding molecular pairs. Finally, collisioninduced, and temperaturedependent polarization is stated in terms a mean, temperaturedependent electron cloud distortion, \( \delta \sigma \left( T \right) . \) Importantly, in order to obtain a viscosity model consistent with available measurements^{22}, we propose that electron cloud distortion, \( \delta \sigma \left( T \right) ,\) decreases linearly with temperature.
Confine attention to classical conditions, assume pairwise intermolecular collisions—see Note (i) in the final section below, neglect nonspherical shape effects^{42} on the (pairwise) intermolecular potential, and focus on simple, nonmetallic liquids, i.e., those composed of nonpolar molecules having nominally spherical, localized electron distributions^{43}. Under these conditions, the attractive potential between colliding molecular pairs is wholly determined by London dispersion^{41,44}.
On time scales longer than the dispersion time scale, \( \tau _d = \mathrm {O} \left( 10^{16} \ \mathrm {s} \right) \)  the scale on which electron distributions oscillate^{45}  but shorter than the intermolecular collision time scale, \( \tau _c = \mathrm {O} \left( 10^{13} \ \mathrm {s} \right) , \) twobody interactions dominate three and higherorder interactions; again, see Note i) in the final section. Due to highfrequency phonon modes, \( \omega > 2 \pi \tau _F^{1} = \mathrm {O} \left( 10^{14} \ \mathrm {s^{1}} \right) , \) we assume that viscosity emerges on an intermediate time scale, \( \tau _v , \) where \( \tau _d<< \tau _v<< \tau _c . \) As shown by^{46}, a modified StokesEinstein relation,
connects diffusion of small and medium sized molecules (in water and carbon tetrachloride^{46}) to the viscosity of the solvent liquids. Here, D is the diffusion coefficient, \( \mu \) is the dynamic viscosity, \( f' \) is a molecularshapedependent factor, and \( n' \) is a correction factor, ranging from approximately 2 to 6, and accounting for a mix of slip and noslip flow conditions on a molecule’s surface.
Importantly, (4) implies that single molecule dynamics can be modeled using the simple, memory free Langevin equation:
where \( \sigma _m = n'f' \sigma / 6 \) is the effective molecular diameter; see, e.g.,^{47}. Here, \( {\mathbf {v}} \left( t \right) \) is the instantaneous velocity of the molecule and \( \mathbf {F_R} \left( t \right) \) is the instantaneous random force on the molecule. Thus, on time scales of order \( \tau _v = \mathrm {O} \left( 10^{15} \ \mathrm {s} \right) , \) and longer, we argue that the rate of work done on an individual molecule by the dispersion force is dissipated by viscous dissipation:
where spatial derivatives are approximated as \( 1 / \sigma , \) the characteristic speed of the nucleus is given by \( u_{nuc} \approx \sqrt{\epsilon / M }, \) the characteristic nuclear displacement over \( \tau _v \) is represented as \( \delta r_{nuc} , \) and the nominal surface area of the molecule is on the order of \( \sigma ^2 . \) Solving (6) for the viscosity then leads to (3).
Dispersion forces determine viscosity and increased nuclear agitation with temperature compress electron clouds, suppressing viscosity
In simple liquids subject to London interactions, the energy scale, \( \epsilon , \) is approximately determined by^{41,44}
where \( h \nu _o \) is the ground state energy of an isotropic quantum oscillator, \( \alpha \) is the polarizability, and \( r_{ab} \) is again the separation between the molecular pair’s nuclei. This expression follows from assuming that pairwise molecular collisions correspond to weak interactions between isotropic quantum oscillators^{41,44}. As an initial consistency check, Supplement 4 compares estimated and experimental kinematic viscosities, \( \nu = \mu / \rho , \) for a set of simple liquids, where \(\nu \) estimates use London’s rigorous second order quantum perturbation model^{41,44}, \( \epsilon = C / r_{{\mathscr {A}} {\mathscr {B}}}^6 \) in (3), and where C is the attractive constant.
Since (7) allows intuitive derivation and interpretation of the viscosity estimate presented here, as well as exposing the apparent central role of electron cloud distortion in viscosity generation, we use (7) to estimate \( \epsilon . \) Polarizability is given approximately by^{41}
where the sum is taken over the principle quantum energy levels of a given molecule, \( \overline{r_i^2} \) is the average squared displacement of the electrons in \( i^{th} \) shell (induced by an external electric field), and \(a_o \) is the Bohr radius. In detail, \( \overline{r_i^2}, \) follows from introduction of Slater orbitals^{41,48}:
where \( Z  S_i \) is the effective nuclear charge of the \( i^{th} \) shell, Z is the nuclear charge, \( S_i \) is the associated screening constant, and \( n_i^* \) the effective principle quantum number.
Focusing on Ne, Ar, Kr, Xe, \( \mathrm {O_2} , \) and \( \mathrm {N_2}, \) we label the sum of mean squared electron displacements as
where
represents the average collisioninduced distortion of all the electrons in a molecule, and where we assume that \( \delta \sigma \left( T \right) \) is temperaturedependent.
Over the narrow temperature ranges on which each of these species exist as a liquid, and based on the observation that liquid viscosities decrease with increasing temperature^{22}, we introduce an ansatz that the mean electron distortion decreases linearly with temperature:
where \( \delta \sigma _v = \delta \sigma \left( T_v \right) , \) is the mean displacement at characteristic temperature, \( T_v = \epsilon / k_B , \) and \( \epsilon _T = \left( T_v  T \right) / T_v .\)
Physically, and in light of (9) and the results below, this guess suggests that electron screening decreases with increasing temperature, consistent with behavior observed in deuterated metals^{49}. Since kinetic energy of both nuclei and electrons increase with rising temperature, where the latter presumably enhances screening, the suppression of screening apparently reflects increased nuclear agitation; intensifying agitation, under spatially packed conditions effectively thins surrounding electron clouds. A similar mechanism has recently been proposed in metallic glasses^{50}, and may underlie atomization of vapor phase molecular clusters as \( T \rightarrow T_v = \epsilon / k_B ,\) where atomization reflects nuclear kinetic energy overtaking intermolecular dispersion forces.
Next, approximate \( \delta \sigma \left( T \right) = \delta \sigma _v \left( 1  \epsilon _T \right) \) as \( \delta \sigma \left( T \right) = \delta \sigma _v \exp \left(  \epsilon _T \right) = \delta \sigma _v e^{1} \exp \left( T^* \right) , \) where \( T^* = T / T_v = T / \left( \epsilon / k_B \right) , \) is the dimensionless temperature defined in the corresponding states correlation. Since the maximum magnitude of \( \epsilon _T \) is on the order of 0.3 for the set of liquids considered, save oxygen, the maximum error introduced by replacing \( \left( 1  \epsilon _T \right) \) with \( \exp \left(  \epsilon _T \right) \) is on the order of 10 %.
Using \( \epsilon _d \) in (7) for \( \epsilon \) in (3), the definition in (11) for the mean electron distortion, and the exponential approximation above for the assumed linear temperature variation in \( \delta \sigma \left( T \right) ,\) leads to an approximate expression for the temperaturedependent viscosity for simple liquids:
where \( C_o = \sqrt{ 243 h \nu _o e^8 \left( \delta \sigma _v / \sigma _o \right) ^8 M /16 } / \sigma _o^2 ,\) and where \( a_o \) is approximated as \( \sigma _o /2 ,\) and \( \sigma _o \) corresponds, e.g., to the molecular diameter at the specie melting point.
Comparison of theoretical and observed viscosities
In order to allow comparison of theoretical, temperaturedependent viscosities, as given by (13), with experimentally measured viscosities, we define the dimensionless viscosity for liquid \( \kappa , \) \( \mu _{\kappa }^* \left( T^* \right) = \mu _{\kappa } \left( T^* \right) / \mu _{m, \kappa } ,\) yielding
where \( c_{\kappa }^* = \mu _{\kappa }^* \left( T_{min, \kappa }^* \right) \exp \left( 4 T_{min, \kappa }^* \right) , \) and where \( \mu _{\kappa }^* \left( T_{min, \kappa }^* \right) \) is the measured dimensionless viscosity for fluid \( \kappa \) at the minimum dimensionless temperature, \( T_{min, \kappa }^* \) at which \( \mu _{\kappa }^* \) is measured. Note, due to the approximations used to obtain the constant \( C_o \) above, plotted viscosity estimates obtained using (13) exhibit the appropriate decay with temperature, but are displaced by a (nominally) fixed magnitude from measured viscosities.
Theoretical and experimental, temperature dependent dimensionless viscosities for the six simple liquids considered here are compared in Figs. 1 and 2. The comparisons lend significant support to our central argument: Decaying liquid viscosity in simple liquids reflects decreased electron screening of the positive nucleus. A similar mechanism may underlie temperaturedependent decay in surface tension coefficients in simple liquids, and may also play a role in viscosity and surface tension variations in, e.g., polar and ionic liquids.
In closing this section, we cite the sources of the experimental viscosity data presented in Figs. 1 and 2, as well as estimated uncertainties, where available. Regarding the latter^{56}, reviews and correlates the data available through 2004, for liquid Ar, \( \mathrm {N_2} ,\) and \( \mathrm {O_2} . \) Using their temperature and pressuredependent correlations for each specie, they define the average absolute deviation, AAD, of any given data set from the correlation, by the (schematic) relationship: \( \mathrm {AAD} =  \left( X_{data}  X_{calc} \right)  / X_{calc} . \) Thus, their estimated AAD’s provide reasonable estimates for the uncertainty in any given experiment. References and estimated uncertainties for the seven sets of viscosity data shown in Figs. 1 and 2 are as folows: Ne^{51}; uncertainty not reported; Ar^{52}, AAD = 0.92 %; Kr^{53}, uncertainty not reported; Xe^{54}, uncertainty = 0.5 %; \( \mathrm {N_2} \)^{55}, AAD = 8.48 %; \( \mathrm {O_2} \)^{55}, AAD = 10.2 % ; \( \mathrm {CH_4} \)^{57}, uncertainty not reported.
Phonons and selfdiffusion
In pursuing our objective of developing a picture of single molecule dynamics, within the framework of Langevin’s equation (5), we confront two additional, connected questions: a) What is the physical origin of the random force. \( \mathbf {F_R} \left( t \right) \)? Typically, \( \mathbf {F_R} \left( t \right) \) is treated as a statistical entity, endowed with physically reasonable statistical properties^{35,36}. In liquidstate dynamics problems, this mathematical approach reflects our poor physical understanding of \( \mathbf {F_R} \left( t \right) .\) b) What is the origin of selfdiffusion, i.e., the random, thermallydriven motion of individual molecules though a liquid? Since \( \mathbf {F_R} \left( t \right) \) drives selfdiffusion, answering either question offers insight into both.
There are two significant experimental clues: a) A series of experiments, carried out in the 1970’s^{38}, measured the selfdiffusion coefficient, \( D_s = D_s \left( T, P \right) ,\) in liquid Ar, Kr, and Xe, over a range of temperatures and a series of fixed pressures, and lead to a (dimensionless) correlation of the following form:
where \( D_s^* = D_s / \sqrt{ \epsilon \sigma ^2 / M} , \) \( T^* = T / \left( \epsilon /k_B \right) , \) and \( P^*= P / \left( \epsilon / \sigma ^ 3 \right) .\) b) More recently, Bolmatov, Brazhkin and Trachenko^{39,40} presented strong evidence that temperaturedependent specific heats, in a large family of liquids, reflect existence of dominant, solidstatelike, equilibrium phonon modes.
Consider the solidlike dynamics of Nmolecule liquidstate systems, over time scales ranging from the Frenkel to the dispersion scales, \( \tau _{F} = 2 \pi / \omega _F = O \left( 10^{14} \ \mathrm {s} \right) \) to \( \tau _d = 2 \pi / \omega _d = O \left( 10^{16} \ \mathrm {s} \right) .\) Using a normal mode analysis  as in solidstate systems  under the assumption that individual molecular oscillations remain small enough to approximate intermolecular potentials as quadratic in the oscillation amplitude, 3N independent, vibrational, i.e., phonon modes are determined. The principal limitation of this model centers on ignoring the random hops of individual molecules. However, based on two separate arguments and results presented below, it appears that molecular hopping, over the spectral range \( \omega _F< \omega < \omega _{d} , \) is limited to a narrow, lowfrequency band of frequencies near the solid state limit: \( \omega \approx \omega _F .\)
On time scales on the order of, and shorter than \( \tau _F , \) and in the absence of singlemoleculescale external perturbations  like those produced by short wavelength neutron scattering beams  individual nuclei undergo small displacements, on the order of a small fraction of \( \sigma . \) Under these conditions, and in terms of the Langevin model—see Table 2—the friction force can be neglected and the dynamics of individual molecules modeled using:
This equation states that on these time scales, individual molecules are subject to the summed effect of all phonon modes extant over the spectrum, \( \omega _F \lesssim \omega \lesssim \omega _d .\) Specifically, the phonon mode having frequency \( \omega _i , \) induces an instantaneous nuclear velocity \( \tilde{{\mathbf {v}}}_i \left( \tau , \omega _i \right) ,\) which, in turn, produces a nuclear displacement  over the small time interval \( \left[ 0,t \right] \)  of \( \Delta {\mathbf {x}}_i \left( \tau , \omega _i \right) = \int _0^t \tilde{{\mathbf {v}}}_i \left( \tau , \omega _i \right) d \tau . \) Since phonon modes are independent, the small displacements, \( \Delta {\mathbf {x}}_i \left( \tau , \omega _i \right) ,\) are likewise. Thus, since \( M \omega _i^2 \Delta {\mathbf {x}}_i \left( t \right) \) corresponds to the \( i^{th} \) instantaneous spring force on the molecule, the sum of the random phononinduced forces corresponds to the right side of (16).
Derivation of the selfdiffusion coefficient, \( D_s \)
From Note f) in the final section, on the solidstatelike time scale, \( \tau _d \lesssim t \lesssim \tau _F ,\) the equation describing nuclear motion, driven by the \( j^{th} \) phonon mode, is given by:
Solving this leads to
Thus, the instantaneous velocity of the nucleus corresponds to the superposition of phononinduced velocity contributions:
so that the dot product, \( {\mathbf {v}} \left( t \right) \cdot {\mathbf {v}} \left( 0 \right) ,\) is given by:
Due to the independence of phonon modes, and in light of (18),
where \( {{\tilde{v}}^2_j} \left( 0; \omega _j \right) = {\tilde{\mathbf {v}}_j} \left( 0, \omega _j \right) \cdot {\tilde{\mathbf {v}}_j} \left( 0, \omega _j \right) .\)
The selfdiffusion coefficient, \( D_s ,\) is given by the integrated velocity autocorrelation function:
or
Integrating then gives:
In order to evaluate the equilibrium average: (i) recall that within a given volume, V, the average number of phonons having frequency \( \omega \) is given by^{58,59}
(ii) at any location in V, assume that the wave vector associated with each mode, over the ensemble, is isotropically oriented, and (iii) due to the nominally continuous distribution of modes, move to a continuum representation of the average in (24):
where \( g \left( \omega \right) \) is the density of modes driving selfdiffusion. Finally, in order to arrive at a theoretical \( D_s \) having the same generic structure as the empirical \( D_s \) in (15), we assume that the density of modes driving selfdiffusion is clustered around a critical frequency, \( \omega _c : \)
As described below, this assumption leads to a detailed, physically consistent explanation of phonondriven selfdiffusion in simple, nonpolar, nonmetallic liquids.
Using (27) in (26), approximating \( \exp {\beta \hslash \omega } 1 \) as \( \exp {\beta \hslash \omega } ,\) and nondimensionalizing using \( D_s^* = D_s / \sqrt{ \epsilon \sigma ^2 / M} , \) \( T^* = T / \left( \epsilon /k_B \right) , \) and \( P^*= P / \left( \epsilon / \sigma ^ 3 \right) , \) finally leads to:
where \( \alpha \) denotes either Ar, Kr, or Xe, and where two undetermined, pressuredependent parameters, \( \langle {\tilde{v}}^2 \left( 0, \omega _c \right) \rangle \) and \( \omega _{c, \alpha } ,\) appear. The first,
is the phononinduced, ensemble averaged, pressuredependent, squared velocity of the molecule, evaluated at the critical phonon frequency,
where \( \omega _c \) is the frequency that induces significant, singleatomscale, random jumps, i.e., selfdiffusion. The physical meaning of these parameters is explored in the next section. Note, approximating \( \exp {\beta \hslash \omega } 1 \) as \( \exp {\beta \hslash \omega } ,\)  again, introduced in order to arrive at a theoretical \( D_s \) having the same form as (15)  is based on the fact that, in liquid Ar, Kr, and Xe, \( \exp {\beta \hslash \omega } = O \left( 10 \right) .\)
Phononinduced hopping speeds and critical frequencies; comparisons with experimental selfdiffusion coefficients
In order to determine \( \langle {\tilde{v}}_{c,\alpha }^2 \rangle \) and \( \omega _{c,\alpha } , \) we use the experimental correlation^{38} (15), leading to
and
Comparisons of temperature and pressuredependent selfdiffusion coefficients, \( D_s^* \left( T^*, P^* \right) , \) predicted by the phononbased model, (28), with experimental measurements^{38} in liquid Ar, Kr, and Xe, are shown in Figs. 3, 4, and 5. Pressuredependent magnitudes of the critical phonon frequency, \( \omega _{c, \alpha },\) driving selfdiffusion, and the root mean square atomic speed, \( \sqrt{ \langle {\tilde{v}}_{c,\alpha }^2 \left( P^* \right) \rangle } , \) induced by these critical phonons, are listed in Table 1. Note that the slight apparent offset between the theoretical correlations and measured selfdiffusion coefficients, observed at \(P* =0.86 ,\) in Figs. 4 and 5, reflects nondimenisonalization of P, T, and \( D_s , \) by the same set of experimental \( \epsilon '\mathrm {s} \) and \( \sigma '\mathrm {s} \)^{41} used to nondimensionalize temperatures and viscosities in Figs. 1 and 2. See^{38} for references to the sources of the \( \epsilon '\mathrm {s} \) and \( \sigma '\mathrm {s} \) used in their study, and note that ca. 1962, variations in experimentally observed magnitudes of these parameters was on the order of 3 %^{38}.
In the next section, we present an alternative derivation of the selfdiffusion coefficient which again assumes that phonons underlie selfdiffusion, but which models the high frequency liquid state, \( \omega _F \lesssim \omega \lesssim \omega _d , \) as a bondfree Einstein solid in which all nuclei vibrate  in cages of surrounding molecules—at or near a fixed (Einstein) frequency, \( \Omega _o . \) This argument leads to a \( D_s^* ,\) (40), having the same generic form as (28), but derived from a significantly different physical viewpoint.
As a preliminary consistency check on the general picture of phonondriven selfdiffusion, leading to the semiempirical expressions for the characteristic hopping frequency, \(\omega _{c,\alpha } ,\) and speed of hopping molecules in \(\langle {\tilde{v}}_{c,\alpha }^2 \rangle , \) (31) and (32), respectively, we highlight the following points:
(a) For small to moderate reduced pressures, \( P^* = O \left( 1 \right) , \) (32) leads to the following approximate equality:
where \( D_{so, \alpha } \) is the scale of the selfdiffusion coefficient. By contrast, as a check on the steps leading from insertion of the integrated single molecule dynamics equation, (16), to the expression for \( D_s , \) written in the form:
the left side of (33) can be obtained by starting with the definition, \( D_{s, \alpha } = \int _0^{\infty } \langle {\mathbf {v}} \left( t' \right) \cdot {\mathbf {v}} \left( 0 \right) \rangle dt' , \) and replacing the upper limit with the characteristic time scale for singlemolecule hops, \( \tau _{hop, \alpha } = \omega _{c, \alpha }^{1} ,\) where the latter captures the assumed deltafunction density of hopinducing phonons near \( \omega _{c, \alpha } ,\) (27). This leads to \( D_{s, \alpha } = \langle {\mathbf {v}} \left( 0 \right) \cdot {\mathbf {v}} \left( 0 \right) \rangle \cdot \omega _{c , \alpha }^{1} = \langle {\tilde{v}}_{c, \alpha }^2 \rangle \cdot \omega _{c , \alpha }^{1} . \)
(b) For all three liquids, Ar, Kr and Xe, estimated critical phonon frequencies, \( \omega _{c, \alpha } \)—which we interpret as the characteristic hopping frequency—lie well within the range of frequencies, \( \omega _{d, \alpha }> \omega _{c, \alpha } > \omega _{F, \alpha } , \) where these liquids maintain solidlike properties^{40}. Estimated \( {\omega _{c, \alpha }}'s \) are approximately six times higher than estimated Frenkel frequencies^{39,40}, \( \omega _{c, \alpha } \approx 6 \omega _{F, \alpha } = 12 \pi G_{\infty , \alpha } / \mu _{\alpha } ,\) and approximately two orders of magnitude smaller than characteristic dispersion frequencies, \( \omega _{d, \alpha } ,\) where \( G_{\infty , \alpha } \) and \( \nu _{\alpha } \) are, respectively, the highfrequency shear modulus^{22,39,40} and dynamic viscosity of specie \( \alpha ,\) and where magnitudes of \( G_{\infty , \alpha } \) are obtained from^{39}, and magnitudes of \( \nu _{\alpha } \) are given in Supplement 4.
(c) Magnitudes of molecular hopping speeds, \( \sqrt{ \langle {\tilde{v}}_{c,\alpha }^2 \rangle }, \) exceed, by roughly a factor of two, both the longitudinal liquidstate sound speed^{60}, \( a_{liq} = \sqrt{ K / \rho } ,\) and the slightly faster longitudinal solidstate sound speed, \( a_{solid} = \sqrt{ a_L^2 + 4/3 a_S } ,\) where K is the bulk modulus and \( a_S = \sqrt{ G_{\infty } / \rho } \) is the shear (transverse) wave speed. Thus, average atomic hopping speeds are well in excess of characteristic liquid and solidstate sound speeds. Equivalently, from (31), the hopinducing phonon energy, \( \hslash \omega _{c, \alpha } \left( P^* \right) ,\) is approximately twice the intermolecular energy scale, \( \epsilon _{\alpha } ,\) and increases (linearly) with pressure.
A second picture of selfdiffusion: the highfrequency liquid state corresponds to a bondfree Einstein solid
The argument in the two previous sections leads to semiempirical expressions for a characteristic hopping frequency, \(\omega _{c,\alpha } ,\) and the speed of hopping molecules in \(\langle {\tilde{v}}_{c,\alpha }^2 \rangle , \) (31) and (32). In order to reinforce and broaden this argument, we outline a second derivation of \( D_s \) that again assumes that selfdiffusion is driven by liquidstate phonon modes. The second argument leads to an alternative, but not inconsistent picture of selfdiffusion: (i) High frequency molecular dynamics in nonpolar liquids, over \( \omega _F \lesssim \omega \lesssim \omega _d , \) can be modeled as those in a bondfree Einstein solid, in which each molecule vibrates as an independent oscillator, trapped in a cage of surrounding, nominally fixed molecules^{62}. (ii) As described below, selfdiffusive hops of individual, initially trapped molecules takes place when the molecule is pushed sufficiently far into a shortlived repulsive state with one of its neighbors.
Four observations provide the basis for this second picture. First, the velocity autocorrelation function (VACF) in LennardJones liquids, e.g., noble liquids, is largely determined by the repulsive part of the LJ potential^{63}. Second, consistent with the fact that repulsive states are shortlived, the short time scale normalized VACF for liquid argon, \( 0 \le t \lesssim \tau _c , \) is wellcaptured by a quadratic shorttime expansion^{22,64}:
where
is the liquidstate Einstein frequency, and where \( \psi \left( t \right) = \langle {\mathbf {v}} \left( t \right) \cdot {\mathbf {v}} \left( 0 \right) \rangle \left[ \langle {\mathbf {v}} \left( 0 \right) \cdot {\mathbf {v}} \left( 0 \right) \rangle \right] ^{1} . \) Third, over the liquidstate frequency spectrum, \( 0 \le \omega \lesssim \omega _F , \) \( \Omega _o \) can be interpreted as the characteristic oscillation frequency of individual liquid state molecules, trapped in a cage of surrounding molecules, all held in their equilibrium positions^{34}. Importantly, while characteristic \( \Omega _o ' \mathrm {s} \) have been calculated in LennardJones liquids^{30}, these assume MaxwellBoltzmann (MB) (kinetic) energy distributions, appropriate to the liquid state portion of the frequency spectrum. Apparently, similar calculations have not been reported over the solid state end of the spectrum, \( \omega _d \lesssim \omega \lesssim \omega _F ,\) where BoseEinstein statistics apply. Significantly, as shown immediately below, use of the liquidappropriate MB distribution in evaluating the averages in (36) leads to incorrect temperature scaling in \( D_s , \) whereas the solidappropriate BoseEinstein (BE) distribution provides the correct scaling. Fourth, assuming that wave speeds, \( v_i = \omega _i / k_i , \ i =1, 2,\ldots , 3N , \) of the 3N independent phonon modes are of the order of the longitudinal liquidstate or solidstate sound speed, \( a_{liq} \) or \( a_{solid} , \) respectively (where again, \( a_{liq} \approx a_{solid} \) ), we find that over the spectrum of available solidstatelike phonon frequencies, \( \omega _F \lesssim \omega _i \lesssim \omega _d , \) phonon wavelengths, \( \lambda _i, \) are all shorter than approximately \( 2 \sigma /3. \) Thus, in contrast to, e.g., crystalline solids, collective, multimolecule oscillations are nonexistent.
To begin the second derivation, truncate the expansion in (35) at the quadratic term, and approximate the quadratic as an exponential, \( 1  \frac{\Omega _{op}^2 t^2}{2!} = \exp \left(  \Omega _{op}^2 t^2 / 2 \right) , \) where \( \Omega _{op} \) denotes the Einstein frequency determined by use of the BE distribution in (36). In order to estimate the error in this approximation, first inspect approximate mean frequencydependent phonon occupation numbers, \( \langle n \left( \omega \right) \rangle = \left[ e^{\hslash \omega / k_B T }  1 \right] ^{1} : \) Doing so, we find that at a characteristic temperature of order \( 10^2 \ \mathrm {K} , \) \( \langle n \left( \omega _F \right) \rangle \sim 10^{5} , \) \( \langle n \left( 10 \omega _F \right) \rangle \sim 10^{44} , \) and \( \langle n \left( \omega _d \right) \rangle \sim \langle n \left( 10^2 \omega _F \right) \rangle \sim e^{1000} \rightarrow 0 .\) Thus, consistent with the first argument leading to (28) above, where it is assumed that the portion of the phonon spectrum driving selfdiffusion lies in the neighborhood of \( \omega _F , \) we assume that the solid statelike Einstein frequency, \( \Omega _{op} , \) is on the order of \( \omega _F . \) In turn, over phonon time scales, \( \tau _d \lesssim t \lesssim \tau _F , \) the relative error, \( 2 \Omega _{op}^2 t^2 / 4! , \) in replacing the quadratic in (35) with an exponential is, at most, on the order of 10 %.
Given the exponential approximation for the shorttime VACF in (35), \(D_s\) can now be expressed as
Importantly, if we assume that liquidstate dynamics determines selfdiffusion, then \( \langle {\mathbf {v}} \left( 0 \right) \cdot {\mathbf {v}} \left( 0 \right) \rangle = 3 k_b T /M, \) and, as noted in^{22} ,\( \Omega _{op} \propto T^{1/4} ;\) in this case, \( D_s \propto T^{3/4} , \) which is clearly inconsistent with the experimentally observed^{38} exponential dependence, \( D_s \propto \exp \left( A^2 / T \right) ,\) shown, for example, in (28) (where \( A^2 \) is a positive real number). By contrast, as shown in the first argument leading to (28) above, solidlike phonon dynamics do provide the appropriate scaling. Thus, repeating the analysis from (17) to (21), and setting \( t=0 ,\) we obtain
The fourth observation above, combined with the fact that \( \beta \hslash \omega \) is large over \( \omega _F \lesssim \omega \lesssim \omega _d , \) suggests that the Einstein model of solid state phonons^{58,62}  which captures incoherent solid state atomic dynamics  is an appropriate model for liquid state phonons. Thus, in order to evaluate \( \langle {\mathbf {v}} \left( 0 \right) \cdot {\mathbf {v}} \left( 0 \right) \rangle ,\) we repeat the integral in (26):
where now, the density of (short wavelength, noncollective) phonon states, \( g \left( \omega \right) ,\) has the same mathematical definition, \( g \left( \omega \right) = \delta \left( \omega  \Omega _{op} \right) , \) but a qualitatively distinct interpretation: Over the spectrum of exclusively short wavelength, noncollective phonon modes available to an \(N\mathrm {molecule} \) liquid state system, \( g \left( \omega \right) \) captures an assumed, nominally fixed density of phonon modes driving selfdiffusive hops of individual molecules. Assuming that the density of all phonon states  which encompasses the narrow band driving selfdiffusion  is reasonably well modeled by the Debye model^{58}, or more generally, by any other solid state phonon density model, then both interpretations of \( g \left( \omega \right) , \) here and above, are completely equivalent.
Again, since \( \beta \hslash \omega>> 1 , \) we can drop the negative one in the denominator in (39), carry out the integral, insert the result in (37), highlight, based on the measurements in^{38}, the temperature and pressuredependence of \( D_s , \) and nondimenisionalize as in the first derivation, to obtain a second (dimensionless) version of \( D_s :\)
where the ’critical phonon frequency’, \( \omega _c ,\) of phonondriven molecular hopping, is now interpreted as the solid state (as opposed to liquid state) Einstein frequency, and where all other terms have the same definitions as before.
Importantly, using two physically distinct arguments, we have derived the same expression for the diffusion coefficient, which, in both cases, has the appropriate \( \exp \left( A^2 /T \right) \) temperature scaling, and which  as described above and below  provides a physically consistent picture of the molecular scale mechanisms driving selfdiffusion.
Electron cloud compression, single molecule hopping, and shearinduced viscosity generation
The argument leading to the theoretical expression for dynamic viscosity, (14), is built on the idea that, in nonmetallic, nonpolar liquids, polarizability  the susceptibility of a molecule’s electron cloud to distort under the action of an external electric field  plays a dominant role in determining the friction force acting on the molecule. Although not apparent in the arguments leading to theoretical selfdiffusion coefficients in (28) and (40), the notion that electron cloud distortion also plays a dominant role here emerges from three observations. First, configurationaveraged instantaneous normal mode (INM) spectra in solid and liquidstate systems show that, at any instant, a significant fraction of interacting molecular pairs in liquids are in a state of mutual repulsion; in corresponding solids, only a small fraction of pairs are in such states^{65}. Second, as noted above, the VACF in LennardJones, e.g., noble liquids, is largely determined by the repulsive part of the LJ potential^{63}. Third, the argument leading to (40) exposes the notion that selfdiffusive hops of single molecules are somehow connected to prehop vibration of the molecule, vibrating (at the solidstate Einstein frequency) within a cage of surrounding molecules. Together, these observations suggest that selfdiffusive hops result from repulsioninducing compression of both the trapped molecule’s electron cloud, as well as that of one or more of its neighbors. [Again, Note (i) in the final section indicates that pairwise interactions are dominant.]
More generally, London dispersion forces comprise the only intermolecular force extant in nonmetallic, nonpolar liquids, and these, in turn, are wholly determined by variations in polarizability^{41}. Thus, building qualitative understanding of the essential connection between electron cloud distortion and single molecule viscous and selfdiffusion forces represents an important task. Computational chemistry and biology^{66} offer a number of techniques for visualizing distortion and modification of electron distributions that accompany various intermolecular interactions^{67,68}. Unfortunately, while dispersive interactions are ubiquitous and must be accommodated to accurately capture, e.g., solvationinduced changes in polyatomic molecular structure^{68}, development of predictive dispersion interaction models remains an open problem^{68}.
Visualization of electron cloud distortion accompanying selfdiffusion and viscosity emergence in liquids, including that in nonmetallic, nonpolar liquids, apparently has not been undertaken. In order to build qualitative understanding of the essential connection between cloud distortion and these fundamental dynamical processes, we introduce two parameters, \( x_{disp} \) and \( x_{diff} ,\) that respectively serve as rough indicators of the speciedependent electron cloud distortions that accompany selfdiffusion and emergence of viscosity.
Considering first electron cloud distortion associated with selfdiffusion, we imagine a trapped molecule, vibrating at or near the solidstate Einstein frequency, \( \Omega _{op} \left( = \omega _c \right) , \) in a cage of surrounding molecules. Again, as argued in the final section, Note i), on all time scales exceeding \( O \left( \tau _d \right) , \) pairwise intermolecular collisions dominate 3body and higherorder collisions. Writing Shrodinger’s equation for the trapped molecule, \( i \hslash \psi _{, t} + \hslash ^2 \nabla ^2 / \left( 2 M \right) = V \psi , \) estimating the scales of the two terms on the left side, where the time scale is determined by the Einstein (= the critical hopping) frequency, \( t \sim \Omega _{op}^{1} = \omega _{c}^{1} , \) we find that the length scale on which quantum uncertainty effects are important, \( x_Q = x_{DeBroglie} \sim \sqrt{ \hslash / 2 M \omega } , \) is small relative to the molecular diameter: \( \sigma : \) \( x_Q / \sigma \sim 10^{2} . \) Thus, at least in the vicinity of \(\omega \sim \Omega _{op} = \omega _{c}, \) nuclear motion is classical.
The apparent mechanism driving molecular hops is sketched in Fig. 6. Since nuclear motion on the \( \tau _F \) time scale is classical, we can apply the classical version of conservation of energy to the interaction between a fixed target (cage) molecule, \( {\mathscr {A}}, \) and a colliding (trapped, vibrating) molecule, \( {\mathscr {B}} .\) On approach toward \( {\mathscr {A}}, \) \( {\mathscr {B}} \) is assumed to have sufficient (relative) kinetic energy and (relative) momentum to allow \( {\mathscr {A}} \) and \( {\mathscr {B}} \) to enter a repulsive state. Applying conservation of energy to \( {\mathscr {B}} ,\) from the instant when maximum electron cloud compression occurs  and the relative velocity of \( {\mathscr {B}} \) is \( {\mathbf {0}} \)  to the instant when the intermolecular separation, \( r_{\mathscr {AB}} , \) equals the LJ potential minimizing separation, \( {\tilde{\sigma }} = 2^{1/6} \sigma ,\) we obtain:
where the intermolecular potential is dominated by repulsion, and where the equation represents the ensemble average dynamics of \( {\mathscr {A}} \) and \( {\mathscr {B}} \) for a single collision. Using \( \langle v_i^2 \rangle = 0 ,\) as well as the relationship \( \delta \sigma _{diff,1} = \left( {\tilde{\sigma }}  r_c \right) /2 ,\) then leads to an estimate for the fractional electron cloud compression, \( x_{diff,1} ,\) that produces single molecule hops:
Resolving the relative motion of \( {\mathscr {A}} \) and \( {\mathscr {B}} \) into normal and tangential directions, the minimum nuclear separation between \( {\mathscr {A}} \) and \( {\mathscr {B}} \) corresponds to \( r_c , \) and \( \delta \sigma _{diff} \) represents the maximum inward displacement, in the normal direction, of each colliding nucleus. See Fig. 6, D. Note, that while the LJ model is wellsuited to noble liquids, other pairwise potentials are available;^{69} highlights other possibilities, as well as limitations of the LJ model. Note too, in order to avoid overestimates of \( \sqrt{ \langle v_c^2 \rangle } \) (by a factor of \( \sqrt{3} ) , \) onedimensional diffusion coefficients are used throughout. For simple isotropic liquids, these can be measured, for example, by measuring displacements of isotopes in a given direction^{70}.
Assuming the validity of the simple energy conservation argument leading to (42), a second estimate of \( \delta \sigma _{diff} ,\) representing a direct connection between selfdiffusion and electron cloud compression, can be derived. First, as described above, evaluation of (34), leads to \( D_{s, \alpha } = \langle {\mathbf {v}} \left( 0 \right) \cdot {\mathbf {v}} \left( 0 \right) \rangle \cdot \omega _{c , \alpha }^{1} = \langle {\tilde{v}}_{c, \alpha }^2 \rangle \cdot \omega _{c , \alpha }^{1} . \) Since again, \( D_s \) scales as \( \sqrt{ \epsilon \sigma ^2 / M ,} \) , then \( \langle v_c^2 \rangle \sim \sqrt{ \epsilon \sigma ^2 / M ,} \omega _s . \) Using this approximation in (42), defining the characteristic frequency of collective, thermallydriven nuclear oscillations as \( \omega _{th} = \sqrt{ \epsilon / \left( M \sigma ^2 \right) } , \) using the fact that, in liquid Ar, Kr, and Xe, \( \omega _{th} = O \left( 10^{11} \mathrm {s^{1}} \right) , \) while \( \omega _c = O \left( 10^{13}  10^{14} \mathrm {s^{1}} \right) , \) and Taylor expanding the \( \left( \cdot \right) ^{1/12} \) term in (42) for small \( \omega _{th}/ \omega _c ,\) we obtain a second estimate for, \( x_{diff,1} ,\) which we label as \( x_{diff,2} :\)
Estimated, pressuredependent magnitudes of \( x_{diff,1} \) and \( x_{diff,2} \) for Liquid Ar, Kr and Xe, are plotted in Fig. 6. Both estimates assume fast, repulsiondriven, collisionless acceleration of a single hopping molecule, initially trapped and oscillating in a cage of surrounding molecules. However, while both estimates also connect \( \langle v_c^2 \rangle \) to the characteristic \( D_s \) scale \( \left( D_s \sim \sqrt{ \epsilon \sigma ^2 /M} \right) , \) the second estimate alone incorporates the critical hopping frequency, \( \omega _c . \) Thus, the consistency of both estimates lends support to the proposed picture of cloudcompressiondriven molecular hopping, as well as further support for the proposed general picture of phonondriven selfdiffusion. Note that the approximate 10 % gap between compression estimates observed for Ar (for \( P^* \ge 0.86 ) \) apparently reflects use of differing magnitudes of \( \epsilon , \) \(\sigma , \) and M, in this article^{41} and in^{38}. For example, depending on the technique used, empirical values of \( \epsilon \) vary by approximately 7 %^{41} .
Tangential electron cloud distortion and single moleculescale viscosity emergence
In liquid state physics, the singlemoleculescale mechanisms that trigger emergence of viscosity and viscous/frictional forces on individual molecules remains an open question. In this section and in the Supplement, focusing on nonpolar, nonmetallic liquids, we present evidence and arguments supporting the following hypotheses, each of which address this question.

(a)
Under the nonrestrictive condition where nonequilibrium microscale momentum currents are produced by continuum scale velocity gradients, viscosity emerges on singlemolecule length scales.

(b)
Under the same conditions, viscosity appears due to sustained tangential distortion of interacting electron clouds.

(c)
Viscosity emerges on time scales intermediate between the dispersion time scale, \( \tau _d = O \left( 10^{16} \mathrm {s} \right) ,\) and the Frenkel scale, \( \tau _F = O \left( 10^{14} \mathrm {s} \right) .\)
Considering hypothesis (a), we first show (Supplement 1) that the Navier–Stokes equations, describing the collective, ensemble average dynamics of \( N\mathrm {molecule}\) (Newtonian) fluid systems can be adapted to singlemoleculescale (SMS) systems. This step provides the necessary theoretical framework for defining an SMS viscosity. Once this step is completed, we then show (Supplement 2)—here in preliminary fashion—that the Green–Kubo viscosity relation^{22,33,35}, applied to singlemoleculescale (ensemble average) variations in the transverse momentum current leads to a parametrically correct relationship between the transverse momentum current correlation function, \( \left\langle P_{xy} \left( 0 \right) P_{xy} \left( t \right) \right\rangle , \) and the dynamic viscosity:
where the meaning of all terms is given in Supplement 2. Together, these steps provide strong evidence that dynamic viscosity, at least in nonpolar liquids satisfying Newtonian constitutive relationships^{33,71}, emerges, and can be defined on single molecule length scales.
The argument supporting hypothesis (b), proceeds as follows. Again, in nonpolar, nonmetallic liquids, London dispersion constitutes the only intermolecular force extant. While dispersion arises as a quantum mechanical perturbation to initially independent ground state wave functions^{41,44}, as described above, London’s model can also be couched in terms of molecular polarizabilities, i.e., electron cloud distortions^{41,44}. Importantly, this picture underlies the simple model used to derive theoretical viscosities, (14), above. Thus, in light of the estimates of electron cloud compression, \( x_{diff,1} \) and \( x_{diff,2} \)  which provide physically distinct, but selfconsistent evidence of cloudcompressioninduced selfdiffusion—we hypothesize that tangential electron cloud distortion underlies the emergence of viscosity. A schematic representation of this hypothesis is shown in Fig. 7B–D.
As a rough quantitative test of this picture, we estimate relative electron cloud distortions, \( x_{disp} , \) induced by persistent nonequilibrium shear stress. Given that surrounding (bath) molecules exist in a state of nearequilibrium, at temperature T and mean kinetic energy, \( 3 k_B T / 2 , \) we anticipate that relative tangential cloud distortions should be of the same order of magnitude as normal distortions, \( x_{diff,1} \) and \( x_{diff,2} ,\) estimated above. Thus, combining (8) and (11), we arrive at:
As shown in Fig. 7, estimated relative, shearinduced electron cloud distortion, for a number of noble and diatomic liquids, are approximately of the same magnitude as \( x_{diff,1} \) and \( x_{diff,2} .\) Dynamically, due to the cohesive nature of viscosity, shearinduced separation of interacting nuclear pairs, \( r_{\mathscr {AB}}, \) exceeds the potentialminimizing separation, \( 2^{1/6} \sigma . \) Finally, note that use of the definition of \( \delta \sigma ,\) given by (11), means that we are approximating the set of leveldependent mean squared electron displacements, \( \overline{r_i^2} , \) as \( \overline{r_1^2} \)^{41}; thus, plotted magnitudes of \( x_{disp} \) represent slight underestimates.
The argument supporting hypothesis c) is presented as Note b) in the final section. We note that derivation^{71} of the singlemoleculescale Stokes (viscous) drag force, D, that appears in the modified Stoke’s–Einstein relation^{46}, (4), requires use of the SMS NavierStokes equations derived in Supplement 1.
In closing this section, we remark that while computational techniques^{66,68,72,73} can be used to test the proposed connections between normal and tangential electron cloud distortion and selfdiffusion and viscosity emergence, we argue that simple, approximate models like those proposed here, are valuable in providing physical insight into the mechanisms that drive selfdiffusion and viscosity emergence.
Time scaledependent models of single molecule dynamics
For nonpolar, spherical atomic liquids like Ar, Kr, and Xe, as well as nonspherical, nonpolar liquids like those examined in^{46}, we propose that single molecule dynamics can be modeled on three distinct time scales: (a) over \( \tau _d \lesssim t \lesssim \tau _F , \) dynamics are solidstatelike and either dissipative or not—see below; (b) over \( \tau _F \lesssim t \lesssim \tau _c , \) dynamics are a mixture of solid and fluidlike and again, either dissipative or not; and c) for \( t > \tau _c , \) dynamics are dissipative and fluidlike. We denote these time scales, respectively, as the solidlike, transitional, and fluidlike regimes.
For clarity, we tabulate in Table 2 the various forms that the single molecule dynamics equation can take. The following general points are highlighted:
(a) Equation structure: The proposed equations are physically explicit versions of the memoryfree and generalized Langevin equation^{22,33,35}:
where the instantaneous molecular force is decomposed into a timedependent external force, \( \mathbf {F_e} \left( t \right) , \) necessary for modeling, e.g., particle scattering problems, a deterministic friction force, \( \mathbf {F_f} \left( t \right) , \) either \(  \int _0^t \kappa \left( t  t' \right) {\mathbf {v}} \left( t' \right) dt' \) or \(  3 \pi \sigma _m \mu {\mathbf {v}} \left( t \right) , \) and a random force, \( \mathbf {F_R} \left( t \right) , \) either determined by the phonon field, \( M \sum _{i=1} \omega _i^2 \int _0^t \tilde{{\mathbf {v}}}_i \left( t , \tau \right) d \tau , \) or by the thermal motion of surrounding molecules, \( \dot{\varvec{\eta }} \left( t \right) .\)
(b) Friction force: The set of equations contrasts the qualitatively distinct dynamics that exist under quiescent conditions, when the continuumscale liquid flows or is stationary, versus the highly dynamic state extant when the target molecule interacts directly with, or lies near an externally introduced particle. In the first case, based on the observation that small molecules follow a slightly modified version of the StokesEinstein drag force law^{46}, we assume that the friction force can be expressed as \(  3 \pi \sigma _m \mu {\mathbf {v}} \left( t \right) , \) where again, \( \sigma _m \) is an effective, shapedependent molecular diameter. This assumption, in turn, assumes that the dynamic viscosity, \( \mu , \) emerges on time scales that are long relative to the fast disperion time scale, \( \tau _d , \) but short relative to the solidliquid crossover time scale, \( \tau _F . \) Based on the observation that temperaturedependent viscosities are wellpredicted by the above model incorporating fastacting dispersion forces, this appears to be a reasonable assumption. Under conditions where, e.g., scattering particles interact with or near the target molecule, numerous experimental observations show that the friction force is history dependent^{20,21,22,33}.
(c) Connecting the phononinduced force to the Brownian force: Kubo’s analysis^{35} can be adapted to show explicitly how the phononinduced force on \( \tau _d \lesssim t \lesssim \tau _F \) can be represented as a Brownian force on \( t \gtrsim \tau _c : \) i) Express the instantaneous phononinduced force (in any of three orthogonal directions) as \( F_{phonon} \left( t \right) = M \sum _{i=1}^n \omega _i^2 \Delta x_i , \) where \( \Delta x_i \left( t \right) = \int _0^t \tilde{{\mathbf {v}}}_i \left( t , \tau \right) d \tau , \) is the nuclear displacement produced by phonon mode i, and \( \omega _i \) the \( \mathrm {i^{th}} \) normal mode frequency; ii) recognize, by (normal mode) construction, that on \( \tau _d \lesssim t \lesssim \tau _F ,\) all \( \Delta x_i ' \mathrm {s} \) are zeromean, independent random displacements; iii) define the sum of displacement variances as \( s_n^2 = \sum _{i=1}^n \sigma _i^2 ,\) where \( \sigma _i^2 = \langle \Delta x_i^2 \rangle \) is the \( \mathrm {i^{th}} \) variance; iv) focusing on time scales on the order of \( \tau _c \) and longer, define a random variable \( Y_n \left( t \right) = F_{phonon} \left( t \right) / s_n ,\) where \( t = \mathrm {O} \left( \tau _c \right) ; \) v) allow n to become large, which corresponds to binning all of the random phononinduced forces acting on \( \tau _d \lesssim t \lesssim \tau _F ;\) vi) by the central limit theorem, the probability density for \( Y_n \left( t \right) \) approaches a Gaussian density, \( p \left( Y \left( t \right) \right) \rightarrow \frac{1}{\sqrt{2 \pi } } \exp { \frac{ Y^2 }{2}} ;\) and vii) on \( t = \mathrm {O} \left( \tau _c \right) , \) argue that phononinduced force components (in each of three orthogonal directions) are deltacorrelated, \( \langle F_{phonon} \left( t \right) F_{phonon} \left( t' \right) \rangle = F_o^2 \delta \left( t  t' \right) , \) where \( F_o^2 \) is the force intensity. In Table 2, this guassian, deltacorrelated remnant of the phononinduced force is labeled as \( {\dot{\varvec{\eta }}} \left( t \right) . \)
(d) The random force, \( \mathbf {F_R} \left( t \right) :\) For the solidlike regime, the arguments from the previous section provide, we believe, substantial support for expressing the random force as \( \mathbf {F_R} \left( t \right) = M \sum _{i=1} \omega _i^2 \int _0^t \tilde{{\mathbf {v}}}_i \left( t , \tau \right) d \tau . \) For the fluidlike regime, the fact that the modified StokesEinstein relation, (4) holds for a large family of molecules^{46}, where again (4) is derivable from the memoryfree Langevin equation (5)^{47}, suggests that \( \mathbf {F_R} \left( t \right) = {\dot{\varvec{\eta }}} \left( t \right) .\) Proposing a reasonable form of \( \mathbf {F_R} \left( t \right) \) over the transition regime, \( \tau _F \lesssim t \lesssim \tau _c , \) remains problematic at this point, however. A mathematically simple assumption, which may not be physically valid, would model \( \mathbf {F_R} \left( t \right) \) as a linear superposition of \( \sum _{i=1} \omega _i^2 \int _0^t \tilde{{\mathbf {v}}}_i \left( t , \tau \right) d \tau \) and \( {\dot{\varvec{\eta }}} \left( t \right) . \) This is an open question, however.
(e) The external force: An external force term only appears for problems in which the spatial scale of the external agent, e.g., a scattering particle or a highenergy photon source (having wavelength on the order of \( \sigma \) or smaller), is on the order of the molecular diameter, \( \sigma . \) To account for such forces, a quantum mechanical model of the interaction is typically required; see, e.g.,^{20,21}.
(f) Physical meaning of the phononinduced force: Over the solidstatelike time scale, \( \tau _d \lesssim t \lesssim \tau _F , \) under conditions where moleculescale external forcing is absent, the phonon field determines: (i) each molecule’s instantaneous velocity, \( {\mathbf {v}} \left( t \right) = \sum _{i} {\tilde{\mathbf {v}}_j} \left( t, \omega _j \right) , \) as well as ii) the instantaneous random force, \( \mathbf {F_R} \left( t \right) =  M \sum _j \omega _j^2 \Delta {\tilde{\mathbf {x}}_j} \left( t \right) =  M \sum _j \omega _j^2 \int _0^t {\tilde{\mathbf {v}}_j} \left( t', \omega _j \right) dt' . \) Thus, the dynamics of individual nuclei: i) can be decomposed into individual contributions produced by each phonon mode: \( M \dot{{\tilde{\mathbf {v}}}_j} =  M \omega _j^2 \int _0^t {\tilde{\mathbf {v}}_j} \left( t', \omega _j \right) dt' ,\) or ii) taken as the resultant of these modes: \( M \dot{{\mathbf {v}}} =  M \sum _j \omega _j^2 \int _0^t {\tilde{\mathbf {v}}_j} \left( t', \omega _j \right) dt' ,\) where time derivatives, denoted by dots, are taken with respect t, on the solid state time scale.
(g) On the weak coupling between continuum scale flow and microscale dynamics: A scaling argument shows that only under extreme circumstances can continuum flow fields produce nonnegligible microscale nonequilibrium mass, momentum and energy currents. Consider, for example, turbulent flow over a mirrorsmooth surface (having asperities on the order of, say, \( 10^{9} \ \mathrm {m} ).\) Taking the ratio of the maximum continuumscale viscous shear stress, evaluated at the surface, \( \tau _{cont} \approx 0.02 \rho U_{\infty }^2 Re_{\delta }^{1/4} \)^{74}, to the characteristic molecularscale shear stress, \( \tau _{molec} \approx 10 \mu a / \sigma , \) leads to \( \tau _{cont} / \tau _{molec} \sim 0.002 U_{\infty } Ma \sigma , \) where \( Re_{\delta } = \rho U_{\infty } \delta / \mu \sim 1 \) is the Reynolds number associated with a turbulent boundary layer of thickness, \( \delta , \) \( U_{\infty } \) is the speed of the flow external to the boundary layer, and \( Ma = U_{\infty } / a ,\) is the associated Mach number. Here, \( \tau _{molec} ,\) which is determined by the transverse momentum current^{22,35}, is most easily estimated using the Stoke’s drag law, \( F_{drag} \approx 3 \pi \sigma \mu a \)^{71}, where the molecular speed is approximated as the sound speed, a. Using the Mach number magnitude, \( Ma \sim 0.3 , \) separating nominally incompressible and compressible flow, leads to the condition: \( \tau _{cont} \sim \tau _{molec} \) when \( U_{\infty } \sigma / nu \sim 10^3 .\) Due the small magnitude of \( \sigma \) for small molecular species, it is found, for Ar, Kr, and Xe, that \( U_{\infty } \) must be on the order of \( 10^6 \ \mathrm {m/s} ,\) or higher for continuumscale dynamics to manifest itself in microscale dynamics.
(h) Development of short time scale collective dynamics models: Under the assumption that fastacting dispersion forces mediate collective dynamics over the elastic, transition, and fluidlike regimes, sum rules^{20,22,33,35} provide a powerful tool for developing hydrodynamic models appropriate to each time scale. Supplement 5 illustrates using a simplified, i.e., nonviscoelastic Navier Stokes model of transition regime collective dynamics. The strategy consists of two steps: (1) propose a model of shorttime scale (ensemble average) molecular hydrodynamics, and (2) constrain the model by satisfying sum rules.
(i) Dominance of pairwise interactions: In many single molecule dynamics problems, as well as in derivation of fieldbased continuum dynamics models^{31,33}, it is important to have solid understanding of the relative importance of simultaneous multimolecule collisions. At any instant, on any time scale exceeding \( \tau _d , \) consider a target (nonpolar, liquidstate) molecule, \( \mathscr {A,} \) surrounded by a set of neighboring molecules, \( {\mathscr {B}}_1 , {\mathscr {B}}_2 ,\ldots , {\mathscr {B}}_m . \) Since the weak dispersive potential, \( \phi ^{(AB_i)} , \) that appears during collision of \( {\mathscr {A}} \) and \( {\mathscr {B}}_i , \) is small relative to the ground state energy, \( E_A^{(0)} + E_{B_i}^{(0)} , \) of adjacent, but unperturbed \( {\mathscr {A}} \) and \( {\mathscr {B}}_i , \) the London collision model^{41,44} is linear and can be readily modified by superposition to account for nbody interactions in which \( {\mathscr {A}} \) simultaneously experiences dispersive interactions with n neighboring molecules. Generalizing Hirschfelder^{41} by assuming a perturbed wave function that is the product of the n unperturbed, isolated wave functions for n colliding molecules, it is readily shown that the approximate, secondorder, dispersive potential has the form,
where \( E_I , \) an empirical constant, is on the order of the ionization energy, \( \alpha \) is the polarizability, \( \mathbf {r_{Ai}} \) is the internuclear distance between molecules \( {\mathscr {A}} \) and \( {\mathscr {B}}_i , \) and \( m=n1 .\)
A ’simultaneous nbody collision’ takes place when the internuclear distances between \( {\mathscr {A}} \) and \(n1\) immediately adjacent molecules are all approximately equal to the minimum of these distances, \( \mathbf {r_{A1}} \approx \mathbf {r_{A2}} \approx ... \approx \mathbf {r_{Ai_{min}}}. \) Writing \( \mathbf {r_{Aj}} = \mathbf {r_{Ai_{min}}} + \Delta \mathbf {r_{Aj}} ,\) forming the ratio \( \mathbf {r_{Aj}} / \mathbf {r_{Ai_{min}}} ,\) and Taylor expanding \( \mathbf {r_{Aj}} , \) we see that for an nbody collision to occur  corresponding to n nonnegligible contributions to \( \phi ^{(n)} \left( \mathbf {r_A} , \mathbf {r_2} ,\ldots , \mathbf {r_{n1}} \right) \) all \( n1 \) molecules must remain within approximately \( 16 \% \) of the minimum separation, \( \mathbf {r_{Ai_{min}}}. \) Thus, while threebody collisions certainly take place, for example, due to this restrictive condition, pairwise collisions dominate. Predicted dynamic viscosities above, which assume dominant pairwise collisions, are consistent with this simple argument.
Conclusions
Unraveling the dynamics of individual atoms and small molecules in liquids represents a centuriesold physics problem. While neutron and lightscattering experiments, as well as molecular dynamics simulations, instantaneous normal mode analyses, and molecular hydrodynamics expose and explain singleparticlescale and collective liquidstate dynamics, the descriptions are largely couched in terms of dynamical correlation functions. In an attempt to expose the essential dynamical elements that determine single molecule motion, at least in nonpolar, nonmetallic liquids, this paper presents physical arguments that suggest: (i) intermolecular dispersion forces and temperaturedependent electron screening determine viscosity, i.e., temperaturedependent intermolecular friction forces, and (ii) a narrow band of phonons, lying near the liquidsolid (Frenkel) transition frequency, drives the random molecular jumps constituting selfdiffusion.
In mechanistic terms, we present preliminary evidence that, in simple liquids, both viscosity and single molecule viscous drag emerge due to small, collisioninduced tangential distortions of individual electron clouds. By contrast, selfdiffusional, singlemolecule hops are produced by collisioninduced compression of interacting molecular clouds; the latter mechanism pushes interacting molecular pairs into shortlived repulsive energy states.
We are hopeful that the preliminary picture of single molecule, liquid state dynamics proposed here promotes further progress in understanding this complex problem.
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This material is based upon research supported by, or in part by, a grant from the US Office of Naval Research under award number N000141812754.
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R.G.K. conceived the models and wrote the paper. J.D. and P.T. located and analyzed previous related work, checked model arguments and calculations, prepared plots, and edited the final manuscript. All authors reviewed the manuscript.
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Keanini, R.G., Dahlberg, J. & Tkacik, P.T. On the physical mechanisms underlying single molecule dynamics in simple liquids. Sci Rep 11, 2528 (2021). https://doi.org/10.1038/s41598021821128
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DOI: https://doi.org/10.1038/s41598021821128
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