ABSTRACT (original article): The velocity autocorrelation function (VACF) encapsulates extensive information about a fluid’s molecular-structural and hydrodynamic properties. We address the following fundamental question: How well can a purely hydrodynamic description recover the molecular features of a fluid as exhibited by the VACF? To this end, we formulate a bona fide hydrodynamic theory of the tagged-particle VACF for simple fluids. Our approach is distinguished from previous efforts in two key ways: collective hydrodynamic modes and tagged-particle self-motion are modeled by linear hydrodynamic equations; the fluid’s spatial velocity power spectrum is identified as a necessary initial condition for the momentum current correlation. This formulation leads to a natural physical interpretation of the VACF as a superposition of products of quasinormal hydrodynamic modes weighted commensurately with the spatial velocity power spectrum, the latter of which appears to physically bridge continuum hydrodynamical behavior and discrete-particle kinetics. The methodology yields VACF calculations quantitatively on par with existing approaches for liquid noble gases and alkali metals. Furthermore, we obtain a new, hydrodynamic form of the self-intermediate scattering function whose description has been extended to low densities where the Schmidt number is of order unity; various calculations are performed for gaseous and supercritical argon to support the general validity of the theory. Excellent quantitative agreement is obtained with recent MD calculations for a dense supercritical Lennard-Jones fluid. CITATION (original article): S. L. Seyler and C. E. Seyler, “Molecular hydrodynamic theory of the velocity autocorrelation function,” The Journal of Chemical Physics, 159(5), 2023 054108. https://doi.org/10.1063/5.0153649
Introduction
Introduction
This Wolfram notebook that provides the full set of necessary data inputs, formulas, and calculation protocol(s) to compute the velocity autocorrelation function (VACF) for simple fluids, either liquid or gas (or supercritical fluids), using the hydrodynamic VACF formulation and framework described in S.L. Seyler & C.E. Seyler. Molecular hydrodynamic theory of the velocity autocorrelation. J Chem Phys (2023). doi:10.1063/5.0153649. The code performs the numerical k-integration indicated in Eq. (7) using the correlation functions defined in Eqs. (13), (14), and (17) and a selectable form of the static spectrum given by either Eqs. (20) or (21). The various parameters appearing in these functions are determined by the set of requirements given in Table I, with specific examples listed in Table II for the liquid cases and in Secs. S3 and S5 of the supplementary material for the gas and supercritical cases; these parameter inputs produced the VACFs in Figs. 1–3 in the article, as well as Figs. S2 and S3 in the supplementary material. Additional notes on how input data and model parameters are determined are provided in the Mathematica code comments.
General Setup and Auxiliary Functions
General Setup and Auxiliary Functions
Base units and input parameter conversion
Base units and input parameter conversion
Define working base units
Define working base units
Preprocessing
Preprocessing
Synopsis of preprocessing functions
Synopsis of preprocessing functions
The functions in this subsection essentially deal with the translation of thermodynamic (, ) and transport inputs (, ) from MD simulation and/or experimental data into the various parameters (, , , , , etc.) that appear in the DR10 hydrodynamic model. These functions are automatically called by the user-facing function VACF when the entire notebook is evaluated from the drop-down menu “Evaluation,” though they may be used individually to inspect the parameters in detail.
ρ
T
η
D
a
υ
γ
γ
s
Define (unnormalized) static spectrum functions and moments
Define (unnormalized) static spectrum functions and moments
Note on calculations that require calculation of thm moment of q(ka): ∞∫02+mkq(ka)dk
Note on calculations that require calculation of moment of : q(ka)dk
th
m
q(ka)
∞
∫
0
2+m
k
The purpose of the function Μ[q,p,a,m] is to evaluate the -moments of the static spectrum, . For example, setting corresponds to the second moment, which is used to match the leading-order term from the short-time VACF expansion; i.e., this is the second frequency-moment condition, req. (5) of the desiderata in Table 1, which we use to determine the value of the molecular parameter in the case of a simple liquid. See the definition of RescaledParameters below for the rescaling calculation.
k
q(ka)
m=2
a
Analytic integrations for generalized Gaussian and generalized Lorentzian static spectrum functions that enter into the Green-Kubo integral for enforcing the zero-frequency sum rule, req. (7). The calculation of the “negative 2nd” moment of the spectrum, i.e., Μ[p,1,-2] comes from needing to calculate one of the terms that results from doing the Green-Kubo time integral of the current correlation functions, i.e., . The other term from the time integration is simply proportional to , the regularization diffusion coefficient. What’s calculated below the result of normalizing the calculation by Μ[p,1,0].
The functional forms given in Eqs. (C6) and (C7) are obtained below for the generalized Gaussian and Lorentzian forms of, respectively.
υ/
2
k
The functional forms given in Eqs. (C6) and (C7) are obtained below for the generalized Gaussian and Lorentzian forms of
q(ka)
In[]:=
(* For generalized Gaussian *)q[k_,p_,a_]:=
-
1
2
p
(k a)
Μ[q,p,1,-2]
Μ[q,p,1,0]
Out[]=
-1/p
4
1
p
Gamma
3
p
In[]:=
(* For generalized Lorentzian *)q[k_,p_,a_]:=
1
1+
p
(k a)
Μ[q,p,1,-2]
Μ[q,p,1,0]
Out[]=
Populate remaining input parameters
Populate remaining input parameters
Compute molecular parameters
Compute molecular parameters
Model Parameters Check
Model Parameters Check
Liquid rubidium
Liquid rubidium
In[]:=
usRb=MakeUnitSystem<|"M"->,"L"->,"T"->|>;inputDataRbLiq=<|"m"->,"ρ"->,"T"->,"η"->,"ηb"->,"σ"->,"Ds"->,"ωE"->,"Γ"->1.8|>;
In[]:=
paramsRbLiq=GenerateParameters[inputDataRbLiq,qGL,14,"KineticModel"->"Einstein","LongitudinalDiffusionFactor"->-1];paramsRbLiq//Dataset[#]&NondimensionalizeParameters[paramsRbLiq,usRb]//Dataset[#]&
Out[]=
Out[]=
Liquid argon
Liquid argon
Gaseous argon
Gaseous argon
SC argon
SC argon
Dense supercritical LJ
Dense supercritical LJ
Correlation Functions
Correlation Functions
Hydrodynamic VACF
Hydrodynamic VACF
Description
Description
The functions in this section essentially deal with the translation of thermodynamic (, ) and transport inputs (, ) from MD simulation and/or experimental data into the various parameters (, , , , etc.) that appear in the DR10 hydrodynamic model. These functions are automatically called by the function VACF when simply running the entire notebook, though they may be used individually to inspect the parameters in detail. The VACF function:
ρ
T
η
D
a
υ
γ
1
.Accepts thermodynamic/transport data set, inputs, as an Association along with the shape parameter of the generalized Gaussian distribution used to model the static spectrum, .
p
q(k)
2
.Accepts an optional parameter “BaseUnits” that allows the user to select a unit set. This option expects an Association that specifies the characteristic mass (“M”), length (“L”), and energy (“E”) as input. Lennard-Jones reduced units for liquid argon are used by default.
3
.Computes, from the thermodynamic/transport data, derived parameters and adds them to the input data set.
4
.Derives the molecular parameters from the user-specified VACF characteristic decay time (i.e., the inverse Einstein frequency) using the rescaling procedure defined in the manuscript.
τ
5
.Inserts the correct parameters into the numerical -space integration functions (currying) and returns an (unevaluated) function of time representing either the full (NQ), longitudinal (NQL) or transverse (NQT) VACF.
k
Hydrodynamic VACF – numerical k-space integration
Hydrodynamic VACF – numerical -space integration
k
User-facing VACF Calculation Function
User-facing VACF Calculation Function
Plot: Liquid rubidium (Fig. 1A)
Plot: Liquid rubidium (Fig. 1A)
Liquid rubidium viscosity data
Liquid rubidium viscosity data
◼
Viscosity data taken from: Andrade, E. N. D. C. & Dobbs, E. R. The viscosities of liquid lithium, rubidium and caesium. Proc. R. Soc. Lond. A Math. Phys. Sci. 211, 12–30 (1952) at C.
o
45
◼
Diffusion coefficient taken from: Van Loef, J. J. Atomic and electronic transport properties and the molar volume of monatomic liquids. Physica 75, 115–132 (1974)
D
s
Define plot settings
Define plot settings
Compute and Plot VACF
Compute and Plot VACF
Plot: Liquid argon (Fig. 1B and Fig. S2)
Plot: Liquid argon (Fig. 1B and Fig. S2)
Liquid argon (bulk) viscosity data
Liquid argon (bulk) viscosity data
Compute and Plot VACF (Fig. 1B)
Compute and Plot VACF (Fig. 1B)
Supplementary Material: parameter tweak to obtain better agreement with “plateau” (Fig. S2)
Supplementary Material: parameter tweak to obtain better agreement with “plateau” (Fig. S2)
Generalized Gaussian (Fig. S2A)
Generalized Gaussian (Fig. S2A)
Generalized Lorentzian (Fig. S2B)
Generalized Lorentzian (Fig. S2B)
Plot: Gaseous argon (Fig. 2A)
Plot: Gaseous argon (Fig. 2A)
Compute and Plot VACF
Compute and Plot VACF
Plot: Supercritical argon (Fig. 2B)
Plot: Supercritical argon (Fig. 2B)
Compute and Plot VACF
Compute and Plot VACF
Plot: Dense supercritical LJ-argon (Fig. 3 and Fig. S3)
Plot: Dense supercritical LJ-argon (Fig. 3 and Fig. S3)
Compute and Plot VACF
Compute and Plot VACF
Pre-calculations used in hybrid model
Pre-calculations used in hybrid model
Lennard-Jones parameters for (liquid) argon:
The characteristic LJ time is
Generate full set of input parameters
Supplemental Calculations (no corresponding figure in main article or Supplementary Material)
Supplemental Calculations (no corresponding figure in main article or Supplementary Material)
Note: the calculation below is not shown in the Supplementary Material nor the main article but is serves as verification that the VACF “dip” is largely controlled by the transverse component (i.e., shear wave modes) and not the longitudinal component (i.e., sound wave modes).
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
Molecular hydrodynamic theory of the velocity autocorrelation function
by Sean L. Seyler & Charles E. Seyler
Wolfram Community, STAFF PICKS, December 24, 2024
https://community.wolfram.com/groups/-/m/t/3344272
by Sean L. Seyler & Charles E. Seyler
Wolfram Community, STAFF PICKS, December 24, 2024
https://community.wolfram.com/groups/-/m/t/3344272