User:MaxwellMolecule/sandbox

Prediction of liquid properties

Methods for predicting liquid properties can be organized by their "scale" of description, that is, the length scales and time scales over which they apply.^[1]^[2]

Macroscopic methods use equations that directly model the large-scale behavior of liquids, such as their thermodynamic properties and flow behavior.
Microscopic methods use equations that model the dynamics of individual molecules.
Mesoscopic methods fall in between, combining elements of both continuum and particle-based models.

Macroscopic

Empirical correlations

Empirical correlations are simple mathematical expressions intended to approximate a liquid's properties over a range of experimental conditions, such as varying temperature and pressure.^[3] They are constructed by fitting simple functional forms to experimental data. For example, the temperature-dependence of liquid viscosity is sometimes approximated by the function $\eta (T)=Ae^{B/T}$ , where $A$ and $B$ are fitting constants.^[4] Empirical correlations allow for extremely efficient estimates of physical properties, which can be useful in thermophysical simulations. However, they require high quality experimental data to obtain a good fit and cannot reliably extrapolate beyond the conditions covered by experiments.

Thermodynamic potentials

Thermodynamic potentials are functions that characterize the equilibrium state of a substance. An example is the Gibbs free energy $G(p,T)$ , which is a function of pressure and temperature. Knowing any one thermodynamic potential ${\mathcal {F}}$ is sufficient to compute all equilibrium properties of a substance, often simply by taking derivatives of ${\mathcal {F}}$ .^[5] Thus, a single correlation for ${\mathcal {F}}$ can replace separate correlations for individual properties.^[6]^[7] Conversely, a variety of experimental measurements (e.g., density, heat capacity, vapor pressure) can be incorporated into the same fit; in principle, this would allow one to predict hard-to-measure properties like heat capacity in terms of other, more readily available measurements (e.g., vapor pressure).^[8]

Hydrodynamics

Hydrodynamic theories describe liquids in terms of space- and time-dependent macroscopic fields, such as density, velocity, and temperature. These fields obey partial differential equations, which can be linear or nonlinear.^[9] Hydrodynamic theories are more general than equilibrium thermodynamic descriptions, which assume that liquids are approximately homogeneous and time-independent. The Navier-Stokes equations are a well-known example: they are partial differential equations giving the time evolution of density, velocity, and temperature of a viscous fluid. There are numerous methods for numerically solving the Navier-Stokes equations and its variants.^[10]^[11]

Mesoscopic

Mesoscopic methods operate on length and time scales between the particle and continuum levels. For this reason, they combine elements of particle-based dynamics and continuum hydrodynamics.^[1]

An example is the lattice Boltzmann method, which models a fluid as a collection of fictitious particles that exist on a lattice.^[1] The particles evolve in time through streaming (straight-line motion) and collisions. Conceptually, it is based on the Boltzmann equation for dilute gases, where the dynamics of a molecule consists of free motion interrupted by discrete binary collisions, but it is also applied to liquids. Despite the analogy with individual molecular trajectories, it is a coarse-grained description that typically operates on length and time scales larger than those of true molecular dynamics (hence the notion of "fictitious" particles).

Other methods that combine elements of continuum and particle-level dynamics include smoothed-particle hydrodynamics,^[12]^[13] dissipative particle dynamics,^[14] and multiparticle collision dynamics.^[15]

Microscopic

Microscopic simulation methods work directly with the equations of motion (classical or quantum) of the constituent molecules.

Classical molecular dynamics

Classical molecular dynamics (MD) simulates liquids using Newton's law of motion; from Newton's second law ( $F=m{\ddot {x}}$ ), the trajectories of molecules can be traced out explicitly and used to compute macroscopic liquid properties like density or viscosity. However, classical MD requires expressions for the intermolecular forces ("F" in Newton's second law). Usually, these must be approximated using experimental data or some other input.^[16]

Ab initio (quantum) molecular dynamics

Ab initio quantum mechanical methods simulate liquids using only the laws of quantum mechanics and fundamental atomic constants.^[17] In contrast with classical molecular dynamics, the intermolecular force fields are an output of the calculation, rather than an input based on experimental measurements or other considerations. In principle, ab initio methods can simulate the properties of a given liquid without any prior experimental data. However, they are very expensive computationally, especially for large molecules with internal structure.

^ ^a ^b ^c Krüger, Timm; Kusumaatmaja, Halim; Kuzmin, Alexandr; Shardt, Orest; Silva, Goncalo; Viggen, Erlend Magnus (2016). The lattice Boltzmann method : principles and practice. Switzerland. ISBN 978-3-319-44649-3. OCLC 963198053.{{cite book}}: CS1 maint: location missing publisher (link)
^ Steinhauser, M. O. (2022). Computational multiscale modeling of fluids and solids : theory and applications. Cham, Switzerland: Springer. ISBN 978-3-030-98954-5. OCLC 1337924123.
^ Poling, Bruce E.; Prausnitz, J. M.; O'Connell, John P. (2001). The properties of gases and liquids. New York: McGraw-Hill. ISBN 0-07-011682-2. OCLC 44712950.
^ Bird, R. Byron; Stewart, Warren E.; Lightfoot, Edwin N. (2007). Transport Phenomena (2nd ed.). John Wiley & Sons, Inc. ISBN 978-0-470-11539-8. Archived from the original on 2020-03-02. Retrieved 2019-09-18.
^ Cite error: The named reference Kardar 2007 p. was invoked but never defined (see the help page).
^ Span, R. (2000). Multiparameter Equations of State: An Accurate Source of Thermodynamic Property Data. Engineering online library. Springer. p. 1. ISBN 978-3-540-67311-8. Retrieved 2023-04-01.
^ Huber, Marcia L.; Lemmon, Eric W.; Bell, Ian H.; McLinden, Mark O. (2022-06-22). "The NIST REFPROP Database for Highly Accurate Properties of Industrially Important Fluids". Industrial & Engineering Chemistry Research. 61 (42). American Chemical Society (ACS): 15449–15472. doi:10.1021/acs.iecr.2c01427. ISSN 0888-5885.
^ Tillner-Roth, Reiner; Friend, Daniel G. (1998). "A Helmholtz Free Energy Formulation of the Thermodynamic Properties of the Mixture {Water + Ammonia}". Journal of Physical and Chemical Reference Data. 27 (1). AIP Publishing: 63–96. doi:10.1063/1.556015. ISSN 0047-2689.
^ Moffatt, H.K. (2015), "Fluid Dynamics", in Nicholas J. Higham; et al. (eds.), The Princeton Companion to Applied Mathematics, Princeton University Press, p. 467–476
^ Wendt, John F.; Anderson, John D., Jr.; Von Karman Institute for Fluid Dynamics (2008). Computational fluid dynamics : an introduction. Berlin: Springer. ISBN 978-3-540-85056-4. OCLC 656397653.{{cite book}}: CS1 maint: multiple names: authors list (link)
^ Pozrikidis, C. (2011). Introduction to theoretical and computational fluid dynamics. New York: Oxford University Press. ISBN 978-0-19-990912-4. OCLC 812917029.
^ Monaghan, J J (2005-07-05). "Smoothed particle hydrodynamics". Reports on Progress in Physics. 68 (8). IOP Publishing: 1703–1759. doi:10.1088/0034-4885/68/8/r01. ISSN 0034-4885.
^ Lind, Steven J.; Rogers, Benedict D.; Stansby, Peter K. (2020). "Review of smoothed particle hydrodynamics: towards converged Lagrangian flow modelling". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 476 (2241). The Royal Society. doi:10.1098/rspa.2019.0801. ISSN 1364-5021.
^ Español, Pep; Warren, Patrick B. (2017-04-21). "Perspective: Dissipative particle dynamics". The Journal of Chemical Physics. 146 (15). AIP Publishing: 150901. doi:10.1063/1.4979514. ISSN 0021-9606.
^ Gompper, G.; Ihle, T.; Kroll, D. M.; Winkler, R. G. "Multi-Particle Collision Dynamics: A Particle-Based Mesoscale Simulation Approach to the Hydrodynamics of Complex Fluids". Advanced Computer Simulation Approaches for Soft Matter Sciences III. Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 1–87. arXiv:0808.2157. doi:10.1007/978-3-540-87706-6_1. ISBN 978-3-540-87705-9.
^ Cite error: The named reference Maitland1981 was invoked but never defined (see the help page).
^ Cite error: The named reference MarxHutter 2012 was invoked but never defined (see the help page).

[Krüger2016-1] Krüger, Timm; Kusumaatmaja, Halim; Kuzmin, Alexandr; Shardt, Orest; Silva, Goncalo; Viggen, Erlend Magnus (2016). The lattice Boltzmann method : principles and practice. Switzerland. ISBN 978-3-319-44649-3. OCLC 963198053.{{cite book}}: CS1 maint: location missing publisher (link)

[Steinhauser2022-2] Steinhauser, M. O. (2022). Computational multiscale modeling of fluids and solids : theory and applications. Cham, Switzerland: Springer. ISBN 978-3-030-98954-5. OCLC 1337924123.

[Poling2001-3] Poling, Bruce E.; Prausnitz, J. M.; O'Connell, John P. (2001). The properties of gases and liquids. New York: McGraw-Hill. ISBN 0-07-011682-2. OCLC 44712950.

[4] Bird, R. Byron; Stewart, Warren E.; Lightfoot, Edwin N. (2007). Transport Phenomena (2nd ed.). John Wiley & Sons, Inc. ISBN 978-0-470-11539-8. Archived from the original on 2020-03-02. Retrieved 2019-09-18.

[Kardar_2007_p.-5] Cite error: The named reference Kardar 2007 p. was invoked but never defined (see the help page).

[Span2000-6] Span, R. (2000). Multiparameter Equations of State: An Accurate Source of Thermodynamic Property Data. Engineering online library. Springer. p. 1. ISBN 978-3-540-67311-8. Retrieved 2023-04-01.

[Huber2022-7] Huber, Marcia L.; Lemmon, Eric W.; Bell, Ian H.; McLinden, Mark O. (2022-06-22). "The NIST REFPROP Database for Highly Accurate Properties of Industrially Important Fluids". Industrial & Engineering Chemistry Research. 61 (42). American Chemical Society (ACS): 15449–15472. doi:10.1021/acs.iecr.2c01427. ISSN 0888-5885.

[Tillner-Roth1998-8] Tillner-Roth, Reiner; Friend, Daniel G. (1998). "A Helmholtz Free Energy Formulation of the Thermodynamic Properties of the Mixture {Water + Ammonia}". Journal of Physical and Chemical Reference Data. 27 (1). AIP Publishing: 63–96. doi:10.1063/1.556015. ISSN 0047-2689.

[PrincetonCompanion-9] Moffatt, H.K. (2015), "Fluid Dynamics", in Nicholas J. Higham; et al. (eds.), The Princeton Companion to Applied Mathematics, Princeton University Press, p. 467–476

[Wendt2008-10] Wendt, John F.; Anderson, John D., Jr.; Von Karman Institute for Fluid Dynamics (2008). Computational fluid dynamics : an introduction. Berlin: Springer. ISBN 978-3-540-85056-4. OCLC 656397653.{{cite book}}: CS1 maint: multiple names: authors list (link)

[Pozrikidis2011-11] Pozrikidis, C. (2011). Introduction to theoretical and computational fluid dynamics. New York: Oxford University Press. ISBN 978-0-19-990912-4. OCLC 812917029.

[Monaghan2005-12] Monaghan, J J (2005-07-05). "Smoothed particle hydrodynamics". Reports on Progress in Physics. 68 (8). IOP Publishing: 1703–1759. doi:10.1088/0034-4885/68/8/r01. ISSN 0034-4885.

[Lind2020-13] Lind, Steven J.; Rogers, Benedict D.; Stansby, Peter K. (2020). "Review of smoothed particle hydrodynamics: towards converged Lagrangian flow modelling". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 476 (2241). The Royal Society. doi:10.1098/rspa.2019.0801. ISSN 1364-5021.

[Español2017-14] Español, Pep; Warren, Patrick B. (2017-04-21). "Perspective: Dissipative particle dynamics". The Journal of Chemical Physics. 146 (15). AIP Publishing: 150901. doi:10.1063/1.4979514. ISSN 0021-9606.

[Gompper2008-15] Gompper, G.; Ihle, T.; Kroll, D. M.; Winkler, R. G. "Multi-Particle Collision Dynamics: A Particle-Based Mesoscale Simulation Approach to the Hydrodynamics of Complex Fluids". Advanced Computer Simulation Approaches for Soft Matter Sciences III. Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 1–87. arXiv:0808.2157. doi:10.1007/978-3-540-87706-6_1. ISBN 978-3-540-87705-9.

[Maitland1981-16] Cite error: The named reference Maitland1981 was invoked but never defined (see the help page).

[MarxHutter_2012-17] Cite error: The named reference MarxHutter 2012 was invoked but never defined (see the help page).

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