# Predicting transport properties of simple fluids using an extended FMSA model

Ignace N Ii Yapi, Oriana Haddad, Mounir Ben Amar, Jean-Philippe Passarello

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Ignace N Ii Yapi, Oriana Haddad, Mounir Ben Amar, Jean-Philippe Passarello. Predicting transport properties of simple fluids using an extended FMSA model. The 33rd European Symposium on Applied Thermodynamics (ESAT 2024), Jun 2024, Edimbourg (Ecosse), United Kingdom. hal-05001947

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https://hal.science/hal-05001947v1

Submitted on 22 Mar 2025

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## 1. Context and Objectives

The fluid transport properties, namely shear viscosity, thermal conductivity and diffusion are important quantities required in engineering design for fluid production, treatment and processing [3]. Computation Fluid Dynamics software require to solve Navier-Stokes equations involving transport properties. In this work, the mean spherical first approximation (FMSA) equation of state of Tang and Lu [1], extended by Dridi and Passarello [2] and recently successfully used to predict rare gas phase equilibria from ab initio interaction potentials is used here to systematically model the transport properties (shear viscosity, self-diffusion coefficient) of simple Mie n-6 fluids and noble gases within the framework of a simplified Coupling Mode Theory (MCT) approach [3].

## 2. Approach for prediction of transport properties based on Egorov [3]

The theory starts from Green-Kubo relations [8]

$$ \eta = \frac{1}{Vk_B T} \int_0^\infty \langle J_{xy}(t) J_{xy}(0) \rangle dt $$

$$ J_{xy}(t) = \sum_i [mv_{ix}(t)v_{iy}(t) + F_{iy}(t)x_i(t)] $$

* $F_{iy}(t), v_{ix}(t)$ and $x_i(t)$ are the x components of force, velocity and position of the i-th particule at time t

### Shear viscosity:

$$ \eta \approx \eta_k + \eta_p $$

Kinetic ≈ Enskog:

$$ \eta_k = \frac{5}{16} \frac{\sqrt{mk_B T}}{\sqrt{\pi \sigma^2 \Omega^{(2,2)}(T)}} $$

Potential: $\eta_p = \int_0^\infty (\eta_B(t) + \eta_{MCT}(t)) dt$

* ✓ Fast collisional events term: independent/uncorrelated binary (B) collisions

* ✓ Slow events term: repeated correlated collisions (MCT)

$$ \eta_B(t) = \eta_p(0) \exp \left[ \frac{\ddot{\eta}_p(0) t^2}{2\eta_p(0)} \right] $$

$$ \eta_p(0) = \frac{2\pi}{15} \rho^2 \int_0^\infty dr g(r) \frac{d}{dr} [r^4 u'(r)] $$

* ■ $\rho$ is the fluid bulk number density

* ■ $u'(r)$ is the first derivative of the potential

* ■ $g(r)$ is the radial distribution function calculated using a simplified exponential approximation using FMSA

* ■ $\ddot{\eta}_p(0)$, can be similarly obtained from $g(r)$ and $u(r)$

$$ \eta_{MCT}(t) = \frac{k_B T}{60\pi^2} \int_0^\infty dk k^4 \left[ \frac{S'(k)}{S(k)} \right]^2 \left\{ \left[ \frac{F(k,t)}{S(k)} \right]^2 - \left[ \frac{F^o(k,t)}{S(k)} \right]^2 \right\} $$

* ■ $S(k)$ is the structure factor

* ■ $F^o(k, t) = S(k) \exp(-\frac{k_B T k^2 t^2}{2mS(k)})$

* ■ $F(k,t)$ is the dynamic structure factor obtained from its Laplace transform of $F(k,z)$ [4]

$$ F(k, z) = \frac{S(k)}{z + \frac{\delta_1(k)}{z + \frac{\delta_2(k)}{z + \tau^{-1}(k)}}} $$

* ■ $\delta_1(k), \delta_2(k)$ and $\tau^{-1}(k)$ are given in [4]

Larger difference at high density → Cage effect

* MCT is needed to provide a good prediction of the phase diagram in the whole range of density

### Self-diffusion coefficient : similar treatment

See Egorov [3]

## 4. Tests for various theoretical potentials

### Mie fluids

<figure><img src="image_1.png" /></figure>

<figure><img src="image_2.png" /></figure>

### Shear viscosity

<figure><img src="image_3.png" /><figcaption>Mie 12-6</figcaption></figure>

### Self-diffusion

<figure><img src="image_4.png" /></figure>

* The model provides good predictions of various Mie fluids viscosity and diffusion over the entire phase diagram and at different temperatures, and is in agreement with experimental argon viscosity data

## Conclusion

<figure><img src="image_5.png" /></figure>

* Use of a theoretical approach based on MCT to calculate transport properties

<figure><img src="image_6.png" /></figure>

* Equation tested and validated by comparison with MD data for various theoretical potentials and real fluids

<figure><img src="image_7.png" /></figure>

* No parameters were adjusted on data

## Perspectives

<figure><img src="image_8.png" /></figure>

* Extension of the approach to thermal conductivity

<figure><img src="image_9.png" /></figure>

* Extension of the model to nanoparticles (colloids) + solvent systems

<figure><img src="image_10.png" /></figure>

* Computational Fluid Dynamics (CFD) application with ANSYS Fluent software

## 6. Test of our approach on noble gas

### Argon

<figure><img src="image_11.png" /></figure>

Pure prediction using the **ab-initio pair potential** of Leonhard and Deiters [7]

## References

[1] Y. Tang, B.C.Y Lu, J. Chem. Phys., 99, 9828 (1993)

[2] S. Dridi, M. Ben Amar, M. Abderraba, J-P. Passarello, Fluid Phase Eq., 562, 113563 (2022)

[3] S.A. Egorov, J. Chem. Phys., 128, 144508 (2008)

[4] S. Bhattacharyya et B. Bagchi. J. Phys, 7885-7892(1998)

[5] C.G. Müller, A. Erich. J. Phys 551-566 (2024)

[6] C. Tegeler, R. Span, W. Wagner, Nist webbook J. Phys. Chem. Ref. Data, (1999)

[7] K. Leonhard and al, M. Phys, 98:20, 1603-1616 (2000)

[8] K. Tankeshwar et al, J. Phys.: Condens. Matter 2 5891(1990)

[9] U. Balucani and M. Zoppi, Clarendon Press, (1994)