# Predicting thermal conductivity and associated autocorrelation function for simple fluids

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 thermal conductivity and associated autocorrelation function for simple fluids. International Conference PPEPPD 2025 -Properties and Phase Equilibria for Product and Process Design, May 2025, Bad Gögging (close to Munich), Germany. hal-05072147

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Submitted on 18 May 2025

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

# Predicting thermal conductivity and associated autocorrelation function (ACF) for simple fluids.

Ignace N°II YAPI¹, Oriana HADDAD¹, Mounir BEN AMAR¹, Jean-Philippe PASSARELLO¹

¹LSPM-CNRS, Laboratoire des Sciences des Procédés et des Matériaux, Université Sorbonne Paris Nord,
99 avenue Jean-Baptiste Clément, 93430 Villetaneuse, France

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

Institut GALILÉE
Université Sorbonne Paris Nord

UNIVERSITÉ
SORBONNE
PARIS NORD

## Introduction

**Background:** This work is a part of a more general study aiming at developing a model predicting transport properties fluids in the full phase diagram.

**Issues:** Analytical developments of a model of thermal conductivity based on Green Kubo and Egorov [1] formalism recently applied by Yapi et al., [2] are very complex. Simplified extension is therefore needed.

**Goals:** We propose here a simple extension of our approach to thermal conductivity basing on similarities with shear viscosity. The model is tested by predicting thermal conductivity of simple fluids (Mie n-6 fluids and noble gases) and its associated ACF.

## Approach for predicting thermal conductivity $\lambda$ and associated autocorrelation functions (ACF) $\lambda(t)$

### Theory, issues and solution

The theory starts from **Green-Kubo relations** between thermal conductivity $\lambda$ and its ACF $\lambda(t)$ [3]:

$$ \lambda = \frac{V}{3k_B T^2} \int_0^\infty \langle J_x^e(t) \cdot J_x^e(0) \rangle dt = \frac{V}{3k_B T^2} \int_0^\infty \lambda(t) dt $$

$$ J_x^e(t) = \sum_i \left[ \frac{p_i^2}{2m} v_{ix}(t) + \frac{1}{2} \sum_{i \neq j} U_{ij} v_{ix}(t) - \frac{1}{2} \sum_{i \neq j} \frac{\partial U_{ij}}{\partial r_{ij}} \hat{r}_{ij} \cdot \mathbf{v}_i(t) \right] $$

* $U(r) = \frac{1}{2} \sum_{i \neq j} U_{ij}$ is the total interaction potential

* $v_{ix}(t)$ is the velocity of the i-th particle at time t

**Problem:** difficult to get an analytical expression for $\lambda(t)$ contrary to $\eta(t)$ shear viscosity ACF.

**Solution:** notice similarities between $\lambda$ and $\eta$

### HS at low densities [4]

From Heyes [4], $\lambda/\lambda_E$ and $\eta/\eta_E$ appears fairly close

### LJ Ar near triple point [5]

At short time, $\lambda(t)$ and $\eta(t)$ appears superimposed

<figure><img src="image_3.png" /></figure>

Also, **Branka and Heyes** [6] showed that $\lambda(t)$ and $\eta(t)$ expressions are very close too

**Hypothesis:** $\lambda(t)$ behaves similarly as $\eta(t)$ at short times

### Modeling shear viscosity $\eta$ and its ACFs

Note that $\eta$ was tested successfully on different Mie fluids and real noble fluids but not for ACFs [2].

$$ \eta = \int_{0}^{\infty} \eta(t) dt = \int_{0}^{\infty} [\eta_{E}(t) + \eta_{B}(t) + \eta_{MCT}(t)] dt $$

### BINARY CONTRIBUTION from Egorov [1]

$$ \eta_B(t) = G_{\infty} \exp \left[ \frac{-t^2}{\tau_{\eta B}^2} \right] \qquad \tau_{\eta B} = \sqrt{\frac{-2G_{\infty}}{\ddot{\eta}_p(0)}} $$

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

* $\ddot{\eta}_p(0)$, can be similarly obtained from $g(r)$ and $u(r)$ [2]

### MCT CONTRIBUTION from Egorov [1]

$$ \eta_E(t) = \eta_E(0) \exp \left[ \frac{-t}{\tau_{\eta E}} \right] \qquad \tau_{\eta E} = \frac{\eta_E}{\eta_E(0)} \frac{5 \sqrt{mk_B T}}{16 \pi \sigma^2 \Omega^{(2,2)}(T)} \rho k_B T $$

### MIE 12-6 FLUIDS

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

Good predictions of viscosity ACFs

### Modeling $\lambda$ and its ACF $\lambda(t)/\lambda(0)$

$\lambda$ is accurately computed using Heyes [4] for HS:

$$ \frac{\lambda}{\lambda_E} = \frac{1}{g(\sigma)} + 1.2b + 0.755g(\sigma)b^2 $$

* $\lambda_E = \frac{75}{64\sigma^2} \sqrt{\frac{k_B^3 T}{\pi m}}$

* $b = \frac{2\pi\sigma^3\rho}{3}$

* $g(\sigma) = 1 + 0.625b + 0.2869b^2 + 0.115b^3$

* an adequate choice of diameter $\sigma$ according to the prescription of Balucani et al. [8] is used:

$$ u(r = \sigma(T)) + \varepsilon = \frac{3}{2} k_B T $$

$\lambda(t)$ assumes the same structure as $\eta(t)$ at short times, i.e without MCT term:

$$ \lambda(t) = \lambda_E(0) \exp \left[ \frac{-t}{\tau_{\lambda E}} \right] + \lambda_B(0) \exp \left[ \frac{-t^2}{\tau_{\lambda B}^2} \right] $$

### Hypothesis

Invoking similarities we simply set:

* $\tau_{\lambda E} = \tau_{\eta E}$

* $\tau_{\lambda B} = \tau_{\eta B}$

Applying $\lambda = \int_0^\infty \lambda(t) dt$:

* $\lambda_E(0) = \frac{\lambda_E}{\tau_{\lambda E}}$

* $\lambda_B(0) = \frac{2(\lambda - \lambda_E(0)\tau_{\lambda E})}{\tau_{\lambda B} \sqrt{\pi}}$

### Reduced quantity for calculations

* $t^* = \tau/\sigma\sqrt{\varepsilon/m}$

* $\rho^* = \rho\sigma^3$

* $T^* = T k_B / \varepsilon$

* $\lambda^* = \lambda\sigma^2/k_B\sqrt{\varepsilon/m}$

## Prediction of conductivity ACF $\lambda(t)$

### MIE 12-6 FLUIDS

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

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

Reasonable predictions, but at low densities more MD data are needed for a full validation.

## Prediction of conductivity $\lambda$ of different Mie n-6 fluids

### MIE 12-6 FLUIDS

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

### MIE 7-6 AND 34-6 FLUIDS

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

The model makes good predictions of various Mie fluids conductivity over the entire phase diagram.

## Test on noble gases

### KRYPTON

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

### NEON

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

$\lambda$ of noble gases are accurately predicted using the **ab-initio pair potential** of Deiters and Sadus [12].

## Conclusion

* ☐ The similarity between $\lambda$ and $\eta$ is used to calculate thermal conductivity and its associated ACF, validated by comparison with MD data.

* ☐ No parameters were adjusted on data.

## Perspectives

* ☐ Model extension to water and Al₂O₃-water nanofluid

* ☐ CFD modeling in a heat exchanger to improve the heat transfer coefficient.

## References

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

[2] I. Yapi, O. Haddad, M. B. Amar, J.P. Passarello, Fluid Phase Equilibria, 114426.(2025)

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

[4] D. M. Heyes, physical Review B, 10, 5677 (1988)

[5] C. Hoheisel, Physics Reports, 3, 111-157 (1994)

[6] A.C Brańka, D. M. Heyes, Physical Review E, 2, 021202. (2004)

[7] K. Meier, A. Laesecke, S. Kabelac, J. Chem. Phys, 8, 3671-3687 (2004)

[8] U. Balucani, R. Vallarui, T. Gaskell, J. Phys.: Condens. Matter, 2, 5015 (1990)

[9] K. Meier, Computer Simulation and Interpretation of the Transport Coefficients of LJ fluids, PhD (2002)

[10] U. G. Chaparro, E.A. Müller, J. Phys. Chem, 551-566 (2024)

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

[12] U.K. Deiters, R.J. Sadus, J. Chem. Phys. 151 (2019)

International Conference PPEPPD 2025 - Properties and Phase Equilibria for Product and Process Design

Contact: jean-philippe.passarello@lspm.cnrs.fr

11 - 15 May 2025 Bad Gögging (close to Munich)