# Simulation of Dipole Moment in Asphaltene Nanocluster

Rustem Z Syunyaev, Victor V Likhatsky, Roman M Balabin

► To cite this version:

Rustem Z Syunyaev, Victor V Likhatsky, Roman M Balabin. Simulation of Dipole Moment in Asphaltene Nanocluster. Engineering school. Cold heavy oil production with sand (CHOPS), DBR Technology Center, Schlumberger Canada, Edmonton, Alberta, Canada. 2010, pp.28. hal-04982657

HAL Id: hal-04982657

https://hal.science/hal-04982657v1

Submitted on 7 Mar 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

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

Distributed under a Creative Commons Attribution 4.0 International License
Simulation of Dipole Moment in
Asphaltene Nanocluster

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

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

Syunyaev R.Z., Likhatsky V.V., and Balabin R.M.

Physics Department of Gubkin Russian State University of Oil and Gas
65, Leninsky Prospekt, Moscow, 119991, B-296, Russia

Speaker e-mail: syunyaev@gubkin.ru, likhatsky@mail.ru, balabinrm@yandex.ru
ABSTRACT

Aggregation of natural macromolecules – e.g., asphaltenes – on
nanoscale is one of key problems of the petroleum colloid science.
Experimental studies of an estimation of energetic and kinetic parameters of
structuring by various physical methods are rather numerous. At the same
time methods of computer modeling are popular. Usually, calculations are
based on application of Lennard-Jones potential for nonpolar particles.

Consideration of dipole moment results in several singularities for
aggregation processes. Monte-Carlo and Molecular Dynamics (MD) methods
are used for estimation the dependence of dipole moment of nanoclusters
from number of primary particles. Well-known Yen's particle is accepted as
primary. Quasispheroidal shape is result of stacking of flat aromatic fragments
of asphaltene molecules. Asphaltenes concern to the most polar substances
in structure of petroleum and have appreciable dipole moment. Earlier the
attention was not attracted to this circumstance. Aggregation of those type
particles becomes actual for analyzing the formation and evolution of
petroleum dispersed systems.
# CONTENT

Aggregation of natural macromolecules - asphaltenes on nanoscale is one of key problems of the petroleum colloid science. Experimental studies of an estimation of energetic and kinetic parameters of structuring by various physical methods are rather numerous. At the same time methods of computer modeling are popular. Usually calculations are based on application of classical Lennard-Jones potential for nonpolar particles.

Consideration of dipole moment results in a number of singularities for aggregation processes. Monte-Carlo and Molecular Dynamics (MD) methods are used for estimation the dependence of dipole moment of nanoclusters from number of primary particles. Well-known Yen's particle is accepted as primary. Quasisppherical shape is result of stacking of flat aromatic fragments of asphaltene molecules. Asphaltenes concern to the most polar substances in structure of petroleum and have appreciable dipole moment. Earlier the attention was not attracted to this circumstance. Aggregation of those type particles becomes actual for analyzing the formation and evolution of petroleum dispersed systems.

Analytical solution within the framework of model of growing cluster at uniting with additional primary particle was received some time back. In approximation of continuous or drop asphaltene phase the power dependence of dipole moment of cluster from number of incorporated particles is shown. The fractal model of growth leads in Kohlrausch-Williams-Watts function.

The Monte-Carlo program complex allows simulating the aggregation process of dipole particles at following assumptions:

* particles are spheres of specified constant radius in the course of simulation.

* particles possess definite dipole moment randomly oriented in space

* particles move freely until the moment of contact

* at moment of contact the electric strength of growing cluster orients dipole of coming new primary particle according to the angle Boltzmann distribution. After sticking calculation of interaction between appeared cluster and next new particle continues

* probability of joining particle to the cluster is independent from their interaction.

In MD method clusters appear owing to sticking primary particles after the certain number of steps.

Results of computer modeling confirmed power law obtained in analytical way. The model allows explaining the experimental results about abnormally large dipole moments of asphaltenes published earlier.
# AFM Images of Asphaltenes
(Suhukum Oilfield)

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

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

Close to Spherical Clusters

Akhmetov I.R.,
Master Thesis, Ufa
– 2007
Scan 3x3 µm
<figure><img src="image_6.png" /><figcaption>Both Continuous and Fractal Structures</figcaption></figure>
# Dipole Moments Data

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

## ASPHALTENES

<table><thead><tr><th>Dipole moment<br>p, Debye</th><th>Solvent</th><th>Oil</th><th>Method</th><th>Author</th></tr></thead><tbody><tr><td>64 Deposit</td><td>Toluene (0.5 - 1.0 wt%)</td><td>West Texas</td><td>Onsager Equation</td><td>Wattana &amp; Fogler, 2005</td></tr><tr><td>60 Crude</td><td>Toluene (0.5 - 1.0 wt%)</td><td>"-"</td><td>"-"</td><td>"-"</td></tr><tr><td>54 Deposit</td><td>"-"</td><td>Louisiana</td><td>"-"</td><td>"-"</td></tr><tr><td>44 Crude</td><td>"-"</td><td>"-"</td><td>"-"</td><td>"-"</td></tr><tr><td>64</td><td>unfractionated</td><td>Cold Lake</td><td>"-"</td><td>"-"</td></tr><tr><td>64.3</td><td>40 CH<sub>2</sub>Cl<sub>2</sub>:60 n-C<sub>5</sub><br>(vol %)</td><td>"-"</td><td>"-"</td><td>"-"</td></tr><tr><td>57.3</td><td>10 CH<sub>2</sub>Cl<sub>2</sub>:90 n-C<sub>5</sub><br>(vol %)</td><td>"-"</td><td>"-"</td><td>"-"</td></tr><tr><td>5-6</td><td>Toluene</td><td>West Siberia</td><td>Onsager- Kirkwood<br>equation</td><td>Syunyaev, 1999</td></tr><tr><td>5.3</td><td>Toluene</td><td>Vacuum<br>Residue</td><td>Onsager Equation</td><td>Maruska &amp; Rao, 1987</td></tr><tr><td>6-7</td><td>Toluene</td><td>Tar</td><td></td><td>Apostolov, 1988</td></tr><tr><td>4-17</td><td>n-C<sub>5</sub></td><td>several<br>crudes from<br>North Sea</td><td>Debye theory</td><td>Halvorsen, 1997</td></tr><tr><td>3-5</td><td>Toluene</td><td>West Siberia</td><td>Debye theory</td><td>Sergienko et al, 1979</td></tr><tr><td>3.3-7</td><td>n-C<sub>5</sub></td><td>8 different oils</td><td>Onsager Equation</td><td>Goual &amp; Firoozabadi, 2002</td></tr><tr><td>4.0-7.7</td><td>n-C<sub>7</sub></td><td>"-"</td><td>"-"</td><td>"-"</td></tr></tbody></table>

## RESINS

<table><tbody><tr><td>2.4-3.2</td><td>n-C<sub>5</sub></td><td>8 different oils</td><td>Onsager Equation</td><td>Goual &amp; Firoozabadi, 2002</td></tr><tr><td>2.5-3.9</td><td>n-C<sub>5</sub>/toluene at<br>changing ratio</td><td>7 different oils</td><td>"-"</td><td>"-"</td></tr><tr><td>3</td><td>Toluene</td><td>Vacuum<br>residue</td><td>Onsager Equation</td><td>Sagitova, 1997</td></tr></tbody></table>
# Electric Strength Lines of Dipole

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

## Dipole moments of Asphaltene Nanoclusters (or distribution of electron density):

**Reasons**
* Heteroatoms (S,O,N),
* Anisotropy of Aromatic Rings etc.

<figure><img src="image_9.png" /><figcaption>Primary Yen Particle, 1962</figcaption></figure>

<figure><img src="image_10.png" /></figure>
# Agglomeration of Primary Yen Particle to Existing N-Cluster

## Parameters of Primary Yen Particle:

* **Quasispherical with Radius**
  $a_0 \sim 1$ nm

* **Dipole Moment**
  $\pi_0 \sim 1$ Debye

* **Radius of N-Cluster** R

* **Angles** $\Theta, \alpha, \varphi$ define mutual orientation of Dipoles

<figure><img src="image_11.png" /></figure>
<figure><img src="image_12.png" /></figure>

$$ \vec{\mathbf{p}}_N = \sum_{i=1}^{N} \vec{\mathbf{p}}_{0i} $$

$$ \vec{\mathbf{p}}_{N+1} = \vec{\mathbf{p}}_N + \vec{\mathbf{p}}_{0N+1} $$

Interaction Energy on a Boundary of N-Cluster

$$ \mathbf{U} = -\vec{\mathbf{p}}_0 \vec{\mathbf{E}}_N $$

Electric Field Strength of N-Cluster

$$ \vec{\mathbf{E}}_N = \frac{1}{4\pi\epsilon\epsilon_0} \left[ \frac{3(\vec{\mathbf{p}}_N \cdot \vec{\mathbf{r}})\vec{\mathbf{r}}}{r^5} - \frac{\vec{\mathbf{p}}_N}{r^3} \right] \qquad \mathbf{E}_N = \frac{1}{4\pi\epsilon_0\epsilon} \cdot \frac{\mathbf{p}_N}{r^3} \sqrt{1 + 3\cos^2 \theta} $$

$$ \vec{\mathbf{E}} = (\mathbf{E}_r, \mathbf{E}_\alpha, \mathbf{E}_\theta) $$
<figure><img src="image_13.png" /></figure>

Dipole Moment of N-Cluster orients along Z-axis in Cartesian System

$$ \vec{p}_N = (0,0,p_N) $$

Agglomerating primary particle $p_0$ reorient following Boltzmann law

$$ e^{-\frac{U}{kT}} = e^{\frac{\vec{p}_0 \cdot \vec{E}_N}{kT}} = e^{\frac{1}{kT}(p_{0x}E_x + p_{0y}E_y + p_{0z}E_z)} $$
<figure><img src="image_14.png" /></figure>

Calculation of averaged values of Projections of agglomerating
particle $p_0$ on X, Y, Z coordinate axes

$$ \langle \mathbf{p}_{0x} \rangle = \mathbf{0} \qquad \langle \mathbf{p}_{0y} \rangle = \mathbf{0} \qquad \langle \mathbf{p}_{0z} \rangle = \frac{4}{15} \mathbf{p}_0 \lambda $$

$$ \text{where} \quad \lambda = \frac{1}{4\pi\epsilon\epsilon_0} \cdot \frac{\mathbf{p}_0\mathbf{p}_N}{\mathrm{kTR}^3} $$

Difference in scalars of vectors

$$ \mathbf{p}_{N+1} - \mathbf{p}_N = \mathbf{p}_{0Z} = \frac{4}{15} \mathbf{p}_0 \lambda $$

Transformation into differential equation

$$ \mathrm{d}\mathbf{p} = \frac{\mathrm{d}\mathbf{p}}{\mathrm{d}N} \mathrm{d}N = \mathbf{p}_{N+1} - \mathbf{p}_N = \frac{4}{15} \mathbf{p}_0 \cdot \frac{\mathbf{p}_0 \mathbf{p}}{4\pi\epsilon\epsilon_0 \mathrm{kTR}^3} $$

Designate new constant

$$ \beta = \frac{4}{15} \cdot \frac{\mathbf{p}_0^2}{4\pi\epsilon\epsilon_0 \mathrm{kTa}_0^3} $$
<figure><img src="image_15.png" /></figure>

# Analytical Solution for Dipole Moment-3

Constant $\beta$ is ratio of dipole-dipole interaction of 2 neighboring Yen initial particles to heat energy

For "continuous asphaltene phase: $R^3 = Na_0^3$

$$ \text{Equation} \qquad \frac{dp}{p} = \beta \frac{dN}{N} $$

**Solution:**
**Power Law**

$$ p = p_N = p_0 N^\beta $$

For fractal case with dimension $d$ $R = a_0 N^{1/d}$

$$ \text{Equation} \qquad \frac{dp}{p} = \beta \frac{dN}{N^d} $$

**Solution:**
**Kohlrausch-Williams-Watts function**

$$ \ln \frac{p}{p_0} = \beta \cdot \frac{d}{3-d} \cdot \left( 1 - N^{\frac{d-3}{d}} \right) $$
N+1 particle agglomerate to N-cluster

<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /></figure>
Definition of Coordinates of N+1
dipole pN+1

<figure><img src="image_18.png" /></figure>

<figure><img src="image_19.png" /></figure>
<figure><img src="image_20.png" /></figure>

<figure><img src="image_21.png" /></figure>

1

<figure><img src="image_22.png" /></figure>

2

$$w \sim e^{\frac{U}{kT}} = e^{\frac{\vec{p}_0 \cdot \vec{E}_N}{kT}}$$

<figure><img src="image_23.png" /></figure>

3

$$\vec{p}_{N+1} = \vec{p}_N + \vec{p}_0$$

<figure><img src="image_24.png" /></figure>

$$\beta = \frac{4}{15} \cdot \frac{1}{4\pi\epsilon_0} \frac{p_0^2}{kT \cdot a_0^3}$$

4
Power Law in MK Simulation

<figure><img src="image_25.png" /></figure>

<figure><img src="image_26.png" /></figure>

$$y = 0.931x^{0.4974}$$

$$R^2 = 0.9973$$
<figure><img src="image_27.png" /></figure>

Qualitatively Coincidence
with prediction

$$ \beta = \frac{4}{15} \cdot \frac{p_0^2}{4\pi\epsilon\epsilon_0 kT a_0^3} $$
Comparison of Results

<figure><img src="image_28.png" /></figure>

<table><thead><tr><td>Analytical Solution</td><td>Simulation</td></tr></thead><tbody><tr><td>1. p<sub>N</sub> ~ N<sup>β</sup></td><td>‡</td></tr><tr><td>2. β ~ 1 / kT</td><td>‡</td></tr><tr><td>3. β ~ p<sub>0</sub><sup>2</sup></td><td>‡</td></tr><tr><td>4. β ~ 1 / r<sub>0</sub><sup>3</sup></td><td>‡</td></tr><tr><td>5. β ∈ [0; 0.5]</td><td>β ∈ [0.5; 1]</td></tr></tbody></table>
<figure><img src="image_29.png" /></figure>

Allen M.P. & Tildesley D.J. Computer Simulation of Liquids - 1987

$$ \vec{f}_{ij} = -\vec{f}_{ji} = 3 \frac{\mathbf{p}_{N_i} \cdot \mathbf{p}_{N_j}}{r_{ij}^4} \left( (\cos \gamma_{ij} - 5 \cos \theta_i \cos \theta_j) \frac{\vec{r}_{ij}}{r_{ij}} + \cos \theta_j \vec{e}_i + \cos \theta_i \vec{e}_j \right) $$

$$ \vec{\tau}_{ij} \neq -\vec{\tau}_{ji} $$

$$ \vec{\tau}_{ij} = - \frac{\mathbf{p}_{N_i} \cdot \mathbf{p}_{N_j}}{r_{ij}^3} \left( \vec{e}_i \times \vec{e}_j - 3 \cos \theta_j \frac{(\vec{e}_i \times \vec{r}_{ij})}{r_{ij}} \right) $$

$$ \vec{\tau}_{ji} = - \frac{\mathbf{p}_{N_i} \cdot \mathbf{p}_{N_j}}{r_{ij}^3} \left( \vec{e}_j \times \vec{e}_i - 3 \cos \theta_i \frac{(\vec{e}_j \times \vec{r}_{ij})}{r_{ij}} \right) $$
Relative Orientation of two Dipole Moments

<figure><img src="image_30.png" /></figure>

<figure><img src="image_31.png" /></figure>
## Relative Orientation of two Dipole Moments

<figure><img src="image_32.png" /></figure>

<figure><img src="image_33.png" /></figure>

РМ Балабин, РЗ Сюняев. Построение кривой стеклования нефти по данным диэлектрической спектроскопии. XIII Всероссийская конференция «Структура и динамика молекулярных систем. Яльчик 2006» 13 (2006) 57-60. [in Russian]
## Relative Orientation of two Dipole Moments

<figure><img src="image_34.png" /></figure>

<figure><img src="image_35.png" /></figure>

РМ Балабин, РЗ Сюняев, Анализ влияния термодинамических параметров на проводимость и вязкость нефти. XIII Всероссийская конференция «Структура и динамика молекулярных систем. Яльчик 2006» 13 (2006) 61-63. [in Russian]
Translational Movement

$$ \vec{r}_i(t + \Delta t) = \vec{r}_i(t) + \frac{D_{ti} \Delta t}{kT} \sum_{j=1}^{n} \vec{f}_{ij} + \vec{G}_i(\Delta t) $$

Rotational Movement

$$ \vec{\phi}_i(t + \Delta t) = \vec{\phi}_i(t) + \frac{D_{ri} \Delta t}{kT} \sum_{j=1}^{n} \vec{\tau}_{ij} + \vec{S}_i(\Delta t) $$

Coordinates of Clusters

$\vec{r}_i(t)$ - Displacement of Dipole

$\vec{\phi}(t)$ - Orientation of Dipole

Diffusion Coefficients

$$ D_{ti} = \frac{kT}{6\pi\eta R_i} $$

$$ D_{ri} = \frac{kT}{4\pi\eta R_i^3} $$

Forces and Torques between Nanoclusters

$\vec{f}_{ij}$

$\vec{\tau}_{ij}$

Influence of Solvent:

Randomly Fluctuated Pushes (Gaussian Distribution)

$$ \langle \vec{G}_i(\Delta t) \rangle = 0 $$

$$ \langle \vec{S}_i(\Delta t) \rangle = 0 $$

$$ \langle \vec{G}_i \vec{G}_j \rangle = 2D_t \Delta t \delta_{ij} $$

$$ \langle \vec{S}_i \vec{S}_j \rangle = 2D_t \Delta t \delta_{ij} $$

$$ \langle \vec{G}_i^2 \rangle = 2D_t \Delta t $$

$$ \langle \vec{S}_i^2 \rangle = 2D_t \Delta t $$

$\delta_{ij}$ - Kronecker Symbol

$$ \delta_{ij} = 0, \text{ if } i \neq j $$

$$ \delta_{ij} = 1, \text{ if } i = j $$
<figure><img src="image_36.png" /></figure>

# MD: Time Evolution

<figure><img src="image_37.png" /></figure>

<figure><img src="image_38.png" /></figure>

<figure><img src="image_39.png" /></figure>

<figure><img src="image_40.png" /></figure>
Power Law in MD Simulation

<figure><img src="image_41.png" /></figure>

<figure><img src="image_42.png" /></figure>

$$y = 1.0699x^{0.5226}$$

$$R^2 = 0.987$$
<figure><img src="image_43.png" /></figure>

# Opportunities of the MD model

## Power Law Exponent by Different Methods

<table><thead><tr><th>Analytical</th><th>MK</th><th>MD</th></tr></thead><tbody><tr><td>0.1 – 0.15</td><td>0.5 – 0.8</td><td>0.4 – 0.8</td></tr></tbody></table>

* Fixed Concentration (realized)

* Variation of Ensembles: NVE, NVT (particularly realized)

* Probability of Agglomeration (not realized)

* Cluster Breakdown (not realized)

* Dipole-Dipole Interactions (realized)

* Dipole-Induction Interactions (not realized)

RZ Syunyaev, RM Balabin, IS Akhatov, Adsorption of petroleum resins and asphaltenes onto reservoir rock sands studied by near infrared (NIR) spectroscopy. 9th Annual International Conference on Petroleum Phase Behavior and Fouling (2008) 85.
# Opportunities of the MD model

## Power Law Exponent by Different Methods

<table><thead><tr><th>Analytical</th><th>MK</th><th>MD</th></tr></thead><tbody><tr><td>0.1 – 0.15</td><td>0.5 – 0.8</td><td>0.4 – 0.8</td></tr></tbody></table>

* Fixed Concentration (realized)

* Variation of Ensembles: NVE, NVT (particularly realized)

* Probability of Agglomeration (not realized)

* Cluster Breakdown (not realized)

* Dipole-Dipole Interactions (realized)

* Dipole-Induction Interactions (not realized)
<figure><img src="image_44.png" /></figure>

Useful references for Simulation

1. D.C. Rapaport, The art of molecular dynamics simulation, 2004.

2. D. Frenkel, B. Smit, Understanding molecular simulation from algorithms to applications, 2002.

3. R.Z. Syunyaev, R.M. Balabin, Frequency Dependence of Oil Conductivity at High Pressure, J Dispersion Sci Technol 28 (2007) 419.

4. R.Z. Syunyaev, R.M. Balabin, Polarization of Fluorescence of Asphaltene Containing Systems, J Dispersion Sci Technol 29 (2008) 1505.

5. M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids, 1987.

6. A.R. Leach, Molecular Modelling Principles and Applications, 2001.

7. V.V. Likhatsky, R.M. Balabin, R.Z. Syunyaev, Calculation of Dipole Moment of Fractal Asphaltene Cluster, J Dispersion Sci Technol.

8. R.M. Balabin, Conformational equilibrium in alanine: Focal-point analysis and ab initio limit, Comput Theor Chem, in preparation.

9. R.M. Balabin, Enthalpy difference between conformations of normal alkanes: effects of basis set and chain length on intramolecular basis set superposition error, Molecular Physics, in preparation.

10. K. Chingin, et al., Optical properties of protonated Rhodamine 19 isomers in solution and in the gas phase, PCCP 12 (2010) 14121.

11. K. Binder, Monte Carlo and Molecular Dynamics Simulations in Polymer Science, 1995.

12. Yu. Petrov, Clusters and Small Particles, 1986. [in Russian]

13. R.M. Balabin, Dispersed Structure of Ethanol-Gasoline Fuel According to Dynamic Light Scattering Method, Journal of Dispersion Science and Technology 29 (2008) 457.

14. R.M. Balabin, On the relation between basis set convergence and electron correlation: a critical test for modern ab initio quantum chemistry on a "mindless" data set, Structural Chemistry.

15. K. Chingin, et al., Rhodamines in the gas phase: cations, neutrals, anions, and adducts with metal cations, PCCP 12 (2010) 11710.

16. R.M. Balabin, Boryl Substitution of Acetaldehyde Makes It an Enol: Inconsistency between Gn/CBS and Ab Initio/DFT Data, Journal of Physical Chemistry A 114 (2010) 3698.

17. R.M. Balabin, R.Z. Safieva, E.I. Lomakina, Universal technique for optimization of neural network training parameters: gasoline near infrared data example, Neural Computing and Applications 18 (2008) 557.

18. T. Schmid, et al., Tip-enhanced Raman spectroscopy and related techniques in studies of biological materials, SPIE 7586 (2010) 758603.

19. L. Opilik, Nanoscale Chemical Analysis of Cell Membrane Constituents Using Tip-Enhanced Raman Spectroscopy, AIP (2010) 1271.

20. R.M. Balabin, Communications: Is quantum chemical treatment of biopolymers accurate? Intramolecular basis set superposition error (BSSE), Journal of Chemical Physics 132 (2010) 231101.

21. R.M. Balabin, The keto-enol equilibrium in substituted acetaldehydes: focal-point analysis and ab initio limit, Molecular Physics.

22. R.M. Balabin, Intermolecular dispersion interactions of normal alkanes with rare gas atoms: van der Waals complexes of n-pentane with helium, neon, and argon, Chemical Physics 352 (2008) 267.

23. K. Chingin, et al., Absorption of the green fluorescent protein chromophore anion in the gas phase studied by a combination of FTICR mass spectrometry with laser-induced photodissociation spectroscopy, International Journal of Mass Spectrometry.

24. А.В. Царев, Р.М. Балабин, С.А. Карпов, Р.З. Сюняев, Оптимизация эксплуатационных и экологических свойств спирто-бензиновых топлив под действием микроволновой обработки, Альтернативная энергетика и экология 8 (2007) 69.

25. A. Valuev, G. Norman, V. Podlipchuk, Molecular Dynamics Method: Theory and Applications, 1989. [in Russian]