1. Predicting phase diagrams of fluids combining
Thermodynamic Perturbation Theory
and
Monte Carlo computer simulations.
Alejandro Gil Villegas
Department of Physical Engineering
University of Guanajuato
México
Thomas Young Centre
November 2015

2. University of Guanajuato
Department of Physical Engineering
Sciences and Engineering Division

3. Sciences and Engineering
Division
Physics
Physical
Engineering
Chemical, Electronic
and Biomedical
Engineering
Bachelor Programs:
Physics (1998)
Physical Engineering (1998)
Chemical Engineering (2010)
Biomedical Engineering (2011)
Postgraduate Programs:
M. Sc. and Ph. D. Physics (1988)
M. Sc. Applied Sciences (2013)

4. Molecular Thermodynamics
Molecular
Simulation
Theory Experiments

5. Associating Fluids
Ludwig Boltzmann, Lectures in gas theory (1895)
Modelos de sistemas asociativos
Michael Wertheim, “Fluids with highly directional attractive forces”
J. Stat. Phys. 35, 19 (1984)

6. Conclusiones
To exclude all knowledge but that which has already been applied
to immediate utility, would be to reduce our faculties to a state of
servitude, and to frustrate the very purposes which we are laboring
to accomplish. No discovery, however remote in its nature from the
subjects of daily observation, can with reason be declared wholly
inapplicable to the benefit of mankind.
Thomas Young
A course of lectures on Natural Philosophy and the Mechanical Arts (1807).
RIFUG 2013 Hooke,Young & Gibbs

7. W. Chapman, K. Gubbins, G. Jackson, M. Radosz, Fluid Phase Equil. 52, 31 (1989)
W. Chapman, K. Gubbins, G. Jackson, M. Radosz, Ind. Eng. Chem. Res. 29, 1709 (1990)
SAFT: Statistical Associating Fluid Theory
Fluidos Asociativ
Michael Wertheim:
Fluids with highly directional attractive
J. Stat. Phys. (1984-1986)
A
NkT
=
Aideal
NkT

Amonomers
NkT

Achain
NkT

Aassoc
NkT
Yukawa:Yukawa:
SAFT-VRSAFT-VR

10. Asphaltenes Biodiesel
Hydrogen

11. OH
σ
Buenrostro, Lira, Gil-Villegas & WuBuenrostro, Lira, Gil-Villegas & Wu
AIChE JAIChE J 50, 255250, 2552 (2004)(2004)
SAFT-VR prediction for asphaltene precipitationSAFT-VR prediction for asphaltene precipitation
from titration data of oil Y3 with n-heptane. Values of the new
parameters are ␺AA/k ϭ 2062.21 K and ␺AR/k ϭ 2359.43 K. No
other assumptions or parameter modifications were used.
The predicted precipitation curves for oil Y3 resulted within
10% error of the experimental data for precipitant/oil ratios Ն 10.
However, for lower ratios, there are larger deviations from the
experimental values, even though the model reproduces the de-
crease of the precipitation as the carbon number of the precipitant
increases. As mentioned earlier, for the prediction of the precipi-
tation in oil Y3, it was not necessary to refit the parameters related
to the mean-attractive energy (the Hamaker constants and the
range of the mean-force potentials). These results can be explained
in terms of the similar properties and composition of C1 and Y3
crude oils and the use of the same values for the asphaltene and
resin properties (Table 1), considering that both oils are produced
To model asphaltene precipitation in
certain T and P conditions, the vapor–liqu
culation of the reservoir fluid is first perfo
equation of state, to obtain the composition
of the liquid phase at equilibrium. Then, us
resin–oil model system described before,
analysis is applied to find out the onset of
tation. When asphaltene precipitation appe
precipitated material is calculated by solvi
equilibrium equations.
Calculation of Medium Properties. F
(2000), the volume-shifted Peng–Robinson
al., 1982; Peng and Robinson, 1976) was
vapor–liquid–equilibrium to obtain the de
tion of the liquid phase. Recommended va
Figure 10. Comparison between experimental and predicted APE for live oil C1.E. Buenrostro, C. Lira, A. Gil-Villegas and J. Wu, AIChE J 50, 2552 (2004).
effective spherical potential given by the following
expression:16
where r is the distance between the centers of mass
(CM) of each molecule with facing parallel aromatic
regions, and Z1, Z2 are the number of atoms in each
molecule. The parameters C1, C2, R, β, and γ were
obtained by fitting eq 1 to the corresponding MM
numerical values (T ) 0 K, in vacuum). Figure 2
displays the A-A, A-R, and R-R intermolecular po-
tentials generated according to the previous discus-
sion.16 The symbols correspond to the MM numerical
values and the continuous curves to the best over-all
fit found with eq 1. Note that within this spherical
approximation for the interactions, asphaltenes and
resins are considered as entities without internal struc-
ture in the MD simulations described further below.
Thus we shall be only concerned with the A-A, A-R,
and R-R spatial distributions in terms of the molecular
CM positions.
The embedding medium is accounted for through a
mean field approximation given by its dielectric con-
stant, ϵ, affecting only the Coulomb and London con-
tributions. The use of an effective medium through its
dielectric constant is known to be questionable for
intermolecular distances of the order of the molecular
size, mainly because it is at these distances where
atomic detail becomes important.18 However, it is still
a valuable first exploratory means to account for the
screening of Coulomb and London interactions due to
the embedding medium for intermediate to large inter-
molecular distances. It is deemed that the initial stages
of aggregation will correlate strongly with the long-
range interactions. As the interacting molecules become
closer to each other the molecular structure of the
solvent is more apparent and local effects become
important. These local interactions may be dominant
in defining the structure of an aggregate, solvation free
energies, possible chemical reactions, etc. Hence, the use
of an effective medium in our case has its aim in looking
at the evolution of aggregation from its initial stages
and extrapolating the behavior of the system in its final
“gross” equilibration (see further below).
Molecular Dynamics Simulations. The intermo-
lecular potentials described in the previous section were
used in a canonical molecular dynamics (MD) simula-
tion for mixtures of asphaltene/resin in different host
media. Our model system consists of NAs and NRe
(18) Smith, P. E.; Pettitt, M. B. J. Phys. Chem. 1994, 98, 9700.
Figure 1. Molecular models used in this work. (a) 3D diagram
of the MM-optimized asphaltene structure (C57H63N1S1) re-
ported in ref 7 showing an aromatic region (continuous lines)
and its aliphatic chain (dashed lines). (b) Transverse view of
the asphaltene structure showing the aromatic region in more
detail. (c) Diagram showing the resin structure (C13H10S1)
reported in ref 5.
V(r) )
Z1Z2
ϵr
(e-Rr
+ C1ϵre-βr
) -
C2
ϵ2
r6
e-γ/r4
(1)
Figure 2. Intermolecular potential curves in a vacuum for
the most favorable relative orientation (see text) between
asphaltene-asphaltene (A-A), asphaltene-resin (A-R), and
resin-resin (R-R). Symbols correspond to MM calculations.
Continuous curve: best fit using eq 1 according to ref 16.

12. Lauric Acid
+ Methanol + Water
Methy Laureate
F. Perdomo, B. Millán, G. Mendoza & A. Gil-VillegasF. Perdomo, B. Millán, G. Mendoza & A. Gil-Villegas
J. Mol. Liq.J. Mol. Liq. 185, 8 (2013).185, 8 (2013).
F. Perdomo, B. Millan, G. Mendoza, A. Gil-Villegas, J. Mol. Liq. 185, 8 (2013)
respect to the theoretical predictions a
with water, since integrals are inaccurate.
compounds: circles for lauric acid, square
for methyl laurate. From Figs. 2B and 3A w
reacting method is consistent with the exp
viations are noted for the biodiesel compo
tailed study, both experimental and the
case. With exception of this point, the agr
experiment is very good.
An expanded version of Fig. 2B is give
retical predictions are given for higher va
this figure, the molar fraction for lauric
R=6. The production of methyl laurate
that enable us to determine the optimal r
tive compositions. This information is of g
of an esterification reactor, and these re
tage of a molecular thermodynamic theor
predictions for compositions that maxim
6. Conclusions
We have presented in this article a the
proach for the prediction of reactive equ
compounds, that was applied to the est
order to test the predictions, experimenta
tion were measured from an experiment
an esterification batch reactor. The molar
Lauric Acid Methanol
Methyl Laurate
Methanol
Lauric Acid
Methyl Laurate
B
Fig. 2. Predicted chemical equilibrium manifold for the esterification of lauric acid with
methanol to produce methyl laureate and water, for atmospheric pressure at experimen-
tal conditions (0.83 atm) and temperature T=338.4 K. Experimental chemical reactions
correspond to reactive relation values R=0.125, 0.166, 0.250 and 1.0 (square, triangle,
circle and rhombus, respectively). Theoretical predictions are given by a dotted and
solid lines in (A) and (B), respectively. Figure (A) corresponds to the four-vertex chemical
equilibrium manifold whereas (B) is the projection onto the lauric acid-methanol-methyl
Fig. 3. Molar fraction of reactants and products (χ) a
(R). In figure (A) dotted, dashed and solid lines corr
acid, methanol and methyl laurate, respectively, wh
noted by circles, squares and triangles, respectively
for lauric acid; in the case of the other two species t
the symbol size. Theoretical predictions are given f
SAFT-VR Reactive System

13. Niels Bohr (1885-1962)
On the Constitution of Atoms and Molecules
Phil. Mag. 26, 1 (1913); 26, 476 (1913); 26, 857 (1913)

14. λB =
h
2πmkT

15. e−βA
=
1
N!λB
3N
drN
e
−βU rN
( )
∫
e
−βAQ
=
1
N!
drN
rN
e−β ˆH
rN
∫

16. rN
e−β ˆH
rN
= e−βEn
φn
*
φn
n
∑
rN
e−β ˆH
rN
= e−βWN
free particles: continuous potentials
hard spheres: discontinuous potentials
φn{ }
lim
λB→0
rN
e−β ˆH
rN
= e−βU
16

17. Eugene Wigner (1902-1995)
Symmetry properties in Quantum Mechanics
Wigner-D matrix
Wigner-Seitz cell
Metallic phase of hydrogen
The Unreasonable Effectiveness of Mathematics
in the Natural Sciences (1960).

18. Wigner-Kirkwood first-order QLJ potential
uLJ r( )= 4ε
σ
r
⎛
⎝⎜
⎞
⎠⎟
12
−
σ
r
⎛
⎝⎜
⎞
⎠⎟
6
⎡
⎣
⎢
⎤
⎦
⎥ u r( )= uLJ r( )+
Λ2
48π2
T *
∇2
uLJ r( )
Λ =
h
σ mε
λB =
Λσ
2πT *
r σ
u r( )
ρ*
T*
WK-1 pair potential for Ne, D2, H2 and He-IV
T* = 0.60
GEMC simulations and Wegner’s expansion
for the LV coexistence curves (WK-1 systems)
V.M. Trejos, A. Martínez, A. Gil-Villegas, J. Chem. Phys. 139, 184505 (2013)

19. Confined fluids
Classical Lennard-Jones system
V.M. Trejos, A. Martínez, A. Gil-Villegas, J. Chem. Phys. 139, 184505 (2013)
Y. Liu, A. Z. Panagiotopoulos, P. Debenedetti, J. Chem. Phys. 132, 144107 (2010)
A. Trokhymchuk, J. Alejandre, J. Chem. Phys. 111, 8510 (1999)

20. Quantum Lennard-Jones system
V.M. Trejos, A. Martínez, A. Gil-Villegas,
J. Chem. Phys. 139, 184505 (2013)

21. i!
∂ψ
∂t
= ˆHψ t → i!β
∂ ˆρ
∂β
= − ˆH ˆρ
x
t
Path Integral Method
R. P. Feynman, Rev. Mod. Phys. 20, 367 (1948)
S. F. Edwards, Proc. Phys. Soc. 85, 613 (1965)
D. Chandler, P. G. Wolynes, J. Chem. Phys. 74, 4078 (1981)

22. i!
∂ψ
∂t
= ˆHψ
ψ ′′x ,t( )= d ′x ′′x∫ e−i ˆH t−t0( ) !
′x ψ ′x ,t0( )
e−i ˆH t−t0( ) !
= e−i ˆHΔt !
e−i ˆHΔt !
e−i ˆHΔt !
⋅⋅⋅e−i ˆHΔt !
Ptimes
" #$$$$$ %$$$$$
′′x e−i ˆH t−t0( ) !
′x =
m
ihΔt
⎛
⎝⎜
⎞
⎠⎟
P
2
dxn∫n=1
P−1
∏ exp
iΔt
!
m
2
xn − xn−1
Δt
⎛
⎝⎜
⎞
⎠⎟
2
−V x( )
n=1
P
∑
⎛
⎝
⎜
⎞
⎠
⎟
⎧
⎨
⎪
⎩⎪
⎫
⎬
⎪
⎭⎪
Δt =
t − t0
P
x
t t +1

23. ∂ ˆρ
∂β
= − ˆH ˆρ
xP e−β ˆH
x0 =
1
Z
mP
π!2
β
⎛
⎝⎜
⎞
⎠⎟
P
2
dxn∫n=1
P−1
∏ exp −
β
P
mP2
2!2
β2
xn − xn−1( )2
+V x( )
⎛
⎝⎜
⎞
⎠⎟
n=1
P
∑
⎧
⎨
⎩
⎫
⎬
⎭
t − t0 = i!β ,
′′x e−i ˆH t−t0( ) !
′x xP e−β ˆH
x0
Δt =
i!β
P
Z =
1
N!
dx x∫ e−β ˆH
x

24. 0
λB

25. rN
e−β ˆH
rN
=
P
λB
⎛
⎝⎜
⎞
⎠⎟
3NP
e−βWNP
t=1
P
∏i=1
N
∏
N-particle fluid N-necklace fluid
P beads
βWNP =
πP
λB
2
ri
t
− ri
t+1
( )
2
t
P
∑
i
N
∑ + βWI

26. t
tt +1
t +1t + 2
t + 2
x
t t +1

27. βWNP =
πP
λB
2
ri
t
− ri
t+1
( )
2
t
P
∑
i
N
∑ + βWI
Crude propagator
Kinetic contribution = Diffraction term
Corrected propagators
pair interactions contribution
Improving kinetic term
Modelling diffraction of thermal waves

28. 105
106
108
0
0.5
1
1.5
2
2.5
-0.4 -0.2 0 0.2 0.4
|0(x)|2
x
1 ⇥ 108
Pasos
0
0.5
1
1.5
2
2.5
-0.4 -0.2 0 0.2 0.4
|0(x)|2
x
1 ⇥ 106
Pasos
0
0.5
1
1.5
2
2.5
-0.4 -0.2 0 0.2 0.4
|0(x)|2
x
1 ⇥ 105
Pasos
φ x( )
2
x
0
0.2
0.4
0.6
0.8
1
-3 -2 -1 0 1 2 3
|0(x)|2
x
1 ⇥ 108
Pasos
0
0.2
0.4
0.6
0.8
1
-3 -2 -1 0 1 2 3
|0(x)|2
x
1 ⇥ 106
Pasos
0
0.2
0.4
0.6
0.8
1
-3 -2 -1 0 1 2 3
|0(x)|2
x
1 ⇥ 105
Pasos
x
φ x( )
2
Free particle in a box
Harmonic oscillator
Non-interacting particles

29. 0
0.5
1
1.5
2
2.5
0 2 4 6 8 10
hEi
Resultado analítico cuántico
Resultado analítico clásico
P=2
P=5
P=20
P=40
E
!ω
β!ω 1
β!ω
1
2
coth
β!ω
2
⎛
⎝⎜
⎞
⎠⎟
P = 2
P = 5
P = 20
P = 40
Harmonic Oscillator

30. 125 necklaces - 6 beads
Quantum Hard Spheres
Conclusions
Quantum Hard Spheres
Matter Out of Equilibrium 2014 Confined fl
r σ
g(r)
1 2 30
1
2
3
4
ρσ 3
= 0.7 , λB σ = 0.3
Semiclassical MC
Path Integral MC
g(σ ) = 0
σeff ≈σ +
λB
2 2

31. Path Integral MC:
L. M. Sesé, R. Ledesma, J. Chem. Phys. 102, 3776 (1995)
C. Serna, A. Gil-Villegas (unpublished)
ZQHS =
1+ηeff +ηeff
2
−ηeff
3
1−ηeff( )
3 , ηeff = ηeff η,λB( )
AQHS =
4ηeff − 3ηeff
2
1−ηeff( )
2
B.-J. Yoon, H. A. Sheraga, J. Chem. Phys. 88, 3923 (1988)
ηeff =
2 + AQHS − 4 + AQHS
3+ AQHS
ηeff = 1+ c1λB( )η + c2λB + c3λB
2
( )η2

32. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
⇢⇤
0
1
2
3
4
5
aQHS
a)
⇤
B = 0.116
⇤
B = 0.3
⇤
B = 0.5
⇤
B = 0.8
Gibson
PIMC
YS
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
⇢⇤
0
2
4
6
8
10
12
ZQHS
b)
⇤
B = 0.116
⇤
B = 0.3
⇤
B = 0.5
⇤
B = 0.8
PIMC

33. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
⇢⇤
6
5
4
3
2
1
0
a1
a)
MC Classical
PIMC B-B
PIMC CM-CM
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
⇢⇤
6
5
4
3
2
1
0
a1
b)
TPT-PI
SAFT-VR
MC Classical
PIMC CM-CM

34. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
⇢⇤
0.5
0.4
0.3
0.2
0.1
0.0
a2
a)
MC Classical
MC Semiclassical
PIMC B-B
PIMC CM-CM
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
⇢⇤
0.5
0.4
0.3
0.2
0.1
0.0
a2
b)
TPT-PI
QLCA
GMCA
LCA
PIMC CM-CM
MC Classical
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
⇢⇤
0.10
0.08
0.06
0.04
0.02
0.00
0.02
0.04
0.06
0.08
a3
TPT-PI
LCA
PIMC CM-CM
MC classical
MC semiclassical

35. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
⇢⇤
7
6
5
4
3
2
1
0
U⇤
SAFT-VRQ
TPT-PI
PIMC QSW

36. Aads
NkT
= a2D − ln
λwallσ
λB
⎛
⎝⎜
⎞
⎠⎟ + βuwall
Abulk
NkT
= a3D
µads = µbulk
Adsorption of CO2 + CH4 and Asphalte
CO2 + CH4/
activated carbon
Asphaltenes/
Dolomite Rock
M. Castro, A. Mart´ınez, A. Gil-Villegas, AST 29, 59 (2011).
M. Castro, J.L. Mendoza, E. Buenrostro-Gonz´alez, S. L´opez-Rodr´ıguez, A. Gil-Villegas, Fluid Pha
M. Sudibandriyo, Z. Pan, J.E. Fitzgerald, R.L. Robinson Jr., K.A.M. Gasem, Langmuir 19, 5323
Thermodynamics 2013 Confined fluids with quantu
SAFT-VR
Experimental data
C/mg/L
AmountAdsorbed/mg/m2
Adsorption Theory
CH4 + CO2 / activated carbon Asphaltenes /Dolomite Rock
bulk phase
adsorbed phase
M. Castro, A. Martínez, A. Gil-Villegas, Ads. Sci. & Techn. 29, 59 (2011)
M. Castro, J. L. Mendoza, E. Buenrostro, S. López, A. Gil-Villegas, Fluid Phase Equil. 286, 113 (2009)
F. del Río, A. Gil-Villegas, J. Phys. Chem. 95, 787 (1991)

37. H2
graphene
Metallic-organic frameworks
Metallic-organic frameworks
Activated carbon
V. M. Trejos, M. Becerra, S. Figueroa-Gerstenmaier, A. Gil-Villegas, Mol. Phys. 112, 2330 (2014)

38. 1.7114 2.8983 19.9291 1.7009 15.9433 0.8165
es for the particle–wall energy parameter ϵw for hydrogen adsorption onto different substrates. Results are given
quantum (Q) adsorption SAFT-VR methods used in this work. Experimental values of the BET specific surface
heats are reported. The different substrates correspond to metal-organic frameworks (MOF), activated carbon
ene.
T(m2
g−1
) T/K Isosteric heat (kJ mol−1
) ϵw(C) (kJ mol−1
) ϵw(Q) (kJ mol−1
)
800 [57] 77 – 2.45 2.24
460 [57] 77 – 2.60 2.25
240 [57] 77 – 2.12 2.34
800 [58] 35 6.00–7.60 [58] 5.04 6.54
40 5.04 6.54
45 5.04 5.88
77 3.23 3.16
93 3.23 3.16
113 3.23 3.16
40 [59] 77 5.14–6.37 [60] 3.64 3.46
123 4.19 4.03
173 4.21 4.23
Quantum H2 1.7114 2.8983 19.9291 1.7009 1
Table 2. Optimised values for the particle–wall energy parameter ϵw for hydrogen adsorption onto different
for the classical (C) and quantum (Q) adsorption SAFT-VR methods used in this work. Experimental values
area (SBET) and isosteric heats are reported. The different substrates correspond to metal-organic framewor
(DAC AX-21), and graphene.
Adsorbent SBET(m2
g−1
) T/K Isosteric heat (kJ mol−1
) ϵw(C) (kJ mol−1
MOF-5 3800 [57] 77 – 2.45
MOF-205 4460 [57] 77 – 2.60
MOF-210 6240 [57] 77 – 2.12
DAC AX-21 2800 [58] 35 6.00–7.60 [58] 5.04
40 5.04
45 5.04
77 3.23
93 3.23
113 3.23
Graphene 640 [59] 77 5.14–6.37 [60] 3.64
123 4.19
173 4.21
V. M. Trejos, M. Becerra, S. Figueroa-Gerstenmaier, A. Gil-Villegas, Mol. Phys. 112, 2330 (2014)

39. 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
0.85
0.90
0.95
1.00
1.05
eff
⇢⇤
=0.1
⇤
B=0.1
⇤
B=0.2
⇤
B=0.3
⇤
B=0.4
⇤
B=0.5
⇤
B=0.6
⇤
B=0.7
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
0.85
0.90
0.95
1.00
1.05
eff
⇢⇤
=0.2
⇤
B=0.1
⇤
B=0.2
⇤
B=0.3
⇤
B=0.4
⇤
B=0.5
⇤
B=0.6
⇤
B=0.7
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
0.85
0.90
0.95
1.00
1.05
eff
⇢⇤
=0.3
⇤
B=0.1
⇤
B=0.2
⇤
B=0.3
⇤
B=0.4
⇤
B=0.5
⇤
B=0.6
⇤
B=0.7
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
0.85
0.90
0.95
1.00
1.05
eff
⇢⇤
=0.4
⇤
B=0.1
⇤
B=0.2
⇤
B=0.3
⇤
B=0.4
⇤
B=0.5
⇤
B=0.6
⇤
B=0.7
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
0.85
0.90
0.95
1.00
1.05
eff
⇢⇤
=0.5
⇤
B=0.1
⇤
B=0.2
⇤
B=0.3
⇤
B=0.4
⇤
B=0.5
⇤
B=0.6
⇤
B=0.7
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
0.85
0.90
0.95
1.00
1.05
eff
⇢⇤
=0.6
⇤
B=0.1
⇤
B=0.2
⇤
B=0.3
⇤
B=0.4
⇤
B=0.5
⇤
B=0.6

40. Exchange Contribution
Indistinguishability
of identical particles
e
−βAQ
=
1
N!
drN
rN
e−β ˆH
rN
∫
Ρ12 rN
= Ρ12 r1 r2 ... rN = ± r2 r1 ... rN
Exchange effects are equivalent to associating ring polymers
D. Chandler & P. G. Wolynes, J. Chem. Phys. 74, 4078 (1981)

41. Felipe Perdomo
Guadalupe Jiménez
Joel
Tavera
César Serna
Jonatan
Suaste
Eduardo Buenrostro,
Mexican Institute of Petroleum
Carlos Ariel Cardona,
National University of Colombia
Gerardo Gutiérrez,
University of Guanajuato
Gerardo
Campos
Mario
Becerra
Victor Manuel Trejos
Francisco
Espinosa
Job
Lino