A Successful Model for the Condensed Phases of Water : TIP4P/2005
Departamento de Quımica Fısica
Universidad Complutense
Madrid, SPAIN
Carlos Vega
Jose Luis F. Abascal
Maria M. Conde
Juan L. Aragones
MODELS OF WATER
SPC/E Berendsen et al. 1987
� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �
� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �
� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �
� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �
� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �
� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �
� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �
� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �
109.5
−2
o
δ /LJ
δ+ δ+
A1o
δ/|e| = 0,4238
σ = 3,16557A
ε/k = 78,2K
µ = 2,35D(1,85)
TIP4P Jorgensen et al. 1983
� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �
� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �
� � �� � �� � �� � �� � �� � �
� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �
� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �
� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �
� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �
δ+ δ+
LJ
0.957
104.5
−2
0.15Α
Αο
ο
ο
δ
δ/|e| = 0,52
σ = 3,154A
ε/k = 78,02K
µ = 2,18D
MODELS OF WATER
TIP5P , Mahoney and Jorgensen , JCP, 2000
δ/|e| = 0,241
σ = 3,12A
ε/k = 80,52K
µ = 2,29D
The phase diagram of water
J.Finney, Phyl.Trans.R.Soc.Lond.B,(2004)
1900 Tammann , 1912 Bridgman, 1968 Whalley , 1998 Finney et al.
Obtaining free energies for the solid (or fluid-solid coexistence)
Cell occupancy W. G. Hoover and F. H. Ree, JCP, 1968
Direct coexistence A. J. C. Ladd and L. V. Woodcock,CPL, 1977.
Constrained fluid lambda-integration
G. Grochola , JCP, 2004, 2005.
D. M. Eike and J. F. Brennecke and E. J. Maginn, JCP, 2005.
Self referential method M. B. Sweatman, PRE, 2005.
Phase switch N. B. Wilding and A. D. Bruce, PRL, 1997, 2000.
Einstein crystal D. Frenkel and A. J. C. Ladd, JCP, 1984.
J. M. Polson and E. Trizac and S. Pronk and D. Frenkel, JCP, 2000.
Water. C.Vega and P. A. Monson, JCP, 1998
Water. E.G.Noya, M.M.Conde and C. Vega , JCP, 2008.
Lattice dynamics Water.G. T. Gao and X. C. Zeng and H. Tanaka, JCP, 2000.
PHASE DIAGRAMS FOR THE TIP4P AND SPC/E MODELSE.Sanz, C.Vega, J.L.F.Abascal and L.G.MacDowell,PRL, 92, 255701, (2004)
Strategy : Free energy calculations + Gibbs Duhem integration (Kofke, 1993)
0.01
0.1
1
10
100 150 200 250 300 350 400 450
P (
GP
a)
T (K)
I
II
VI
VIII VII
liquidV
III
0.01
0.1
1
10
150 200 250 300 350 400 450
P (
GP
a)
T (K)
I
II
VI
VIII VII
liquid
V
III
0.01
0.1
1
10
100 150 200 250 300 350 400 450
P+
0.1
(G
Pa)
T (K)
I
II
VI
VIII VII
liquid
TIP4P Experiment SPC/E
The ideas leading to TIP4P/2005
J.L.F.Abascal and C.Vega, JCP, 123, 234505, (2005)
The model takes from TIP4P the geometry of the charge distribution,
since it reproduces correctly the phase diagram of water (due to a good
balance between dipolar and quadrupolar forces ).
The model takes from SPC/E the idea of reproducing the vaporization
enthalpy after including the vaporization correction.
The model takes from TIP5P the idea of using the maximum in density of
water as a target property.
The model also includes as target properties the density of several ice
polymorphs and the melting point of ice Ih.
Potential models of water
Model dOH H-O-H σ (ε/kB) qH dOM dOL
TIP3P 0.957 104.5 3.150 76.5 0.417 0 -
TIP4P 0.957 104.5 3.154 78.0 0.52 0.15 -
TIP4P/2005 0.957 104.5 3.158 93.2 0.556 0.155 -
TIP5P 0.957 104.5 3.120 80.5 0.241 - 0.70
Does such a small change make any difference ?
Ten properties to be analyzed
1. VLE and critical point.
2. Surface tension.
3. Densities of the different ice polymorphs.
4. Phase diagram calculations.
5. Melting temperature.
6. TMD. α, and κT .
7. Structure of water and ice Ih.
8. EOS at high pressures.
9. Self-diffusion coefficient.
10. Dielectric constant.
Award 0 to 3 points for each property (3=best)
Vapour-liquid equilibrium and surface tension
300
350
400
450
500
550
600
650
0 0.2 0.4 0.6 0.8 1
T (
K)
ρ(g/cm3)
ExptlTIP4P/2005
TIP4PTIP3PTIP5P
200 300 400 500 600 700T / K
0
20
40
60
80
σ
/ (m
N /
m)
Experiment TIP3PTIP5PTIP4PTIP4P/2005
σ =Lz
2[pN − pT ] (1)
σ = lım∆S→0
−kT
2∆S
(
ln⟨
exp(−∆U+/kT )⟩
− ln⟨
exp(−∆U−/kT )⟩)
(2)
TAM. G. J. Gloor , G. Jackson , F. J. Blas and E. de Miguel, JCP, 123, 134703, (2005)
Density predictions for ices
-0.1
-0.05
0
0.05
0.1
ρ−ρ
TIP5P SPC TIP4P TIP3P TIP4P/2005
exp
6.5% 2.2% 2.6% 3.5% 0.9%
IIIV
XIIc
VIXII
IhIX
VIII
Phase diagram of TIP3P and TIP5P
-500
0
500
1000
140 160 180 200 220 240 260 280 300
p (
MP
a)
T (K)
liquid
Ih
II
V
liquidIh
II
TIP3PExpt
-500
0
500
1000
1500
2000
2500
3000
3500
4000
220 240 260 280 300 320 340 360p (
MP
a)
T (K)
liquid
Ih
II
VI
liquidVI
VII
TIP5PExpt
Phase diagram of TIP4P/2005
100 200 300 400 500
T(K)100
1000
10000
1e+05
p(b
ar) Liquid
II
III
I
V
VI
VIIVIII
100 200 300 400 500
T (K)
100
1000
10000
1e+05
p (
bar)
VIIPlastic crystal
(XV?)VIII
VI
V
Ih
IILiquid
III
J.L.Aragones,M.M.Conde,E.G.Noya and C.Vega,Phys.Chem.Chem.Phys., in press,
2008.
Melting properties of ice Ih
Model TIP3P SPC/E TIP4P TIP4P/2005 TIP5P Exptl
Tm(K) 146 215 232 252 274 273.15
ρl 1.017 1.011 1.002 0.993 0.987 0.999
ρIh 0.947 0.950 0.940 0.921 0.967 0.917
∆Hm 0.30 0.74 1.05 1.16 1.75 1.44
dp/dT -66 -126 -160 -135 -708 -137
Tm/Tc 0.25 0.337 0.394 0.394 0.525 0.422
DIRECT DETERMINATION OF THE FLUID-SOLID EQUILIBRIA
R.G. Fernandez, J.L.F.Abascal and C.Vega, JCP, 124, 144506, (2006)
Melting temperatures obtained by direct coexistence are in agreement with those
obtained from free energy calculations
Temperature of maximum density
150 200 250 300 350T (K)
0,97
0,98
0,99
1
1,01
1,02
1,03
1,04
ρ (g
/cm
3 )
TIP3P
TIP4P/2005
TIP5P
TIP4P
O-O radial distribution function : water and ice Ih
2 4 6 8 10r / Å
0
0.5
1
1.5
2
2.5
3
3.5
g (
r)
Experiment TIP4PTIP4P/2005
2 4 6 8 10r / Å
0
1
2
3
4
5
6
7
8
9
g (
r)
Experiment TIP4PTIP4P/2005
Equation of state and diffusion coefficient of water
1.2 1.25 1.3 1.35 1.4
ρ (g/cm3)
10000
12500
15000
17500
20000
22500
p (
bar)
Exp.
TIP4P/2005TIP4PTIP3PTIP5P
1
2
4
8
3.1 3.2 3.3 3.4 3.5 3.6
109 D
(m
2 /s)
103/T (K-1)
TIP3PTIP4PTIP5P
TIP4P/2005Exptl
Property TIP3P TIP4P TIP4P/2005 TIP5P
1. VLE, Tc 1 2 3 0
2. Surface tension 1 2 3 0
3. ρ ices 0 2 3 1
4. Phase diagram 0 2 3 1
5. Tm melting prop. 0 1 2.5 2.5
6. TTMD, α, κT 0 1 3 2
7. Structure 0 1 2.5 2.5
8. EOS (high p) 2 1 3 0
9. D 0 1 3 2
10. ε 2 0 1 3
Total 6 13 27 14
Viscosity of water at room T and p
Model Viscosity (centi Poises)
TIP3P 0,33
SPC/E 0,73
TIP4P/2005 0,86
Experiment 0,89
Variation of the TMD with p, and EOS for supercooled water
0 250 500 750 1000 1250 1500p / bar
220
230
240
250
260
270
280
T /
K TIP4P/2005Experimental
150 175 200 225 250 275 300 325 350 375 400
T / K0,85
0,875
0,9
0,925
0,95
0,975
1
1,025
1,05
den
sit
y /
g c
m -3
TIP4P/2005Experiment
Experiment (confined water): F. Mallamace et al., PNAS, 104, 18387, (2007)
Calculations: H.L.Pi, C.Vega, et al., Mol.Phys. , submitted (J.J.Weiss special issue)
240 260 280 300 320 340 360 380
T (K)
30
40
50
60
70
80
9010
5 κ T (M
Pa
-1)
Expt (100 MPa)
Expt (0.1 MPa)
TIP4P/2005
240 260 280 300 320 340 360 380
T (K)
30
40
50
60
70
80
90
105 κ T
(MP
a-1)
Expt.
TIP4P
SPC/E
TIP5P
0.1 MPa
Isothermal compressibility
Conclusions
The TIP4P/2005 model (designed to be used with Ewald sums) keeps thegeometry of TIP4P, incorporates the polarization correction of Berendsenet al. (as SPC/E) and incorporates the TMD as a target property (as TIP5P)
Since the model does not include polarizability it fails in describing thedielectric constant of water and the properties of the vapor phase.
The model provides a quite good description of the phase diagram ofwater, density of ices, TMD of water , melting point , (by design). Themodel describes quite well D, EOS at high p, VLE, Tc, σ, η, kT andTMD(p).
The model provides an overall improvement (27 points) in the descriptionof water with respect to TIP3P (6 points) , TIP4P (13 points), TIP5P (14points) and SPC/E (21 points). The model is probably close to the best ofwhat can be achieved by a rigid, non-polarizable model , with a LJ centerand three charges.
References.
The model.
J.L.F.Abascal and C.Vega,J.Chem.Phys., 123, 234505, (2005)
Free energy calculations and phase diagram determination.
C.Vega, E.Sanz, J.L.F.Abascal and E.G.Noya , Determination of
phase diagrams via computer simulation: methodology and applications to
water, electrolytes and proteins , J.Phys.Condens.Matter 20, 153101,
(2008).
Comparing TIP4P/2005 with other water models.
C. Vega, J. L. F. Abascal , M. M. Conde and J. L. Aragones What ice
can teach us about water interactions: a critical comparison of the
performance of different water models, Faraday Discussions, 141, 251,
(2009). DOI: 10.1039/b805531a
Dipole and quadrupole moments
Model µ Qxx Qyy Qzz QT µ/QT
SPC 2.27 2.12 -1.82 -0.29 1.97 1.15
SPC/E 2.35 2.19 -1.88 -0.30 2.03 1.15
TIP3P 2.35 1.76 -1.68 -0.08 1.72 1.36
TIP4P 2.18 2.20 -2.09 -0.11 2.15 1.01
TIP4P/2005 2.30 2.36 -2.23 -0.13 2.30 1.00
TIP5P 2.29 1.65 -1.48 -0.17 1.56 1.46
Gas(Expt.) 1.85 2.63 -2.50 -0.13 2.56 0.72
Computing the free energy for solids
A=A Ein−idCM
Λ )3+kTln(V/
Solid
Λ )3−kTln(V/
Solid with one fixedparticle
Ideal Einstein moleculewith one fixed particle
A=A Ein−mol−id
Ideal Einstein molecule
∆Α + ∆Α1 2 ∆Α + ∆Α1 2* *
∆Α 3*
Ideal Einstein crystalwith fixed CM
Solid with fixed CM
Solid
(b)(a)
The Einstein molecule approach
EINSTEINMOLECULEMETHOD
CARRIER( Atom O fixed )
NVT ( T )
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2
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� �������� 1
�� �������� 2
ANISOTROPIC NpT
U lattice
��� ��� ���� 1
��� � !�! "�" #�# $�$ 2
∆A 1
EVALUATING
∆A 2
EVALUATING% %% %& && &'�' (�( )�) *�* 2
+�+ ,�,- -- -. .. .
1/�/�/ 0�0�0
U sol
U Ein,or
Ein−mol−idU
U Ein,or
Ein−mol−idU 1 11 11 12 22 22 23�3 4�4 56 2
7�789 99 9: :: :
1;�; <�<
U sol=0
E.G.Noya, M.M.Conde and C.Vega, J.Chem.Phys., 129, 104704, (2008)