On the validity of the Stokes–Einstein relation for various water force fields

Figures & data

Table 1. List of studies for which MD-calculated values for self-diffusivities and shear viscosities are available, along with the corresponding exponents for the fractional Stokes-Einstein relation that are calculated in the current study.

			Intercept	slope (s)	Intercept	slope (t)
Authors	Water Model	T range [K]	(D vs. T/n)	(D vs. T/n)	(D/T vs. n)	(D/T vs. n)
Dehaoui et al. [11]	Experimental	238–273	−30.355	0.812	−31.328	−0.800
Dehaoui et al. [11]	Experimental	274–373	−31.968	0.950	−32.192	−0.941
Harris [49]	Experimental	238–273	N/A	N/A	−5.297	−0.668
Harris [49]	Experimental	274–623	N/A	N/A	−4.977	−0.943
This work (SSE ^(*) included)	SPC/E	220–270	−31.949	0.946	−32.223	−0.941
This work (SSE included)	SPC/E	280–560	−31.877	0.942	−32.071	−0.924
This work (SSE included)	TIP4P/2005	220–270	−30.921	0.855	−31.689	−0.848
This work (SSE included)	TIP4P/2005	280–310	−31.119	0.883	−31.628	−0.861
This work ^(**) (SSE included)	TIP4P/2005	280–623	−31.793	0.934	−32.017	−0.913
Guevara-Carrion et al. [30]	SPC/E	280–373.15	−31.194	0.898	−31.597	−0.873
Moultos et al. [32]	SPC/E	298.15–623.15	−31.872	0.945	−32.044	−0.925
Park et al. [34]	SPC/E	243–273	−30.135	0.819	−31.038	−0.801
Park et al. [34]	SPC/E	273–550	−31.873	0.941	−32.064	−0.921
Guevara-Carrion et al. [30]	TIP4P/2005	273.15–400.15	−30.804	0.862	−31.373	−0.831
Moultos et al. [32]	TIP4P/2005	298.15–623.15	−32.439	0.977	−32.509	−0.968
Shvab & Sadus [33]	TIP4P/2005	670	−8.9073	0.798	−10.223	−0.798
Jiang et al. [25]	TIP4P/2005	298.15–523.15	−10.8148	0.918	−11.0311	−0.885
Koster et al. [35]	TIP4P/2005	280–340	−11.115	0.944	−11.356	−0.932
Kawasaki & Kim [38]	TIP4P/2005	190–260	−32.706	1.029	−32.542	−1.027
Kawasaki & Kim [38]	TIP4P/2005	260–300	−30.963	0.811	−32.032	−0.820
de Hijes et al. [29]	TIP4P/2005	200–270	−30.443	0.814	−31.428	−0.807
de Hijes et al. [29]	TIP4P/2005	280–300	−31.623	0.923	−32.011	−0.915
Shvab & Sadus [33]	TIP4P/2005f	670	−9.8614	0.859	−10.782	−0.859
Guillaud et al. [27]	TIP4P/2005f	225–273	−10.178	0.859	−10.916	−0.851
Guillaud et al. [27]	TIP4P/2005f	273–360	−11.245	0.950	−11.462	−0.941
Koster et al. [35]	Huang et al.	280–340	−6.008	0.553	−8.005	−0.480
Koster et al. [35]	TIP4P-TPSS	280–340	−13.433	1.110	−12.972	−1.132
Koster et al. [35]	TIP4P-TPSS-D3	280–340	−9.583	0.826	−10.319	−0.792
Jiang et al. [25]	HBP	298.15–523.15	−31.618	0.920	−31.898	−0.896
Jiang et al. [25]	BK3	298.15–523.15	−32.173	0.957	−32.335	−0.945

^(*) SSE: System Size Effects

^(**) MD data in the range 220–310 K are from the current work, while for higher temperatures the MD data are from Moultos et al. [32].

Figure 1. (a) Density plotted as a function of temperature for SPC/E (denoted with triangles) and TIP4P/2005 (denoted with circles) water models. Blue colour denotes values at 1 bar, while red colour at 1,000 bar. Solid lines (for T > 273 K) denote values from NIST [62] while dashed lines (for T < 273 K) experimental values reported in [63]. The dashed-dotted vertical black line indicates the temperature equal to 273.15 K. The inset of the lower-left side is a blow-up of the temperature range 180–380 K. (b) Comparison of densities for TIP4P/2005 calculated in the current study (denoted with circles) and in the work of Abascal and Vega [64] (denoted with crosses).

Figure 2. Arrhenius plots for: (a) the water shear viscosities, and (b) the water self-diffusion coefficients. MD results from SPC/E water model are denoted with triangles and circles, while MD results from TIP4P/2005 are denoted with crosses and x’s. Blue colour in symbols or lines denotes values at 1 bar, while red colour at 1,000 bar. The dashed-dotted vertical black line indicates temperatures equal to 273.15 K. Uncertainties are reported in the Supplemental Information. In the water shear viscosity plot (a) solid lines denote values from NIST [62], while the dashed black line denotes the experimental measurements by Dehaoui et al. [11]. In the water self-diffusion coefficient plot (b) the dashed lines denote experimental measurements from: Price et al. [9] (1 bar, T < 273 K), Krynicki et al. [8] (1 bar, T > 273 K), Prielmeir et al. [4] (1,000 bar, T < 273 K) and Krynicki et al. [8] (1,000 bar, T > 273 K).

Figure 3. Self-diffusion coefficients of water as a function of the inverse temperature at 1 bar (blue circles) and 1,000 bar (red triangles) for two water models: (a) SPC/E, and (b) TIP4P/2005. Black diamonds denote the MD results of Moultos et al. [32] for T > 273 K. The dashed lines correspond to correlations of all the MD data of the current study. Colour code of dashed lines. Arrhenius (ARH) law: green line; Vogel–Fulcher–Tamann (VFT) equation: magenta line; Speedy – Angel (SA) power law: cyan line, and Mode Coupling Theory (MCT): black line. The dashed-dotted vertical black line indicates temperatures equal to 273.15 K. Uncertainties are reported in the Supplemental Information.

Table 2. Parameters for the MD self-diffusion coefficient of water calculated using different correlations and % average absolute deviation (% AAD) between experimental data and correlations. The MD data are corrected to account for system size effects.

Correlation	Do (m^2/s)	α (K)	β (K)	γ (−)	% AAD
SPC/E water model (220–560 K)
ARH-type	$0.4625\times {10}^{-6}$	$1.510\times {10}^3$	na	na	30.64
VFT-type	na	$1.2308\times {10}^3$	39.7380	14.9288	18.86
SA-type	$1.399\times {10}^{-8}$	na	na	1.6881	9.55
MCT-type	$2.1508\times {10}^{-12}$	na	na	1.6426	13.77
TIP4P/2005 water model (220–623 K)
ARH-type	$0.6218\times {10}^{-6}$	$1.681\times {10}^3$	na	na	49.14
VFT-type	na	$2.043\times {10}^3$	−48.3683	13.9386	13.94
SA-type	$1.2673\times {10}^{-8}$	na	na	1.8713	9.86
MCT-type	$0.7026\times {10}^{-12}$	na	na	1.8332	14.86

na: not applicable

Figure 4. Comparison of the various calculated fractional exponents, t, with the experimental value for two cases: (a) temperatures below 274 K and (b) temperatures above 274 K. Data sources are the same as in Table 1. The vertical dashed red lines indicate the corresponding values obtained from the experimental data reported by Dehaoui et al. [11].

Figure 5. Testing of the Stokes-Einstein relation $D\sim (T/\eta )$ for the SPC/E water model: (a) This work, and (b) Other studies. The symbols denote MD results and the dashed lines are from experimental data (Dehaoui et al. [11]) fitted by the fractional Stokes-Einstein relation $D\sim (T/\eta {)}^s$. The dashed-dotted black line indicates the experimental data slope for T > 273.15 K. The dashed black line indicates the experimental data slope for T < 273.15 K. The vertical black dotted line indicates temperatures equal to 273.15 K.

Figure 6. Testing of the Stokes-Einstein relation $D\sim (T/\eta )$ for the TIP4P/2005 water model: (a) This work, and (b) Other studies. The symbols denote MD results and the dashed lines are from experimental data (Dehaoui et al. [11]) fitted by the fractional Stokes-Einstein relation $D\sim (T/\eta {)}^s$. The dashed-dotted black line indicates the experimental data slope for T > 273.15 K. The dashed black line indicates the experimental data slope for T < 273.15 K. The vertical black dotted line indicates temperatures equal to 273.15 K.

Figure 7. Testing of the Stokes-Einstein relation $D\sim (T/\eta )$ for various water models. The symbols denote published MD results and the dashed lines are from experimental data (Dehaoui et al. [11]) fitted by the fractional Stokes-Einstein relation $D\sim (T/\eta {)}^s$.

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