Kirkwood-Buff Integrals for Aqueous Urea Solutions Based upon the Quantum Chemical Electrostatic Potential and Interaction Energies

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1 Supporting Information for Kirkwood-Buff Integrals for Aqueous Urea Solutions Based upon the Quantum Chemical Electrostatic Potential and Interaction Energies Shuntaro Chiba, 1* Tadaomi Furuta, 2 and Seishi Shimizu 3 1 Education Academy of Computational Life Sciences, Tokyo Institute of Technology, J Nagatsuta-cho, Midori-ku, Yokohama Japan 2 School of Life Science and Technology, Tokyo Institute of Technology, B Nagatsuta-cho, Midori-ku, Yokohama , Japan. 3 York Structural Biology Laboratory, Department of Chemistry, University of York, Heslington, York YO10 5DD, United Kingdom Contents S1. Experimental thermodynamic quantities and derivatives of molar fraction activity... S2 S2. Parameters for the urea molecule... S4 S3. Radial distribution functions and Kirkwood-Buff Integrals... S7 S4. Thermodynamic quantities calculated with AMOEBA without polarizable effect... S27 References... S28 S1

2 S1. Experimental thermodynamic quantities and derivatives of molar fraction activity It is necessary to calculate derivatives of the molar fraction activity to derive KBIs. We assumed that G m E could be expressed using the Redlich-Kister polynomial 1 as Chitra et al. 2 did to derive KBIs of urea. We expressed G m E up to the 2 nd order Redlich-Kister polynomial as a function of urea mole fraction, x i : G m E = x i (1 x i )(A + B(1 2x i ) + C(1 2x i ) 2 ), (S1) where A, B, and C are fitting parameters. Eq. (7) and the condition, ln f c (0) = 0, yields the following relationship: ln f c = 1 k B T [ 2(A + 3B + 5C)x c + (A + 9B + 29C)x c 2 4(B + 8C)x c Cx c 4 ]. (S2) By fitting Eq. S2 to experimental values 3 using NonlinearModelFit function in Mathematica version 10.0, 4 we obtained kj mol -1, kj mol -1, and 4.50 kj mol -1 for A, B, and C with enough small p-values of order. The adjusted R 2 of the experimental and fitted ln f c was We also tried the 3 rd and 4 th order Redlich-Kister polynomials for the fitting. Although the correlation between experimental and fitted ln f c improved, p-values of fitted parameters deteriorated to 10-3 and 10-1 order, respectively, indicating overfitting. Hence, we confirmed the 2 nd order form was appropriate. The molar fraction activity thus obtained and other experimental quantities are given in Table S1. S2

3 Table S1. Experimental thermodynamic quantities and corresponding KB integrals. c a d a mol L -1 g ml -1 V m,u a cm 3 mol -1 V m,w a cm 3 mol -1 κ T a GPa -1 f uu a Fitted a G m E b kj mol -1 G uu b cm 3 mol -1 G uw b cm 3 mol -1 G ww b cm 3 mol -1 N uu b N uw b N wu b N ww b a) The original Experimental values were obtained from the literature: mole concentration, c; density, d; 5 partial molar volume, V m,i ; 6 isothermal compressibility, κ T. 7 Data for each value were fitted to a polynomial as a function of concentration. f uu was obtained by fitting experimental activity coefficients to a model 3 (see section 2.1 KBIs and thermodynamic quantities). b) Based on experimental values, the mole excess Gibbs energy, G E m, and KB integrals, G ij were calculated. The excess coordination numbers, N ij were calculated via N ij = ρ j G ij. S3

4 S2. Parameters for the urea molecule The following parameters were used to calculate the Kirkwood-Buff integrals and related thermodynamic quantities. The vdw parameters were not optimized in the following parameters. The units for energy and length are given in kcal mol -1 and angstrom, respectively. Note that the unit for out-of-plane bend was set to This made urea structure almost plane. Atom name: Type Class Symbol Description Atomic Mass Valence C nonplanarurea O nonplanarurea N nonplanarurea H nonplanarurea Bond Stretching Parameters: Atom Classes K(S) Length Angle Bending Parameters: Atom Classes K(B) Angle Stretch-Bend Parameters: Atom Classes K(SB)-1 K(SB) Out-of-Plane Bend Parameters: Atom Classes K(OPB) S4

5 Torsional Parameters: Atom Classes 1-Fold 2-Fold 3-Fold 4-Fold 5-Fold 6-Fold Additional Pi-Orbital Torsion Parameters : Atom Class 2-Fold van der Waals Parameters : Atom Class Size Epsilon Reduction Atomic Multipole Parameters : Atom Type Coordinate Frame Definition Multipole Moments Bisector Z-then-X Z-then-X Z-then-X S5

6 Atomic Dipole Polarizability Parameters : Atom Type Alpha Damp Group Atom Types S6

7 S3. Radial distribution functions and Kirkwood-Buff Integrals Using sampling structures obtained from molecular dynamics simulation, radial distribution functions (RDF) and Kirkwood-Buff integrals (KBI) were calculated. In Figure S1-S4, the RDFs and KBIs for AMOEBA, GAFF, optimized AMOEBA, and KBFF for mol L -1 are shown. The KBIs calculated by averaging the values of KBIs from the range of nm are tabulated in Table S2 with statistical uncertainties calculated from individual production runs. The solution density, d, the partial molar volume, V m,i, and the excess coordination number, N ij were calculated using the calculated KBIs using Eqs (3), (5)-(7). The molar excess Gibbs E energy of the mixture, G m was calculated with the same procedure as outlined in Section S1. These thermodynamic quantities for AMOEBA, GAFF, optimized AMOEBA, and KBFF are tabulated in Table S3-S6. Comparing these parameters given by AMOEBA and the vdw-optimized AMOEBA, improvement of the latter can be observed in Figure S5 in terms of almost all the thermodynamic quantities. The exceptions were partial molar volume and isothermal compressibility, whose deviation from experimental values increased slightly. S7

8 Figure S1 S8

9 Figure S1 (Continued) S9

10 Figure S1 (Continued) S10

11 Figure S1 (Continued) Figure S1. Radial distribution functions (RDFs), enlarged RDFs, and Kirkwood-Buff integrals (KBIs) for urea-urea (uu), urea-water (uw), and water-water (ww) pairs at various urea concentrations calculated using AMOEBA. Green dash and cyan chain lines represent the standard error, respectively. For short r, some figures of the enlarged RDFs lack RDF curves because no data to plot curves in the range of the Y-axis were available. The KBIs were calculated with the DKBI method. S11

12 Figure S2 S12

13 Figure S2 (Continued) S13

14 Figure S2 (Continued) S14

15 Figure S2 (Continued) Figure S2. Radial distribution functions (RDFs), enlarged RDFs, and Kirkwood-Buff integrals (KBIs) for urea-urea (uu), urea-water (uw), and water-water (ww) pairs at various urea concentrations calculated using GAFF. Green dash and cyan chain lines represent the standard error, respectively. For short r, some figures of the enlarged RDFs lack RDF curves because no data to plot curves in the range of the Y-axis were available. The KBIs were calculated with the DKBI method. S15

16 Figure S3 S16

17 Figure S3 (Continued) S17

18 Figure S3 (Continued) Figure S3. Radial distribution functions (RDFs), enlarged RDFs, and Kirkwood-Buff integrals (KBIs) for urea-urea (uu), urea-water (uw), and water-water (ww) pairs at various urea concentrations calculated using the vdw-optimized AMOEBA. Green dash and cyan chain lines represent the standard error, respectively. For short r, some figures of the enlarged RDFs lack RDF curves because no data to plot curves in the range of the Y-axis were available. The KBIs were calculated with the DKBI method. The RDF and KBI of neat water is the same as those determined with AMOEBA. S18

19 Figure S4 S19

20 Figure S4 (Continued) S20

21 Figure S4 (Continued) S21

22 Figure S4 (Continued) Figure S4. Radial distribution functions (RDFs), enlarged RDFs, and Kirkwood-Buff integrals (KBIs) for urea-urea (uu), urea-water (uw), and water-water (ww) pairs at various urea concentrations calculated using KBFF for urea. Green dash and cyan chain lines represent the standard error, respectively. For short r, some figures of the enlarged RDFs lack RDF curves because no data to plot curves in the range of the Y-axis were available. The KBIs were calculated with the DKBI method. S22

23 Table S2. Calculated KB integrals using AMOEBA, GAFF, optimized AMOEBA, and KBFF. c AMOEBA c GAFF c mol L -1 a G uu SE G uw SE G ww SE G uu SE G uw SE G ww SE (Continued) c vdw-optimized AMOEBA c KBFF c mol L -1 a G uu SE G uw SE G ww SE G uu SE G uw SE G ww SE KBIs were calculated using the direct KBI method. The standard errors (SEs) were calculated from the individual production runs (see Section 3.2). a) c stands for mole concentration b) Numbers of urea and water molecules calculated using the experimental mole concentration, c and density, d in Table 1, which were used to build simulation systems. c) All the KB integrals (G ij ) and standard errors (SEs) are given in cm 3 mol -1. S23

24 Table S3. Calculated thermodynamic quantities with AMOEBA. c mol L -1 d a g ml -1 V m,u cm 3 mol -1 V m,w cm 3 mol -1 f uu G m E b kj mol -1 κ T Eq. (4) GPa -1 κ T c GPa -1 N uu N uw N wu N ww a) The standard error calculated from the individual production runs was less than 10-4 g ml -1. b) Using calculated KBIs and Eq. S2, the fitting parameters, A (34.61 kj mol -1 ), B ( kj mol -1 ), and C (9.52 kj mol -1 ), were obtained. The maximum p-value (0.068) of the fitted parameters was higher compared to that obtained when we fitted Eq. S2 to the experimental values, indicating over fitting. c) These values were calculated via the volume fluctuation formula (( V 2 V 2 ) k B T V ). Table S4. Calculated thermodynamic quantities with GAFF. c mol L -1 d a g ml -1 V m,u cm 3 mol -1 V m,w cm 3 mol -1 f uu G m E b kj mol -1 κ T Eq. (4) GPa -1 κ T GPa -1 N uu N uw N wu N ww a) The standard error calculated from the individual production runs was less than 10-4 g ml -1. b) Using calculated KBIs and Eq. S2, the fitting parameters, A (30.15 kj mol -1 ), B ( kj mol -1 ), and C (8.77 kj mol -1 ), were obtained. The maximum p-value of the fitted parameters was c) These values were calculated via the volume fluctuation formula (( V 2 V 2 ) k B T V ). S24

25 Table S5. Calculated thermodynamic quantities with optimized AMOEBA. c mol L -1 d a g ml -1 V m,u cm 3 mol -1 V m,w cm 3 mol -1 f uu G m E b kj mol -1 κ T Eq. (4) GPa -1 κ T GPa -1 N uu N uw N wu N ww a) The standard error calculated from the individual production runs was less than 10-4 g ml -1. b) Using calculated KBIs and Eq. S2, the fitting parameters, A (7.65 kj mol -1 ), B (-5.18 kj mol -1 ), and C (3.08 kj mol -1 ), were obtained. The maximum p-value (0.59) of the fitted parameters was higher compared to that obtained when we fitted Eq. S2, indicating over fitting. And G m E are not reliable. c) These values were calculated via the volume fluctuation formula (( V 2 V 2 ) k B T V ). Table S6. Calculated thermodynamic quantities with KBFF. c mol L -1 d a g ml -1 V m,u cm 3 mol -1 V m,w cm 3 mol -1 f uu G m E b kj mol -1 κ T Eq. (4) GPa -1 κ T GPa -1 N uu N uw N wu N ww a) The standard error calculated from the individual production runs was less than g ml -1. b) Using calculated KBIs and Eq. S2, the fitting parameters, A (25.89 kj mol -1 ), B ( kj mol -1 ), and C (7.57 kj mol -1 ), were obtained. The maximum p-value (0.09) of the fitted parameters was higher compared to that obtained when we fitted Eq. S2, indicating over fitting and that G m E are not reliable. c) These values were calculated via the volume fluctuation formula (( V 2 V 2 ) k B T V ). S25

excess coordination numbers, N ij, as a function of urea concentration, are plotted against molar concentration of

26 Figure S5. Comparison of calculated thermodynamic quantities for 0 to 7.83 mol L -1. Solution density, d, partial molar volume, V m,i, activity derivative, f uu, isothermal compressibility, κ T, and excess coordination numbers, N ij, as a function of urea concentration, are plotted against molar concentration of urea. κ T was calculated via Eq. (4) or the volume fluctuation formula (( V 2 V 2 ) k B T V ). 8 The latter is plotted with a thin line marked with an arrow. S26

S4. Thermodynamic quantities calculated with AMOEBA without polarizable effect We estimated how much the polarizable effect of AMOEBA contributed to the improvement of reproducing thermodynamic

83-mol L -1 solution systems for 1 ns without incorporating polarizability, where interaction energy was calculated without induced dipole moments.

27 S4. Thermodynamic quantities calculated with AMOEBA without polarizable effect We estimated how much the polarizable effect of AMOEBA contributed to the improvement of reproducing thermodynamic quantities. We conducted MD simulations for and 7.83-mol L -1 solution systems for 1 ns without incorporating polarizability, where interaction energy was calculated without induced dipole moments. The density of each system reached plateau by the first 0.5 ns and the following 0.5-ns trajectory was used to calculate the average system density. The resulting system densities significantly deviated from the densities calculated with the induced dipole and experimental values, i.e., ± and ± g ml -1 for and 7.83-mol L -1 solution systems, respectively. (The experimental densities and calculated densities calculated with the induced dipole are shown in Table S1 and S3, respectively). Lack of the induced dipole would weaken favorable interaction of molecules, resulting in expansion of the system. Figure S2 shows a plot of the AMOEBA energies against the QC energies indicating that the incorporation of the induced dipole corrected underestimated favorable interaction energies. Figure S6. Interaction energy for urea-urea, urea-water, and water-water pairs calculated with AMOEBA is plotted against those calculated with MP2/aug-cc-pVTZ using the BSSE correction. Blue triangle and red circle AMOEBA s correspond to interaction energies calculated without and with induced dipole. S27

28 References 1. Redlich, O.; Kister, A. T., Algebraic Representation of Thermodynamic Properties and the Classification of Solutions. Ind. Eng. Chem. 1948, 40, Chitra, R.; Smith, P. E., Molecular Association in Solution: A Kirkwood Buff Analysis of Sodium Chloride, Ammonium Sulfate, Guanidinium Chloride, Urea, and 2,2,2-Trifluoroethanol in Water. J. Phys. Chem. B 2002, 106, Stokes, R., Thermodynamics of aqueous urea solutions. Aust. J. Chem. 1967, 20, Mathematica, Version 10.0, Wolfram Research, Inc., Champaign, IL, Kawahara, K.; Tanford, C., Viscosity and Density of Aqueous Solutions of Urea and Guanidine Hydrochloride. J. Biol. Chem. 1966, 241, Gucker, F. T.; Gage, F. W.; Moser, C. E., The Densities of Aqueous Solutions of Urea at 25 and 30 and the Apparent Molal Volume of Urea1. J. Am. Chem. Soc. 1938, 60, Endo, H., The Adiabatic Compressibility of Nonelectrolyte Aqueous Solutions in Relation to the Structures of Water and Solutions. Bull. Chem. Soc. Jpn. 1973, 46, Allen, M. P.; Tildesley, D. J., Computer Simulation of Liquids. Oxford University Press: S28

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