Calculation of Physical Properties of the Methanol-Water Mixture Using Molecular Dynamics Simulation

Transcription

1 Iranan Journal of Chemcal Engneerng Vol. 6, No. 4 (Autumn), 29, IAChE Calculaton of Physcal Propertes of the Methanol-Water Mxture Usng Molecular Dynamcs Smulaton N. Farhadan, M. Sharaty-Nassar 2 - Academc Member of Azad Unversty, Damghan Branch., Iran 2- Assocaton Professor of Chemcal Engneerng School of Engneerng Unversty of Tehran- Iran. Abstract In ths study some propertes of the methanol-water mxture such as dffusvty, densty, vscosty, and hydrogen bondng were calculated at dfferent temperatures and atmospherc pressure usng molecular dynamcs smulatons (MDS). The results were compared wth the avalable expermental data as well as some theoretcal models; overall ndcatng a good agreement. Ths shows the useful and effectve applcaton of MDS for determnaton of physcal propertes. Keywords: Molecular dynamcs smulaton, Ensten equaton, Dffuson coeffcent, Hydrogen bond - Introducton As molecular smulatons are consdered useful tools for the nterpretaton of expermental data, two methods namely Monte Carlo (MC) and Molecular Dynamcs (MD), can be appled. Ths allows us to determne macroscopc propertes by evaluatng exactly a theoretcal model of molecular behavor usng a computer program. [, 2] Molecular smulaton consders small sze systems, at a typcal scale of a few nanometers, and determnes ther behavour from a careful computaton of the nteractons between ther components. These computatons take much more computer tme than classcal thermodynamc models (. e. from a few hours to several weeks of computng tme, dependng on system sze). As such, molecular smulaton s now consdered n the chemcal ndustry as capable of fllng the exstng gap between expermental data and engneerng models n varous crcumstances such as unknown chemcals, extreme condtons of temperature or pressure, and toxc compounds. The frst part of ths artcle provdes a bref descrpton of Molecular dynamc smulaton Correspondng author: na.farhadan@ gmal.com 62

2 Calculaton of Physcal Propertes of the Methanol-Water Mxture Usng Molecular Dynamc Smulaton and how t can be analyzed for approvng expermental data. In the second part, dfferent propertes of the system wll be calculated by molecular dynamc smulaton. Results wll then be compared wth avalable expermental data and theoretcal models. 2- Molecular dynamcs method 2.- What s molecular dynamcs method? Molecular dynamcs smulaton nvolves ntegraton of Newton s equaton of moton for each atom n an ensemble n order to generate nformaton on the varaton n the poston, velocty, and acceleraton of each partcle as a functon of tme. The equatons solved are [3]: f m a () f U (2) r Where f s the force exerted on the th atom, m s ts mass, and a s ts acceleraton. The force actng on each atom s obtaned from a potental functon U, such as that depcted below [3]: b kj k U (r r ) ( ) 2 2 dhedrals eq 2 jk eq 2 j j jk jk angle eq k cos(n( )) A B q q j j 2 6 j rj rj 4rj (3) Where the frst three terms reflect ntramolecular nteractons between covalently bonded atoms and the last term descrbes the non-bonded nteractons (van der Waals and electrostatc). The constants k b, k è, and k ö are force constants for the covalent bonds, bond angles, and dhedrals, respectvely, r j are the dstances between atoms and j, A j and B j are the Lenard Jones parameters, and q s a partal charge [3]. To calculate the propertes of a real system, t s convenent to defne an ensemble. An ensemble can be regarded as an magnary collecton of a very large number of systems n dfferent quantum states wth common macroscopc attrbutes. The ensemble average of any dynamc property (A) can be obtaned from the relatonshp: A A p (4) Where A s the value of A n quantum state, p represents the probablty of observng the -th state, and the angled brackets denote an ensemble average. The tme averaged propertes of the real system are related to the ensemble average by nvokng the followng assumpton: A t A (5) The above equvalence of the tme average and ensemble average s called the Ergodc hypothess. Form of p s determned by the macroscopc propertes that are common to all of the systems of the ensemble Calculaton of physcal propertes Predcton of transport propertes of fluds of ndustral nterest s one of the man motvatons to perform MD smulatons [4]. Iranan Journal of Chemcal Engneerng, Vol. 6, No. 4 63

3 Farhadan, Sharaty-Nassar As t s not possble to observe ndvdual atoms or molecules drectly, varous models are used to descrbe and/or predct the propertes of a system wth molecular dynamc methods. On the other hand, t s possble to calculate macroscopc propertes of the mxtures by calculatng the nteracton of the molecules. The partal densty s one macroscopc property that could be calculated usng molecular dynamcs smulaton. Also transport propertes lke the dffuson coeffcent and shear vscosty can be computed drectly from an MD smulaton too [5]. In an equlbrum molecular dynamcs (EMD) smulaton, the selfdffuson coeffcent D S of component s computed by takng the slope of the mean square dsplacement (MSD) at a long perod of tme: D S = 2dN lm t N = ( r ( t) r ()) 2 (6) Where N s the number of molecules of component, d s the spatal dmenson of the system, t s the tme, and r s the center of mass of molecule of component. Equvalently, D S s gven by the tme ntegral of the velocty autocorrelaton functon: N v ( ) t v = S D = () (7) dn Where v s the center of mass velocty of molecule of component. Equaton (6) s known as the Ensten equaton and equaton (7) s often referred to as the Green-Kubo relaton. [6] Also shear vscosty can be calculated usng the followng equaton: V η = σ xy ( t) σ xy () dt (8) k T B Where σ xy s the pressure tensor of the components. k B s the Boltzman constant and T s the temperature of the system GROMACS Software GROMACS s an acronym for GROnngen MAchne for Chemcal Smulaton. It s the latest release of a versatle and very well known optmzed package for molecular smulaton. The package s a collecton of programs and lbrares for the smulaton of molecular dynamcs and the subsequent analyss of trajectory data. Although t s prmarly targeted at bologcal molecules wth complex bonded nteractons, the very effectve mplementaton of nonbonded force calculatons makes GROMACS sutable for all knds of molecular dynamcs smulatons based on par potentals. Apart from normal potental functons lke Lenard-Jones, Buckngham and Coulomb, t s possble to use arbtrary forms of nteractons wth splne-nterpolated tables [7]. The GROMACS molecular dynamcs code contans several algorthms that have been developed n the Berendsen group. The leapfrog ntegrator s used n two versons, for molecular dynamcs and stochastc dynamcs [8]. 3. Result and Dscusson 3.-System defnton In Gromacs software, the system s defned 64 Iranan Journal of Chemcal Engneerng, Vol. 6, No. 4

4 Calculaton of Physcal Propertes of the Methanol-Water Mxture Usng Molecular Dynamc Smulaton by ts sze and shape, the number and types of molecules t contans, and the coordnates and veloctes of all atoms. Veloctes may also be generated from a Maxwellan dstrbuton [9]. In ths study MD smulaton was used to examne some physcal and transport propertes of the mxture of methanol water. Ths system conssts of 26 molecules of water and 26 molecules of methanol. Smulatons are carred out usng a fully atomstc system wth the maxmum sze 4.6 * 2.3 *2.3 nm. The box s shown n Fg.. Gromacs software was used for the MD calculatons. Smulaton was performed usng GROMOS 96 [, ] force feld for 2 ps. MD smulatons are performed at atmospherc pressure wth four dfferent temperatures 273, 283, 288, 293 k at the sothermal-sobarc ensemble (NPT). Table shows the Lenard-Jones parameters and partal charge for methanol molecule. Water molecules are modeled as SPC water model wth ther related parameters [] Dffuson coeffcent of the methanol n the mxture As the frst step, dffuson coeffcent of methanol molecules n a mxture of methanol-water has been studed by MD method. For ths reason mean square dsplacement (MSD) of methanol molecules n the mxture was calculated by Gromacs software. Fg. 2 shows MSD values versus tme for methanol molecules dffusng n the methanol-water mxture at four temperatures 273, 283, 288 and 293 k. These graphs show that the behavor of MSD versus tme for <t<2 ps s lnear, so Ensten equaton can descrbe the dffuson behavor of methanol. Calculated Dffuson coeffcents of methanol n the mxture by Ensten equaton are shown n Table and Fg. 3. Fgure. Computer generated mage of a methanol-water box Iranan Journal of Chemcal Engneerng, Vol. 6, No. 4 65

5 Farhadan, Sharaty-Nassar Table. The partal charge and Lenard-Jones parameters for methanol molecule [] Atom type Charge Lenard Jones parameters C6 C2 CH e e-6 O e e-6 H.4 Root Mean Square Devaton Dsplacement (RMSD) RMSD (nm^2) K 288 K 283 K 273 K tme (ps) Fgure 2. Root Mean square dsplacement of methanol molecules at dfferent temperatures Calculated Dffuson Coeffcent 3 MD Cal. Wlke-Chang chang Eq. Eq. Dffuson Coeffcent * ^5 (Cm^2/s) Temperature Fgure 3. The effect of temperature on dffuson coeffcent. 66 Iranan Journal of Chemcal Engneerng, Vol. 6, No. 4

6 Calculaton of Physcal Propertes of the Methanol-Water Mxture Usng Molecular Dynamc Smulaton To evaluate the calculated results, the dffuson coeffcent of ths mxture was calculated by Wlke- Chang equaton [2]. Results are shown n Table and Fg 3: D AB (7.3 8 V )( M.6 A ) / 2 B (9) The avalable expermental data for dffuson coeffcent of methanol n water are shown n Table 2 [3]. The averaged error between MD results and expermental value s about T 2%, whle ths s about % between expermental value and Eq Calculaton of densty In the second step, the densty of methanolwater mxture was calculated at dfferent temperatures and dfferent ponts n the box. The results are shown n Table 3 and Fg. 4 and 5. The comparson between expermental [3] and calculated results shows a good agreement between these data and the averaged error s less than %. Table 2. Calculated Dffuson Coeffcent of Methanol n Water by MD method Dffuson coeffcent Dffuson coeffcent Expermental Temperature (K) by MD (cm 2 /s) by Eq. 9 (cm 2 /s) value [] * * * -5.8* * -5.42* * * -5.6* -5.8 * -5 Table 3. Calculated and expermental densty n dfferent temperatures Temperature (K) Cal. Densty (kg/m^3) Exp. Densty (kg/m^3) [] Error % Iranan Journal of Chemcal Engneerng, Vol. 6, No. 4 67

7 Farhadan, Sharaty-Nassar Calculated and Expermental Densty 96 Densty (kg/m^3) MD EXP Temperature (K) Fgure 4. The effect of temperature on densty for methanol-water mxture Local Densty n the box T=273 K T=283 K T=288 K T= 293 K Densty (kg /m^3) z Box axs (nm (nm) ) Fgure 5. Calculated densty n dfferent ponts n the box by MD method 3.4- Shear vscosty One of the mportant propertes of the mxtures s vscosty. It s possble to calculate ths macroscopc property of the mxture by calculatng the nteracton of the molecules n mcroscopc scale. Here the vscosty of the methanol-water mxture was calculated by MD smulaton and the results are compared wth expermental data. As t was shown n Table 4 and Fg. 7, vscosty s decreased by ncreasng temperature, the same as expermental value. 68 Iranan Journal of Chemcal Engneerng, Vol. 6, No. 4

8 Calculaton of Physcal Propertes of the Methanol-Water Mxture Usng Molecular Dynamc Smulaton Mxture Vscosty K 283 K 288 K 293 K 3 K 3 Vscosty (cp) tme (ps) Fgure 6. Vscosty of the methanol-water mxture at dfferent temperatures Table 4. Average calculated and expermental vscosty of the methanol-water mxture Temperature 273 K 283 K 288 K 293 K 3 K MD Calculated value Expermental value [2] Error % Expermental & Calculated vscosty of the mxture.5 MD vscosty (cp).5 Exp temperature (K) Fgure 7. The effect of temperature on vscosty for methanol-water mxture Iranan Journal of Chemcal Engneerng, Vol. 6, No. 4 69

9 Farhadan, Sharaty-Nassar 3.5- Hydrogen Bond Many thermodynamcal propertes (e.g., meltng and bolng pont) of the mxture depend on the strength of the hydrogen bonds (H-bond) formed among molecules. The H-bond s a specal attractve nteracton parameter that exsts between an electronegatve atom and a hydrogen atom bonded to another electronegatve atom. Snce water has two hydrogen atoms and one oxygen atom, t can form an H-bond between the Oxygen and the H atoms. Each water molecule can form up to four hydrogen bonds at the same tme (two through ts two lone pars, and two through ts two hydrogen atoms). The average number of H-bonds per molecule calculated n bulk water vares approxmately from 2.3 to 3.8 accordng to the water model and the way used to defne the H-bond. [4, 5] In ths study the number of hydrogen atoms between water molecules and water-methanol molecules were calculated. The results, shown n Fgs. 8 and 9, ndcate that the number of hydrogen bonds between water molecules s from 2 to 3 but t decreases for water and methanol molecules down to. The hydrogen bond dstrbuton s also plotted n the box, where the most probable ones form near the center, Fg. 4. Concluson In ths paper molecular dynamc theoretcal method s presented to calculate densty, shear vscosty, hydrogen bondng and dffuson coeffcent for the mxture of methanol-water. Gromacs software was used for ths purpose. The smulaton was performed up to 2 ps. The results n Table and Fg. 3 clearly show that the calculated dffuson coeffcent by MD method ndcate a good agreement and an almost smlar trend wth those obtaned from the Wlke-Chang equaton, both of whch are approved by expermental data. These values, however ncrease lnearly wth temperature (D~T). Hydrogen bond of water molecules K 283 K 288 K 293 K 3 K 3 Probablty tme (ps) Fgure 8. Hydrogen bond of water molecules n the mxture 7 Iranan Journal of Chemcal Engneerng, Vol. 6, No. 4

10 Calculaton of Physcal Propertes of the Methanol-Water Mxture Usng Molecular Dynamc Smulaton Hydrogen bond of the mxture K 283 K 288 K 293 K 3 K Probablty tme (ps) Fgure 9. Hydrogen bond of water and methanol molecules n the mxture 3 Hydrogen Bond Dstrbuton 273 K 283 K 288 K 293 K Probablty z Dstance axs (nm) (nm) Fgure. Hydrogen bond dstrbuton of methanol-water mxture n the box Densty and vscosty smlar to that of dffuson coeffcent are affected by temperature. The densty decreases wth temperature wth a decreasng devaton between MD and expermental data. Ths shows that MD s much more relable at hgher temperatures. In the case of vscosty, t also decreases wth temperature wth a decreasng devaton. After that, t decreases but wth a lower devaton of MD wth those of the expermental data. The reference temperature for MD calculaton n the nput fles of the smulaton s 3 K, so some of the calculated propertes such as vscosty and densty at the ponts near ths temperature have the mnmum devaton from the expermental data. The MD calculaton for pure water n ths study estmated H-bond numberng between 2 to 3 whch s n a good agreement wth the prevous studes [4, 5]. Ths number decreases consderably for the case of watermethanol molecules. Ths s due to the lower polarty of methanol molecules, compared to Iranan Journal of Chemcal Engneerng, Vol. 6, No. 4 7

11 Farhadan, Sharaty-Nassar that of water molecules. Hydrogen bond formaton has the most dstrbuton probablty at the center of the box and the temperature has no effect on dstrbuton. In the center of the box, there are water and methanol molecules n all the drectons, so the hydrogen bond has the maxmum dstrbuton n ths part of the system, but n the ponts near the sdewall of the system the number of molecules decreases so the hydrogen bondng has the mnmum dstrbuton n these ponts. Overall, the results show the relablty of the MD method to calculate the physcal propertes of the systems. One can conclude that, ths method may be used to determne the macroscopc propertes of the systems, nstead of practcal methods n the laboratory. Nomenclature T Temperature (K) M B Solvent molecular weght (kg/kmol) ϕ Solvent accumulaton coeffcent (-) μ Vscosty (kg/m.s) V A solute molar volume at bolng pont (m 3 /kmol) References [] Montcella, L., Kandasamy, S. K., Perole, X., Larson, R. G., The MARTINI coarse graned force feld: extenson to protens., J. Chem. Theory and Comput., 4, , (28). [2] Sadus., R. J., Molecular smulaton of flud, theory, algorthms and object-orentaton., Elsever Scence B.V., (999). [3] Xang, T., Anderson, B., Lposomal drug Transport: A molecular perspectve from molecular dynamcs smulatons n lpd blayers, Advanced Drug Delvery Revews, 58, , (23). [4] Ungerer, P., Neto-Dragh, C., Rousseau, B., Ahunbay, G., Lachet, V., Molecular smulaton of the thermophyscal propertes of fluds: From understandng toward quanttatve predctons, Journal of Molecular Lquds, 34, 7 89, (27). [5] Krshna, R., Wesselngh, J. A., The Maxwell-Stefan approach to mass transfer, Chem. Eng. Sc., 52, 86-9, (997). [6] Dubbeldam, D., Snurr, R. Q., Recent developments n the molecular modelng of dffuson n nanoporous materals, Molecular Smulaton Journal, Vol. 33, (27). [7] Lndahl, E., Hess, B. and Van der Spoel, D. GROMACS 3.: a package for molecular smulaton and trajectory analyss, J Mol Model, 7:36 37, (2). [8] Van Gunsteren, WF, Berendsen, HJC, A leap frog-algorthm for stochastc dynamcs, Mol Smul :73 85, (988). [9] Spoel, D.V., Lndahl, E., Hess, B., Groenhof, G., Mark, A., Berendsen, H.J.C., GROMACS: Fast, Flexble, and Free., J. Comput. Chem., 26, 7 78, (25). [] Gunsteren W.F., Krugerg, V. and Berendsen, J.C., Computer Smulaton of molecular dynamcs methodology, appled and perspectves n chemstry, Angew Chem Int Ed Engl, 29, , (99). [] Gunsteren W.F., Krugerg, V. and Brlleter, P., SR. and et al, The GROMOS 96 manual and user gude, Bomos and Hochschulverlag AG an der, ETH ZA/4rch, Gronngen, (996). [2] Treybal., R., Mass transfer phenomena, McGraw-Hll Company, thrd ed., (89). [3] Perry, R. H., Perry s chemcal engneer s handbook, McGraw-Hll Company, ffth ed., (984). [4] Jorgensen, W. L., Madura, J. D. Optmzed Intermolecular Potental Functons for lqud Hydrocarbons, Mol. Phys. 985, 56, (38). [5] Zelkewcz, J. Structural propertes of water: Comparson of the SPC, SPCE, TIP4P, and TIP5P models of water, J. Chem. Phys, 23, 45, (25). 72 Iranan Journal of Chemcal Engneerng, Vol. 6, No. 4