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J IRAN CHEM SOC DOI 10.1007/s13738-016-0952-5

ORIGINAL PAPER

Glycerol revisited molecular dynamic simulations of structural, dynamical, and thermodynamic properties Monireh B. Moghaddam1 · Elaheh K. Goharshadi1,2   · Fatemeh Moosavi1 

Received: 3 February 2016 / Accepted: 28 July 2016 © Iranian Chemical Society 2016

Abstract We performed molecular dynamics simulations to investigate the properties of glycerol for a wide range of temperatures at standard pressure. We calculated structural (radial distribution functions and pair potential of mean force), dynamical (mean square displacement and transport properties), and thermodynamic (density, thermal expansion, and Hildebrand solubility parameter) properties of glycerol. The results of structural properties showed that the correlation between glycerol atoms weakens as temperature increases. The values of mean square displacement showed that changing temperature has a strong influence on mobility of glycerol atoms. The values of diffusion coefficient and viscosity are remarkably close to the experimental values over the whole range of temperatures studied. The simulation results provide a reasonable estimation of density with percent error of 0.40 %. The simulated values of Hildebrand solubility parameter of glycerol decrease with raising temperature because the cohesive forces weaken. To the best of our knowledge, this work for the first time calculates the potential of mean force, viscosity, and Hildebrand solubility parameter of glycerol by MD simulation. Keywords  Molecular dynamics simulation · Glycerol · Viscosity · Solubility parameters

* Elaheh K. Goharshadi [email protected] 1

Department of Chemistry, Faculty of Sciences, Ferdowsi University of Mashhad, Mashhad 91775‑1436, Iran

2

Center of Nano Research, Ferdowsi University of Mashhad, Mashhad 91775‑1436, Iran



Introduction Glycerol, a sugar alcohol, plays a very important role in many technological and scientific applications [1] including in personal care, food, tobacco, polymer, and pharmaceutical materials [2]. It is also used as a raw material in chemical syntheses [3], in medicinal applications [4], and as a substitute for sweeteners. Computer simulations can provide information on structural, thermodynamic, and transport properties for various liquids such as glycerol [5]. The first computer simulation study of glycerol was performed by Root and Stillinger [6] in an attempt to characterize the structure of its liquid and amorphous solid state. They employed “united atoms” approximation wherein each of the three carbon atoms of glycerol and their directly attached H atoms is treated as a single interacting site. They used a molecular model for two temperatures 202.4 and 303.2 K. The small size of the system (32 molecules) and the short simulation time of less than 1 ps were the limitations of their work. United atom models of glycerol were used in several subsequent studies [5, 7]. Root and Berne [5] carried out a molecular dynamics (MD) simulation with reversible reference system propagator algorithm on liquid glycerol to characterize the effect of pressure on hydrogen-bond formation at room temperature. They studied liquid glycerol at series of five densities ranging from 1.28 to 1.38 g cm−3, corresponding to pressures from about atmospheric pressure to 0.7 GPa. The first computer simulation study using a full atomistic model for glycerol was performed by Chelli et al. [8]. They studied different properties of glycerol in the crystal, liquid, and glass states including structural, thermodynamic, and dynamical properties in a wide temperature range [8–10]. They adopted the general-purpose AMBER force field. Blieck et al. [11], using MD simulation, studied structural (characteristic

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times and static structure factor) and dynamical (diffusion coefficient) properties of liquid glycerol at 400 K. Busselez et al. [12] performed the MD simulations of liquid glycerol confined in a realistic model of a cylindrical silica nanopore. They analyzed the influence of the hydrophilic surface and the geometrical confinement on the structure, hydrogen-bond lifetime, rotational and translational molecular dynamics. They performed MD simulations using the allatom model designed for glycerol, which is an extension of the electrostatic and van der Waals point charge potential of Chelli et al. [8] based on the AMBER force field. They simulated glycerol in a temperature range from 400 to 250 K and from 450 to 270 K for the bulk and for the confined systems, respectively. The bulk system comprised of 108 glycerol molecules in a cubic cell of size about 24 Å. The simulation box for the confined system consisted of 130 glycerol molecules in order to retrieve the average density of the bulk which was obtained as 1.22 g cm−3 from simulation of the bulk system at 300 K and 1 bar. This value is close to the experimental density (1.25 g cm−3). The MD simulations of glycerol at 1 atm and temperatures in the range of 300–460 K, using five different force fields (AMBER, CHARMM22, OPLS1, OPLS2, and OPLS3), were performed by Jahn et al. [1]. They studied thermodynamic (density, thermal expansion coefficient, and specific heat), dynamical (diffusion coefficient), and structural properties of glycerol by the LAMMPS software package. We performed MD simulations by DL_POLY package using AMBER force field based on Blieck et al. [11] work to investigate the structural (radial distribution function and pair potential of mean force), dynamical properties (mean square displacement and transport properties), and thermodynamic properties (density, thermal expansion, and solubility parameter) of glycerol for 293.15–333.15 K at standard pressure.

Computational details All simulations were performed by the DL_POLY MD package [13] using AMBER force field for glycerol because of its simple analytical form. The force field parameters for non-bonded and bonded atoms of glycerol were reported by Blieck et al. [14], and density functional theory performed using GAUSSIAN03 [15] at B3LYP/631g(d) level of theory. In addition, the partial atomic NBO charges were computed at the same level of theory. The Newton’s equations of motion were integrated using the velocity Verlet algorithm with a time step 1 fs. The cutoff radius was chosen at 15 Å. The system studied consisted of 512 flexible glycerol molecules in a cubic box by applying periodic boundary conditions. All simulations were initially run for 0.5 ns in NPT ensemble at 1 bar and desired

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J IRAN CHEM SOC

temperature to reach the equilibrium and then switched to NVE ensemble for 0.5 ns. The pressure was set to standard pressure by Berendsen barostat [16] with a relaxation time of 1.0 ps. The temperature was maintained constant during simulation using Berendsen thermostat [16] with a relaxation time of 0.2 ps.

Results and discussion Structural properties To obtain a better understanding of the structure of glycerol, the radial distribution functions (RDF) of all atoms were computed at different temperatures. The RDF denoted as g(r) is an indication of local molecular structure and defined mathematically as [17]:

g(r) =

�N(r, �r)� 1 2 NρV (r, �r)

(1)

where angular brackets indicate a temporal average, N(r, Δr) is the number of atoms within a spherical shell of radius Δr. N is the total number of atoms in the system, ρ is number density, and V(r, Δr) is volume of the shell. The center-of-mass RDFs for all atoms of glycerol at various temperatures are shown in Fig. 1. Figure 1a shows that the first peak for all atoms at about 0.96 Å. The second and third peaks occur at 1.09 and 1.41 Å, respectively. As the inset of Fig. 1a shows by increasing temperature, the peak height of the first peak decreases but the maximum remains at the same position. Hence, the correlation between glycerol atoms weakens by increasing temperature slightly. From the RDFs, we calculated the effective pair potential of mean force, W(r), to calculate an average force over all the configurations of glycerol at 293.15 K (Fig. 1b) which is defined as [18]:

W (r) = −kT ln g(r)

(2)

Figure 1b shows when RDF essentially is zero; the corresponding effective pair potential is large and positive for small pair separation. When RDF is larger than unity, W(r) is negative. The W(r) profile in Fig. 1b exhibits a local potential energy minimum of −1.7 at r = 1.09 Å. The negative part of the potential of mean force shows an effective attraction between two glycerol molecules. Hence, there is a high probability of two glycerol molecules being at distances near the minimum of the potential curve. When the RDF is less than unity, W(r) is positive. This is due to the strong repulsive forces at small intermolecular distances. The structural properties of liquid glycerol were also investigated in terms of various site–site pair radial

J IRAN CHEM SOC

(a)

6

for the O–H1 function is the distance characteristic of strong hydrogen bonding. The number of hydrogen bonds per glycerol molecule was calculated from the area under the first gO–H1 peak at various temperatures. From our simulations, the number of stable hydrogen bonds per glycerol molecule at 293.15, 303.15, 313.15, 323.15, and 333.15 K is 5.92, 5.93, 5.87, 5.87, and 5.90, respectively.

5.4

T= 293.15 K 303.15 313.15 323.15 333.15

5.2 5.0

g (r)

5 4

4.8 4.6 4.4

g (r)

4.2 4.0 0.94

3

0.95

0.96

0.97

0.98

0.99

r (A°)

2

Dynamical properties

1

The molecular mobility of glycerol molecules can be analyzed using the mean square displacement (MSD) which is defined as:

0 0

2

4

6

8

r (A°)

MSD =

(b) 12 8

W (r) / kT

(3)

n=1

10

6 4 2 0 -2 -4

N  1  �[�rn (to + t) − �rn (to ) ]2 N

0

1

2

3

4

5

6

7

8

r (A°)

Fig.  1  a RDFs at different temperatures (inset shows the first peak, clearly), b the effective pair potential of mean force at 293.15 K

distribution functions namely gC–C, gC–H, gC–O, gC–H1, and gO–H1 at different temperatures (Fig. 2a–e). For example, gC–C is defined as the probability of finding two carbon atoms of two different glycerol molecules separated by the distance rC–C. The RDF for the C–C pair (Fig. 2a) shows that the first peak of glycerol molecule is about 4.5 Å whereas a second peak is positioned at about 5.5 Å. The inset of Fig. 2b shows a slight broadening of the first peak with increasing temperature without changing the position of the peak. The first peak for the C–C, C–H, C–O, and C–H1 (H1 is connected hydrogen atom to O) is located at 4.5, 3.5, 3.5, and 3 Å, respectively. As Fig. 1a, e shows, the position and intensity of the first peak of gC–C and gO–H1 agrees well with that of Chelli et al. [8]. Figure 2 shows that C–C atoms have the strongest correlation, followed by C–H and C–O atoms and then C–H1 atoms for all temperatures. Finally, all RDFs show that weak ordering persists beyond 8 Å excepting the carbon–carbon pair. The position of the first peak of gO–H is often used to define hydrogenbond criterion [5]. The large intensity of the peak at ~2.5 Å

where rn (t) is the position vector of atom n at time t. to is an initial time step. The MSDs as a function of time at different temperatures are given in Fig. 3. The plot has linear portion at long times for all temperatures. As this figure shows, the most displacement is observed for the highest temperature because the interaction between glycerol molecules weakens by increasing temperature. The MSD of glycerol after 200 ps at 293.15 K and 333.15 is 1.4877 and 2.9258 Å2, respectively, i.e., with the increase of 40 K in temperature after 200 ps, the MSD of glycerol increased approximately 1.44. Figure 4a shows the MSDs of C, H, O, and H1 atoms of glycerol as a function of time at 293.15 K. According to the results of RDFs (Fig. 2), we expect that the C and H1 atoms have the lowest and the highest MSD or mobility, respectively. Figure 4a shows the increasing sequence of mobility between glycerol atoms followed by H1, O, H, and then C atoms for all temperatures, respectively. The reason for this observation is that H1 atom has hydrogen bonding between two glycerol molecules so it has the highest mobility. To prove this claim, Fig. 4b shows the displacement of glycerol atoms. As Fig. 4b shows the displacement of H1 and C atoms is the highest and the lowest, respectively. Therefore, Fig. 4b confirms the results of RDFs and MSDs. The limiting slope of MSD versus simulation time, considered for time intervals sufficiently long, is related to self-diffusion coefficient, D [19, 20]:   dMSD 1 D = lim (4) s t→∞ dt T where s depends on the space dimensionality (6 for three dimensions and 4 for two dimensions). The self-diffusion coefficient for glycerol is obtained by fitting the slope of the linear region of Fig. 3 (Eq. 4). Our simulation results for D are shown in Fig. 5 and compared with the corresponding simulation and experimental data. As Fig. 5 shows, there is a rather satisfactory agreement between the

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J IRAN CHEM SOC

(a)

2.5 2.0

1.2 1.0

g C-O (r)

1.5

g (r)

(c)1.4

T= 293.15 K 303.15 313.15 323.15 333.15

1.0

0.8 0.6 0.4

0.5 0.0

0.2 0

2

4

6

8

10

12

14

0.0

16

0

2

4

6

8

r (A )

(b) 1.4

(d) 1.2

1.2

1.0

12

14

16

0.8

0.8

g C-H1 (r)

g C-H (r)

1.0 1.32 1.30

0.6

1.28

0.4

0.6 0.4

1.26 1.24

0.2 0.0

10

r (Ao)

o

0.2

1.22 1.20

0

2

4

6

3.5

3.6

8

10

3.7

12

3.8

0.0

3.9

14

16

0

2

4

6

r( Ao)

8

o

10

12

14

16

r( A )

(e) 1.6 1.4 1.2

g O-H1 (r)

1.0 0.8 0.6 0.4 0.2 0.0 0

2

4

6

8

10

12

14

16

o

r (A )

Fig. 2  RDFs of glycerol atoms a C–C, b C–H, c C–O, d C–H1 (H1 is connected hydrogen atom to O) and e O–H1 at various temperatures (inset shows the first peak)

simulated results and experimental data [21, 22] with percent error within 23.36 and 25.55 %, respectively. Percent error between the Jahn᾽s et al. work [1] and experimental values is within 120.23 and 184.81 %, respectively. The temperature dependence of diffusion coefficient of glycerol can be described by Arrhenius-type equation [23]:   −Ea D = Do exp (5) kT

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where Ea and Do are migration energy and the maximum diffusion coefficient, respectively. The simulated values for self-diffusion coefficient were fitted with the Arrheniustype equation (R2  = 0.9991). The migration energy of 57.98 and 49.65 kJ mol−1 was calculated using our work and Jahn᾽s et al. MD simulation [1] over the temperature range 293.15–323.15 K, respectively. The migration energy of this work MD simulation is in a reasonable agreement

J IRAN CHEM SOC 5e-11

5

T= 293.15 K 303.15 313.15 323.15 333.15

3

3e-11

D ( m2 s-1)

o2

MSDall atoms ( A )

4

2

2e-11

1e-11

1

0

Our work Sim. [1] Exp. [21] Exp. [22]

4e-11

0

0

100

200

300

400

290

500

300

310

t (ps)

C H O H1

3.5

330

340

4.0

Our work Exp. [26] Exp. [25]

3.5

2.5

log η (mPa s)

MSD (Ao2)

3.0

320

Fig. 5  Self-diffusion coefficient calculated based on MD simulation, simulation data from ref. [1], and experimental data [21, 22] for glycerol

Fig. 3  MSD of all glycerol atoms at different temperatures

(a) 4.0

T (K)

2.0 1.5 1.0

3.0

2.5

2.0

0.5 1.5

0.0

0

100

200

300

400

500

t (ps)

Displacement per atom

(b)

1.0 290

300

310

320

330

340

T (K)

34

Fig. 6  The logarithm of viscosity of glycerol as a function of temperature

33 32

percent errors 13.44 % and 21.01  % with the experimental data of  migration energy [21, 22], respectively. The simplest relation describing shear viscosity can be evaluated using a relationship between shear viscosity and diffusion coefficient [24]:

31 30 29 28 50

100

150

Configuration Fig.  4  a MSDs and b displacement of glycerol atoms at 293.15 K

with the experimental values (57.36 kJ mol−1 [21] and 62.86 kJ mol−1 [22]) with percent errors 1.08 % and 7.79 %, respectively. The simulation data from ref. [19] showed the

η=K

ρT D

(6)

where K is a proportionality constant and is written as:

K=

ηo Do ρ o To

(7)

where ηo is the value of viscosity at a given diffusion coefficient, Do, density, ρo, and temperature (To) for a reference material. In this work, the value of K was calculated by

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J IRAN CHEM SOC 1.28

0.0008

-3 ρ (g cm )

1.26

Thermal Expansion Coeff.( K-1)

Our work Sim. [1] corr. [28] Exp. [29]

1.24

1.22

1.20

1.18

0.0006

0.0004

0.0002

0.0000

290

300

310

320

330

Our work Sim. [1] EXP. [29]

340

290

300

T (K)

Fig. 7  Densities calculated based on MD simulation, simulation data from ref. [1], correlation [28], and experimental data [29] for glycerol as a function of temperature

experimental values for the reference material of glycerol. Figure 6 shows the logarithm of viscosity of glycerol as a function of temperature. A good agreement with experiment is an indication of accuracy of our method [25, 26]. Figure  6 demonstrates viscosity decreases logarithmically with increasing temperature. As temperature increases, the intermolecular interactions between the glycerol molecules weaken and therefore the viscosity decreases. The effect of temperature on viscosity of glycerol was described by Arrhenius equation [27]:   Evis η = A exp (8) RT where Evis is activation energy and A is called the pre-exponential factor. The Evis shows the energy barrier that must be overcome before the fluid flows. The simulated values for viscosity were fitted well with the Arrhenius equation (R2  = 0.9981). The activation energy of 51.52 kJ mol−1 was calculated for glycerol over the temperature range 293.15–333.15 K, respectively. The percent errors for the activation energy are 7.61 % and 11.69 % with respect to the experimental values of 55.76 kJ mol−1 [25] and 58.33 kJ mol−1 [26]. The high value of activation energy shows glycerol is very viscous. Hence, the movement of glycerol molecules is a temperature-activated process. Thermodynamic properties Figure  7 shows the computed, the corresponding simulation data from ref. [1], correlation [28], and experimental [29] densities for glycerol as a function of temperature. For all temperatures, our simulated densities are in a reasonable agreement with the corresponding correlations

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310

T(K)

320

330

340

Fig. 8  Thermal expansion coefficient calculated based on our MD simulation, simulation data from ref. [1], and experimental data [29] for glycerol as a function of temperature at 1 bar

and experimental values with percent error within 0.35 and 0.39 %, respectively. The simulation data from ref. [1] have percent error within 3.6 and 3.75 % with the correlation and experimental data, respectively. This difference may be due to different MD softwares used in our and Jahn᾽s et al. [1] work. Figure  8 shows the computed, the corresponding simulation data from ref. [1], and experimental thermal expansion coefficient for glycerol as a function of temperature at 1 bar. The deviation between the current simulations with experimental values was within 27.74 % that is an indication of inaccuracy. Percent error between Jahn᾽s et al. [1] work and experimental value is within 42.44 %. The only difference between Jahn᾽s et al. and our work is the used software. Solubility parameter, δ, is a numerical value indicating relative solvency behavior of a specific solvent [30]. Hildebrand [31] proposed the square root of cohesive energy density as a numerical value for the solvency behavior of a specific solvent:

δ=



�Hm − RT Vm

1/2

(9)

where ΔHm is molar enthalpy of vaporization and R is gas constant. Vm stands for molar volume. The simulated and experimental [32] values of solubility parameter for glycerol are given in Table 1. As Table 1 shows, there is a satisfactory agreement with the simulated results and corresponding experimental value with percent error within 6.98 % at 298.15 K. This table shows, by raising temperature, the solubility parameter of glycerol decreases because the cohesive forces weaken with raising temperature.

J IRAN CHEM SOC Table 1  Simulated and experimental [32] value of Hildebrand solubility parameters for glycerol T/K

We acknowledge the HPC center of FUM since the MD simulations were performed there.

δ (MPa1/2) Exp. [32]

MD data

298.15 303.15 313.15

36.1 – –

38.62 37.88 37.22

323.15



36.97

“–” means the corresponding experimental value does not exist

Conclusions We performed extensive classical MD simulations for glycerol at different temperatures. Several structural, dynamical, and thermodynamic properties of glycerol were computed. To the best of our knowledge, for the first the potential of mean force, viscosity, and Hildebrand solubility parameter of glycerol were calculated using MD simulation. It seems to us this work has the following main conclusions: • The structural properties of glycerol were investigated by evaluating all atoms and various site–site pair radial distribution functions at different temperatures and effective pair potential of mean force. The negative part of the potential of mean force at distances near the minimum of the potential curve indicates an effective attraction between two glycerol molecules. The results showed that the correlation between glycerol atoms weakens as temperature increases. • The molecular mobility of glycerol atoms was calculated by evaluating the MSD at different temperatures. The highest displacement is observed at the highest studied temperature. The present study also showed that with increase of 40 K in temperature after 200 ps, the MSD of glycerol increased approximately 1.44 times. • The viscosity of glycerol decreases logarithmically with increasing temperature. A good agreement with experimental data is an indication of accuracy of our method [25, 26]. The Arrhenius-type equation for viscosity shows the activation energy is 51.52 kJ mol−1. • For all temperatures, the simulated densities are in a reasonable agreement with the experimental values within 0.4 %. In contrast, the simulation data from ref. [1] have percent error within 3.6 and 3.75 % with the correlation and experiment, respectively. • The Hildebrand solubility parameter for glycerol decreases with raising temperature. Acknowledgments The authors express their gratitude to Ferdowsi University of Mashhad (FUM) for support of this Project (3/23036).

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