1 Introduction
Ions have significant impact on the structures and dynamics of water in aqueous solution, which have been extensively explored by the experimental and theoretical methods.1, 2, 3, 4, 5, 6, 7, 8, 9, 10 The relevant phenomena are brought into sharp focus in recent years, such as ion pairing5, 6 and ion specific effect on the biological systems.1, 7, 8, 9 In recent years, the ion disturbance on the motion of the water molecules has been extensively explored by the modern spectroscopy techniques3, 4, 10, 11, 12, 13 and the theoretical methods.14, 15 But the dynamic properties of ion itself in aqueous solutions are not well addressed at the microscopic level, for example ammonium ion.
The dynamics of ammonium ion and ammonium moieties plays an important role in the chemical, environmental, and biological processes.16, 17, 18 Ammonium transport across the membranes is a crucial life-process for growth of plants, fungi, and bacteria. The transport mechanism of ammonium ion and ammonia in the protein channels of the ubiquitous ammonium transporter/methylamine permease/rhesus (Amt/MEP/Rh) family is far from understood.18 The previous measurements of the nuclear magnetic resonance (NMR) suggested that the observed solvent dependence of the rotational mobility of ammonium ion showed a poor correlation with the hydrodynamic Stokes-Einstein-Debye (SED) model.19, 20, 21, 22, 23 There is stronger short-range friction between ammonium ion and water due to the multiple strong hydrogen bonds (HBs) than that between ammonium ion and methanol. However, NH4+ rotates rather fast in aqueous solutions comparing to that in other solvents such as methanol. This is unexpected for the hydrodynamic theories. The further simulations with the classical, non-additive, and electronic structure methods show that a discontinuous jump rotation possibly contributes to this unexpected fast rotation in aqueous solution.23, 24, 25, 26, 27, 28, 29, 30, 31 However, the rotational mechanism of ammonium ion at the molecular level, is not well clarified intuitively and quantitatively until now.
The extended jump model (EJM) developed from Ivanov jump model, 32, 33 shows that the rotation of water in aqueous solutions is determined by a large-amplitude angular jump and a less significant diffusive “frame diffusion”.14, 15, 32, 33 It has been applied to explore the rotational mechanism of water in the electrolyte solutions and on the hydrophobic interfaces combined with infrared (IR) spectroscopy and molecular dynamics simulations.32
In this work, the fast rotation of NH4+ in previous experiments was also observed in our simulations. The rotation of ammonium ion in NH4Cl aqueous solutions also follows the EJM like water molecule. It is mainly due to the large-amplitude angular jump during the HB switching processes of NH4+ from one acceptor to another in aqueous solutions. Two characteristic jump angles observed in our simulations are 50° and 65° for water and ammonium ion, respectively. Water prefers the former over the later, but conversely for ammonium ion.
2 Methods
2.1 Rotational correlation function
The rotational correlation function, C2(t), of water along OH bond vector or NH4+ along NH bond vector, u, can be expressed as:
where P2 is the second-rank Legendre polynomial. C2(t) is usually employed to obtain the rotational relaxation time in simulations, which corresponds to the NMR and the ultrafast IR spectrum measurements.32 The rotational correlation functions can be decomposed into two sub-processes (Fig. 1), a short-time libration and a long-time rotational relaxation. The rotational relaxation times of two processes can be approximately obtained by fitting the following function:
图1
图1
Fig 1
Rotation correlation functions (C(t)), the HB correlation functions (1―CW-W(t)) and the rotation correlation functions of HB frame (Cf(t)) of the water and ammonium for the HB exchange between water molecules in NH4Cl solution at 0.5 mol·L-1
where A is a prefactor of the exponential functions. $\tau $1 is the librational time. $\tau $2 is the rotational relaxation time related to the measurements of the ultrafast IR spectrum.4, 12, 13, 32 The integration value of C2(t) within a long-time window is usually used for the rotational relaxation time by NMR measurements. The parameters in Eq.(2), $\tau $1 and $\tau $2 were determined from a fit of equation (2) within 0-8 ps as previous works.4, 12, 13, 32
2.2 Extended jump model
The jump rotation of water or ammonium is triggered by the HB switching from initial HB acceptor to a new one (Fig. 2).32 Before a HB exchange (t < 0 for the time window of the HB exchange process), O*H* or N*H* rotates diffusively with its HB frame axis of O*···Oa or N*···Oa. The average direction is assumed to be the same as the frame axis, neglecting the libra-tion of O*H* or N*H* in local HB frame axis. When a new HB acceptor is available, a fast HB switching happens. H* jumps from initial acceptor to a new one within the local frame of OaO*Ob or OaN*Ob (Fig. 2, Oa and Ob: the HB initial and final acceptors (oxygen atoms of water); H*: the HB donating hydrogen). When a new HB forms, the time window of the HB exchange process takes at t > 0. The middle time between the ending moment of old HB and the beginning time of new HB, is taken as the time origin (t = 0) within the HB exchange process. The jump angle $\phi $ is the angle between initial HB frame vector and the new HB frame vector at t = 0 (Fig. 2).
图2
图2
Fig 2
Definitions of the geometric variables along the HB exchange path
$\theta $ is the angle between the (N*H*O*H*) vector and the bisector plane of OaN*Ob(OaO*Ob). $\phi $i-j is defined as the jump angle (the angle between the vector O*Oa and vector O*Ob at t=0) during the HB exchange from initial HB acceptor i to final HB acceptor j. Within the local HB frame of NH4+-water, the Z axis is defined as the vector of N*H* direction. The X axis is defined as the perpendicular vector of Z axis in the plane of HN*H*.
Based on the assumptions above, the rotational correlation function of molecule along the vector u (O*H* and N*H* bond vectors for water and ammonium ion, respectively), C2(t), can be further decomposed into two sub-processes after an initial fast libration decay (Fig. 2):32 (1) a sudden large-amplitude angular jump rotation of uv (uv is the O*H* or N*H* bond vector within the local HB frame uf, which is the O*···Oa or N*···Oa vector of HB pair) and (2) a slow diffusive rotation of the frame vector uf. The correlation functions of two sub-processes (a jump rotational correlation function CJ(t) and a frame rotational correlation function Cf(t)) are both assumed to be exponential.
In NH4Cl aqueous solutions, the possible HB donors are the hydrogen atoms of water and NH4+. The possible HB acceptors are the water oxygen and chloride ion. Four HB exchange processes can be defined according to the types of initial and final HB acceptors: from water to water (W-W), from water to Cl- (W-Cl), from Cl- to water (Cl-W), and from Cl- to Cl- (Cl-Cl). The HB exchange can be traced along the trajectories of simulations, if the HB criteria are defined. The HB criteria: RO*Oa (RO*Ob) < 0.350 nm, RH*Oa (RH*Ob) < 0.245 nm and the HB angle $\theta $H*O*Ob < 30° for water-water HB as previous work, 32 RO*Oa (RO*Ob) < 0.345 nm, RH*Oa (RH*Oa) < 0.235 nm and the angle θH*O*Ob < 30° for the NH4+-water HB according to the radial distribution functions.34 Rij is the distance between two atoms i and j of two HB connected molecules. θH*O*Ob is the angle between the vectors O*H* and O*Ob.
For a HB exchange of O*H* or N*H* from initial acceptor i to new acceptor j (Fig. 2), the rotational relaxation time $\tau $i-j according to Eq.(3), can be written as
$\tau _{i {-} j}^{\rm{J}}$is the jump rotational time and $\tau _{i {-} j}^{\rm{f}}$ is the diffusive rotational time of HB frame. The rotational relaxation time ($\tau$EJM) of water or ammonium along the O*H* or N*H* bond is equal to the weighted contributions from different types of HB exchanges according to the EJM.
Ai-j and $\tau _{i {-} j}$are the fraction and the rotational time of O*H* or N*H* with an initial acceptor i and a final acceptor j.
According to the Ivanov model, 33 the jump rotational relaxation time $\tau_{i {-} j}^{\rm{J}}$ is derived from
fi-j($\phi $i-j) is a function of jump angle $\phi $i-j defined in Fig. 2. $\phi $i-j is the average value of jump angle during the HB exchange from initial HB acceptor i to final acceptor j. The jump angle has a unsymmetrical distribution around the average value, 32 so a numerical integration value Fi-j($\phi $) over $\phi $i-j from 0 to π, is adopted in this work instead of the average value in previous work, 32
P($\phi $i-j) is the possibility of the HB exchange from i to j with jump angle $\phi $.32
The HB relaxation time, $\tau_{i {-} j}^0$, can be derived from a HB exchange correlation function within the time range of 0-8 ps according to the stable state pictures.15, 32 The strict HB criteria is used for $\tau_{i {-} j}^0$, so that the HB exchange events really happened and excluded the transient breaking of hydrogen pair.32 Two water molecules form HB, if RO*Oa (RO*Ob) < 0.31 nm, RH*Oa (RH*Ob) < 0.20 nm and the HB angle θH*OaOb < 20°. The HB between water and NH4+ must satisfy the conditions, RN*Oa (RN*Ob) < 0.30 nm, RH*Oa (RH*Ob) < 0.20 nm, and the HB angle θH*N*Oa (θH*N*Ob) < 20° (Fig. 2) according to the pair radial distribution functions between ammonium nitrogen and water oxygen.34
The frame rotational time $\tau_{i {-} j}^{\rm{f}}$ is determined from the rotational correlation function of HB frame axis O*···Oa or N*···Oa with equation (2) with the same time window as $\tau_{i {-} j}^{\rm{0}}$.
2.3 Simulation protocol
The cubic bulks were constructed for pure water and NH4Cl aqueous solutions by inserting the water and ions into the empty box randomly, then the simulations were performed. 2000 waters were filled into each sample box, then NH4+ and Cl- were inserted into the box until the concentration of solution is approximately equal to 0.5, 1, 2, and 5 mol·L-1. The SPC/E water model35[]was used and ammonium force field was taken from the previous publication by Jungwirth and his workfellows (Model I).34 In their work, the properties, such as the solvation structures, ion clustering tendency in ammonium halide aqueous solutions, were well presented with the current combined potentials in this work.34 Another force field of ammonium (Model II)26 was used to explore the effect of force field in simulations on the rotation of ammonium ion. The jump behaviors of ammonium ion and water are similar for two models (see Section 3.1). A little shorter HB relaxation time is observed for Model II than that for Model I. Additionally, previous work suggests that a classical force field is sufficient for the jump rotation behavior of water.14 The jump rotation was also found in ab initio simulations.28, 29 The quantum effect on the rotation of ammonium ion should be further discussed, but the rotational mechanism of ammonium ion is expected to be not changed.
The bond lengths and angles of water and NH4+ were fixed at the equilibrium values by the SHAKE algorithm.36 The Lorentz-Berthelot combination rules37 were used for the Lennard-Jones interactions. For each sample, a 2 ns isothermal-isobaric ensemble (NPT) equilibration simulation was carried out to generate the proper size of the simulation box, followed by a 2 ns microcanonical ensemble (NVE) simulation to calculate the dynamic properties. For each NPT simulation, the bulk systems were weakly coupled to a bath with the Nose-Hoover thermostats38, 39 at the goal temperature with the relaxation time of 0.1 ps. The weak coupling Berendsen scheme was used to control the system pressure at 1.01×105 Pa with the coupling time constant of 1 ps.40 The equations of motion were integrated using the velocity Verlet integration scheme37 and a time step of 2 fs. The long-range Coulombic forces were calculated using the particle-mesh Ewald method.41 The non-bonded van der Waals interactions were truncated at 1.2 nm using the switching functions. Minimum image conditions were used.37 The simulation configures were saved every 20 fs. All simulations were performed with the Tinker simulation code.42
3 Results and discussion
3.1 Jump rotation of NH4+
The rotational diffusion constant DR can be derived from the mean square displacements (MSDs) of the labeled molecule or ion in the solutions, according to the Einstein relation, 43
${\varphi _i}(t) = \int_0^t \delta {\varphi _i}(t')dt', \delta {\varphi _i}(t')$ is the rotational vector of NH bond vector Pi(t) within the interval of $[t', t' + \delta t]$ with mode δ$\theta $. The rotational diffusive constants of ammonium ion along Z, X, and Y axes defined in Fig. 2, are 0.088 × 1012, 0.085, and 0.092 rad2·s-1. The average value is 0.088 rad2·s-1, which is in reasonable agreement with the experimental value (0.075 × 1012 rad2·s-1).19, 20 This suggests that the force field used here is reasonable to describe the rotational dynamic behavior of water and ammonium ion.
According to the ideal diffusive fluid model, 44 the nth-rank rotational time of NH4+ ($\tau _n $) along the NH vector would satisfy the relationship with the rotational diffusion constant DR,
The ratios $\tau _1 / \tau _2$ and $\tau _1 / \tau _3$ should be equal to the constant values, 3 and 6. However they are only 1.97 and 2.71 at 0.50 mol∙L-1, which is much lower than the ideal constants. The similar case is also found for water with the values of 1.97 and 2.78. The rotation of NH4+ apparently does not follow the diffusive Brownian motion picture.44, 45
Does ammonium ion also follow the EJM like water?32, 33 An instantaneous and large-amplitude angular jump of water is motivated by the HB exchanges. The HB exchange can be considered as a chemical reaction from the reactant state O*H*···Oa to the product state O*H*···Ob. This process passes through an dangling or bifurcated HB state of O*H*, which is a transition state with a very short lifetime, much shorter than the stable single HB reactant and product states.32, 33 Indeed in the NH4Cl solution at 1 mol∙L-1, a very fast decay of the HB state correlation function to the value below 0.1 is observed within 0.1 ps for ammonium ion hydrogen with the dangling, and bifurcated HBs comparing to the single HB state (Fig. 3). The lifetime of NH4+ hydrogen with dangling HB state and bifurcated HBs is evidently shorter than those of water correspondingly. The facts above suggest that a transition state passes during the HB exchanges of ammonium ion like water molecule.
图3
图3
Fig 3
HB state correlation functions of the hydrogen atom of water and NH4+
dangling state with HB number n=0, single HB state with HB number n=1, and bifurcated HB state with HB number n=2
For ammonium ion, we can construct a similar HB exchange reaction system as water molecule. This exchange reaction systemis made of three molecules (ammonium ion, its initial and final HB acceptors). The configurations of remaining water molecules and ions in bulk solution are considered as the average background effect. The reaction coordinates of the HB exchange (Fig. 2), which are the distance between N* and Oa (RN*Oa), the distance between N* and Ob (RN*Oa) and the angle ($\theta $) between the N*H* bond and bisector plane of OaN*Ob, are analyzed from more than 250000 successful HB exchanging events. Among the possible HB exchanges in NH4Cl solution at 0.5 mol·L-1, the fraction of W-W HB exchange of N*H* (Fig. 1) has the highest value, 0.93 (the definitions of HB exchange type in method part).
An intuitive picture is presented in Fig. 4 for the HB exchange reaction of ammonium. The departure of initial acceptor Oa and the arrival of new acceptor Ob take place cooperatively during the HB exchange process. A sudden and large-amplitude angular rotation of N*H* within the local HB frame of OaN*Ob is observed at t = 0. It is reasonable to assume that the local HB frame of OaN*Ob does not change during the transient jump rotation of N*H*.32 This is suggested by the value of $\phi $ (definition in Fig. 2), which nearly does not change during the W-W HB exchange (Fig. 4). Jump angle populations of water and NH4+ are shown in Fig. 5. For water, a main peak locates at 50° and a shoulder peak at 65º. However, only one peak is visible at 65° for the jump angle of NH4+. The jump angle distribution of NH4+ is almost symmetrical around the average value. The different features for the jump angles of water and NH4+ suggest that the configurations at the transition state (t = 0) are different. RN*Oa and RN*Ob are not sensitive to the configuration of the transition state due to the strong HB interaction, so the jump angle is mainly determined by the distance (ROaOb) of HB acceptors, Oa and Ob. The possibility of jump angle at about 500 is much lower for ammonium ion than that for water, if a hydrogen bond is formed between the initial HB acceptor water and final HB acceptor water. The distances of N*···Oa, N*···Ob, and Oa···Ob, are about 0.31, 0.31, and 0.28 nm for N*H* (Figs. 2 and 4). The average jump angle at 65° is identified for both water
图4
图4
Fig 4
Reaction coordinates Rij, $\theta $, and $ \bf{\phi} $ along the path of the W-W HB exchange of ammonium ion and water
图5
图5
Fig 5
Jump angle populations of W-W HB switching for water and NH4+ ion
P(θ) with Models I and II at 0.5 mol·L-1; the populations of jump angle for the HB exchange between two water molecules with HB each other (P1(θ)) and without HB each other P2(θ)r
and ammonium ion for the HB exchange event without hydrogen bond between two HB acceptor water molecules at the transition 46 state. The distances of N*···Oa, N*···Ob, and Oa···Ob are about 0.31, 0.31, and 0.35 nm for anmonium ion, about 0.33, 0.33, and 0.38 nm for the water. The possibility of the HB exchange between two HB water molecules without formation of HB each other, is higher for ammonium ion than that for water. This is mainly due to less available new HB acceptor around N*H* than O*H*.
3.2 Contributions from the jump and frame rotations
The rotational relaxation times of water and ammonium directly from equations (1, 2) and from the EJM with equations (3)-(5), are presented in Fig. 6. The tendency of the rotational relaxation times with concentration is well reproduced by the EJM. The rotational relaxation times are a little underestimated by the EJM relative to the direct measurements. The results from EJM suggest that the water rotation along the OH vector can be divided into two sub-processes. One is the large-amplitude and fast jump rotation. The other is the diffusive rotation of HB frame O*Oa. In NH4Cl solution at the lowest concentration (0.5 mol·L-1) of the samples in this work, the jump rotational times of water and ammonium ion are 3.81 and 4.57 ps, respectively. Their frame rotational times are 5.96 and 9.30 ps, respectively. The jump and diffusive rotational times rise to 7.30 and 18.59 ps at 5 mol·L-1. The contribution from the jump rotation to the rotational mobility of ammonium increases with concentration. The rotation time of ammonium ion and water is mainly determined by the jump rotation over the whole range of concentration.
图6
图6
Fig 6
$\tau _2 $ and $\tau _{\rm{EJM}} $ in the NH4Cl solutions at different concentrations
As for the underestimation of the EJM, this is mainly due to the invalidation of its assumption and its intrinsic limitation. For the jump rotation, the average direction used in the EJM is assumed to be the same as the frame vector, neglecting the libration of O*H* or N*H* in local HB frame. The jump rotation term in fact covers a part of fast libration contribution in its long-time rotation contribution in equation (2). Additionally, the EJM does not consider the rotations of ammonium and water in local basins suggested in reference.50 The fast HB exchange in local basin has bigger contribution on the fast decay of the rotational correlation function than the long-time decay.47 A faster jump rotation is expected for the EJM, given its intrinsic limitations above.
4 Conclusions
The rotational mechanism of ammonium ion is explored by MD simulations and the extended jump model. The fast rotation of NH4+ observed in previous experiments is due to the sudden jump rotation. The jump rotation of NH4+ is motivated by the HB exchanges. The rotational behavior can be approximately described with the EJM like water in aqueous solutions. The rotational correlation function can be decomposed into a jump rotation and a diffuse rotation of HB frame based on the EJM. The contribution from the fast jump rotation is bigger than the diffusive part. The jump rotation becomes more and more important with concentration. Comparing to the HB exchange of water, the HB exchange of ammonium ion between two water molecules forming HB each other has a lower possibility than water.
参考文献
