This assumption is the more real, the larger the difference in the rigidity of intra- and intermolecular. Computer Simulation of Water Clusterization on Chlorine ...
Colloid Journal, Vol. 64, No. 2, 2002, pp. 243–251. Translated from Kolloidnyi Zhurnal, Vol. 64, No. 2, 2002, pp. 270–279. Original Russian Text Copyright © 2002 by Shevkunov.
Computer Simulation of Water Clusterization on Chlorine Ions: 2. Microstructure S. V. Shevkunov St. Petersburg State Technical University, ul. Politekhnicheskaya 29, St. Petersburg, 195251 Russia Received December 28, 2000
Abstract—The molecular structure of Cl–(H2O)n clusters, n = 1–60, in equilibrium with vapor, and the cluster with n = 500 was studied by the Monte Carlo method. The first hydrated layer of a cluster is formed in unsaturated water vapors. The second hydrated layer begins to be formed in saturated vapor. The position of hydrated layers is not changed with an increase in cluster size and coincides with the position of the hydrated layers of ions in aqueous solutions of weak electrolytes. Orientational order in a cluster also has the layered structure. The orientation of molecules between the layers is random. The stability of the first layer is ensured only due to direct interactions with ions, whereas the stability of subsequent layers is due to cooperative interactions between molecules and between molecules and ions. As temperature decreases, the effect of ion displacement to the cluster surface becomes stronger.
1. INTERACTIONS IN A SYSTEM Strictly speaking, molecular motion in clusters at room and lower temperatures should be described by the methods of quantum statistics. Quantum-mechanical uncertainty in the position of water molecules with the mass of m = 3.013 × 10–26 kg at T = 300 K is estimated by the thermal de Broglie wavelength h δr = ------------------------- ≈ 0.024 nm, 2πmk B T
(1)
where h is Planck’s constant. The value of δr from Eq. (1) is by an order of magnitude smaller than characteristic molecular sizes. This allows us to describe the translational degrees of freedom using methods of classical statistical mechanics. Similar estimate of quantum-mechanical uncertainty in the orientation of molecules yields h δϕ = ----------------------- ≈ 1.28 rad = 73°, 2πIk B T
(2)
where I = 1.024 × 10–47 kg m2 is one of the principal moments of inertia of water molecule. It is seen from Eq. (2) that the rotational motion of molecules bears an essentially quantum character and should be described by the methods of quantum statistics. Nevertheless, it is possible, remaining within the framework of classical statistics, to approach fundamentally correct description of rotation, if we introduce corrections to the angular part of Hamiltonian decreasing the degree of its anisotropy. Such an effective interaction potential is essentially temperature-dependent pseudo-potential. All intermolecular potentials restored from the experimental data on the energy and structure of condensed substances are the pseudo-potentials of such kind. At the same time, potential curves constructed using the
methods of quantum chemistry involve the application of quantum statistics; the method of Feynman path integrals can be mentioned as an example [1–3]. Real molecules are not absolutely rigid and can be deformed during the formation of intermolecular bonds in a cluster. Intermolecular potentials are affected by these perturbations. There are two crucially different trends for describing this situation. The first trend consists in treating atoms composing molecule as individual strongly interacting heavy particles; the methods of classical statistics with the model interaction potential between these particles [4], in particular, the models with “soft” molecules [5] as well as with the potential calculated using the density functional for electron shells [6, 7] or the methods of quantum chemistry [8] are applied for describing intramolecular motion. The second trend is the approach to molecule as an integral particle taking into account its deformation in the field of other particles in an implicit form in intermolecular interaction potential. The first trend requires much larger computational resources at each step of a procedure and makes an impression of a more consistent approach, albeit essentially it is less substantiated. Indeed, the molecular motion in the condensed medium is strongly cooperative. The use of the methods of classical statistics is warranted only for the low-frequency part of the spectrum of cooperative modes with vibrational quanta hν Ⰶ kBT. Intermediate modes hν ~ kBT should be described by the methods of quantum statistics, whereas the high-frequency modes hν Ⰷ kBT should be excluded from consideration, because they are not excited modes. The application of numerical methods of classical statistics is based on the assumption that the contribution of intermediate modes is small. This assumption is the more real, the larger the difference in the rigidity of intra- and intermolecular
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interactions, i.e., the more molecules retain their individuality and chemical inertness in the condensed phase. Rare gases and isoelectronic molecules including water molecule are most properly satisfy this condition. The low- frequency modes can be described by the methods of classical statistics, while the high-frequency modes are “frozen.” The model of rigid intramolecular interactions is most corresponded to these requirements. The introduction of the models of “soft” molecules can result in a qualitatively incorrect behavior of a system. For example, the vibration frequency in the ground quantum state of the OH complex is equal to 3735 cm–1 [9] that, at T = 300 K, corresponds to negligibly low probability of the excitation of this mode, exp(–hν/kBT) = 1.66 × 10–8. Formal introduction of intramolecular motions into the description based on the methods of classical statistics would imply its complete excitation. The distortion of the energy distribution over the degrees of freedom inevitably affects all characteristics and first the heat capacity and diffusion coefficient, which appeared to be significantly overestimated. The heat capacity is directly connected with the phase transitions and the free energy of a system. In turn, the probability of macroscopic events in a molecular system exponentially depends on free energy. In this study, we employed many-point intermolecular potentials with the rigid geometry of force centers. The combined approach is used for describing the interactions in water cluster. The interaction between water molecules is described by the ST2 potential, while the interaction of molecules with Cl– ion is based on the geometry of charge distribution in the electron shell of a molecule adopted in the SPC model [10]. In contrast to the SPC model, the distribution of force centers in the ST2 model is the bulk distribution corresponding to the tendency in forming tetrahedral geometry of hydrogen bonds [11–16]. This tetrahedral structure is manifested itself already in the elementary act of proton capture by real water molecule. In the hydroxonium ion H3O+, three protons do not lie in one plane with oxygen atom [17]. The arrangement of force centers in the ST2 model reflects the directionality of hydrogen bonds (which are not reduced to simple Coulomb interaction) rather than the charge distribution in the electron shell of a molecule. At the same time, it is the Coulomb component of interactions that increases in the strong electric field of atomic ion. The SPC model with the plane geometry of force centers is in principle closer to the charge distribution in real water molecule. Therefore, the arrangement of charges in the SPC model is fundamentally closer to the geometry of interaction between water and simplest ions. At the same time, it is most likely that the ST2 potential more correctly reflects the character of interaction in small ensembles of particles as, for example, in the interaction between water molecule and the only hydrogen atom. We expect that this tendency will be retained in extremely small clusters containing 3–5 molecules.
In the model under consideration, the interaction between water molecules is continuously degenerated into five-center pair ST2 potential. In the Coulomb ion field EI, the energy of polarization was explicitly accounted for in the following form: 2
w–ion
U pol
EI = – α w -----, 2
(3)
where αw = 1.44 × 10–3 nm3 is the isotropic part of the polarization factor of free water molecule [9]. Interactions between the dipoles induced on molecules and the field of molecules are also taken into account. The pattern of this interaction as is follows. The field of the jth molecule in the point of the ith molecule is added from (1) the electric fields of four point charges q k positioned in the points in accordance with the geometry of the ST2 potential [11–16] (1)
4
(1) E ij
=
qk j i ------------------- ( r0 – rk ) j i 3 k = 1 r0 – rk
∑
(4)
and the field of induced dipole ( p i r ij )r ij pi (2) -, E ij = – ----------+ 3 ----------------------5 3 ( r ij ) ( r ij ) ind
ind
(5)
j
where r 0 is the position of Lennard-Jones center of the jth molecule and p i = αwEI is the vector of dipole moment of the ith molecule induced by the filed of ion EI. The secondary polarization of molecules in the filed of induced dipoles is not taken into account due to its smallness. The energy of interaction between the induced molecules with the proper electric field and the induced dipoles of other water molecules is calculated as ind
1 w–w U ind = – --2
∑ ( 2E
(1) ij
(2)
+ E ij )p j . ind
(6)
i, j
The interaction between the induced dipoles and proper field of molecule at distances of a few angstroms is several times weaker; the interaction between the induced molecules by the order of magnitude is weaker than that calculated by Eq. (3) and equal to the hundredths of kBT. The energy of secondary polarization in the field of induced dipoles at the same distances is the thousandths of kBT and rapidly decays with distance; hence, it is not explicitly taken into account in this model. The Cl– ion interacting with water molecules is simulated as the source of the Lennard-Jones potential with the depth εwI = 6.75 × 10–15 erg, the radius of action σwI = 0.393 nm, and the negative point elementary charge in the center. When calculating the energy of interaction between the ion charge and water molecule, it is assumed that the point sources of the electric field of a molecule are located in accordance with the SPC COLLOID JOURNAL
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model: the negative point charge q 1 = –3.8804 × 10–10 CGSE is fixed at the oxygen atom in the same point as the source of the Lennard-Jones potential, and two pos(2) itive point charges q k = +1.9402 × 10–10 CGSE, k = 2.3 are fixed at a distance of 0.1 nm from oxygen atom at the beams incidenting at an angle 109°28′ from its center. The polarization of Cl– ion in the field of water molecules is accounted for explicitly. For this purpose, the strength of electric field of all point charges of the SPC molecules is added in the point of Cl– ion position: 3
Ew ( R ) =
∑∑ i
(2)
qk i ------------------( R – r k ), i 3 k = 1 R – rk
(7)
where R is the position of the center of Cl– ion. The energy of Cl– ion polarization in the field of molecules is calculated as 2
ion–w
U pol
–E = – α -----w- , 2
(8)
where α– = 3.59 × 10–3 nm3 is the coefficient of Cl– ion polarization [9]. Secondary effects of ion polarization in the field of induced dipoles are not taken into account explicitly. The numerical values of ion–water interaction parameters are assumed to be such that to obtain the values of free energy and enthalpy of several first reactions of water molecule addition to the cluster on ion that are as much as closer to the experimental values [18]; they also allow us to reproduce the position of the first maxima of atom-atom correlation functions of ion–water. When describing the ion–water interactions, we did not use more complex model developed by us [19–27], but took advantage of the traditional, simpler model. This model makes it possible to reproduce the experimental values of the positions of maxima of the correlation functions of ion–oxygen and ion–hydrogen molecules; however, the entropy of two first addition reactions is underestimated as compared to the experimental values by about kB. In a more complex model [19–27] accounting for the partial screening of the ion charge at contact distances, we succeeded in exact reproducing the experimental value of entropy; however, an agreement with experiment with respect to the ion–hydrogen correlation function is deteriorated. In this study, while choosing the model of interactions, we preferred data on the correlation functions and free energy of the first addition reactions between molecule and ion, because it is the free energy but not the enthalpy that precisely determine the probability of events in a molecular ensemble. In the models studied until now, the agreement with the experimental data on free energy was not controlled and is reported here for the first time. The enthalpy of the first two reactions differs from the experimental value [18] by the order of kBT; however, this does not affect the probability of cluster formation. As is seen from data listed in table, COLLOID JOURNAL
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Free Gibbs energy ∆G and enthalpy ∆H of the addition reaction of water molecule Cl–(H2O)n – 1 + H2O = Cl–(H2O)n in kcal mol–1 at T = 298 K calculated by the Monte Carlo method and recalculated to the standard vapor pressure p = 1 atm. Experimental data in the last two columns are taken from [18] n
–∆G
–∆H
–∆Gexp
–∆Hexp
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
8.20 6.49 5.00 4.22 3.62 3.12 2.72 2.42 2.11 1.89 1.69 1.53 1.42 1.31 1.25 1.19 1.16
14.00 12.80 11.40 11.12 10.75 10.34 10.06 10.00 9.61 9.59 9.44 9.48 9.53 9.52 9.49 9.60 9.82
8.2 6.5 4.5 3.4 – – – – – – – – – – – – –
13.1 12.7 11.7 11.1 – – – – – – – – – – – – –
this discrepancy in enthalpy is balanced beginning with the third reaction. 2. RESULTS OF NUMERICAL SIMULATION The analysis of the cluster stricture with the account of thermal fluctuations is performed using the correlation functions of various types. In this study, we calculated the ion–water and water–water correlation functions. Correlation functions were calculated at each step of the Marcovian process by the direct counting of the number of atoms included into corresponding spherical layer at a preset distance from the other atom or the amount of molecules, whose orientation angles are located in corresponding angular intervals. We report here only the small part of calculated curves. However, the inferences about the cluster structure are based on the analysis of all actual data including those remained beyond the framework of this publication. Figure 1 shows the correlation functions of Cl– ion– oxygen atom of water molecules in a Cl–(H2O)n cluster at room temperature within the wide range of cluster sizes. The last curve for n = 500 can be considered as calculated in the bulk phase. Oscillations of correlation function reflect clearly pronounced layered structure of a cluster. The first hydrated layer is at a distance of 0.325 nm from the ion center and has the thickness of about 0.05 nm. The density of molecules in this layer reaches its maximal value that is about three times as
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gCl,O, nm–3 1 2 3 4
100
50
0
0.2
0.4
0.6
0.8
r, nm
Fig. 1. Correlation functions of ion–oxygen atom of water molecule in the ël–(H2O)n clusters of different sizes in equilibrium with vapor at 298 K and various values of 〈n〉: (1) 0.599, (2) 8.48, (3) 19.4, and (4) 500. In the last case, cluster size was fixed. Functions are normalized to the average number of molecules in a cluster and present the average local density of molecules at corresponding distances from the ion. The dashed line denotes the overall density in liquid water under standard conditions.
large as the overall density in a liquid water and twice as large as in the first hydrated layer around electroneutral water molecules. The second layer is located at a distance of 0.52 nm and is twice as thick. Maximal density in this layer is about two times larger than the overall density in water. On cluster growing, the first coordination number increases up to n = 5, in general, only at the initial stage. Its further increase slows down greatly but it is not stopped even for clusters of several tens of molecules. At n = 500, the first coordination number obtained by the integration of correlation function within the limits of the first minimum at a distance of 0.395 nm is equal to 6.15. Using the method of correlation dynamics with the periodic boundary conditions, Driesner and Cummings [28] obtained the value of 0.314 nm for the radius of the first hydrated layer of Cl– ion in the bulk phase in the “soft” BJH model of water molecules at T = 300 K, 0.381 nm for the first minimum, and 7.3 for the first coordination number. In the same work, the obtained values of these parameters were equal, respectively, to 0.322 nm, 0.406 nm, and 6.6 in the “rigid” SPC model and to 0.322 nm, 0.400 nm, and 8.4, in the modified “soft” SPC model. In [29], for the SPC model with slightly different potential of ion– water interaction, the correlation functions were obtained by the method of molecular dynamics differing from those reported in [28] by lower degree of structurization. The first maximum was located at a distance of 0.330 nm, and the second maximum was almost unobserved. The uncertainty in the position of the second maximum greatly complicates the determination of the first coordination number. The value of 9.0
was obtained in [29] assuming that the first minimum is located at a distance of 0.500 nm. Probably, the last two values are overestimated. In [30], the second maximum of correlation function was observed at a distance of 0.51 nm, whereas the first minimum, at a distance of about 0.38 nm. From data presented in [30] in a graphic form and with allowance for this position of the first minimum, the first coordination number is equal to 6.0. The range of 0.316–0.320 nm for the position of the first maximum was obtained in [31] using the TIP4P model; judging from the reported graphs, the position for the second maximum equals 0.49 nm. The first minimum is located at a distance of 0.38 nm. Experimental values for the first coordination number of Cl– ion in aqueous electrolytes obtained by measuring neutron scattering demonstrate wide scatter and vary from 5.0 to 8.0 [32–34]. Comparison with these data indicates that our results are consistent (within the limits of uncertainty related to the use of different models of interaction) with the data of other authors and experimental data obtained for electrolytes. Some difference from the results of electrolyte simulation is observed in the degree of structurization of correlation functions. Our correlation functions demonstrate more clearly pronounced oscillations. The smoothest correlation functions were obtained in [29]. The reason for these variations is in the various models of interaction. In the presented calculations, interactions between water molecules are described by the ST2 potential where the attempt was made to describe the hydrogen bonds in more detail. Potentials with the geometry of the SPC type were employed in [28–31]. It is known from previous studies that the SPC potential in pure water leads to exaggeratedly smooth correlation functions as compared to the experimental data. At a pressure below the saturated vapor pressure, most molecules are present in the first hydrated layer. At the pressure of saturated vapor, about 20% of molecules belongs to the second hydrated layer and the consequent layers are absent. It can be assumed that the formation of the second and consequent layers occurs mainly in the supersaturated vapor. The density of molecules in the first layer increases further after the formation of the second layer, although this growth slows down considerably in clusters with n > 20. The analysis of energy characteristics of the cluster shows that, beginning with this size, the mechanism of molecule retention in the cluster changes qualitatively from the retention due to direct interactions between the molecules and the ion to the binding predominantly by the interactions between the molecules. As cluster grows further, the sign of interactions between the molecules in point n = 3 changes from positive antibinding to negative binding; at n > 20, the mean interaction energy between the molecule and other molecules in the cluster exceeds (by the absolute value) the energy of its interaction with the ion. The stability in clusters of such a size is ensured mainly due to the binding interactions between the molecules; however, the ion field still plays COLLOID JOURNAL
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the role of the stabilizing factor up to the size corresponding to inversion ninv in spite of its decreasing part in the general balance of interactions in cluster. Note the constancy at which maxima of correlation functions retain their positions upon cluster growth within the entire size range. Function maxima are not shifted, but only increase in the height according to an increasing number of molecules in a system. Similar stability in maximum position on varying the content of admixtures in electrolyte solution (Na+Cl–) was mentioned in [30]. Seemingly, this property is of universal character and is associated with the strong interactions between the ion and the molecules compared with interactions between the molecules in the hydrated shells next to the ion. The energy of interaction between the molecules in the first hydrated shell is, on the average, positive and by the absolute value is about fifty times lower than the interaction with the ion. In the second hydrated shell, the interactions of both types are negative and their values are comparable; however, the motion of molecules toward the ion is associated with the distortions in a relatively strong structure of the first shell. Therefore, on adding new molecules, all rearrangements in the second shell occur along the surface of a cluster and do not change the distance to the ion that forms its ion–molecule correlation function. The distance between the first and second maxima of the correlation function is equal to 0.175 nm that is about a factor of 1.5 shorter than the distance of 0.276 nm in the stable configuration between two water molecules with the ST2 interaction potential. Evidently, the molecules of the second layer are partly incorporated between the molecules of the first layer but are not placed above these molecules. This rule is just a tendency and is exhibited, on the average, in sufficiently large statistical samplings. The formation of a rigid crystal-like structure is not observed. Individual hydrated layers and, moreover, correlations in the positions of molecules in these layers in general are hardly distinguishable in a single molecular configuration. The behavior of ion–atoms of hydrogen molecule correlation function with the change in cluster size (Fig. 2) is qualitatively similar to that of previous correlation function. The first maximum is located at a distance of 0.227 nm; the second, at 0.355 nm. The values of 0.219 and 0.346 nm in the BJH model, 0.224 and 0.366 nm in the rigid SPC model, and 0.222 and 0.363 nm in the “soft” model with the SPC geometry were obtained for the same maxima in the bulk phase [28]. The positions of the same maxima are estimated in [31] as equal to 0.22 and 0.36 nm; in [30], these data are shown in graphical form from which it is difficult to ascertain the exact positions of maxima, although the profile of correlation functions coincides qualitatively with that obtained by us in a cluster; in [29], information on correlation functions of this type is absent. The ratio of the heights of maxima in the bulk phase in [31] is almost the same as in our calculations: the first maximum in a cluster was higher than the second one. HowCOLLOID JOURNAL
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gCl,H, nm–3 1 2 3 4
100
50
0
0.2
0.4
0.6
0.8
r, nm
Fig. 2. The same as in Fig. 1 and for the same cluster size, but for hydrogen atoms of water molecules. At the same normalization as in Fig. 1, functions present the half-local density of hydrogen atoms.
ever, we obtained the height ratio of 3 : 1, whereas in [31] it was estimated as approximately 2 : 1. As in the case of ion–oxygen atom correlation function, the tendency to more distinctly expressed maxima was also observed in [31] that arises from more detailed account of hydrogen binding in the ST2 model. Both maxima are formed by the hydrogen atoms of the molecules of the first hydrated layer. The distance between the protons in the ST2 molecule is equal to 0.1633 nm, whereas the distance between the first and the second maxima of the correlation function is 0.128 nm. This means that, in the first hydrated layer, the dipole moments of molecules are oriented at an angle of 52° with respect to the ion. This angle is close to 54°44′ between the vector of dipole moment and the direction along the O–H bond in the ST2 molecule; its value manifests that the molecules in the first hydrated layer are oriented by one of their hydrogens toward the ion so that the O–H line almost coincides with the direction toward the ion. If the molecules had been represented by the point dipoles, their dipole moments would be oriented exactly toward the ion. The deviation of 52° is indicative of the prevalence of higher multipole moments of molecules in the interactions with the ion at distances corresponding to the first hydrated layer. The positions of maxima of correlation function are stable and virtually are not changed with the cluster growth except for the extremely small size n = 1. In a cluster with one molecule, the second maximum is much closer to the ion than in larger clusters, at a distance of 0.336 nm. Shorter distance between the first and the second maxima corresponds to smaller angle between the vector of dipole moment of a molecule and the direction toward the ion. This fact indicates that the cooperative interactions in the first hydrated layer lead to the additional rotation of the dipole moments of mol-
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gCl,O, nm–3 100 1 2 75
50
25
0
0.2
0.4
0.6
0.8
r, nm
Fig. 3. The same correlation functions as in Fig. 1 for two values of temperature: (1) 298 K, 〈n〉 = 8.48 and (2) 210 K, 〈n〉 = 8.90.
ecules from the direction toward the ion by about 10°; thereby, the orientation of a molecule approaches that corresponding to the minimum of energy of interaction between the single molecule and the ion, that is, corresponding to almost linear Cl––H–O bond. The molecules of the second and third hydrated layers form two almost merging maxima at distances of 0.44 and 0.58 nm (Fig. 2); the first of these maxima appears together with the second hydrated layer, while the second maximum develops synchronously with the third layer shown in Fig. 1. Both maxima are formed by the atoms of hydrogen molecules belonging to different hydrated layers of a cluster. The analysis of the positions of maxima allows us to conclude that the average orientation of the dipole moments of molecules in the second and third hydrated layers still markedly deviates from the direction toward the ion center and is about 60° and 70°, respectively; however, the reason for this deviation is qualitatively different than in the first hydrated layer and consists in the strengthening of hydrogen bonds between neighbor molecules with simultaneous weakening of the orienting action of the ion electric field accompanied by thermal fluctuations. Even in the fourth layer, the average angle between the dipole moments of molecules and the direction toward the ion is about 75° at a distance of 0.85 nm (Fig. 1) that cannot be explained by their interactions with higher multipoles of molecules. If the protons of molecules of the first hydrated layer cause the appearance of two maxima at the ion–hydrogen correlation function with one of them being closer and the other farther than the first maximum at the ion–oxygen correlation function, each successive layer generates only one of such maxima that is always closer to the ion than corresponding maximum for oxygen atom. Together with data on relatively large values of the angle between the dipole moments of molecules and the direction toward the ion,
this implies that, in this case, those orientations of molecules prevail for which the plane of H–O–H molecule is normal to the plane corresponding to the Cl––O line and the vector of the dipole moment of a molecule. Note that, in occasional intermediate positions between the hydrated layers, molecules are oriented almost randomly, thus indicating the cooperative character of orienting mechanisms in a cluster. Each hydrated layer is structured as a whole such that the incorporated molecules are oriented by their dipole moments toward the ion predominantly at acute angles only due to interaction with neighbor molecules of the layer. In this case, nowhere in a cluster, the dipole moments of molecules are oriented strictly toward the ion, as could be expected in the simplest model, provided that the particles with point dipoles are substituted for water molecules. Hydrogen bonds cause the disorientation of molecules between the hydrated layers, which is unfavorable for the viewpoint of the energy of interaction with the ion. However, the number of such molecules is small and they do not affect substantially the total energy balance of a system. Evidently, the layered structure of orientations in a cluster is a definite compromise between the orienting field of an ion and the geometry imposed by hydrogen bonds. In all likelihood, the formation of permittivity in polar liquids is based qualitatively on the same cooperative mechanism of increasing the orienting effect of field. The dependence of the structure of the hydrated shells on temperature is of special interest in view of the problem of chlorine accumulation in the stratosphere. On cooling to the temperature of about 210 K, one can expect the structural changes in clusters similar to those occurring during a water freezing. However, it is wellknown that the crystallization point of clusters depends on their sizes and, for clusters of several tens or hundreds of molecules lies in the temperature range typical of polar stratosphere. In addition, the action of strong electric field of the ion incorporated into the cluster can fundamentally change the scenario, and the crystallization per se could not take place at all. It is precisely such a picture that seems to us most probably results from the analysis of our data. The state of cluster with the size of at least up to several hundreds of molecules with the incorporated ion should most likely be considered as liquid than crystal-like even at temperatures typical of polar stratosphere. Figures 3–5 show the ion–water correlation functions at room temperature (298 K) and 210 K typical of polar stratosphere. Comparison of these data indicates that, on cooling, oscillations of correlation functions become more intense. The first hydrated layer shifts negligibly toward the ion, while thermal fluctuations in the molecule orientations are slightly weaken. This is manifested by the deeper and broader minimum dividing two first maxima in Fig. 4. The radius of the second hydrated layer decreases on cooling by 0.02 nm (Fig. 3). The most probable position of the protons of molecules of the second layer shifts by the same distance toward the ion; however, the COLLOID JOURNAL
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maximum for protons is not splitted (Fig. 4), thus indicating that the orientation of the plane of H–O–H molecules is not changed. Thus, no qualitative changes in the molecular order take place on cooling, although the degree of ordering noticeably increases. Significant differences between the correlation functions in clusters and macroscopic solution are observed only in extremely small clusters. In a cluster of two molecules and an ion, one of the statistically advantageous (under these conditions) ion positions is in the middle between molecules. In this case, the distance between molecules is about 0.6 nm, and the maximum is formed in the correlation function at this distance (Fig. 5). Such a linear arrangement of two molecules in the first hydrated layer is energy more advantageous than the compact arrangement, because the dipole moments of molecules are oriented, in this case, predominantly toward the ion and are pushed apart; however, in the linear configuration, the distance between molecules is maximal and the interaction energy is minimal. In addition, there is also other statistically significant cluster configuration that originates much sharper maximum at the intermolecular distance of 0.28 nm (Fig. 5). This maximum is determined by strong hydrogen bond between two molecules arising at a contact distance and being stronger than the repulsion of the dipoles of molecules. In such a configuration, an ion and two water molecules form almost isosceles triangle. The hydrogen bond operates in a relatively narrow range of distances beyond which the molecule dipoles oriented in the ion field are predominantly pushed apart. Therefore, the first maximum is much sharper than the second. It is the prevalence of hydrogen bonding over electrostatic interaction between the molecules that is precisely responsible for the similarity of correlation functions of molecules in a cluster and macroscopic liquid phase. On the contrary, in extremely small clusters, the first hydrated layer is not completed, and in such a “rarefied” system, the longrange electrostatic interactions have more chances in the formation of its structure. The growth of cluster results in an increase in the first maximum of the correlation function and the fast shift of the second maximum toward the ion by a distance of 0.445 nm corresponding to the radius of the second hydrated layer of a molecule. The problem of the localization of the chlorine ion in water clusters was discussed in [35]. The main conclusion is that the position of ion in a cluster is unstable and the explicit account of the polarizability of chlorine ion leads to the displacement of ion to the cluster surface, the effect of displacement being dependent on the polarization factor of an ion. There is some uncertainty in data on the value of the polarization factor of Cl– ion due to the different procedures and conditions of its measurement. The measurement of the polarizability of free Cl– ion gives the value of 2.8 × 10–3 nm3 [9]. At the same time, measuring the ion polarizability in electrolytes resulted in the value of 3.59 × 10–3 nm3 [36]. In our COLLOID JOURNAL
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gCl,H, nm–3 100 1 2 75
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0
0.2
0.4
0.6
0.8
r, nm
Fig. 4. The same as in Fig. 3 for the correlation functions of ion–hydrogen atom of water molecule.
gO,O, nm–3 1 2
15
10
5
0
0.2
0.4
0.6
0.8
r, nm
Fig. 5. Correlation functions of oxygen atom–oxygen atom of water molecule in extremely small clusters at various temperatures: (1) 298 K, 〈n〉 = 2.48 and (2) 210 K, 〈n〉 = 2.50. The functions are normalized to 〈n〉 – 1 and present the average local density at corresponding distances from molecule in a cluster.
calculations, we preferred the polarizability measured in [36]. It was shown [35] that the effect of ion displacement from the cluster vanishes at room temperature upon decreasing polarization factor below ≈2 × 10–3 nm3. It was of interest to elucidate what will be the behavior of the cluster (modeled by us) on the ion with and without the polarizability. The analysis of the array of molecular configurations obtained in the process of simulation made it possible to conclude that the effect of the displacement of Cl– ion to the surface significantly depends on temperature, being increased on cooling. In the state of “switched-off” ion polarizability, we observed that, at 298 K, the ion is predominantly
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SHEVKUNOV
Fig. 6. Typical configuration of Cl–(H2O)60 cluster at 210 K. Chlorine ion is marked by the dark color.
positioned inside the cluster; at 210 K, mainly at the cluster surface. With account of ion polarizability, it is always displaced from the cluster to its surface. One of such configurations is shown in Fig. 6. Energy preferable position of ion at the cluster periphery is explained by the nonlinear effects of polarization. Cluster that is located sideways of ion turned out to be polarized completely in one direction. In turn, the field of polarized cluster polarizes the ion, the ion energy being negative and growing “quadratically” with the strength of cluster field. The effect of displacement to the surface is retained only for clusters of intermediate sizes. Obviously, beginning with a certain size, further cluster growth sideways from the ion becomes energy less advantageous than the enveloping of the ion from all sides, since the molecules added from the one side appeared to be too far from the ion. The simulation demonstrated that, in cluster composed of 500 molecules with fixed spherical boundaries, the displacement of ion to the surface is not accompanied by a marked decrease in energy. 3. CONCLUSION Results of the simulation of the microstructure of water clusters on the negative chlorine ions can be summed up as follows. The position of the hydrated layers in a cluster resembles that of solvation layers of chlorine ions in aqueous solutions of weak electrolytes. The chlorine ion in unsaturated vapor carries one hydrated layer composed of 5 to 7 molecules. The growth of next hydrated layers occurs in the supersaturated vapor. Clearly pronounced orientational order is observed in the hydrated layers; however, in neither one of the layers, molecules are oriented by their dipole moments
directly toward the ion. In the first layer, the deviation from this direction is attributed to the prevalence of interactions between the ions and higher multipole moments of a molecule; in the subsequent layers, to strong hydrogen bonds between the molecules. In the intermediate positions, the orientation of molecules between the layers is random. The stability of the first hydrated layer is ensured only by the direct interactions with the ion; in the second layer, the interactions with the ion and between the molecules are comparable by their values, both of them being of binding character; in the subsequent layers, the role of ion gradually vanishes. Cooling to the temperatures that are typical of polar stratosphere does not lead to the qualitative changes in the structure of clusters, although the degree of ordering increases. In clusters with the chlorine ion, the effect of displacement of ion from the cluster is observed; however, it vanishes as cluster size increases to several hundreds of molecules. This effect becomes more pronounced on a decrease in temperature and, at stratospheric temperatures, it is observed even for the ion with the zero-valued polarization factor. The complementarity of cluster structure formed on the chlorine ion to the structure of stable condensed bulk phase of water should facilitate the incorporation of cluster as a whole into the subsurface layers of ice microparticles in the stratosphere and the absorption of ions by the microparticles. ACKNOWLEDGMENTS This work was supported by INTAS, project INTAS 99-01162. COLLOID JOURNAL
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COMPUTER SIMULATION OF WATER CLUSTERIZATION ON CHLORINE IONS. 2.
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