Thermodynamic properties of clusters: isobar heat capacity ( a ), isothermal compressibility ( b ). For other designations, see Fig. 1. 

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... thermodynamic equilibrium; for this time interval, we calculated the analogs of thermodynamic properties [15]. The properties themselves may be determined from the analogs by passing from the molecular dynamic cluster to the thermodynamic one using the procedure described in [16] involving the addition of the “interacting” environment to the cluster. The energy E tot of the most stable forms of water clusters with N ≤ 20 at T = 300 K calculated by the Monte Carlo method [23] fluctuates within − 9.6 ≤ E tot ≤ − 7.8 kcal/mole. The energy of the “quasiequilibrium” formations of water molecules (2 ≤ N ≤ 20) in our calculations varied from − 8.8 to − 5.5 kcal/mole. On the short ( t < 1 ns) time interval, the clusters under study were spheroids, which subsequently transformed into a pulled cloud of irregular form. Note that E tot of the water aggregates and aggregates with a CO impurity approach each other at large N (Fig. 1 a ), whereas E tot of the clusters containing CO 2 have considerably higher values at large N . The energies E a − w of interaction between the impurity (CO and CO 2 ) and water molecules in the cluster start to approach each other with increasing N (Fig. 1 b ). The different values of E a − w for CO and CO 2 molecules at low N are explained by the differences in size and interaction with water molecules. Calculation of the isobaric heat capacity c p involves determination of the isochor heat capacity c v , adiabatic compressibility β s , and the derivative (∂ P / ∂ T ) V . Heat capacity c v was calculated in terms of kinetic energy fluctuations [9], and β s was evaluated in terms of pressure fluctuations [19]. Calculation of the derivative (∂ P / ∂ T ) V demands the calculation of the cross fluctuations of pressure and potential energy [25]. Heat capacity c p of the clusters under study decreases with increasing N , approaching the experimental [26] value of c p for pure water (Fig. 2 a ). For N > 12, c p of pure water clusters and aggregates with CO approach each other, whereas c p of the clusters with CO 2 is still higher. Isothermal compressibility was determined in terms of the known thermodynamic ...

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... the whole range of N , the pure water clusters and the aggregates with CO 2 have lower values of T than the corresponding experimental value [26] for water ( Fig. 2 b ). The clusters with CO are characterized by higher compressibility, since their size ( N > 4) is considerably larger than that of the analogous pure water aggregates or clusters with CO 2 . Determination of the surface tension of clusters by relation (3) eventually demands knowledge of the density of states f (ω) for vibrations of molecular systems. Figure 3 a shows as an example the calculated autocorrelation functions of rate (AFR) for water molecules and the CO molecule in the cluster with N = 6. For the impurity molecule, AFR decays faster and has a less extended frequency spectrum (Fig. 3 b ) than AFR of water molecules. The spectral function f (ω) for ω = 0 is proportional to the self-diffusion coefficient D . The coefficient D of the impurity molecule is higher (3.81 ⋅ 10 − 9 m 2 /s) than D of the water molecule (2.88 ⋅ 10 − 9 m 2 /s). Both spectra depicted in Fig. 3 b have the form typical for the spectra of the individual vibrations of molecules in liquids. The liquid-like spectra of AFR are characteristic of all aggregates under study, due to which we have ample reason to state that the clusters are in the liquid state. The resulting size dependence of the surface tension γ of the clusters is shown in Fig. 4 a . The same figure gives the experimental value of γ ∞ of pure water at T = 300 K for the planar interface [26]. Monotonous growth of surface tension with increasing N was observed for pure water clusters and aggregates with a CO molecule. Approximation of the dependence γ( N ) for clusters with CO 2 was done by extrapolating the corresponding curve for N ≤ 10 to large values of N . The real dependence γ( N ) for this case is shown in Fig. 4 b (dots). The same figure shows an approximation of the dependence by the second-order polynomial. The quantity T/c p 0 is a thermal criterion of stability for an extended phase. It may be represented as one of the criteria of stability for systems with a limited number of degrees of freedom [27]. The dependence c p ( N ) in general reflects (Fig. 2 a ) increased thermal stability of the clusters when the constituent molecules increase in number. As for the bulk phase, for clusters one can apply the mechanical criterion of stability in the form of β − T 1 > 0 [27]. The dependence β T ( N ) (Fig. 2 b ) tends to grow with N for both pure water clusters and for aggregates with CO. This means that the mechanical stability of such clusters decreases with their size. For aggregates with CO 2 , the mechanical stability changes insignificantly. The monotonously increasing dependence γ( N ) (Fig. 4 a ) was used to estimate the size of the critical seed. Let us consider that for all clusters under study, γ → γ ∞ for N → ∞ . Assuming that the surface tension of the seed of the critical size does not differ widely (within 5%) from γ ∞ [28], we extrapolate the curves given in Fig. 4 a to the region of large N until they intersect the line γ = γ ∞ . The perpendiculars from the intersection points to the abscissa axis define the size of the critical seeds: N ∗ = 57 for pure water clusters and water clusters with CO; N ∗ = 74 for aggregates with CO 2 . The criterion of cluster integrity is the positive and finite value of its surface tension γ . The modeling showed that there is high probability that the cluster with a CO 2 molecule will not reach the critical size. This is clearly indicated by the dependence γ( N ) (Fig. 4 b ). The function γ( N ) has a maximum whose position is fixed at N = 10 and then quickly decreases. When γ decreases to values close to zero, it is highly probable that the integrity of the aggregate will be broken, i.e., that the aggregate will fractionate into smaller clusters or molecules will evaporate from the formation under analysis. To examine this question in more detail, we investigated the evolution of a system of clusters using a separate visualization program. In the course of calculation, the molecular coordinates were recorded every 10,000 time steps (10 ps). Moreover, additional calculations were carried out for clusters of 18 water molecules and an impurity: CO 2 in one case and CO in the other. The new calculations differ from the former ones in that the impurity molecule was initially placed outside the aggregate but not at the center. In both cases, the impurity molecules were trapped by water clusters, but by the moment of time 3 ns, the virtual cluster with CO 2 split into two water clusters (with 6 and 12 molecules) and the CO 2 monomer, whereas the cluster with CO retained its integrity. The work of formation of a critical seed may be defined as ...

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... the whole range of N , the pure water clusters and the aggregates with CO 2 have lower values of T than the corresponding experimental value [26] for water ( Fig. 2 b ). The clusters with CO are characterized by higher compressibility, since their size ( N > 4) is considerably larger than that of the analogous pure water aggregates or clusters with CO 2 . Determination of the surface tension of clusters by relation (3) eventually demands knowledge of the density of states f (ω) for vibrations of molecular systems. Figure 3 a shows as an example the calculated autocorrelation functions of rate (AFR) for water molecules and the CO molecule in the cluster with N = 6. For the impurity molecule, AFR decays faster and has a less extended frequency spectrum (Fig. 3 b ) than AFR of water molecules. The spectral function f (ω) for ω = 0 is proportional to the self-diffusion coefficient D . The coefficient D of the impurity molecule is higher (3.81 ⋅ 10 − 9 m 2 /s) than D of the water molecule (2.88 ⋅ 10 − 9 m 2 /s). Both spectra depicted in Fig. 3 b have the form typical for the spectra of the individual vibrations of molecules in liquids. The liquid-like spectra of AFR are characteristic of all aggregates under study, due to which we have ample reason to state that the clusters are in the liquid state. The resulting size dependence of the surface tension γ of the clusters is shown in Fig. 4 a . The same figure gives the experimental value of γ ∞ of pure water at T = 300 K for the planar interface [26]. Monotonous growth of surface tension with increasing N was observed for pure water clusters and aggregates with a CO molecule. Approximation of the dependence γ( N ) for clusters with CO 2 was done by extrapolating the corresponding curve for N ≤ 10 to large values of N . The real dependence γ( N ) for this case is shown in Fig. 4 b (dots). The same figure shows an approximation of the dependence by the second-order polynomial. The quantity T/c p 0 is a thermal criterion of stability for an extended phase. It may be represented as one of the criteria of stability for systems with a limited number of degrees of freedom [27]. The dependence c p ( N ) in general reflects (Fig. 2 a ) increased thermal stability of the clusters when the constituent molecules increase in number. As for the bulk phase, for clusters one can apply the mechanical criterion of stability in the form of β − T 1 > 0 [27]. The dependence β T ( N ) (Fig. 2 b ) tends to grow with N for both pure water clusters and for aggregates with CO. This means that the mechanical stability of such clusters decreases with their size. For aggregates with CO 2 , the mechanical stability changes insignificantly. The monotonously increasing dependence γ( N ) (Fig. 4 a ) was used to estimate the size of the critical seed. Let us consider that for all clusters under study, γ → γ ∞ for N → ∞ . Assuming that the surface tension of the seed of the critical size does not differ widely (within 5%) from γ ∞ [28], we extrapolate the curves given in Fig. 4 a to the region of large N until they intersect the line γ = γ ∞ . The perpendiculars from the intersection points to the abscissa axis define the size of the critical seeds: N ∗ = 57 for pure water clusters and water clusters with CO; N ∗ = 74 for aggregates with CO 2 . The criterion of cluster integrity is the positive and finite value of its surface tension γ . The modeling showed that there is high probability that the cluster with a CO 2 molecule will not reach the critical size. This is clearly indicated by the dependence γ( N ) (Fig. 4 b ). The function γ( N ) has a maximum whose position is fixed at N = 10 and then quickly decreases. When γ decreases to values close to zero, it is highly probable that the integrity of the aggregate will be broken, i.e., that the aggregate will fractionate into smaller clusters or molecules will evaporate from the formation under analysis. To examine this question in more detail, we investigated the evolution of a system of clusters using a separate visualization program. In the course of calculation, the molecular coordinates were recorded every 10,000 time steps (10 ps). Moreover, additional calculations were carried out for clusters of 18 water molecules and an impurity: CO 2 in one case and CO in the other. The new calculations differ from the former ones in that the impurity molecule was initially placed outside the aggregate but not at the center. In both cases, the impurity molecules were trapped by water clusters, but by the moment of time 3 ns, the virtual cluster with CO 2 split into two water clusters (with 6 and 12 molecules) and the CO 2 monomer, whereas the cluster with CO retained its integrity. The work of formation of a critical seed may be defined as ...

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... the whole range of N , the pure water clusters and the aggregates with CO 2 have lower values of T than the corresponding experimental value [26] for water ( Fig. 2 b ). The clusters with CO are characterized by higher compressibility, since their size ( N > 4) is considerably larger than that of the analogous pure water aggregates or clusters with CO 2 . Determination of the surface tension of clusters by relation (3) eventually demands knowledge of the density of states f (ω) for vibrations of molecular systems. Figure 3 a shows as an example the calculated autocorrelation functions of rate (AFR) for water molecules and the CO molecule in the cluster with N = 6. For the impurity molecule, AFR decays faster and has a less extended frequency spectrum (Fig. 3 b ) than AFR of water molecules. The spectral function f (ω) for ω = 0 is proportional to the self-diffusion coefficient D . The coefficient D of the impurity molecule is higher (3.81 ⋅ 10 − 9 m 2 /s) than D of the water molecule (2.88 ⋅ 10 − 9 m 2 /s). Both spectra depicted in Fig. 3 b have the form typical for the spectra of the individual vibrations of molecules in liquids. The liquid-like spectra of AFR are characteristic of all aggregates under study, due to which we have ample reason to state that the clusters are in the liquid state. The resulting size dependence of the surface tension γ of the clusters is shown in Fig. 4 a . The same figure gives the experimental value of γ ∞ of pure water at T = 300 K for the planar interface [26]. Monotonous growth of surface tension with increasing N was observed for pure water clusters and aggregates with a CO molecule. Approximation of the dependence γ( N ) for clusters with CO 2 was done by extrapolating the corresponding curve for N ≤ 10 to large values of N . The real dependence γ( N ) for this case is shown in Fig. 4 b (dots). The same figure shows an approximation of the dependence by the second-order polynomial. The quantity T/c p 0 is a thermal criterion of stability for an extended phase. It may be represented as one of the criteria of stability for systems with a limited number of degrees of freedom [27]. The dependence c p ( N ) in general reflects (Fig. 2 a ) increased thermal stability of the clusters when the constituent molecules increase in number. As for the bulk phase, for clusters one can apply the mechanical criterion of stability in the form of β − T 1 > 0 [27]. The dependence β T ( N ) (Fig. 2 b ) tends to grow with N for both pure water clusters and for aggregates with CO. This means that the mechanical stability of such clusters decreases with their size. For aggregates with CO 2 , the mechanical stability changes insignificantly. The monotonously increasing dependence γ( N ) (Fig. 4 a ) was used to estimate the size of the critical seed. Let us consider that for all clusters under study, γ → γ ∞ for N → ∞ . Assuming that the surface tension of the seed of the critical size does not differ widely (within 5%) from γ ∞ [28], we extrapolate the curves given in Fig. 4 a to the region of large N until they intersect the line γ = γ ∞ . The perpendiculars from the intersection points to the abscissa axis define the size of the critical seeds: N ∗ = 57 for pure water clusters and water clusters with CO; N ∗ = 74 for aggregates with CO 2 . The criterion of cluster integrity is the positive and finite value of its surface tension γ . The modeling showed that there is high probability that the cluster with a CO 2 molecule will not reach the critical size. This is clearly indicated by the dependence γ( N ) (Fig. 4 b ). The function γ( N ) has a maximum whose position is fixed at N = 10 and then quickly decreases. When γ decreases to values close to zero, it is highly probable that the integrity of the aggregate will be broken, i.e., that the aggregate will fractionate into smaller clusters or molecules will evaporate from the formation under analysis. To examine this question in more detail, we investigated the evolution of a system of clusters using a separate visualization program. In the course of calculation, the molecular coordinates were recorded every 10,000 time steps (10 ps). Moreover, additional calculations were carried out for clusters of 18 water molecules and an impurity: CO 2 in one case and CO in the other. The new calculations differ from the former ones in that the impurity molecule was initially placed outside the aggregate but not at the center. In both cases, the impurity molecules were trapped by water clusters, but by the moment of time 3 ns, the virtual cluster with CO 2 split into two water clusters (with 6 and 12 molecules) and the CO 2 monomer, whereas the cluster with CO retained its integrity. The work of formation of a critical seed may be defined as ...